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Department for Industrial Economics and Technology Management

Pricing Credit Derivatives

Harald Martin Myhre Arve Ree

Amund Westbye

Trondheim, 8 December 2003

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Foreword

We are students in the final year of our Master education in Industrial Economics and Technology Management at Norwegian University of Science and Technology. This paper is the result of our project during the autumn 2003, and it corresponds to half study load for each of us. The main aim of the project is to form a strong foundation for a Master Thesis within the same study field. We have chosen to write about credit risk and credit derivatives because it is a relatively new and rapidly growing area where consensus on standard models has yet to be reached. Another attractive feature, especially with the structural model we concentrate on, is that the principles are easily applicable to other areas within quantitative finance. The CD at the back cover of this paper contains the model we have implemented. We would like to thank our supervisor, Assistant Professor Sjur Westgaard at the Department for Industrial Economics and Technology Management, and Professor Håvard Rue at the Department of Mathematical Sciences, for useful help throughout the process.

Trondheim, Norway, 8 December2003

Harald Martin Myhre Arve Ree Amund Westbye

Harald Martin Myhre: [email protected] Arve Ree: [email protected] Amund Westbye: [email protected]

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Abstract

Defaults frequently result in unexpected and severe losses for parties involved with the defaulted company, as has been demonstrated with the big financial scandals of Enron and WorldCom recently. Traditionally the only means investors and banks have had to elude the major consequences of bankruptcies has been to diversify their investments and require collateral. In this aspect the rapid growth of credit derivatives has revolutionized the financial industry and made it possible to both transfer credit risk to a third-party, as well as taking on credit risk in otherwise unfunded positions. The main purpose of this paper is to gain a thorough understanding of credit derivatives through modeling the underlying credit risk. We try to incorporate both the financial aspects and motivations behind the use of credit derivatives, as well as the more fundamental mathematics behind the modeling and pricing of the underlying securities and derivatives. In particular, we investigate and implement a model proposed by Zhou (1997a), which has its foundation in the classical models proposed by Merton and Black and Cox in the 1970’s. The model allows for jumps in the market value of firms’ assets, and we will show that this added complexity produces results that resemble observed features in the market considerably better than the original framework. We do not assess empirical data, but show how Markov chain Monte Carlo could be used to estimate the model. Finally, we explain the implications of different assumptions and combinations of input parameters.

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Table of Contents FOREWORD ........................................................................................................................................ II ABSTRACT.......................................................................................................................................... III CONTENTS ......................................................................................................................................... IV FIGURES ...............................................................................................................................................V 1 INTRODUCTION......................................................................................................................... 1

1.1 OUTLINE ................................................................................................................................ 2 2 CREDIT DERIVATIVES OVERVIEW....................................................................................... 3

2.1 RAISON D’ÊTRE ...................................................................................................................... 3 2.2 THE MARKET ......................................................................................................................... 5 2.3 LEGISLATION AND CONTRACTS ............................................................................................... 6

2.3.1 Default .............................................................................................................................. 7 2.3.2 Recovery ........................................................................................................................... 7

2.4 PRODUCTS.............................................................................................................................. 8 2.4.1 Asset Swap ........................................................................................................................ 8 2.4.2 Total Return Swap ............................................................................................................. 8 2.4.3 Credit Default Swap .........................................................................................................10 2.4.4 Credit-Linked Notes..........................................................................................................11 2.4.5 Collateralized Debt Obligation .........................................................................................11 2.4.6 Exotic Structures ..............................................................................................................12

3 CREDIT RISK MODELING.......................................................................................................13 3.1 STRUCTURAL MODELS ...........................................................................................................13

3.1.1 The Merton Model (1974).................................................................................................14 3.1.2 The Black and Cox First Passage Time Model...................................................................16 3.1.3 Practical Implementations of Structural Models................................................................17

3.2 REDUCED FORM MODELS........................................................................................................19 3.2.1 Intensity Based Modeling..................................................................................................19 3.2.2 Recovery Modeling...........................................................................................................21 3.2.3 Credit Rating Modeling ....................................................................................................22 3.2.4 Jarrow and Turnbull (1995)..............................................................................................23 3.2.5 Jarrow, Lando and Turnbull (1997) ..................................................................................25 3.2.6 Duffie and Singleton (1999) ..............................................................................................27

3.3 HYBRID MODELS ...................................................................................................................28 3.3.1 Giesecke (2001)................................................................................................................28

4 ZHOU’S MODEL ........................................................................................................................31 4.1 THE MODEL ..........................................................................................................................31 4.2 CLOSED-FORM SOLUTION ......................................................................................................33 4.3 IMPLEMENTATION..................................................................................................................37

4.3.1 Primer on Monte Carlo Methods.......................................................................................37 4.3.2 Solution............................................................................................................................38 4.3.3 Implementation Issues ......................................................................................................39

4.4 PRICING CREDIT DEFAULT SWAPS ..........................................................................................41 4.5 RESULTS ...............................................................................................................................41 4.6 CALIBRATION ........................................................................................................................47

4.6.1 Markov Chain Monte Carlo ..............................................................................................49 4.6.2 Calibration Methodology..................................................................................................50

4.7 SUGGESTED EXTENSIONS .......................................................................................................52 5 CONCLUDING REMARKS .......................................................................................................55 6 BIBLIOGRAPHY ........................................................................................................................56 A ZHOU.C DLL PRORAM CODE................................................................................................... I B VBA PROGRAM CODE FOR CLOSED FORM SOLUTION ................................................VII C VBA CODE FOR CHANGING DIFFUSION VOLATILITY AND JUMP VOLATILITY ..... IX D MCMC RESULTS .....................................................................................................................XII E R CODE FOR MCMC............................................................................................................. XIV

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Figures

FIGURE 2.1 ASSET SWAP ........................................................................................................................... 8 FIGURE 2.2 TOTAL RATE OF RETURN SWAP ................................................................................................ 9 FIGURE 2.3 CREDIT DEFAULT SWAP..........................................................................................................10 FIGURE 2.4 CREDIT LINKED NOTES............................................................................................................11 FIGURE 3.1 JARROW AND TURNBULL (1995).............................................................................................24 FIGURE 3.2 SHORT-TERM CREDIT SPREADS IN THE CASE OF INCOMPLETE INFORMATION (DUFFIE AND LANDO

2001) .............................................................................................................................................28 FIGURE 4.1 CONVERGENCE OF MONTE CARLO SIMULATION.......................................................................40 FIGURE 4.2 VARYING NUMBER OF TIME STEPS IN MONTE CARLO SIMULATION ...........................................40 FIGURE 4.3 VARYING JUMP INTENSITY AND JUMP VOLATILITY WHILE KEEPING TOTAL ASSET VOLATILITY

CONSTANT ......................................................................................................................................42 FIGURE 4.4 VARYING JUMP AND DIFFUSION VOLATILITY WHILE KEEPING TOTAL ASSET VOLATILITY AND JUMP

INTENSITY CONSTANT .....................................................................................................................43 FIGURE 4.5 VARYING THE INITIAL DEBT TO ASSET VALUE ..........................................................................44 FIGURE 4.6 VARYING THE RISK-FREE INTEREST RATE ................................................................................44 FIGURE 4.7 CORRESPONDING SIMULATION AND CLOSED-FORM SOLUTION WHILE VARYING JUMP AND

DIFFUSION VOLATILITY AND KEEPING TOTAL ASSET VOLATILITY AND JUMP INTENSITY CONSTANT.....45 FIGURE 4.8 CORRESPONDING BOND SPREAD WITH CREDIT DEFAULT SWAP SPREAD .....................................46 FIGURE 4.9 CONVERGENCE OF ALGORITHM CALCULATING FIRM VALUE. INITIAL FIRM VALUE IS SET TO 5000,

FACE VALUE OF DEBT TO 1500, EQUITY TO 1550 AND OTHER PARAMETERS TO THE VALUES USED IN

CHAPTER 4.5...................................................................................................................................48 FIGURE 4.10 TRACE PLOT OF DRIFT DIFFUSION PARAMETER IN MCMC ESTIMATION ...................................52

Tables

TABLE 2.1 FUNDING EXAMPLE: INITIAL REVENUES ..................................................................................... 4 TABLE 2.2 JP MORGAN REVENUES WITH CDS............................................................................................ 5 TABLE 2.3 EBC REVENUES WITH CDS...................................................................................................... 5 TABLE 2.4 MARKET SHARE BY INSTRUMENT (SCHÖNBUCHER 2003)........................................................... 6 TABLE 2.5 MARKET SHARE BY INSTITUTION (LEHMAN BROTHERS 2001).................................................... 6 TABLE 2.6 TOTAL RETURN SWAP: INDEPENDENT BANKS ............................................................................ 9 TABLE 2.7 TOTAL RETURN SWAP: BANKS ENGAGING IN A TRS................................................................... 9 TABLE 3.1 PAYOFFS FROM THE MERTON MODEL ......................................................................................14 TABLE 3.2 AVERAGE ONE-YEAR RATING MIGRATION RATES 1985-2001 (MOODY’S 2002)...........................22 TABLE 3.3 INFORMATION WITH GIESECKE ................................................................................................29 TABLE 4.1 DESCRIPTION OF PARAMETERS USED IN ZHOU'S MODEL .............................................................31


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1 Introduction

Financial risk is a measure of adverse changes in a financial position as a result of changing environment. Broadly, financial risk may be divided into two categories: Market risk stems from general economic variations while credit risk1 is due to uncertainty and changed perceptions of a debt issuer’s credit quality. (Giesecke 2001) Credit risk may be further divided into components that have to be carefully considered when deciding on the complexity and accuracy of credit models. Most obvious is the arrival risk as a term for whether a default will occur or not, often measured by the probability of default. Secondly, there is uncertainty about the timing of the default, which expectation typically varies significantly between large, stable firms and risky ventures. A third factor, which may often be just as important as the arrival and timing risk, is the recovery risk. It is related to the uncertainty about the payoff that creditors will receive after default. Finally, there is default correlation risk which is the risk of several joint defaults in a time period. (Schönbucher 2003) Although continuous-time finance has been used to value defaultable securities since the initial proposal of Black and Scholes (1973) it is particularly in the last decade that this area of research has received attention. Banks have devoted more attention to this task. The European Monetary Union increased liquidity and competition and made credit risk the key determinant of price differences. Furthermore, the historically low interest rates have forced investors to accept more credit risk to maintain high yields. Most important has been the transition where internal risk models are becoming increasingly accepted as a basis for regulatory capital prescriptions. A system where internal risk models are accepted as to determine adequate capital reserves will give significant advantages to banks that have such models in place. With the introduction of credit derivatives in the early 1990’s investors have been able to separate and trade credit risk according to their own risk profile. The rapid growth of credit derivatives and the many still remaining challenges regarding standardization of pricing models have motivated this paper with the objective summarized below.

This paper explores the vast area of credit derivatives and the literature on modeling credit risk. We aim to gain insight in the financial as well as the mathematical foundation of credit risk and derivatives, in particular by investigating one specific model in depth.

1 Credit risk and default risk are used synonymously throughout this paper.


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1.1 Outline

The rest of the paper is structured as follows: Chapter 2 gives to a general overview of credit derivatives and explains the motivation for the use of credit derivatives with background in market needs and regulations. Additionally, we give an in-depth analysis of some of the most popular credit products available in market. In chapter 3 we move on to discuss credit risk models proposed in the literature and compare those to some models actually being used by credit rating agencies and investment banks. The models can be separated into two main categories: structural models and reduced form models. The structural models were initially proposed by Black and Scholes (1973) and Merton (1974) and model the evolvement of firm value as a diffusion process where default is occurring when the firm value goes below a barrier. The reduced form approach contains models with a variety of foundations. However, what they all have in common is that the link to the firms’ fundamentals is abandoned, and the time of default is modeled as a totally inaccessible stopping time with an exogenously given intensity. Finally, in chapter 4 we analyze a jump-diffusion model proposed by Zhou (1997a), which incorporates features from both approaches mentioned above and mitigates the differences between model output and empirical data. For this particular model we derive a closed-form solution for the pricing of equity in a special case. By building a model with Monte Carlo in the C programming language we create a solid and flexible framework that easily adapts to problems faced by real world situations. Though it may be too detailed for practical purposes, we focus on understanding the exact way pricing of securities and credit derivatives should be done, and incorporate it into our model. We have focused on implementing the model in such a way that it in particular gives a thorough analysis on how the model parameters influence the output. Estimating these parameters is non-trivial because the value of a firm’s assets is neither observable in the market, nor it can be retrieved directly from the balance sheet. We suggest methods for estimating the firm value from equity data and thereby calibrating the model.


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2 Credit Derivatives Overview

Credit derivatives were introduced to the market in the beginning of the 1990’s. Despite their short history their use have grown rapidly. Credit derivatives are now used not only by banks, but also by various funds, insurance companies and even corporations, in order to separate and trade credit risk. The term credit derivatives is being used for a broad range of securities. Schönbucher (2003) gives one definition of credit derivatives which covers the different securities presented in this paper:

A credit derivative is a derivative security whose payoff is materially affected by credit risk. (Schönbucher 2003, p 8)

In the following we will explain the motivation for the use of credit derivatives and give an overview of the market. Moreover, we briefly explain the most general legislation governing the use of credit derivatives, before we in depth explain the most important credit derivatives currently available.

2.1 Raison d’être

The primary purpose of credit derivatives is to enable investors to transfer and repackage credit risk (Das 2000). Basic credit derivatives provide an efficient way to replicate in a derivative form the credit risk that would otherwise be present in a standard cash instrument. More exotic credit derivatives enable the credit profile of a particular asset or group of assets to be split up and redistributed into a more concentrated or diluted form according to the various risk preferences of investors. (Lehman Brothers 2001) By exploiting these features, investors can use credit derivatives to quantify their risk and highlight credit risk concentrations (SunGard 2002). Other motives for entering into credit derivatives contracts spans from hedging, speculation and diversifying, to taxation issues (Schönbucher 2003; Tavakoli 2001). In general there exists no equivalent replication of a credit derivative2. In other words, there are no good alternatives if one wants to trade credit risk. The lack of equivalents adds to the difficulty of pricing credit derivatives, because otherwise one could have used replication pricing. But lack of alternatives makes the market expand, since people have to use them if they want to obtain the benefits from credit derivatives. Below we explain why regulation also has induced a need for credit derivatives.

2 Some features may be obtained through the use of e.g. special purpose vehicles, see chapter 2.4.5.


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Most countries with internationally active banks have adapted their legislation to an international standard, the July 1988 Basel Capital Accord, Basel I. This is introduced by the Basel Committee on Banking Supervision, also called the BIS group after the Bank for International Settlements which facilitates the meetings. (Bank for International Settlements 2003) Through this accord banks are required to hold capital to protect against unexpected losses, and the agreement links the amount of capital to the risk involved for the lending banks. However, the categories of risk are only differentiated in broad categories: banks have to set aside as much capital against a loan to Microsoft as to a Hungarian dotcom, as much against a loan to the US as one to South Korea (Economist 2002). Credit derivatives provide a way to ease the capital requirements for banks. If a bank has lent to the private sector, it would normally need to set aside 100% BIS risk weight times 8% of the face value. However, if it buys protection from an OECD bank in form of a credit derivative, it only needs to hold 20% * 8% = 1.6%. (Lehman Brothers 2001; Tavakoli 2001) Let us look at an example. Liberty Corporation, rated AA-, is seeking $100 mill of financing over five years priced at LIBOR + 75 bps. Both JP Morgan Chase (JPMC) and European Banking Corporation (EBC), rated A+, have been approached to be a single lender in the transaction. JPMC is able to fund the loan at LIBOR – 10 bps, while EBC’s cost of capital is LIBOR + 25 bps. Both banks are subject to the requirements from BIS, namely they must hold 8% against risk-weighted assets times 100% weight for corporations. Assume LIBOR is currently 3%. Each bank may fund Liberty with the below indicated return on capitals. JP Morgan Chase European Banking Corp. Interest Income 3,750,000 $ 3,750,000 $ Interest Expense -2,900,000 $ -3,250,000 $ Net Interest Margin (1-2) 850,000 $ 500,000 $ Return on Capital 10.63% 6.25%

Table 2.1 Funding example: Initial revenues

Now suppose JPMC enters into a credit default swap (CDS)3 with EBC, where EBC takes on the credit risk, and thus need to hold 100% of 8% of notional, in return for a 60 bps spread. Now the spreadsheets become:

3 In effect a credit risk insurance, discussed in chapter 2.4.3.


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Table 2.2 JP Morgan revenues with CDS

JPMC Interest Income 3,750,000 $ Interest Expense -2,900,000 $ Net Interest Margin 850,000 $ CDS Fee -600,000 $ Net Income 250,000 $ Return on Capital 15.63%

Table 2.3 EBC Revenues with CDS

EBC CDS Fee 600 $ Return on Capital 7.5%

Both banks are better off with the credit default swap, and the customer was not affected. This simple principle illustrates how the BIS regulations have helped to propel the use of credit derivatives. The above example also illustrates another point. Today’s regulations are not sophisticated enough since they treat totally different risks the same way. Therefore, the BIS group is working on an upgrade, the Basel II. The changes will not be effective until January 2007, but enough is ready to know that the impact will be substantial. The Basel II proposals are not clarifications or amendments, but complete revision of the Capital Accord of 1988 (Kesdee 2003). The Basel II will decrease the capital requirements for many classes of credit risk, offset by a totally new charge for operational risk. That leaves the overall minimum regulatory capital in the banking system about the same as now. The details for credit derivatives usage await the final agreement. (Economist 2003) The numerical example above also illustrates another motivation for the swap. This may arise when the bank for some reason consider itself obliged to give a loan, for instance to a loyal and important customer. In this case the bank may provide even a risky loan. The bank can then insure itself from the risk by entering an insurance position like the one exemplified above, without the knowledge of the customer and the according implications on the customer relationship.

2.2 The Market

The credit derivatives market was created in London and New York in the early 1990’s, propelled by the Based I accord. After the first half of 2003 the notional outstanding amounted to 2.69 trillion dollars, and trade grew by 25% in the first six months of 2003 (ISDA 2003).4 The market share by instrument type as of 2002:

4 Beware that the statistics as trade is mainly OTC, and the statistics based on surveys.


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Instrument Share Credit default swaps (including First to Default swap) 67% Synthetic balance sheet Collateralized Loan Obligations 12% Tranched portfolio default swaps 9% Credit-linked notes, asset repackaging, asset swaps 7% Credit spread options 2% Managed synthetic Collateralized Debt Obligations 2% Total return swaps 1% Hybrid credit derivatives 0.2%

Table 2.4 Market share by instrument (Schönbucher 2003)

We see that credit default swaps are by far the most traded credit derivative. This is a relatively plain structure, and other varieties may gain market share as the market matures. Below are the market shares based on trade participants5. Counterparty Protection Buyer Protection Seller Banks 63% 47% Securities firms 18% 16% Insurance companies 7% 23% Corporations 6% 3% Hedge funds 3% 5% Mutual funds 1% 2% Pension funds 1% 3% Government/Export credit agencies 1% 1%

Table 2.5 Market share by institution (Lehman Brothers 2001)

The variety of applications attracts a variety of market participants. One obstacle for a even broader use is the difficulties regarding pricing. Since there is no widespread pricing standard for credit derivatives of either type, the actors need to be of a certain size to have the needed competence and resources. (Tavakoli 2001)

2.3 Legislation and Contracts

There has been a debate over whether credit derivatives are to be considered insurance, and the legislation for them is in general incompletely developed. (Das 2000) In order to facilitate over the counter (OTC) trade, the International Swaps and Derivatives Association Inc., ISDA, has authored the Sovereign Master Credit Derivatives Confirmation Agreement. This is a template for contracts that parties may voluntarily refer to in their contracts. Today, 95% of credit derivative contracts are based on the standardized ISDA framework (Schüler 2001). 5 Again caution must be taken since the numbers are based on surveys, but they provide a general picture.


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2.3.1 Default Normally in the credit derivative contract, a credit event will trigger the equivalent to the payment of an insurance payment. Therefore the terminology used is crucial. Even the meaning of default is not fully straightforward. ISDA define the notion of default to be one of failure to pay or deliver, breach of agreement, or credit support default (ISDA 1992). Schönbucher (2003) gives some more other examples of credit events that may trigger a credit derivative payment, including restructuring and rating downgrades to or below a specified level. Though the practitioner needs to be aware of the exact juridical definitions, we take the notion of default as given in this paper. However, in calibrating the model, one will need to pay attention if one uses market default data.

2.3.2 Recovery After a default another important issue arise, namely recovery. Here is recovery used as the percentage of notional a creditor receives after a credit event. If the recovery had been 100%, a default had been no issue at all. In practice this value depends on many factors, thus making it very difficult to predict, even for similar companies. How to model recovery in a credit risk framework will be further discussed in chapter 3.2.2. In fact, recovery after bankruptcy may be a lengthy process involving many parties and, thus, major costs. As far as credit derivatives are concerned, they are normally settled within weeks of default, giving the protection buyer the contingent payoff within a predictable time period. However, difficulties arise as uncertainty about recovery still remains after the default. Most often the credit derivative contract should contain information on how to deal with these problems, but there are also situations where a third-party will enter into the process and give a fair value of recovery. Now consider a company currently in, or on the way into, financial distress, resulting in a low bond value. Suppose one of the bondholders buy credit protection on the asset through a CDS contract. This bondholder will now have incentive to force the firm into bankruptcy, for example by refusing to extend or renegotiate the loan contract in situations he would have otherwise done it to avoid immediate default. The reason lies in the fact that the CDS will pay off according to the face value of the debt and not on the difference between the real value of the bond and the recovery. Additionally he will also get away without paying the fixed contingent payments until the maturity of the CDS. Obviously this may raise ethical questions and give incentives to declare otherwise evitable bankruptcies. Debt is issued with different seniorities giving the creditor varying security in the case of default. After a default the most senior bonds are supposed to pay off completely before more junior debts are paid off. In practice there are many legal aspects of a bankruptcy resulting in that this absolute priority rule is not upheld. Nevertheless, expected recovery for senior debt is still substantially higher than for junior debt.


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2.4 Products

This chapter is devoted to an overview of some important credit derivatives products. By far credit default swaps are the mostly traded credit derivative. As shown in the market chapter they constitute approximately two thirds of the market when we include basket and first-to-default derivatives. Still there exist a large variety of products accommodated to certain situations and we introduce here some of the basic structures.

2.4.1 Asset Swap The asset swap package converts the payoffs of a defaultable (coupon) bond into a payoff stream of LIBOR plus a spread. This spread is chosen to give the whole package a value of par. Thus, we see that the contract is a combination of a defaultable bond and an interest-rate swap contract.

Asset swap buyer

Defaultable bond

Asset swap seller Coupon payments

LIBOR + asset swap spread

Coupon payments

Figure 2.1 Asset swap

In its strictest form the asset swap is not a credit derivative. The reason is that the payments of the swap will continue even if a credit event occurs. Hence, it is merely a way of securing cash flows than separating and trading credit risk. The market for asset swaps is very liquid and frequently it is the underlying asset for other derivatives, such as asset swaptions. There exist many additional features that can be included in the contract to give it the desired payoffs.

2.4.2 Total Return Swap The total return swap (TRS), also called total rate of return swap (TRORS), adds another perspective to the asset swap. With the TRS the counterparties agree to exchange all cash flows from two different investments, normally one of them defaultable and the other one considered not to be. Basically this structure does an exchange of the payoffs without legally transferring the ownership of the asset.


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TRS payer (A)

Defaultable reference asset

TRS receiver (B) Total rate of return

Asset payments LIBOR + TRS spread Mark-to-market

Figure 2.2 Total rate of return swap

The TRS payer gives all the payments of a defaultable reference asset to the TRS receiver, at the same time as he receives LIBOR + TRS-spread. Additionally the contract is marked to market at regular intervals. If the reference asset has appreciated, party A must pay the difference to B, and if the asset has depreciated B must pay the difference to A. The default event is handled by the latter case, as that will correspond to a depreciated value of the asset, and the TRS receiver must pay A the difference between the pre-default value and the recovery value. As an example of a TRS we may take the following situation described in Tavakoli (2001). Consider two banks with different ratings and funding costs that want to invest in a BBB rated asset Table 2.6 gives an overview of the costs, payments and earnings for these two actors when the purchase is committed independently. Both banks have a 100% BIS risk weight. Bank Rating Purchased Asset Risk Weight Coupon Payments Funding Cost Net Spread

Payer: AA BBB 100% LIBOR + 65 bps LIBOR – 15 bps 80 bps Receiver: A- BBB 100% LIBOR + 65 bps LIBOR + 30 bps 35 bps

Table 2.6 Total return swap: Independent banks

By entering a TRS with the A- bank, the AA bank is able to give the A- bank a more favorable funding cost. Contemporaneously AA is hedging both its credit and market risk, and can reduce its capital charge on the transaction from 100% to 20%6 while still remaining the legal owner of the asset. For many investors this might be the only possibility to attain a short position in specific assets due to the fact that directly shorting of defaultable bonds often is impossible. Hence, the TRS receiver benefits from lower funding costs. The TRS payer benefits from lower joint default probability between the receiver and the reference asset, thus reducing its risk weight in the position. (Table 2.7) Bank Rating Purchased Asset Risk Weight Coupon Payments Funding Cost Net Spread

Payer: AA A+ 20% LIBOR + 15 bps LIBOR – 15 bps 30 bps Receiver: A- BBB 100% LIBOR + 65 bps LIBOR + 15 bps 30 bps

Table 2.7 Total return swap: Banks engaging in a TRS

6 Assuming that both banks are OECD banks.


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The TRS can be used to defer losses because of the off-balance sheet feature of the contract. An investor with an unrecognized loss in a bond position can defer the loss by entering a TRS and then recognize the loss at maturity of the contract. (Tavakoli 2001) For the TRS receiver, a TRS might be applied in a variety of ways in practice, but the primarily use is that of financing. There are no initial investments, with the exception of a collateral, which makes it possible for party B to take advantage of leverage. Some total return swaps can be regarded as a new asset with specific maturity and features currently not available in the market. In other cases the contract might be the only way B can invest in the reference asset due to legal restrictions. (Tavakoli 2001)

2.4.3 Credit Default Swap A credit default swap (CDS) is an exchange of a fee for a payment in case a credit default event occurs on a reference asset. Other names for the CDS include credit default options and default swaps. A CDS will make it possible to separate and trade purely in the credit risk of different underlying assets, e.g. loans, bonds and receivables. With the aid of this contract it is possible for a company to hedge its credit risk exposure at the same time as an investor might speculate in the reference asset's future. The contract consists of two parties; one protection buyer and one protection seller. The buyer is paying a fee up front or a fee amortized over the life of the security. In the case of a credit event the seller will have to pay a contingent payment. This payment can take many forms and will need a thorough specification in the contract. The most common alternatives include physical delivery of the reference asset against repayment at par, cash settlement based on post-default market value and a pre-agreed fixed payoff, more commonly called a digital default swap. Neither of the parties need to have a funded position in the reference asset, but still the physical settlement is widely used as default payment.

Protection buyer

Reference asset

Protection seller Fee

Contingent payment upon default

Figure 2.3 Credit default swap

Notice the importance of the correlation between the protection seller and the reference asset. If these two parties are perfectly correlated, the CDS will be of reduced value, as a default on the reference asset will imply a default on the seller as well. Preferably we would like a protection seller with zero or negative correlation with the reference asset and as high credit rating as possible. In practice these criteria would have to be relaxed in order to make the contract affordable.


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The variations between different CDS contracts might be large and the specifications of the contract might be adapted to the needs of the counterparties. ISDA has introduced a set of common specifications that serve as a standard for credit default swaps, and there exists a relatively liquid market for these instruments. Most credit default swaps are quoted with five years to maturity.

2.4.4 Credit-Linked Notes Credit-linked notes (CLN) are a fixed-income security with an embedded credit derivative. There are several versions of credit-linked notes, but we will here focus on the family of credit-linked structured notes, and more specifically on credit default linked notes (Das 2000), which embed a CDS with the fixed-income security.

CDS

Issuer

Derivatives dealer

Investor

Principal/ recovery rate Coupon

Investment

Figure 2.4 Credit linked notes

The issuer of the CLN is normally a high rated trust (AA or AAA). It is paying the investor a fixed or floating coupon based on the chosen reference asset. If there are no credit events the investor receives the principal of maturity, otherwise he will receive the recovery rate or any other pre-agreed default payment version. At the same time the issuer sells credit protection to a derivatives dealer through a CDS contract. The payments from the CDS contract are further used to create coupon payments to the investor. Hence, the issuer has created a synthetic bond, which gives the investor the opportunity to obtain a position without making the direct investment in the reference asset. Das (2000) lists some of the main difficulties of direct investment, especially in high-yield and emerging markets. Firstly there are regulatory issues and high transaction costs related to foreign investments. Secondly there is often a lack of underlying securities and liquid markets corresponding to the investors’ preferences. The credit-linked notes open the market to a broader range of investors. Moreover, as buying a CLN is similar to a fully collateralized sale of credit protection, the number of investors that can sell default protection is significantly increased.

2.4.5 Collateralized Debt Obligation Collateralized debt obligations (CDO) are used to securitize portfolios of defaultable assets. The portfolio of assets is transferred into a specially created company, a special purpose vehicle (SPV). The SPV is then issuing notes with loss layers, which are used in


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the case of a default during the existence of the CDO. When the CDO is liquidated the senior tranches are paid off first, and further the other notes according to their ranking. The tranches of lower seniority serve as protection for the tranches of higher seniority. CDOs allow investors to invest in notes that they would otherwise not be allowed to invest in and adapt the desired risk profile. The two most common forms of collateralised debt obligations are arbitrage CDOs and balance sheet CDOs. The former aims to arbitrage the price between the components of the underlying portfolio with the sale price of the CDO-notes while the latter tries to free up regulatory capital tied up in the underlying loan7 or bond8 portfolio.

2.4.6 Exotic Structures New credit derivatives have popped up regularly with more advanced features than the plain vanilla structures introduced so far. The easiest of these include options and forwards on a credit default swap. Furthermore we have asset swap switches, different knock-in-options and credit spread options. One of the most promising and interesting structures is the purchase of protection on a basket of credits. For a portfolio manager this means that he can purchase a first-to-default structure on several assets instead of buying them separately. Such a diversified basket will obviously be cheaper than a CDS on each of the reference assets. We refer to Tavakoli (2001) and Das (2000) for a more thorough presentation and analysis of these exotic structures.

7 Collateralized Loan Obligation (CLO) 8 Collateralized Bond Obligation (CBO)


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3 Credit Risk Modeling

There are two main approaches to credit risk modeling. The first approach includes the firm-value or structural models initially proposed by Black and Scholes (1973) and Merton (1974). In these models the value of the firm follows a diffusion process. Default is modeled as when the value of the firm goes below a boundary, usually a function of the capitalization of the company. The main advantage of the structural approach is that it is based on solid economic arguments and it models default in terms of fundamental firm variables. Critics against this approach point out that time of default will be a predictable stopping time because of the continuity of the process. Therefore, as time to maturity goes to zero, credit spreads should also approach zero which again is not consistent with empirical evidence. The second approach is the intensity based or reduced form models. In this approach the link to the firms’ fundamentals is abandoned, and the time of default is modeled as a totally inaccessible stopping time with a default intensity. The advantage of this modeling approach is in particular its tractability and empirically good performance in that the models are easier calibrated to fit observed data than structural models are. Reduced form models, however, do not formulate economic arguments about why a firm defaults. One rather takes the default event and its stochastic structure, and frequently the recovery process as well, as exogenously given. The basic underlying assumption for the models described in the following is that there exists a unique equivalent martingale measure °P making the markets for default-free and risky debt complete and arbitrage-free. By using this technique pricing is done by discounting expected values of payoffs under °P leading to greatly simplified valuation problems. (Neftci 2000; Johannes and Polson 2002) In the following we will present the basic theory behind the two approaches. In particular, we will investigate the theoretical structural models and correspond them to the way models are actually implemented in practice. Further we describe reduced form models and how default intensities and recovery rates are modeled, before we discuss the most referenced models. Finally, we give a brief overview of hybrid models, which incorporate features from both approaches.

3.1 Structural Models

In a structural model the starting point is the value of the assets, V, of a firm that has issued bonds. This value is assumed to change stochastically, usually following a lognormal diffusion process


14

dV Vdt VdWµ σ= +

where µ is the expected drift, s is the firm’s volatility and dW is a standard Brownian motion. The value of the firm is then used to value all claims on the firm as derivative securities with the assets as the underlying. A default can either be triggered only at maturity of the debt if V is insufficient to repay the debt, as in the Merton model, or it can be modeled as a down-and-out barrier, as in equity options allowing the firm to default as soon as V hits a pre-specified barrier. Below we investigate the two most famous models describing these events, namely the Merton model and the Black and Cox model. Although this kind of models has proven very useful in addressing the quantitatively important principles of pricing credit risk, they have been less successful in practical applications. This owes to the difficulty of modeling realistic boundary conditions. The boundaries include both the conditions under which default occurs, and in the event of default, the division of the value of the firm among claimants.

3.1.1 The Merton Model (1974) In the classical Merton approach a firm is financed by a zero-coupon bond with face value K and maturity date T. Default can only occur at maturity of the debt. Assets Bonds Equity No Default

TV K≥ K TV K− Default

TV K< TV 0

Table 3.1 Payoffs from the Merton model

According to Black and Scholes (1973) by issuing debt, equity holders sell the firm’s assets while keeping a European call option on the firm’s value. The strike is equal to the face value of the debt. Similarly, the debt can be viewed as a default free bond and a short European put on the firm’s value, with strike equal to the face value of the debt. Thus, we have the following payoffs to the firm’s liabilities at time T, where ( ), ,B V t T is the price of the risky bond at time t and maturity T, and S(T) the value of equity:

( , , ) min( , ( )) max(0, ( ))B V T T K V T K K V T= = − −

( ) max(0, ( ) )S T V T K= −

Although V is not a traded asset, the stock as a derivative of V is. Assuming no arbitrage opportunities Merton shows that the valuation of derivatives of V will be independent of investor’s risk preferences. It therefore exists an equivalent probability measure °P under which V is a martingale if discounted at the continuously compounded riskless rate r(t).If we assume that interest rates are constant r(t)=r>0 we can use the classic Black-Scholes formula to derive the price of the equity and the risky bond:


15

1 2(0) ( , , (0, ), , (0)) (0) ( ) (0, ) ( )c vS BS T B T r V V N d B T N dσ= = −

1 2(0, ) (0, ) ( , , , , (0)) (0) ( ) (0, ) ( )p vB T B T BS T K r V V N d B T N dσ= − = − +

where

(0, ) rTB T Ke−= ,

2

1

(0) 1ln( ) ( )2 V

V

V r TKd

T

σ

σ

+ += and

2 1 Vd d Tσ= −

From the equation for the risky bond B and the risk free bond B we can easily compute the bond and credit spread:

1 1

(0, ) (0)( ) ( )

(0, ) v rT

B T VN d T N d

B T Keσ −= − − −

1 1(0)

ln( ( ) ( )( )

V rT

VN d T N d

KeCS TT

σ −− + −= −

We further find the actual and risk-neutral probabilities of default by solving the initial

SDE ( )

( )( ) v

dV tdt dW t

V tµ σ= + which has a unique solution

21( ) ( )

2( ) (0) v vt W tV t V e

µ σ σ− +=

[ ]2

21

( ) ( )2

1ln ( )

(0) 2( ) (0) ( )

v vv

t W t

V

KT

VP V T K P V e K P W T

µ σ σµ σ

σ

− +

− − < = < = <

As W(t) is normally distributed with mean zero and variance T we get

[ ]21

ln ( )(0) 2

( )v

V

KT

VP V T K N

T

µ σ

σ

− −

< =

Similarly to obtain the risk-neutral probabilities we substitute µ with r to get

° [ ] 2( ) ( )P V T K N d< = −

To estimate the default barrier, K, one can use balance sheet data. In the classical Merton model, K is assumed to be equal to the face value of debt. r can be estimated from default-free bonds (Treasury bonds), though some models give better empirical results when interbank rates are used (Schönbucher 2003). (0)V and Vσ can be estimated by


16

first observing the equity value (0)S and the volatility Sσ . We can then find the parameters (0)V and Vσ by solving two equations. The first is obtained from the equity pricing formula in Merton’s model above, and the second by applying Ito’s formula to the equity value:

1(0) ( ) (0)S vS N d Vσ σ=

Extensions to the Merton Model In the original Merton setting, all bonds are considered to be pure discount bonds. A simple extension to the Merton model is to treat coupon bonds as a portfolio of zero coupon bonds each of which can be priced with the original Merton model. A more precise treatment of risky coupon bonds would be the Geske (1977) model. Geske assumes that equity holders have a compound option on the firm’s value, and that they decide to pay the coupon or not. If the equity holders do not pay the coupon, the firm will default and the default boundary is thus modeled endogenously. The value of the option equity holders have can be expressed in terms of multivariate normal distribution with dimensions depending on the number of coupon payments. The Merton model has also been extended to callable and convertible bonds, variable rate bonds and bonds with different seniorities. In JP Morgan’s E2C model, which we will treat later, they use the original and simple Merton model and extend it with a variable default barrier. The KMV model is also loosely based on the original Merton framework.

3.1.2 The Black and Cox First Passage Time Model The Merton model described in the previous section does not allow bankruptcy before the maturity of the bond. In order to allow this Black and Cox (1976) modified the Merton model to allow bondholders to force bankruptcy during the lifetime of the security. These types of models are often referred to as first time passage or first passage time models. In addition to allowing the debt to default before maturity Black and Cox also assume that the shareholders receive a continuous dividend payment proportional to the firm value V , which gives the following SDE under °P :

°( ) ( )( ) ( ) ( )dV t V t r dt V t dW tκ σ= − +

0κ > represents the dividend rate. Interest rate risk was not included in the original Black and Cox model so interest rates are constant, r(t)=r>0. Also, unlike the Merton model, the default barrier is made time dependent. For the firm to default before maturity of the debt, the firm’s value must hit a default barrier ( )T tKe γ− − . Otherwise the firm can default at maturity if the firm’s value is less or equal to the face value of the debt, ( )V T D≤ . The default time is modeled by

inf{ 0 : ( ) ( )}t V t K tτ = > ≤

In the case with constant interest rates, Black and Cox provide a closed-form solution.


17

Black and Cox also provide an argument that allows capital structures with different seniority of the debt. The price of a junior bond can be derived from the price of a senior bond by assuming that the junior bondholders will only be paid off after the senior bondholders have been paid, similar to the waterfall in securitization. If D=S+J is the total debt owed at maturity to senior bondholders, S, and junior bondholders, J, combined, then the price of the senior bond can be found as if there were no junior debt

( ,0, )B S T . Then the price of the junior debt will be the difference between the bonds considering the total debt and the bonds only considering the senior debt

( , , ) ( , , ) ( , , )B J t T B D t T B S t T= −

This priority assumption holds for for instance CDOs and securitization products. But as, mentioned in previous chapters, absolute priority is not upheld in corporate bankruptcy proceedings this assumption does not hold for corporate bonds. All other first time passage models as described below therefore assign a specific function for recovery with different parameters for different seniorities.

Extensions to the Black and Cox model Longstaff and Schwartz (1995) extend the Black and Cox model to allow interest rates to be stochastic, with dynamics as proposed by Vasicek (1977). Moreover, they do not require the recovery rate to be the boundary value, but rather use an exogenously given recovery rate w . They can then price different seniorities by assigning different recovery rates to the different seniorities (Cossin and Pirotte 2001). Coupon bonds are modeled as a portfolio of zero coupon bonds. As in the illustration above by Giesecke they assume the default boundary to be constant and exogenously given. In their implementation they set the barrier equal to the par value of the bonds. The Longstaff and Schwartz model has great advantages over the Merton model as it relaxes the assumptions made. It has great flexibility and allows correlation between the Brownian motion of the firm value and the risk-free interest rate in Vasicek. However, there is no explicit solution available, and Longstaff and Schwartz only provide an approximation to the solution. Another problem with the model is that it allows a payout upon default that might be greater than the firm value at default. Ammann (1999) gives a good summary of possible extensions to fist time passage models. These extensions include models with different interest rate modeling like Cox-Ingersoll-Ross (1985), different default boundaries and models with jump-processes like the Zhou (1997a) model we implement in this paper.

3.1.3 Practical Implementations of Structural Models Below we present two models implementations that are being used by leading companies providing credit risk analysis and pricing of credit derivatives; Moody’s KMV model and JP Morgan’s E2C model.


18

Moody’s KMV The most successful commercial variant of structural models is the KMV model that loosely uses the original Merton framework (Soudaram 2001). Like the Merton model KMV uses Black and Scholes to compute the asset volatility from the equity volatility. For the default barrier, K, KMV’s approach is based on the empirical observation that default tends to occur when the market value of the firm’s assets falls below a point that typically lies below the face value of all debt and above the book value of short-term liabilities. The default barrier used is the sum of the short-term liabilities plus 50% of the long-term liabilities. Given the firm value, the asset volatility and the default barrier, KMV calculates how many standard deviation moves that would result in the firm value falling below the default barrier. The result is called distance to default, d. The distance to default is used to look up in a database to identify the proportion of firms with the same d that actually defaulted within a year. The result is the expected default frequency, which is the main output of the KMV model. The main reason for using the distance to default to look up in a database, instead of just calculating the default probability directly assuming V is normally distributed, is because the distribution of V in reality has excess kurtosis.9 Since a default usually occurs when the firm value drops significantly kurtosis becomes very important and the normal distribution is thus a poor approximation. (Soudaram 2001)

JP Morgan’s E2C Model Another successful practical implementation of the Merton model is the Credit Grades E2C10 model, developed by JP Morgan. The model is currently being used to manage high yield credit derivative inventory and monitor the investment derivative book. As in the Merton model the asset volatility is derived from the volatility of equity. Further the asset value is assumed to have zero drift, i.e. 0µ = , as one assumes that debt would be issued to keep the leverage level steady over time. The default barrier is assumed to follow a lognormal distribution with percentage standard deviation ϑ . The values of K and ϑ are determined by an empirical study and set to 0.5 and 0.3 respectively, for the general case. The probability of no default can then be calculated as

ln( ) ( ) ln( ) ( )(0, )

( ) 2 ( ) 2d A T d A T

B T N d NA T A T

= − − ⋅ − −

where, 2 2 2( ) VA T Tσ ϑ= + ,

2(0)Vd e

Kϑ= and

9 The distribution of the asset values is “fat-tailed” compared to the normal distribution. 10 Equity-to-Credit


19

(0) (0)V S K= +

The credit spread is calculated by integrating the default probability density and applying a constant asset specific recovery Vθ (Lardy 2002):

0

0

( , ) 1 (0, )( ) (1 )

( , )

Trt

V Trt

e d B t T B TCS T

e B t T dtθ

−

−

− + −= −

∫

∫

The power of this model lies in the simple expressions above combined with robust approximations leaving us with only observable parameters. Compared to historical results the model performs well and it is widely used in practice. The E2C-model resembles the economical intuition behind the model proposed by Giesecke (2001), which will be discussed in chapter 3.3.1.

3.2 Reduced form models

Das (2001) classifies the reduced form models into three broad categories. Default models use stochastic processes to model default directly. Spread models decompose risk into credit and recovery risk and solely model the spread without reference to its components. Credit rating models depict the evolution of a firm as changes in its credit rating. We start this chapter by giving an introduction to the basic theory behind default intensities and recovery rates. Further we move on to a general discussion on credit rating modeling. In particular we will discuss the Markov chain models proposed by Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997) and the framework developed by Duffie and Singleton (1999). Of other contributors, we mention Schönbucher (1998) who presents a tree model for defaultable bond prices. This model serves as an implementation framework for a wide range of intensity based models and accommodates for comparison of these models’ properties as well as several different recovery setups. There exists a wide range of published papers on reduced form models and we show to Bohn (2000) for a more thorough study of the literature available.

3.2.1 Intensity Based Modeling Schönbucher (2003) and Giesecke (2002) give an overview of different properties and limitations of the exogenous intensity process. The most common approach is to model the intensity as a Poisson process, which is a discrete statistical process used in a great variety of ways. Most commonly Poisson


20

processes are used in connection with a counting process ( ){ }, 0N t t ≥ having an intensity rate 0λ > . The number of events in any interval of length t is then Poisson distributed with mean tλ . (Ross 2000)

( ) ( ){ } ( ) n=0,1,... s,t 0

!

nt t

P N s t N s nne

λ λ−+ − = = ∀ ≥

As for the simulation of default processes, we are normally concerned with the first jump, interpreting it as default. Hence, the time of default, t , is exponentially distributed with parameter λ (Walpole et al. 1998) and the intensity rate is the conditional default arrival rate given no default (Giesecke 2002). However, this setup can easily be extended when working on more advanced products like for example a portfolio of securities where we might be interested in the structure of several defaults. Constant intensity implies flat term structure of credit spreads. In order to develop more sophisticated models allowing for time-varying intensity, we will need an inhomogeneous Poisson process (as opposed to the homogeneous process described above) with time-dependent intensity ( )tλ . If the counting process N(t) has independent increments for s<t we have (Ross 2000):

( ) ( ){ }( )

1, 0

! ( )

s t

s

y dyns tP N s t N s n n

ns

y dy e

λ

λ

+

− ∫+

+ − = = ≥ ∫

Giesecke (2002) also extends this intensity to be a function of the current information, ω , available in the market A further generalization is applicable through the Cox process. The probability of a jump in the interval [s, s + t] is proportional to the realized value of ( )tλ at time t:

( ) ( ){ }1 ( )tP N s t N s X tλ+ − = =

The intensity is now allowed to be random under the restriction that conditional on the actual realization of λ , N is an inhomogeneous Poisson process. Accordingly, the random intensity process can be a function of some state variables, like for example spot interest rates and credit ratings. The Cox process is not measurable in any way, but it has a background process, which determines the long-term average of the Cox process. (Lando 1998) This far we have only considered modeling of single firms. Often one may be interested in modeling default correlations due to firms’ dependence on each other and economic factors. A natural way of incorporating this feature is to allow the firms’ intensity processes to be correlated through time. Alternatively it is possible to introduce jumps in the intensities, either as a consequence of default of other firms, or as a result of joint


21

credit events. We will not cover default correlations in this paper, but we refer to Giesecke (2002) and Schönbucher (2003) for a more in-depth treatment.

3.2.2 Recovery Modeling In most structural form models recovery is given endogenously. However, in reduced form models we need to make an assumption on the behavior of the recovery process. Modeling of recovery is in many cases just as important as finding the time of default and can have a significant effect on the pricing of credit derivatives (Schönbucher 2003). It is obviously difficult to model the real recoveries that will happen in a bankruptcy process, but there exist approaches that give an approximation of the real world. In its simplest form it is assumed that recovery will be zero. Even though zero recovery (ZR) is not realistic in most cases, it is an easy approach and often serves as a benchmark for other recovery models. Houwelingen (2003) gives a summary of three recovery models found in the literature; recovery of treasury, recovery of market value and recovery of face value. With recovery of treasury (RT) a bond at default will be replaced by a fraction of a similar, but default-free bond. Obviously RT requires that the price of default-free bond can be determined, but for coupon bonds it is normally not a problem. The RT model is computational easy and is a convenient way of improving from zero recovery. The main drawback with the model is that it gives unrealistic spread curves for low credit qualities (Schönbucher 2003). With recovery of market value (RMV) the bond is assumed to loose a fractional amount, q, of its market value. In the case of default the value of our claim has been reduced to (1 )q− of its original market value without any other default effects. The last model considered is recovery of face value (RFV), or recovery of par (RP) as denoted by Schönbucher. At default the claimant recovers a fraction of the face value of the bond. This model closely correlates to what is happening in the case of bankruptcy. According to the priority rule, loans with higher seniority will be paid off before those with lower. However, since absolutely priority seldom holds in reality, seniority claimants will receive a payoff of a certain fraction of par (not par), while the junior claims will be paid off with a smaller fraction afterwards. RMV and RFV are considered the best models for recovery. When it comes to pricing, RMV is the easiest to handle, though the extra work still might have to be carried out when pricing certain credit derivatives. Schönbucher (2003) gives an extensive discussion on the pros and cons of the recovery models described and how they are mathematically priced.


22

3.2.3 Credit Rating Modeling There exist several agencies that evaluate and give credit ratings to defaultable companies. The most familiar ones include Standard and Poor’s, Moody’s KMV and Fitch. Companies are classified according to their credit quality and these ratings are updated as new market information is revealed. However, there is normally a time lag, sometimes months, from the credit information reaches the market until the rating agencies update the ratings. Even though the delay in updating a company’s rating makes them less suited for practical purposes, they are frequently used as a first classification of the obligor. Most credit rating models are based on discrete or continuous-time Markov chains with transition matrices equal to historical average rating transition frequencies as published by the rating agencies. Below is an example of a transition matrix based on average ratings between 1985 and 2001.

Europe Rating to: Aaa Aa A Baa Ba B Caa-C Default WR

Aaa 86,34% 8,21% 0,19% 0,00% 0,00% 0,00% 0,00% 0,00% 5,26% Aa 0,76% 86,71% 9,13% 0,10% 0,00% 0,00% 0,00% 0,00% 3,30% A 0,00% 5,05% 84,80% 3,63% 0,10% 0,02% 0,00% 0,02% 6,39% Baa 0,74% 0,25% 4,82% 78,83% 2,86% 1,16% 0,04% 0,00% 11,31% Ba 0,00% 0,00% 0,64% 10,52% 71,40% 9,29% 0,68% 0,25% 7,22% B 0,00% 0,00% 0,33% 1,03% 9,40% 65,52% 8,28% 3,29% 12,17%

Rating from:

Caa-C 0,00% 0,00% 0,00% 0,00% 0,00% 22,41% 48,58% 14,53% 14,47%

Table 3.2 Average one-year rating migration rates 1985-2001 (Moody’s 2002)

In general the transition probability matrix Q(t,T) for the time interval [t,T] is given as

11 12 1

21 22 2

1 2

( , ) ( , ) ... ( , )( , ) ( , ) ... ( , )

( , )

( , ) ( , ) ... ( , )

K

K

K K KK

q t T q t T q t Tq t T q t T q t T

Q t T

q t T q t T q t T

=

M M O M

The element qij(t,T) is the probability that the rating process is in state j at time T given that it was in state i at time t. Formally we can write

[ ] { }( , ) ( ) | ( ) , 1,..., ,ijq t T P R T j R t i i j K t T= = = ∀ ∈ ≤

where R(t) is the credit rating at time t. Schönbucher (2003) assumes that the credit spreads are constant throughout time and that they are the same for all bonds in the same credit class. Specifically, a bond can only change its spread through a rating transition. The setup rendered below further assumes zero recovery, but Schönbucher also shows how this assumption can be relaxed to cover RT or RP. Given zero recovery and an initial rating of R(0) by time 0, the price of the defaultable bond will be


23

(0) (0)(0, ) (0, )(1 (0, ))R R KB T B T q T= −

Hence, the price of the risky bond is given by the default-free bond times the survival probability. Despite their apparent simplicity credit rating models have several major drawbacks which make them a limited reflection of the real world. Firstly, the transition matrix is based on historical (backward looking) average, though it has been verified that the implied historical default rate overstates the true default rate. Furthermore empirical studies show that especially downgrading of companies is followed by an increase in the credit spread (Cossin and Pirotte 2001). As a result the credit agencies might be reluctant to change the credit rating of a company, even though it could be from a low status in class A to a very strong status in BBB. If there existed a continuous scale instead, the consequences of minor updates in the credit status would not imply a jump in the credit spread. Finally, Schönbucher (2003) points out that calibration is the hardest problem related to these models. There is a big difference between the historical transition matrices published by the agencies and the credit spreads actually observed in the market. Partly the problem is a consequence of the fact that all bonds of a credit rating class are assumed to have the same credit term structure and partly because the ratings take little account of recoveries, which might differ significantly between firms. Reduced form models are built on the assumption that default hits the market by surprise. The probability of this event is a hazard rate and is frequently called the intensity rate

( ),tλ ω , which is a function of the time t and the state of the world ω . That means that the default is not an effect of the firm’s assets or liabilities, but rather the result of an exogenous process governed by selected state variables. The economic interpretation of this kind of models is not as good as for the structural default models when it comes to explaining why a default occurs. Nevertheless the tractability is good and the models provide better fit to historical data. The reason lies in the fact that reduced form models are calibrated to data from rating agencies and financial time series. (Cossin and Pirotte 2001)

3.2.4 Jarrow and Turnbull (1995) Jarrow and Turnbull (1995) assume the capital structure of a firm to be irrelevant for the default process, which in turn is based on an exogenous intensity model. They identify two classes of underlying zero-coupon bonds needed in their model. Firstly, there exist default-free zero-coupon bonds for all maturities, where ( , )B t T is the time t value of the bond paying a dollar at maturity, T. Secondly, we have defaultable zero-coupon bonds of all maturities for a company XYZ where ( ),B t T is the time t value of the bond promising a dollar at maturity T.


24

They further give a foreign currency analogy in pricing the risky debt of XYZ. The payments from XYZ, ( ),foreignB t T 11, are denoted in a hypothetic XYZ-currency which is linked to the dollar by a hypothetic exchange rate, e(t). Putting these expressions together we have an intuitive decomposition of the XYZ zero-coupon bond:

( ) ( ), , ( )fB t T B t T e t= (0.1)

As long as company XYZ is not in default, e(t) will be 1 as each promised XYZ-dollar is worth exactly 1 dollar. In the case of default, e(t) reduces to the recovery rate. As a result of this analogy to foreign currency, techniques from foreign currency option pricing can be applied in order to price credit derivatives involving credit risk. Jarrow and Turnbull first present their model in two-periodic discrete time with

{ }0,1, 2t ∈ . At time ti the probability for default under the equivalent martingale measure in the period [ti,ti+1] is iP λµ=% leaving us with the possibilities that default has occurred and that it has not, respectively state b and n. In addition they give the default-free spot interest rate a stochastic evolution with two possible states in the next time period: u and d.

1 Bf(0,2)

1 Bfd,n(1,2)

d d Bfd,b(1,2)

1 Bfu,n(1,2)

d dBfu,b(1,2)

°11 p−

°1p

°1p

d

d

d

d

1

1

0 2 1

°11 p−

t

° °0 0(1 )(1 )pπ− −

° °0 0(1 )pπ−

° °0 0(1 )pπ −

° °0 0pπ

Figure 3.1 Jarrow and Turnbull (1995)

The risk-neutral probability of an up-move is denoted by °iπ and accordingly for a down-move 1- °

iπ . For simplicity they assume that the spot interest rate process and the bankruptcy process are independent under the equivalent measure. Finally they construct

11 This amount is default-free in the hypothetical currency. Further in the text we will shorten the

denotation to Bf(t,T)


25

an exogenously given recovery process, which pays of a fraction, d, of notional. This corresponds to the recovery of treasury model. To show the result of their model Jarrow and Turnbull construct a two-periodic binomial tree and extends it to cover more periods and finally the continuous case which we render here to show the intuition behind the model. As mentioned we model two processes which each have two possible outcomes. The result is four different states at time 1, depending on the up-down move of the interest rate and whether bankruptcy has been declared or not. The upper part of the boxes in the figure gives the exchange rate and the lower part expresses the value ( ),B t T as given by equation (0.1). Note that when default has occurred there is no uncertainty about the future as the bond in any case only pays out the recovery rate at maturity and the short-term rate is given for the next period. Using risk-adjusted pricing techniques we obtain the following bond prices at time 1:

( ) ( ) ( ), ,1,2 1, 2 1, 2fu b u b uB B Bδ δ= =

( ) ( ) ° °, , 1 11,2 1, 2 (1 ) (1, 2)f

u n u n uB B p p Bδ = = + −

( ) ( ) ( ), ,1, 2 1, 2 1, 2fd b d b dB B Bδ δ= =

( ) ( ) ° °, , 1 11, 2 1, 2 (1 ) (1,2)f

d n d n dB B p p Bδ = = + −

where Bu(1,2) and Bd(1,2) are the spot prices of bonds after an up or down move in the interest rate. Applying the same technique we obtain the defaultable bond price at time 0:

( ) ° ° ( ) ° ° ( )° ° ( ) ° ° ( )

0 0, ,0 0

0 0, ,0 0

0, 2 1, 2 (1 ) 1,2

(1 ) 1, 2 (1 )(1 ) 1, 2

u b u n

d b d n

B p B p B

p B p B

π π

π π

= + − +

− + − −

Further in their article Jarrow and Turnbull relax some of the assumptions outlined above. The suggested relaxations include introduction of correlation between the bankruptcy- and default-free-free interest rate processes, inclusion of traded equity on company XYZ and finally estimating the continuous-time limit of their model.

3.2.5 Jarrow, Lando and Turnbull (1997) The model of Jarrow et al. (1997) is a Markov model for the term structure of credit spreads and is an extension of Jarrow and Turnbull’s paper from 1995. The life of a company is seen as a journey through possible credit rating states with one absorbing state, identical to bankruptcy. These transition parameters can be observed from credit rating agencies and are assumed to give a good measure of the creditworthiness of the company. Even though ratings are known to be rather slow in reacting to new market information they are accepted to be a major variable, on which credit spreads can be based. (Cossin and Pirotte 2001)


26

The K x K discrete, finite-state Markov chain ( ): 0t tη τ≤ ≤ transition matrix is specified as

11 12 1

21 22 2

1,1 1,2 1,

...

...= ...

...0 0 ... 1

K

K

K K K K

q q qq q q

q q q− − −

Q=

.

where 0ijq ≥ for all i, j, i j≠ . The (i,j)th entry is the true probability of going from state i to state j in one time step. The highest credit rating is defined to correspond to state 1 and the lowest credit rating is in state K-1. State K is the absorbing state that corresponds to default. Now historical transition matrices can be used as Q. Under the assumption of no arbitrage opportunities we have to calculate a similar transition matrix under the equivalent probability measure such that the credit rating process is given by % ( ), 1 ( ) i ijijq t t t q i jπ+ = ∀ ≠ in which ( )i tπ is a deterministic function of time. The risk-adjusted probability of default after time T is

° ( ) % ( )Q * , 1 ( , )i

t iKijj K

T q t T q t Tτ≠

> = = −∑

where { }* inf : ss t Kτ η= ≥ = represents the first time of bankruptcy. Furthermore

( ) °, ( , )( (1 ) ( * ))iitB t T B t T Q Tδ δ τ= + − >

is the value of a zero-coupon bond issued by a firm currently rated in credit class i. The term structure of the default-free spot rate can be modeled with any appropriate model, like for example Black, Derman and Toy (1990), Cox, Ingersoll and Ross (1985) or Heath, Jarrow and Morton (1992), as it is assumed to be independent of the credit rating process. Jarrow, Lando and Turnbull also extends the credit ratings and default probabilities to apply in the continuous case. Recovery is modeled as a fraction of face value and is considered to be an exogenously given constant, δ . Jarrow et al. suggest that this constant can be regulated for different seniority debts in order to incorporate recovery differences into their model. As in Jarrow and Turnbull (1995) they assume the recovery rate to be independent from the stochastic process driving the term structures, however, this is relaxed by Das and Tufano (1995). Das (2001) gives a step-by-step example on how the model can be used and calibrated on the basis of real data. The model’s simplicity also gives rise to some of its major drawbacks. The assumption that recovery rates are constant is in most cases not applicable in the real world. The


27

same also applies to the modeling of intensities, which are set to be constant over time and also equal for all bonds in the same rating class. (Schmidt 2001) Lando (1998) makes a further generalization by allowing default intensities and transitions to be governed by a Cox-process and the hazard rate to be dependent on the risk-free short interest rate. The result is a more flexible model, which can depend on different state variables of interest, but infers a computational cost.

3.2.6 Duffie and Singleton (1999) Duffie and Singleton follow another approach than the two models already mentioned. Their model has its foundation in Madan and Unal (1994). Under the risk-neutral probability measure they denote the hazard rate, ht, as an exogenous process governing the default process and Lt the expected fractional loss conditional on information available up to time t. Over a small, discrete interval t∆ , a default will then occur with the risk-neutral probability th t∆ . The recovery to creditors stochastic and is assumed to follow a recovery of market value (RMV) model, though Duffie and Singleton also give a discussion on other recovery models. RMV allows for correlation between the hazard rate, loss process and the term structure of the default-free interest rate. Their objective is to be able to describe the price of a defaultable bond given with the following equation

( ) °0

0

T

uR duPB T Xe

− ∫ =

E

where X is the face value of the bond. In the following we will use X=1 for simplicity and R is a default-adjusted short-rate process, replacing the usual short-term interest rate. Cossin and Pirotte (2001) gives an intuitive explanation to the relation between these interest rate processes and shows that it must satisfy

[ ]1 1(1 ) (1 )

1 1h t L h t

R r= ∆ − + − ∆

+ +

If we take the limit and let 0t∆ → , we get R R hL≈ + . Under certain circumstances in the continuous case this representation of R is precise. (Cossin and Pirotte 2001) Duffie and Singleton also propose another factor in this equation, denoted as liquidity effects, l , in order to account for carrying cost. The expression for the default-adjusted interest rate then becomes R r hL= + + l . The processes ht and Lt are often difficult to identify separately using defaultable bond yields. However, Duffie and Singleton state that for some derivatives this does not imply


28

any problems. As an example they explain how bond option prices can be used to separate these parameters and how this can be used to price a credit-spread put option.

3.3 Hybrid Models

Recently there have been proposed models incorporating features from both the structural and the reduced form methodology. In chapter 4 we will consider the model proposed by Zhou (1997a) that can be regarded as a hybrid model. First, however, we will give a brief overview of a hybrid model proposed by Giesecke (2001) with a slightly different foundation.

3.3.1 Giesecke (2001) Giesecke’s model (2001), like Zhou (1997a), has its foundation in the classical structural approach. The structural model face especially two difficulties that are not consistent with the real market. Firstly, short-term credit spreads above zero is not possible. As one knows there is no default today, one is not interested in paying a default premium for bonds with short maturities. Secondly, in these models bond prices converge continuously to their recovery value, though in the market price jumps are frequently observed in the case of default. Giesecke (2001) presents an extended model, which faces these problems through the assumption of incomplete information. Duffie and Lando (2001) established a model where firm’s assets cannot be observed perfectly by investors as they only get incomplete and noisy reports on the firm’s true assets. This way unexpected default can happen, as in reduced form models. However, they still assume that investors have complete information about the default barrier, so that a company will be forced into bankruptcy when the barrier is triggered:

Figure 3.2 Short-term credit spreads in the case of incomplete information (Duffie and Lando 2001)


29

Giesecke (2001) generalizes the model further as he allows for uncertainty about the barrier level as well as the firm’s assets. Surprisingly, though, a different kind of incomplete information results in a different term-structure. The reason is that we observe the historical low of the assets as time passes by and accordingly get an upper bound on the barrier. The table below summarizes the different kind of and combinations of uncertainties about firm asset and the barrier, and shows the resulting term-structure and short-term credit spreads.

Information Incomplete

Barrier

Complete

Assets At Historical Low

Above Historical Low

Both

Term Structure

Hump-shaped

Hump-shaped

Decreasing Hump-shaped

Decreasing

Short Spreads

Zero Non-zero Non-zero Zero Non-zero

Table 3.3 Information with Giesecke

Giesecke (2001) illustrates the Black and Cox model and provides a simpler solution by considering the default barrier constant ( )K t K= as in the Merton model. Giesecke defines a running minimum log-asset process ( ) for 0M t t ≥

21( ) min(( ) )

2 v v ss tM t s Wµ σ σ

≤= − +

where M is the historical low of the log-asset value. Then the default probability becomes

[ ] ( ) ln(0)K

P T P M tV

τ

< = ≤

.

To derive this probability he uses the fact that ( )M t is inverse Gaussian and obtains

[ ]

2

2

2 22( )ln(0)( ) ln ln ( )

(0) (0)1

v

v

KVv v

v v

K KT T

V VP T N e N

T T

µ σ

σ

µ σ µ στ

σ σ

−

− − + −

≤ = − +

Giesecke develops a reduced form framework that can be applied to all credit models predicated on unexpected default. This does not only apply to the intensity based models, but also Giesecke’s own model. Through this framework he is able to derive closed-form solutions under specific conditions and also to value credit spreads by making use of compensators. Compensators may be explained by the Doob-Meyer decomposition


30

theorem. (Neftci 2000) It states that a continuous-time process can be decomposed into a martingale and a drifting process, which in our case is the compensator. Hence, the compensator simply equals the expected intensity of the indicator process. The model is economically intuitive as it reflects the true world in which investors seldom have complete information. It also makes sense when we take into account the recent surprising bankruptcies at Enron and WorldCom where arrival of new information gave unexpected news about firms’ assets and liabilities. The model requires the following input: daily equity prices, equity volatility forecasts, reported liabilities and interest rates. These data may be obtained in the market, thus making the model easy to calibrate. Other variables, like firm volatility and initial firm value derives from the initial parameters with option pricing formulas.


31

4 Zhou’s Model

There are some major drawbacks with the structural and reduced form approaches to modeling credit risk. This chapter will be devoted to present a model proposed by Zhou (1997a; 2001), which tries to incorporate the advantages of both approaches described above. The model can just as well be categorized as a second-generation structural model as a hybrid model, due to its foundation built on the economically sound structural framework. At the same time it allows for better fit to real data by allowing jumps in the firm value to occur. We will first present the foundation of the model and then move on to a closed-form solution valid under a certain assumption. Further, we give a Monte Carlo solution for the general case as proposed by Zhou. Next, we present and correspond the results from the two solution approaches and use the implementation to price a credit default swap. Finally, we suggest how the model can be calibrated by the use of Markov chain Monte Carlo and possible further extensions in order to make it fit real market data.

4.1 The Model

The model is described by a jump-diffusion process predicting the market value, V, of a firm’s total assets according to the following process

1( ) ( 1)t

t

dVdt dZ dY

Vµ λν σ= − + + Π −

where the variables have the interpretation stated in the table below. It is assumed that the firm is funded solely by equity and zero-coupon bonds.

tV Total market value of firm, i.e. assets + debt

Z1 A standard Brownian motion

dY A Poisson process (allowing one jump)

λ Intensity parameter for the Poisson process dY

σ Volatility of the firm’s assets

Π Jump amplitude with ( )E Π = 1ν + and 2ln( ) ~ ( )N π πµ σΠ +

ν 2

2: [ 1] exp( ) 1E πσπν µ= Π − = + −

µ Expected drift in the value of the firm

dt An infinitesimal time step

Table 4.1 Description of parameters used in Zhou's model


32

dZ1, dY and Π are mutually independent. This process can be interpreted as follows; The firm value is subject to a standard Brownian motion for all infinitesimal time steps dt. Jumps occur with an intensity λ and expected amplitude ν . Given that µ is the expected drift in the diffusion process, we need to compensate the asset process by subtracting λν to obtain the desired expected drift. The firm defaults on all its obligations immediately if its value falls on or below a time-dependent positive threshold value Kt. Zhou assumes that K is a deterministic function that follows 0

tK e Kϕ= . By doing so, ϕ can be interpreted as the growth rate of the firm’s liabilities (Zhou 2001). However, we will in the following set 0ϕ = implying that K is a constant equal to Kt. Now, define Xt as the ratio of the firm’s value to the barrier: t

t

Vt KX = implying default

when 1tX ≤ . Given a default at time t , the bondholder receives a fraction 1 ( )w Xτ− of face value at maturity time, T, where w(Xt) is a function determining the write-down of the bond. Normally w(Xt) is a non-increasing function of X, so that the lower the firm value at default, the less the bondholder receives. Zhou uses the function w(X)=w0-w1X with w0 and w1 as non-negative constants to determine the write-down. In order to make the model more realistic, one may use different write-down functions for different grades of seniority. In reality, the write-down functions will differ for even the same seniority class of bonds and for different time periods (Zhou 2001). According to Zhou (1997a) the capital asset pricing model (CAPM) is still assumed to hold for equilibrium security returns while the jump component is purely firm specific and uncorrelated with the market. Consequently the ß for the jumps must equal 0, again implying that there is no risk premium for the jump risk.12 Giesecke and Goldberg (2003) claim that Zhou’s jump-diffusion model adds too much complexity into the problem in order to incorporate the capability of positive short-term spreads. One of the major drawbacks with Zhou’s model is its complexity when it comes to calibrating the model to market data. We will come back to this issue in section 4.6, in which we describe in detail how it may be done. Another problem discussed by Giesecke (2001) is that jumps do not necessarily imply default as is the case with reduced form models. Actually the jump may not be large enough to force the firm value down to or beyond the barrier making it hard to correlate default rates actually observed in the market with the jump intensity in Zhou’s model. On the other side the model has several nice features. First of all it has the flexibility needed in order to accommodate for the variety of term structures observed in the market, including above zero short-term credit spreads. Also the model is consistent with the fact

12 According to CAPM the risk premium = ß(rj-rf)


33

that there is often observed a jump in the bond value at the moment of default. Finally, the model differs from the structural models as it correlates the recovery rate to the firm value at default, whereby generating a random recovery rate endogenously.

4.2 Closed-Form Solution

Zhou (1997a) provides a closed-form solution for valuing debt under certain conditions. The main difference from the general model is that the debt cannot default before maturity. As all debt has to be paid at maturity, the default barrier has to be equal to the face value of debt. In the following we derive a closed-form solution for valuing equity under the same assumptions. Analogous to Zhou when calculating the risk-neutral price of a bond we assume that investors do not receive a risk premium for the jump risk. Under these assumptions the put-call parity holds if there are no bankruptcy costs, i.e. the value of the firm is equal to the risk-neutral value of the equity and debt. When there are bankruptcy costs there will still not exist arbitrage opportunities since the firm’s assets are not traded. We want to calculate the expected value of the equity under the risk-neutral probability measure °P . We know that the firm will default if the firm’s value is below the face value of debt at maturity. As we define /X V K= we know that the firm will default if

1TX ≤ and from a risk-neutrality argument we have for K = 1 = face value of debt:

( ) ° ( )( ), max( ,0)rT PTS X T e E KX K−= −

Under °P the original SDE

( ) ( )ln 1dX

r dt dZ dYX

λν σ= − + + ∏ −

can be rewritten as: 2ln( ) ( / 2 ) ln( )d X r dt dZ dYσ λν σ= − − + + ∏

We further define

° ( ) ° °| : ( | ) (ln( ) ln( ) | )PT T TG X P X X P X Xξ ξ ξ= ≥ = ≥

which is the probability that TX ξ≥ given the current X. In the following 1ξ = due to the discussion above. To find the probability

° ( )1|PTG X we first find the distribution of ln( )X for a given

number of jumps. Given a certain number of jumps, i, before maturity, ln( )TX will be normally distributed with the mean, ndµ , including the drift of the jump component


34

multiplied by the number of jumps. Similarly the variance, 2ndσ , will be the total variance

combining the diffusion and jump component: 2

2 2 2

ln( )2nd

nd

X r T i

T i

π

π

σµ λν µ

σ σ σ

= + − − +

= +

We let TY be the number of jumps from time 0 to T to get

22 2ln( | , ) ln( ) ( ) ,

2T TX X Y i N X r T i T iπ πσ

λν µ σ σ

= + − − + +

∼

The probability of a certain number of jumps is given by the Poisson distribution

° ( )( )

!

T i

T

e TP Y i

i

λ λ−

= =

We can then multiply the probabilities of a certain number of jumps with the normal distribution given the number of jumps and sum up.

° ( ) ° ( )

° ( ) ° ( )0

2 20

1| ln( ) ln(1) | ln( )

ln( ) ln(1| ,

ln(1) ln( ) ( )( ) 2!

PT T

T T Ti

T i T

i

G X P X X

P Y i P X X Y i

X r T ie TN

i T i

λ π

π

σλν µλ

σ σ

∞

=

−∞

=

= ≥

= = ⋅ ≥ =

− − − − − = ⋅ −

+

∑

∑

(0.2)

Merton (1974) consider equity as an option on the firm’s asset with strike equal to the total debt. Similarly, the

°PTG we have calculated represents the probability that the option

(equity) is in the money at expiration. It is important to notice, however, that this probability also includes the cases were the firm’s value falls below the barrier in the interval 0 to T and rises again to be above the barrier at time T. To calculate the value of equity at time 0 we need to find the expected value of the equity at time T. We find this by calculating the equity value given no default at time T and multiplying it with the probability of no default.

° ( ) °| 1 (1| )P P

T T TE X X G X≥ ⋅


35

Above we found that ln( )X is normally distributed with ndµ and ndσ . For each of the possible number of jumps i we define

° ( ) ° ( )

( )

2

2

2

2

2

2

ln( ) 0

2 2

ln( ) 0

2 2

2

2

2

2

| 1, 1|

2

2

2

1

nd nd

T

ndndnd

T

ndnd

nd nd

ndnd

ndnd

P Pi T T T T

zz

X

z

X

z

z

nd nd

nd

nd nd

nd

r

h E X X Y i G X

dze e

dze e

dze e

e N

e N

Xe

µ σ

σσµ

σµ

µ σ

σµ

σµ

σλν

π

π

π

µ σσ

µ σσ

∞−+

=

−∞+ −

=

∞+ −

− −=

+

+

− −

= ≥ =

= ⋅

=

=

− −= −

+=

=

∫

∫

∫

2 2

22 2

2

2 2

ln( )2

T iT i

X r T i T iN

T i

ππ

σ σ π πµ

π

σλν µ σ σ

σ σ

++ +

+ − − + + +

+

We will then sum for the possible number of jumps to get the expected value up to n jumps.

° ( ) °

2 2 2

22 2

2 2

2 20

| 1 (1| )

ln( )2( )

!

V v

P PT T T

VT i vT i r T i

i v

E X X G X

X r T i T ie T

Xe Ni T i

ππ

σ σ σ π πλ λν µ

π

σλν µ σ σ

λ

σ σ

+− − − + +∞

=

≥ ⋅

+ − − + + +

= +

∑(0.3)


36

We can now combine (0.2) and (0.3) to calculate the risk-neutral value of the equity:

( ) ° ( )( )° ( ) ( )( ) ° ( )( )° ( ) ° ( ) ° ( ) ° ( )( )° ( ) ° ( ) ° ( )( )

2 22

22 2

2 2

2

, max( ,0)

| 1 1| 1 1|

| 1 1| 1| 1 1|

| 1 1| 1|

ln( )2( )

!

rT PT

rT P PT T T T

rT P P P PT T T T T

rT P P PT T T T

T iT i r T i

rT

S X T e E KX K

e E X X X G X

e E X X G X E X G X

e E X X G X G X

X r T i T ie T

Xe Ne i T

ππ

σ σσ π πλ λν µ

σλν µ σ σ

λ

σ

−

−

−

−

+− − − + +

−

= −

= > − >

= > − >

= > −

+ − − + + +

= +

2 22

2

0

2 20

2

2 2

ln(1) ln( ) ( )( ) 2!

ln( )2( )

!

i

T i T

i

T iT i r T i

rT

i

X r T ie TN

i T i

X r T ie T

Xe Ni

e

ππ

π

λ π

π

σ σσ πλ λν µ

σ

σλν µλ

σ σ

σλν µ

λ

∞

=

−∞

=

+− − − + +

−

− − − − − − −

+

+ − − +

=

∑

∑

2 2

2 2

0

2 2

ln( ) ln( ) ( )2

i

T

T i

T i

X r T iN

T i

π

π

π

π

σ σ

σ σ

σξ λν µ

σ σ

∞

=

+ +

+

− − − − − − − +

∑

We see that our result is somewhat similar to the result in the E2C model described previously. In the E2C model the firm’s assets follow a diffusion process and the default barrier jumps at maturity. This is pretty much the same as in our result where a jump will trigger a default in the same way. However, in our result there can be several jumps, which will be significant if λ is large (Craine et al. 2000). Similarly to the E2C model we could also have expanded our result to calculate an equity hedge for credit. With our result the equity can be viewed as a European call option on the firm’s assets as in Merton’s model. In the general framework, in which the firm can default before the maturity of the debt, the analogous option is a path dependent option or more specifically a down-and-out call option. It is obvious that with the same strikes a European call will be more valuable than a down-and out call. Similarly the debt will be less valuable in the closed-form solution provided by Zhou. It is evident that both equity and debt as options on the firm’s value is in fact path dependent and that fist passage time modeling, as done by Black and Cox (1976) for a


37

pure diffusion process, provides a more realistic representation of reality than the Merton model. However, when the underlying path includes jumps, the options become so complex that they most often do not have a closed-form solution. In our simulation described in the next section we will include the calculation of equity prices under risk-neutrality as suggested by Zhou (1997b).

4.3 Implementation

We implement Zhou’s model by the use of Monte Carlo methods. Below we will give a primer on Monte Carlo methods before moving on to a discussion on the solution of the model and the algorithm we use in the implementation. Finally, we discuss some of the practical concerns related to the implementation, including the choice of programming language and parameter values.

4.3.1 Primer on Monte Carlo Methods Monte Carlo (MC) simulation is a powerful and straightforward technique frequently used to solve financial problems. The most common applications include high dimensions integration and path-dependent problems. Let X be a discrete random vector with a set of possible values xj and a probability density function given by { } , 1jP j= ≥X x . Now suppose that we are interested in calculating

( ) { }1

( )j jj

E h h Pθ∞

=

= = = ∑X x X x

for some specified function h. Frequently h(x) is hard to calculate exactly and we have to turn to approximation. Simulation is then a powerful approach. (Ross 2000) By the central limit theorem and the law of large numbers (Glasserman 2003) one is ensured that an estimate based on several simulations is approaching its correct value as the number of simulations is increasing to infinity. Let X1, X2, …, Xn be independent and identical distributed random vectors from the given mass function. Our Monte Carlo estimate is then given by

$1

1lim ( )

n

ini

hn

θ→∞

=

≈ ∑ X

and the standard deviation of the estimate is inversely proportional to the square root of the number of simulations:

1n

nθσ σ=


38

In order to reduce the error by half, the number of simulations has to be increased four times. As a result, Monte Carlo is a computational intensive method, though it enjoys great flexibility making it applicable to a wide range of problems. One of the major strengths of Monte Carlo methods is related to path-dependent problems. Through the simulation of stochastic processes, one does not only get the final result but also the development path for the process. For instance is MC a common way of valuing Asian options as they are dependent on the path a stock takes as well as its final value. Also when it comes to modeling credit risk MC forms an intuitive method in simulating the value of a firm’s assets, as will be described in the next chapter.

4.3.2 Solution Though there under certain conditions exist closed-form solutions for credit risk models, one often needs numerical methods in order to find approximate solutions. Monte Carlo methods serve as a powerful option despite its computational disadvantage. In most cases it is an easy way of implementing and experimenting with models, and the addition of new features to the model is normally straightforward. Zhou (2001) describes a Monte Carlo approach to valuing a defaultable bond and we here render the most important steps of the algorithm:

1) Divide the time interval [0,T] into n equal sub periods. 2) Let

0

*tX X= .

For j=1 to m - Generate the independent random variables xi p,i, yi according to the

following distributions: 2

2,2i

T Tx N r

n nσ

φ λν σ

− − −

∼

( )2,i N π ππ µ σ∼

T0 with prob. 1-

nT

1 with prob. n

iyλ

λ

∼

- Calculate * *1ln( ) ln( ) for i=1,...,n

i it t i i iX X x y π−= + +

- Find the smallest integer i n≤ such that 1it

X ≤ . If such an i exists, let

the write-down *( )ij tW w X= 13. Otherwise Wj=0.

3) Calculate the value of the defaultable bond according to the expression:

( )

1

1( , ) 1

MrT

jj

B X T e wM

−

=

= −

∑ .

13 We base our write-down function on Zhou’s suggestion, but impose a liability restriction in order to

avoid recovery rates below 0. Our write-down function is therefore w(X)=min{w0-w1X,1}.


39

The complete computer code can be found in the appendix and run from the accompanying CD.

4.3.3 Implementation Issues Due to the fact that Monte Carlo methods are computer intensive we choose to implement the model in the C programming language and Visual Basic for Applications (VBA). With our speed test it turns out that the same algorithm runs 97% faster in C than in VBA. This number varies with the implementation and tuning, but it hints to the speed potential of C. The choice of C for another compiled language14 is due to the fact that C is a wide spread industry standard.15 Therefore, we implement the number crunching algorithms using the C compiler Bloodshed Dev-C++ version 4.9.8.3 Beta (Bloodshed 2003) on the Windows platform. In order to create user-interfaces and display graphics, we chose the industry standard spreadsheet, Microsoft Excel with VBA. VBA includes an option to link external programs with it by using dynamically linked libraries (dll). One of the major concerns regarding C is that the built-in random number generator for uniform deviates has poor performance (Press et al. 2002). The uniform deviates are essential in Monte Carlo simulation and stochastic modeling. As a consequence we use Mersenne Twister to implement a uniform random number generator (Mersenne Twister 2002). The Zhou-algorithm also makes use of standard normal distributed numbers, so in order to sample from this distribution we use the Box-Muller algorithm described in Glasserman (2003). Our first runs of the model are designed to verify that the principle of Monte Carlo works as expected. Figure 4.1 shows how the bond value converges towards the correct number as the number of simulations increases. In general the uncertainty is given by the standard error ˆ /M M Mε σ= (Jäckel 2001)16. ˆMσ is normally uncertain and could be subject to further analysis17. We normally use

50000 in this paper, as a trade-off between accuracy and speed.

14 The code is pre-compiled to binary file format, in contrast to VBA code, which runs as it is typed. 15 C is a broadly speaking a subset of C++ programming language, where the main advantage of the latter is

its support for object orientation – a feature we did not need. 16 Jäckel uses N, but in order to follow Zhou (1997a) and our code we use M for number of simulations. 17 See for example Ross (2000) for a quick method on how to estimate ˆNσ and the number of simulations.


40

0,577

0,579

0,581

0,583

0,585

0,587

0,589

0 10 000 20 000 30 000 40 000 50 000 60 000 70 000 80 000 90 000 100 000

Number of simulations m

Bo

nd

val

ue

The maturity time is split into n intervals to emulate continuous time. We are interested in keeping this number as low as possible because the Monte Carlo simulation time is close to linearly dependent of n. Our variation of time steps yields Figure 4.2.

0,250

0,300

0,350

0,400

0,450

0,500

0,550

0,600

0,650

0 50 100 150 200 250 300 350 400 450 500

No of t i me steps

X = V/K

rf Mat. T Time step

No. MC

w_0 w_1 λ πµ 2

πσ 2.Diffσ Xσ

2.0 5% Varying Varying 50000 1.4 1.0 1.0 0.0 0.25 0.0225 0.035 Figure 4.2 Varying number of time steps in Monte Carlo simulation

With a low number of time steps the bond value is a few percents higher than at the converged part of the graph. The intuition comes as the value process is not the regular Brownian motion looking path, but rather a straight line. As such, the implementation is embedded in just one step and we include those paths that would have otherwise passed the default barrier in situations with higher resolution. When n passes approximately 50, the graph converges. We use n=100 further in this paper.

X = V/K

rf Mat. T Time step

No. MC w_0 w_1 λ πµ 2

πσ 2.Diffσ Xσ

2.0 5% 10 100 Varying 1.4 1.0 0.05 0.0 0.25 0.0225 0.035

Figure 4.1 Convergence of Monte Carlo simulation


41

4.4 Pricing Credit Default Swaps

As previously described, the credit default swap involves the exchange of regular payments at fixed intervals for a contingent payment in the case of a credit event. Zhou (1997a) sets the first steps into pricing a CDS and we use it as a basis when we derive an expression for the credit default spread, sCDS. Let G(Xt) be the default contingent paid at default. If we let A(t ,T) be the present value of G(Xt) it follows that it will be equal to the discounted expected write-down under the risk-adjusted probability measure:

( ) ( ) ° ( ), r PTA T e E G X Iτ

τ ττ −≤= ⋅ ⋅

By using the results from the Monte Carlo simulation we know the time of default, t , and the write-down, ( )w Xτ of the bond. W(XT)=0 when no default occurs so the present value of the contingent payment turns into:

( ) ( ) ( )( )1 (1 )r TrA X e e w Xτττ τ

− −−= − −

As we know t , it is also trivial to calculate the number of contingent legs that are paid on the CDS before default or maturity; { }( , , ) min ,C T p T p pτ τ= ⋅ ⋅ where p is the number of contingent payments per year. For simplicity we assume that the protection buyer must pay the periodic fee for the whole outstanding period if default happens between two payment dates. The present value of the default payment must equal the present value of all regular payments. In order to calculate the periodic payment, K, we use the well known amortization formula:

( ) ( ) ( )( )( , , )

, 1, ,

1 C T p

A T dK C d A

d d τ

τ ⋅ −=

⋅ −

where d is the periodic discount factor given by rpd e

−= . Finally, we calculate the CDS

spread by using amortization on all payments during a year:

( ) ( )( )

( )( )( , , )

, , 1 1( , )

1 1

p pCDS

C T p

d K C d A d ds A T

d d ττ

⋅ ⋅ − −= =

− −

In the following chapter we will make use of this formula during the simulation in order to estimate the credit spread.

4.5 Results

Here we present some findings that shed light of the mechanics of the model. In general we use Zhou’s (2001) values as reasonable initial values and keep the short-term risk-free


42

interest rate constant. Below we give an analysis of the output when varying a limited number of parameters simultaneously. By keeping the volatility of the assets and diffusion process constant over time,

2 2 2X π πσ σ λ σ= + ⋅ , we may investigate the importance of varying πλ and 2

πσ while keeping their product constant.

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10

Maturity

Bon

d sp

read Jump_vol=1

Jump_intensity=0,0125

Jump_vol=0,25Jump_intensity=0,05

Jump_vol=0,00325Jump_intensity=3,84615384615385

X = V/K

rf Mat. T Time step

No. MC W0 W1 λ πµ 2

πσ 2.Diffσ 2

Xσ

2.0 5% Varying 100 50000 1.4 1.0 Varying 0.0 Varying 2,25% 3,5%

Figure 4.3 Varying jump intensity and jump volatility while keeping total asset volatility constant

For the graph with high default intensity and low jump volatility we clearly see the reason behind the criticism of the structural models. At short maturities, the jumps are too small to cause sudden default, thereby leading to zero short-term bond spread. For longer maturities however, the frequent jumps add to the diffusion volatility and increase the spread to the other curves. Jump intensity equal to 0.05 creates a high spread throughout the period. This is because the balance of frequent jumps with considerable amplitude creates jumps that are large enough to cause a default, while they also happen sufficiently frequently. If we instead hold lambda constant and distribute the still fixed total volatility between the jump volatility and Brownian motion component we get the graph represented below.


43

0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8 9 10

Maturity

Bo

nd

sp

read

Diffusion_vol=0,001Jump_vol=0,68



X = V/K

rf Mat. T Time step


πσ 2.Diffσ 2

Xσ

2.0 5% Varying 100 50000 1.4 1.0 0.05 0.0 Varying Varying 3,5%

Figure 4.4 Varying jump and diffusion volatility while keeping total asset volatility and jump intensity constant

Again we notice that when the variance of the jump component is close to zero the short-term bond spread is close to zero. Due to the relationship between firm value at default and the write-down it is of major importance whether the default is caused by a jump or by the diffusion process. Values below the barrier may only be reached by jumps and lead to high write-down and low recovery. Under the observations made here, short-maturity bonds have usually lower expected recovery rates than bonds with longer maturity (Zhou 2001). At the other extreme we have maximum variance in the jump component. Here the Brownian motion is weak, and we observe a declining bond spread. In most textbooks the term structure of interest rates are considered to be upward sloping. However, as Zhou (2001) points out, there exist hump-shaped credit curves and even downward sloping term structures. Especially bonds with high credit risk and low credit rating (B-rated or BB-rated) may have a curve following the latter structure. The reason lies in the way default risk is regarded for these firms. They are rated as junk bonds today, leaving them with a high bond spread and a high default probability. If they still survive in the long run it may be interpreted as a sign that the company has sound performance and thereby deserves a lower bond spread given that they survive the first critical years.


44

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10

Time

Bo

nd

sp

read

X_0=1.3

X_0=1.70

X_0=2.50

X = V/K

rf Mat. T Time step


πσ 2.Diffσ 2

Xσ

Varying 5% Varying 100 50000 1.4 1.0 0.05 0.0 25% 2.25% 3.5%

Figure 4.5 Varying the initial debt to asset value

X0 is the firm value to default threshold ratio. Varying the initial ratio has substantial impact on the bond spread. Higher X0 yields lower bond spreads as the initial probability of default is lower. When the debt ratio is high, i.e. X0 is low, the bond spread decreases again if the maturity is longer than approximately two years. This is because the probability of default is large in the beginning of the period. If the firm is still alive after a few years, it is more likely that it will make it till the maturity of the bond. These results are sensitive to the choice of the other parameters and thus we cannot draw any conclusions based on the results presented here.

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10

Maturity

Bo

nd

sp

read r= 0,02

r= 0,06

r= 0,15

X = V/K

rf Mat. T Time step

No. MC

W0 W1 λ πµ 2

πσ 2.Diffσ 2

Xσ

2.0 Varying Varying 100 50000 1.4 1.0 0.05 0.0 25% 2,25% 3,5%

Figure 4.6 Varying the risk-free interest rate


45

Now we vary the risk-free interest rate. At short maturities the rate has little impact and all graphs give approximately similar bond spreads. The reason is that at short maturity, the spread is mainly caused by the jump term of the process making the spread less dependent on the influence of r on the process drift. More interesting is what happens at longer maturities. With high interest rates, the expected drift of the assets is high. This drift will diminish the possibilities for default, leading to low bonds spreads at longer maturities. For low rates we get the opposite effect, where the long maturity augments the possibility for default and thus increases the bond spread. It may also be inviting to think that the spread is influenced by the discount rate of the bond, but the following calculation shows that this assumption is not correct:

[ ]( )[ ]( )( )

( ) [ ]( )

[ ]( )

ln( ), 1

ln 1

ln ln 1

ln 1

rT

rT

rT

Bs r B e E W

Te E W

s rT

e E Wr

T TE W

T

−

−

−

−= − = −

− −= −

− −= − −

−= −

where s is the bond spread and E[W] is the simulated write-down. In order to illustrate the difference between the simulation and the closed-form solution we have derived, we review Figure 4.4 again with the closed-form solutions included.

-5

15

35

55

75

95

115

0 1 2 3 4 5 6 7 8 9 10

Maturity

Bo

nd

sp

read




Closed_f. Diff.vol=0,001J.vol=0,68

Closed_f. Diff.vol.=0,0175J.vol=0,35

Closed_f. Diff.vol.=0,034J.vol=0,02

X = V/K

rf Mat. T Time step

No. MC

W0 W1 λ πµ 2

πσ 2.Diffσ 2

Xσ

2.0 5% Varying 100 50000 1.4 1.0 0.05 0.0 Varying Varying 3,5%

Figure 4.7 Corresponding simulation and closed-form solution while varying jump and diffusion volatility and keeping total asset volatility and jump intensity constant.


46

The closed-form solution is similar to Merton in that it only accepts default at the end of the period; Tτ = . Economically this means that the shareholder’s option on the firm value has become a regular call option, in contrast to a knock-out barrier option in the simulation. The uppermost graphs are calculated from a large πσ , displaying a large error compared to the simulated bond spread. This illustrates one of the advantages from a first passage time model compared to one of Merton type. The middle closed-form graph seems to fit quite well with the simulation, while the one with a higher diffusion build up error with maturity. The large errors produced illustrate the effect of erroneously modeling equity as a European option. There are two sources of error when using the closed-form. The lack of path dependency results in fewer defaults and a lower bond spread. This is also shown for the two data sets with lowest jump volatility. However, raising the jump volatility yields higher bond spreads with closed-form solution than with simulation. Firms are allowed to continue past the default barrier until maturity of the bond, resulting in higher write-downs than would have otherwise occurred. With higher jump volatility, the asset values are allowed to drift well below the barrier and thereby increasing the bond spread above the simulated spread.

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9 10

Maturity

Bo

nd

sp

read

/ CD

S s

pre

ad

Bond_spread:Diffusion_vol=0,001Jump_vol=0,68

CDS_spread:Diffusion_vol=0,001Jump_vol=0,68





X = V/K

rf Mat. T Time step

No. MC

w_0 w_1 λ πµ 2

πσ 2.Diffσ Xσ CDS

year K

2.0 5% Varying 100 50000 1.4 1.0 0.05 0.0 Varying Varying 3,5% 2 1

Figure 4.8 Corresponding bond spread with credit default swap spread

From Figure 4.8 we see that the credit spread is always higher than the bond spread. This result is as expected and follows by a closer investigation of what happens at default. A bondholder will receive the discounted recovery value from the maturity of the bond. With the CDS the situation is slightly different as the protection buyer is paid off immediately. As shown in section 4.4 he will receive the difference between notional and


47

the recovery value at maturity, thus leaving him with a higher payoff than the bond obligor.

4.6 Calibration

There are several unknown parameters that need to be estimated in the model, but we will focus on the estimation of five of them { }, , , ,π πµ σ µ σ λ . To our knowledge, no attempt has been made to estimate the parameters of the Zhou model and test them empirically. As opposed to the jump-diffusion models proposed by Merton (1976) and Ball and Torous (1985), Zhou is modeling the drift of firm’s assets and not the stock value. This issue raises the complexity of the calibration problem significantly. The reason lies in the fact that asset values cannot be inferred from the firm’s balance sheet and neither be observed in the market. As such, there exist no market data that can be applied directly to the calibration problem though there are still a great number of securities that can be used, of which we will discuss four; bond prices, CDS-prices, option prices, and stock prices. Bond prices and CDS-prices are only quoted for a few time intervals, thus providing limited information. The most probable solution is that several sets of parameters match our scarce data, but that none matches well with the data in general. Next, option prices are not quoted for long enough maturities, giving no information on what the market expects in the whole maturity period of the our simulation. If we instead of calibrating our model to the forward-implied volatility calibrate it to historical volatility, we can use the above data as well as equity prices to an accurate calibration. It is important to notice, however, that the parameters will change over time. Analogous to equity derivatives it would be more correct to use the implied volatility than the historical one. Schönbucher (2003) describes how firm value and firm volatility can be inferred from information on share prices, implied volatility on shares and the capitalization of the firm. He applies the Black and Scholes option pricing framework, though requiring that assets are purely governed by a diffusion process and that there is no path dependence. These assumptions could have been relaxed separately and solved respectively by the Merton (1976) or Black and Cox (1976) frameworks. However, the Zhou model comprises of jumps as well as path dependency when it comes to default detection. Consequently, the link between equity and assets are invalidated and more advanced techniques are required. Instead several of the data sources described above have to be combined. It is beyond the scope of this paper to go into detail, but such a model could be built by defining graphical


48

models specifying conditional independence. BUGS18 is a widely used program for analyzing and solving complex statistical problems without an exact solution. In order to estimate our model we choose an approximation. Still we make use of equity data, but we run the Zhou algorithm iteratively to find the real initial asset value. The algorithm is like the following:

1. Choose an arbitrary initial value for the assets, V0 2. Do until error is less than pre-specified level

- Run Zhou-simulation with Vi as firm value - Calculate new value by using the bond value from the simulation

Vi+1 = BondValue * FaceValueDebt + Equity

Figure 4.9 Convergence of algorithm calculating firm value. Initial firm value is set to 5000, face value of debt to 1500, equity to 1550 and other parameters to the values used in chapter 4.5.

This algorithm converges within a few simulations regardless of the initial choice of V0. (See Figure 4.9) Using this technique we have the market value of the debt today and can use it to calculate time series with asset values from historical share prices. The firm value will still change over time and it may be necessary to run the algorithm on fixed intervals. We assume that debt is continuously renewed everything equal and as such, that market value of debt is constant. Under the assumption that we have a time series of asset values, there exist a variety of ways the model could be calibrated. Traditionally jump-diffusion models have been estimated by using the maximum likelihood (ML) approach [Walpole et al. (1998); Johnson and Wichern (2002)]. Still ML is largely criticized and proved erroneous by empirical tests made by Honoré (1998), Craine et al. (2000) and Lin and Huang (2001). The problem lies in their likelihood function, which under certain conditions is

18 Bayesian Inference using Gibbs Sampling

0

1000

2000

3000

4000

5000

6000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Number of iterations

Firm

value


49

unbounded. Results show that the estimator ignores the jump process and splits the data into high and low volatility regimes for the jump and diffusion process respectively. The jump probability therefore turns out to be overestimated while the jump volatility is underestimated. Honoré (1998) suggests limiting the parameter set to avoid unbounded situations or to add a second jump term with constant jump volatility. In the following we will, however, turn to the use of Bayesian inference statistics and Markov Chain Monte Carlo simulation.

4.6.1 Markov Chain Monte Carlo Let us turn to an estimation method with a wide number of possible applications within finance as well as for other statistical purposes. By the use of Markov chain Monte Carlo (MCMC) one calculates a time-reversible Markov chain with a transition kernel ( )|q y x for proposing the new candidate y when the chain is in state x. In the following we present the underlying concepts of MCMC for the continuous case and show how these can be applied in estimating our model. (Johannes and Polson 2002) Under the assumption that the Markov chain is irreducible, aperiodic and recurrent it can be shown that the stationary distribution of the Markov chain converges to the posterior distribution for a set of unknown, stochastic parameters, T . The first step is to estimate a prior distribution for these parameters. Usually there is no certain information on these distributions, but for the most general MCMC algorithms it hardly imposes any problems. Nevertheless, as we will show, there exist certain conditions on the knowledge of priors that make a more efficient algorithm called the Gibbs sampler possible. Next, we need the likelihood function for T observed data points, V, given a set of parameters:

( ) ( )1

| |T

ii

V Vπ=

Θ = Θ∏l .

Finally, we need an expression for the density function for all unknown parameters given a set of observed data:

( ) ( ) ( )( ) ( )

||

|

VV

V d

ππ

π

Θ ΘΘ =

Θ Θ Θ∫l

l.

The integral, however, is generally non-trivial to solve analytically. As a consequence we use the Bayesian framework to calculate a proportional expression for the posterior distribution. Due to the fact that ( )Vπ is unknown and not dependent on T we form a posterior distribution that is not normalized, but which we could sample from:

( ) ( ) ( )( ) ( ) ( )|

| |V

V VV

ππ π

πΘ Θ

Θ = ∝ Θ Θl l


50

In cases where the likelihood function and the priors give a conditional posterior distribution that can be expressed in closed-form, one can make use of the Gibbs sampler due to the phenomenon of conditional conjugacy. A more general algorithm available is the Metropolis-Hastings algorithm, which does not require knowledge of all the conditional distributions in advance. The algorithm will be discussed in the following as we propose our estimation model.

4.6.2 Calibration Methodology In this section we suggest a methodology for estimating the model under the assumption that we have historical realization of the firm values. Based on Lin and Huang (2001) we assume the priors to have the distributions motivated below:

• The drift parameters are given normal distribution: ( )20,Nµ µ σ∼ and

( )21,Nπ πµ µ σ∼

• The standard deviation is always greater than zero. In the literature it is normal to use the inverse of the variance to estimate the prior distribution. We suggest using a chi-squared distribution with ν grades of freedom: 2 2s h

σσ σ νχ∼ and

2 2s hππ π νχ∼ where 2

1hσ σ

= and 2

1hπ

πσ=

• The probability of a jump through a certain time period is λ∆ and must be between 0 and 1. A proper choice for prior distribution will then be: ( )0 0,Betaλ α β∆ ∼

In order to avoid under- and overflow during the simulation it is a convenient standard to use log-scale when calculating the priors. There are a number of parameters in the prior distributions that are assumed to be predefined. These parameters have to be tuned to get an acceptance rate boosting the performance of the algorithm. Normally tuning the rate is not critical, but a recommended acceptance rate lies between 25 and 50%. Further, we assume mutually independence between the prior distributions. Hence, the prior density kernel can be described as:

( ) ( ) ( ) ( ) ( ) ( )π ππ π µ π µ π σ π σ π λΘ = ∆ In addition we need a proposal distribution ( )( 1) ( )|g gq +Θ Θ in order to make use of Metropolis-Hastings, which serves as a basis for our algorithm rendered below.


51

1. Set the initial parameters { }(0) , , , ,π πµ σ µ σ λΘ = and the parameters of the prior

distributions { }2 20 1 0 0, , , , , , ,s h s hσ σ π πσ σ α β .

2. Initialize a matrix, X, to store Θ from each step in the simulation. 3. For g = 1 to G + burn-in period

- Draw { }( )1, 2,...,5i Uniform∼ to decide which variable to update.

- Set ( ) ( 1)g gi i

−− −Θ = Θ where –i indicates all variables but the one that is

updated. - Calculate the asset values from the algorithm presented in the previous

section. - Generate a candidate value ( )* ( )| g

i i i iq −Θ Θ Θ∼ .

- Calculate ( ) ( ) ( )( ) ( )

* ( 1) *( 1) *

( 1) * ( 1)

|, exp min log ,0

|

gg

g g

q

q

πα

π

−−

− −

Θ Θ Θ Θ Θ = Θ Θ Θ .

- Draw ( )0,1U Uniform∼

- If ( )( 1) *,gU α −< Θ Θ then set ( ) *gi iΘ = Θ .

- Else set ( ) ( 1)g gi i

−Θ = Θ . - If g > burn-in periods then update matrix X.

4. Calculate average and standard deviation for each parameter using X. By choosing symmetric proposal distributions the acceptance probability turns into

( ) ( )( )

*( 1) *

( 1), min log ,0g

ge

πα

π−

−

Θ Θ Θ = Θ as ( ) ( )( 1) * * ( 1)| |g gq q− −Θ Θ = Θ Θ , greatly

simplifying the algorithm. For some of the distributions we are able to take advantage of conditional conjugacy and derive the posterior distribution and hence, use the Gibbs sampler. Lin and Huang (2001) show how the prior distributions above multiplied with the likelihood function can be turned into specific posterior distributions which can be drawn from directly. We implement an MCMC with the background in Lin and Huang (2001) by using the software package and programming language R.19 It is a basically a free, limited version of S-PLUS, which is widely used for statistical purposes due to its simplicity when it comes to data manipulation and graphical display. (Venables and Smith 2003) The full code and results from the estimation is found in the appendix (D).

19 R can be regarded as an implementation of the S language.


52

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

T r a c e p lo t d r i f t d i f f u s i o n

Ite r a t i o n s

Figure 4.10 Trace plot of drift diffusion parameter in MCMC estimation

A trace plot of each of the simulated parameters is a necessary tool to determine the efficiency and state of the process. If we consider the trace plot of the jump drift variable we see that the plot oscillates around an equilibrium, which is a typical pattern for this kind of plots. Still there is no guarantee of convergence20 and in many situations a large data sample is needed. It is normal to remove the first steps, the burn-in period, of the MCMC process to be able to calculate the empirical expectation and variance practically independent of the initial parameters.

4.7 Suggested Extensions

Even though the results presented in previous chapters capture the basic features observed in the market, there is a long way until we can conclude that the model represents the real world in a satisfactory manner. First of all, as we have also discussed in the previous chapter, the model has to be calibrated to historical and current data. Estimating the parameters is a lengthy process and involves computer intensive methods and a large amount of data. Moreover, as also mentioned by Zhou, the model lacks empirical testing with statistical data. The flexible framework makes the model suitable for most observed features regarding default probabilities, recovery rates and credit spreads. However, we await extensive research on how well the models adapt to the mentioned features.

20 It is impossible to prove convergence, but the suggested techniques give an idea of the state of the

process.


53

When it comes to the model itself, Zhou (1997a) suggests relaxing the assumption on constant short-term interest rate with a stochastic interest rate. The risk-free interest rate is then assumed to follow a diffusion process following the stochastic differential equation proposed by Vasicek (1977) 21

( ) 2dr r dt dZζ β η= − +

where dZ2 is a standard Brownian motion correlated with dZ1 in the jump-diffusion process of the assets by dtρ . Both dZ1 and dZ2 are still independent of dY and Π . As a result Zhou presents the following Feynman-Kac22 solution to the bond value under the risk-adjusted probability measure:

~

0( , , ) exp ( (1 ( ))

T

rdtP

T TB X r T E I w X Iτ τ τ

−

> ≤

∫ = + −

Using stochastic interest rate would add another parameter to our model, as there would be a need to estimate the correlation between the asset diffusion process and the interest rate. There are also reasons to believe that the volatility will be time-dependent and not constant as assumed in our implementation. A firm which is on the way to enter, or is already entering financial distress, is likely to have a different volatility than it had previous to this situation. Hand and Jacka (1998) give an overview of candidate models for stochastic volatility. Clearly such an extension will add further complexity to the model, but Craine et al. (2000) shows that the calibration of such a stochastic volatility jump-diffusion model yields substantially improved results compared to the estimation of a jump-diffusion process. Such a model will allow for volatility clustering without producing non-reasonable values of the jump intensity. Research shows that the jump intensity tends be overestimated in jump-diffusion models without a stochastic volatility term as data is arranged into a high and low volatility regimes (Craine et at. 2000; Honoré 1998; Lin and Huang 2001). Another possible extension to the model is to alter the write-down function we are currently using, in order to capture more advanced and realistic features that occur at default. It may also be necessary to use different functions depending on the seniority of the debt that is priced. Taking these suggested extensions into account it is obvious that the flexibility and power of Monte Carlo simulations make the inclusion of new features in the model easy. It may also be interesting to price other securities and derivatives than the zero-coupon bonds

21 Other interest rate processes could also have been used. 22 The Feynman-Kac theorem connects a class of conditional expectations of future payoffs under risk-

neutral measure and certain partial differential equations. (Jäckel 2001)


54

and credit default swaps we implement, and they only demand minor changes in the code in order to incorporate the desired structures. For example it would be trivial to use the framework to price coupon bonds or exotic structures with the same underlying. Another issue that is eligible for further investigation is the risk premium of jump risk, which is assumed to be zero due to the non-systematic nature of jumps. In the market this is not necessarily reality and Zhou’s framework may be a way to observe the actual risk premium demanded in the market. Even though the jumps are completely asset-specific and uncorrelated with the market, Merton (1976) claims that it in practice will be impossible to diversify the jump risk. Given that investors will not be able to diversify the jump risk there will be a premium for this risk. As jumps in asset value are not directly observable we propose to estimate this premium via the equity markets and equity option markets. If we use down-and-out equity options when estimating this risk premium it would be analogous to the equity with regard to the asset value. We would assume that the risk premium for jump risk with equity as the underlying would be equal to the jump risk when the asset value is the underlying.


55

5 Concluding Remarks

This paper gives an overview of the vast area of credit derivatives and different products in the market. There exist two main approaches used to model credit risk; the structural approach based on the diffusion of a firm’s value and the reduced form approach based on the hazard rate. The two approaches are theorized by presenting models proposed in the literature. In particular, we discuss and implement a jump-diffusion model proposed by Zhou (1997a). Through a closed-form solution and a Monte Carlo simulation we price defaultable bonds and credit default swaps. From the results obtained by varying input parameters we see that the model is capable of resembling observed features of bond and credit spreads, with declining, increasing and hump-shaped term structures. We introduce methodology for estimating the model, but still the model awaits empirical testing and the investigation of possible extensions by including stochastic volatility and short-term interest rate. In this paper we only model individual securities. To model and price the more advanced credit derivatives briefly presented, the modeler have to accommodate for default correlation and default dependency. Though demanding, these are necessary features in order to develop models suitable for credit analysis and risk management. Reduced form models have a clear advantage over structural models in that they easily can be calibrated to prevent arbitrage opportunities. The advantages of structural models on the other hand are that they are based on solid economic arguments and provide a link between equity and credit. This link is exploited in models like the E2C model where equity is used as a proxy hedge for credit. The use of equity to hedge credit or vice versa called capital structure arbitrage (Euromoney 2002) is an area that has received much attention the last couple of years. However, as we point out, a caveat with the models currently used for this purpose is that they model equity as an in-the-money European call while it should be modeled as a down-and-out barrier option. The general model by Zhou, though significantly more difficult to calibrate and more computer intensive, would mitigate the problem of defaults before maturity in current models. For further research we therefore also suggest empirical tests to verify Zhou’s model with regard to capital structure arbitrage.


56

6 Bibliography

Ammann, Manuel (1999), “Pricing Derivative Credit Risk”, Berlin: Springer-Verlag Ball, Clifford A. and Walter N. Torous (1985), “On Jumps in Common Stock Prices and

Their Impact on Call Option Pricing”, Journal of Finance, 40, 155-173 Bank for International Settlements (2003), http://www.bis.org/bcbs/aboutbcbs.htm Black, Fischer and John C. Cox (1976), “Valuing Corporate Securities: Some Effects of

Bond Indenture Provisions”, Journal of Finance, 31, 361-367 Black, Fischer and Myron Scholes (1973), “The Pricing of Options and Corporate

Liabilities”, The Journal of Political Economy, 81, 637-654 Black, Fischer, Emanuel Derman and William Toy (1990), “A One-Factor Model of

Interest Rates and Its Application to Treasury Bond Options”, Financial Analysts Journal, 46(11)

Bloodshed Software (2003), http://www.bloodshed.net Bohn, Jeffrey R. (2000), “A Survey of Contingent-Claims Approaches to Risky Debt

Valuation”, The Journal of Risk Finance, Spring 2000, Vol 1, p. 53-71 Cossin, Didier and Hugues Pirotte (2001), “Advanced Credit Risk Analysis”, Chichester,

UK: John Wiley & Sons Ltd. Cox, John C., Jonathan E. Ingersoll Jr. and Stephen A. Ross (1985), “A Theory of the

Term Structure of Interest Rates”, Econometrica, 53, 385-408 Craine, Roger, Lars A. Lochstoer and Knut Syrtveit (2000), “Estimation of a Stochastic-

Volatility Jump-Diffusion Model”, Working Paper, University of California at Berkeley

Das, Sanjiv Ranjan (2001), “Pricing Credit Derivatives”, Working Paper, Santa Clara

University Das, Sanjiv Ranjan, and Peter Tufano (1995), “Pricing Credit-Sensitive Debt When

Interest Rates, Credit Ratings and Credit Spreads are Stochastic”, Working paper, Harvard University

Das, Satyajit (2000), “Credit Derivatives and Credit Linked Notes”, Singapore: John

Wiley & Sons Ltd. Duffie, Darrell and David Lando (2001), “Term structures of credit spreads with

incomplete accounting information”, Econometrica, 69, 633-664


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Duffie, Darrell and Kenneth J. Singleton (1999), “Modeling Term Structures of Defaultable Bonds”, The Review of Financial Studies, 12, 687-720

Economist (2002), “The regulator who isn't there”, The Economist, May 16th 2002 Economist (2003), “Deep impact”, The Economist, May 8h 2002 Euromoney (2002), “And Now for Capital Structure Arbitrage”, Euromoney, December

2002 Geske, Robert L. (1977), “The Valuation of Corporate Liabilities as Compound Options”,

Journal of Financial and Quantitative Analysis, 12, 541-552. Giesecke, Kay (2001), “Default and information”, Working Paper, Cornell University Giesecke, Kay (2002), “Credit risk modeling and valuation: an introduction”, Working

paper, Cornell University Giesecke, Kay and Lisa R. Goldberg (2003), “Forecasting default in the face of

uncertainty”, Working paper, Cornell University & Barra, Inc. Glasserman, Paul (2003), “Monte Carlo Methods in Financial Engineering”, New York:

Springer-Verlag Hand, David J. and Saul D. Jacka (1998), “Statistics in Finance”, London: Arnold

Applications of Statistics Series Heath, David, Robert Jarrow and Andrew Morton (1992), “Bond Pricing and the Term

Structure of Interest Rates: A New Methodology for Contingent Claims Valuation”, Econometrica, 60, 77-105

Honoré, Peter (1998), “Pitfalls in Estimating Jump-Diffusion Models”, Working Paper,

The Aarhus School of Business Houwelingen, Patrick (2003), “Empirical studies on Credit Markets”, PhD thesis.

Erasmus University Rotterdam ISDA (1992), http://www.isda.org/publications/1992masterlc.pdf ISDA (2003), http://www.isda.org/statistics/recent.html Jarrow, Robert A. and Stuart M. Turnbull (1995), “Pricing Derivatives on Financial

Securities Subject to Credit Risk”, The Journal of Finance 50, No. 1 (Summer), 53-85

Jarrow, Robert A., David Lando and Stuart M. Turnbull (1997), “A Markov Model for

the Term Structure of Credit Risk Spreads”, The Review of Financial Studies, 10, 481-523


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Johannes, Michael and Nicholas Polson (2002), “MCMC Methods for Financial Econometrics”, Working paper, To appear in Handbook of Financial Econometrics

Johnson, Richard A. and Dean W. Wichern (2002), “Applied Multivariate Statistical

Analysis”, Upper Saddle River, USA: Prentice Hall, Inc., 5th edition Jäckel, Peter (2001), “Monte Carlo Methods in Finance”, Chichester, UK: John Wiley &

Sons Ltd Kesdee (2003), http://www.kesdee.com/html/cobasel.html Lando, David (1998), “On Cox processes and credit risky securities”, Working paper,

University of Copenhagen Lardy, Jean-Pierre (2000), “E2C: A Simple Model to Assess Default Probabilities from

Equity Markets”, JP Morgan Credit Derivatives Conference, January 16, 2002 Lehman Brothers (2001), “Credit Derivatives Explained – Market, Products and

Regulations”, Structured Credit Research Lin, Shinn-Juh and Ming-Tui Huang (2001), “Estimating Jump-Diffusion Models using

the MCMC Simulation”, Working Paper, National Tsing Hua University, Taiwan Longstaff, Francis A. and Eduardo S. Schwartz (1995), “A Simple Approach to Valuing

Risky Fixed and Floating Rate Debt”, The Journal of Finance, 50, 789-819 Madan, Dilip B. and Haluk Unal (1994), “Pricing the Risks of Default”, Working paper,

University of Maryland Mersenne Twister (2002), http://www.math.keio.ac.jp/matumoto/emt.html Merton, Robert C. (1974), “On the pricing of corporate debt: The risk structure of interest

rates”, Journal of Finance, 29, 449-470 Merton, Robert C. (1976), “Option Pricing when Underlying Stock Returns Are

Discontinuous”, Journal of Financial Economics, 3, 125-144 Moody’s (2002), “Default & Recovery Rates of European Corporate Bond Issuers, 1985-

2001”, Moody’s Investor Service, Global Credit Research, July 2002 Neftci, Salih N. (2000), “An Introduction to the Mathematics of Financial Derivatives”,

San Diego: Academic Press Press, William H., Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2002),

“Numerical Recipes in C++”, Cambridge: Cambridge University Press Ross, Sheldon M. (2000), “Introduction to Probability Models”, San Diego: Harcourt/

Academic Press


59

Schmidt, Thorsten (2001), “Credit Risk – An Introduction to Credit Risk Modelling with respect to Credit Derivatives Pricing”, Talk presented in the Frankfurt Mathfinance Colloquium, Goethe University

Schönbucher, Philipp J. (1998), “A Tree Implementation of a Credit Spread Model for

Credit Derivatives”, Department of Statistics, Bonn University Schönbucher, Philipp J. (2003), “Credit Derivatives Pricing Models”, Chichester, UK:

John Wiley & Sons Ltd Schüler, Marcus (2001), “Credit Derivatives – The Driving Force in Credit Markets”, JP

Morgan presentation Soudaram, Rangarajan K. (2001), “The Merton/KMV Approach to Pricing Credit Risk”,

Extra Credit SunGard (2003), http://www.sungard.com/products_and_services/stars/panorama/

information/albournedec2001.pdf Tavakoli, Janet M. (2001), “Credit Derivatives & Synthetic Structures”, New York: John

Wiley & Sons Ltd Vasicek, Oldrich (1977), “An equilibrium characterization of the term structure”, Journal

of Financial Economics, 5, 117-161 Venables, W. N. and D. M. Smith (2003), “An Introduction to R”, http://www.r-

project.org Walpole, Ronald E., Myers, Raymond H., Myers, Sharon L. (1998), “Probability and

Statistics for Engineers and Scientists”, New Jersey, USA: Prentice Hall International, Inc, 6th edition

Zhou, Chunsheng (1997a), “A Jump-Diffusion Approach to Modeling Credit Risk and

Valuing Defaultable Securities”, Board of Governors of the Federal Reserve System Zhou, Chunsheng (1997b), “Path-Dependent Option Valuation When the underlying path

is Discontinuous”, Board of Governors of the Federal Reserve System Zhou, Chunsheng (2001), “The term structure of credit spreads with jump risk”, Journal

of Banking & Finance, 25, 2015-2040


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A Zhou.c dll proram code #include "zhou.h" #include <windows.h> #include <stdio.h> #include <stdlib.h> #include <math.h> /* This C code file is developed with the free Bloodshed Dev-C++ (and C) compiler under Microsoft Windows XP Pro*/ /* Possible improvements for the C code: - The NAG C library is available to students at NTNU. This library is heavily tested and used. Thus we could have sped up CPU time the quality assured the code by using the NAG library. - The random function we used was the best we found, and we have tested it both for accuracy (uniform distribution) and stress (10E9). It has passed all tests. Still there might be faster algorithms available, and possibly with less digits (which are not crucial for an MC). We chose not to test further since the performance is acceptable anyway. Our random() returns in the interval [0,1], which is out of range for one func. We solved this with an if (while) for simplicity, but we could also have modified the random() slightly. */ /* x(i) function defined in Zhou (2001): */ double x_i ( double discountRate, double sigmaDiff_2, double lambda, double nu, int n, double timeStop) { double mu_i, sigmaDiff_2_i; mu_i = (discountRate - (sigmaDiff_2 / 2) - (lambda * nu)) * timeStop / n; sigmaDiff_2_i = sigmaDiff_2 * timeStop / n; return dllRandNorm (mu_i, sqrt (sigmaDiff_2_i)); } /* Calculate y = 0 or y * pi directly: */ double yPi (double lambda, double timeStop, int n, double mu_i, double sigma_pi_2) { if (genrand() > lambda * timeStop / n) { return 0; } else { // Return y_ * pi_i: return dllRandNorm (mu_i, sqrt(sigma_pi_2)); } } /* Simple write-down function as described in Zhou (2001): */ double writedown (double w_0, double w_1, double x) { return min (w_0 - w_1 * x, 1); } DLLIMPORT double _stdcall dllPlotting( double discountRate, double sigmaDiff_2, double lambda, double nu, double timeStop,


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double mu_pi, double sigma_pi_2, double X_0, double w_0, double w_1, int n, // for LnX int m, // for W double cdsPayments, double LnX[], double W[], double Out[], double K) { int i, j, k; double x, pi, y, X_t; double yPi_i = 0; double npvW_j, cds_j, defaultTime_j; // Initialize: double bondValue = 0; double bondSpread = 0; double sumW = 0; double sumNpvW = 0; double sumCds = 0; double sumCdsSpread = 0; double cdsValue = 0; double cdsAmt = 0; double payments = 0; double spreadCds = 0; double dblN = 0; double npvCdsW_j = 0; double equity_j = 0; double equity = 0; double rateFactor = exp(-discountRate / cdsPayments); // Be sure nu is correct: nu = exp(mu_pi + sigma_pi_2/2) - 1; for (j = 0; j < m; j++) { // Run from 0 to j-1 // Initialize: LnX[0] = log(X_0); i = 1; while (i < n && LnX[i - 1] > 0) // Run from 0 to n-1 { x = x_i (discountRate, sigmaDiff_2, lambda, nu, n, timeStop); // Merge Zhou's y and pi in one go: yPi_i = yPi(lambda, timeStop, n, mu_pi, sigma_pi_2); LnX[i] = LnX[i - 1] + x + yPi_i; i++; } // Mark end of this run with flag for use in VBA: LnX[i] = -999; // If default: if (LnX[i-1] <= 0 ) { W[j] = writedown(w_0, w_1, exp(LnX[i - 1])); // Calculate equity: equity_j = 0; } else { // Not deafult: W[j] = 0; // Calculate equity: equity_j = exp(LnX[i-1]) * K - 1; } sumW += W[j];


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dblN = n; defaultTime_j = (timeStop / dblN) * (i - 1); // Years npvW_j = exp (-discountRate * timeStop) * W[j]; sumNpvW += npvW_j; // Paid at T npvCdsW_j = (1 - exp(-discountRate * (timeStop - defaultTime_j)) * (1-W[j])) * exp(-discountRate * defaultTime_j); payments = min (cdsPayments * timeStop, ceil (defaultTime_j * cdsPayments)); // Test for error values (if zero, we get div 0 later): if (payments < 0.0001) { // Seems to work with 0.0001 payments = 0.0001; } cdsAmt = ((1/rateFactor) * npvCdsW_j * (1 - rateFactor))

/ (1.0 - pow(rateFactor, payments)); spreadCds = cdsAmt * rateFactor * (1 - pow(rateFactor, cdsPayments)) / (1.0 - rateFactor); cdsValue += cdsAmt; sumCdsSpread += spreadCds; equity += equity_j; } //bondValue = exp (-discountRate * timeStop) - sumNpvW / m; // See dllBondValue bondValue = exp (-discountRate * timeStop) * (1 - sumW / m); bondSpread = -log(bondValue) / timeStop - discountRate; bondSpread *= 10000; // Find average: sumCdsSpread /= m; sumCdsSpread *= 10000; cdsValue /= m; equity /= m; // Discount: equity *= exp(-discountRate * timeStop); // Add to vector for vba to read: Out[0] = bondSpread; Out[1] = cdsValue; Out[2] = sumCdsSpread; Out[3] = equity; return bondValue; } /* Box-Muller algorithm to draw from Normal distribution: */ /* Returns: random ~ N(0, 1) */ /* Uses: math.h */ /* Vars for use with dllRandSNorm and dllRandNorm: */ double randSNorm2 = 0; int rand1 = 1;


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const float PI = 3.1415926535; double dllRandSNorm() { /*******************/ // Box-Muller: /*******************/ double u1 = genrand(); // Avoid div zero. (This happens extremely rarely): while (u1 <= 0.0 || u1 >= 1.0) { double u1 = genrand(); } double r = -2 * log (u1); double v = 2 * PI * genrand(); // This accepts randoms of 0 or 1 /* Calculate two vars at one go, and save the other: */ randSNorm2 = sqrt(r) * sin (v); return sqrt(r) * cos(v); } // Returns: random ~ N(mu, sigma) double dllRandNorm (double mu, double sigma) { if (rand1 == 1) { rand1 = 0; // False return (mu + dllRandSNorm() * sigma); } else { rand1 = 1; // True return (mu + randSNorm2 * sigma); } } /***********************************************/ /* S t a n d a r d d l l f u n c t i o n : */ /***********************************************/ /* Necessary method for the dll: */ BOOL APIENTRY DllMain (HINSTANCE hInst /* Library instance handle. */ , DWORD reason /* Reason this function is being called. */ , LPVOID reserved /* Not used. */ ) { switch (reason) { case DLL_PROCESS_ATTACH: break; case DLL_PROCESS_DETACH: break; case DLL_THREAD_ATTACH: break; case DLL_THREAD_DETACH: break; } /* Returns TRUE on success, FALSE on failure */ return TRUE; } /*************************************/ /* R A N D O M G E N E R A T O R : */ /*************************************/ /* A C-program for MT19937: Real number version */ /* genrand() generates one pseudorandom real number (double) */


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/* which is uniformly distributed on [0,1]-interval, for each */ /* call. sgenrand(seed) set initial values to the working area */ /* of 624 words. Before genrand(), sgenrand(seed) must be */ /* called once. (seed is any 32-bit integer except for 0). */ /* Integer generator is obtained by modifying two lines. */ /* Coded by Takuji Nishimura, considering the suggestions by */ /* Topher Cooper and Marc Rieffel in July-Aug. 1997. */ /* This library is free software; you can redistribute it and/or */ /* modify it under the terms of the GNU Library General Public */ /* License as published by the Free Software Foundation; either */ /* version 2 of the License, or (at your option) any later */ /* version. */ /* This library is distributed in the hope that it will be useful, */ /* but WITHOUT ANY WARRANTY; without even the implied warranty of */ /* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. */ /* See the GNU Library General Public License for more details. */ /* You should have received a copy of the GNU Library General */ /* Public License along with this library; if not, write to the */ /* Free Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA */ /* 02111-1307 USA */ /* Copyright (C) 1997 Makoto Matsumoto and Takuji Nishimura. */ /* Any feedback is very welcome. For any question, comments, */ /* see http://www.math.keio.ac.jp/matumoto/emt.html or email */ /* [email protected] */ //#include "mt19937.h" //#include <windows.h> //#include <stdlib.h> //#include <math.h> //#include <stdio.h> /* Period parameters */ #define N 624 #define M 397 #define MATRIX_A 0x9908b0df /* constant vector a */ #define UPPER_MASK 0x80000000 /* most significant w-r bits */ #define LOWER_MASK 0x7fffffff /* least significant r bits */ /* Tempering parameters */ #define TEMPERING_MASK_B 0x9d2c5680 #define TEMPERING_MASK_C 0xefc60000 #define TEMPERING_SHIFT_U(y) (y >> 11) #define TEMPERING_SHIFT_S(y) (y << 7) #define TEMPERING_SHIFT_T(y) (y << 15) #define TEMPERING_SHIFT_L(y) (y >> 18) static unsigned long mt[N]; /* the array for the state vector */ static int mti=N+1; /* mti==N+1 means mt[N] is not initialized */ /* initializing the array with a NONZERO seed */ void sgenrand(seed) unsigned long seed; { /* setting initial seeds to mt[N] using */ /* the generator Line 25 of Table 1 in */ /* [KNUTH 1981, The Art of Computer Programming */ /* Vol. 2 (2nd Ed.), pp102] */ mt[0]= seed & 0xffffffff; for (mti=1; mti<N; mti++) mt[mti] = (69069 * mt[mti-1]) & 0xffffffff; } /* generating reals */ /* unsigned long */ /* for integer generation */ DLLIMPORT double _stdcall genrand() { unsigned long y; static unsigned long mag01[2]={0x0, MATRIX_A}; /* mag01[x] = x * MATRIX_A for x=0,1 */


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if (mti >= N) { /* generate N words at one time */ int kk; if (mti == N+1) /* if sgenrand() has not been called, */ sgenrand(4357); /* a default initial seed is used */ for (kk=0;kk<N-M;kk++) { y = (mt[kk]&UPPER_MASK)|(mt[kk+1]&LOWER_MASK); mt[kk] = mt[kk+M] ^ (y >> 1) ^ mag01[y & 0x1]; } for (;kk<N-1;kk++) { y = (mt[kk]&UPPER_MASK)|(mt[kk+1]&LOWER_MASK); mt[kk] = mt[kk+(M-N)] ^ (y >> 1) ^ mag01[y & 0x1]; } y = (mt[N-1]&UPPER_MASK)|(mt[0]&LOWER_MASK); mt[N-1] = mt[M-1] ^ (y >> 1) ^ mag01[y & 0x1]; mti = 0; } y = mt[mti++]; y ^= TEMPERING_SHIFT_U(y); y ^= TEMPERING_SHIFT_S(y) & TEMPERING_MASK_B; y ^= TEMPERING_SHIFT_T(y) & TEMPERING_MASK_C; y ^= TEMPERING_SHIFT_L(y); return ( (double)y / (unsigned long)0xffffffff ); /* reals */ /* return y; */ /* for integer generation */ }


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B VBA Program Code for Closed Form Solution Public Sub calcClosedForm(discountRate, _ sigmaDiff_2, lambda, timeStop, _ mu_pi, sigma_pi_2, _ X_0, w_0, w_1, n, m, cdsPayments) ' Check values: If sigmaDiff_2 <= 0 Then 'Div by zero MsgBox "Sigma diffusion must be above zero" Range("d18").Activate Exit Sub End If 'Vars are set as parts of the complete expressions in 'order to make the computation simpler. 'Refer to the paper for a decription of the algorithm. Dim mu_nd As Double ' subscript _nd: Normal Distributed, see report Dim sigma_nd_2 As Double ' subscript _2: squared Dim jumpProb As Double Dim driftPart As Double Dim norm_n As Double Dim norm_n_G As Double Dim normProb As Double Dim normProb_G As Double Dim expValue_S As Double Dim expValue_B As Double Dim survProb As Double Dim stockValue As Double Dim bondValue As Double Dim bondSpread As Double Dim norm_n_br As Double Dim normProb_br As Double Dim normProb_F_br As Double Dim norm_n_F_br As Double Dim survProbF_br As Double Dim nu As Double 'Initialize: expValue_B = 0 expValue_S = 0 survProb = 0 survProbF_br = 0 pubBondValue = 0 pubBondSpread = 0 pubStockValue = 0 nu = Exp(mu_pi + sigma_pi_2 / 2) - 1 Dim i As Long Dim infinity As Long infinity = 150 ' Max is 170 For i = 0 To infinity mu_nd = (discountRate - (sigmaDiff_2 / 2) - lambda * nu) * timeStop _ + i * mu_pi ' Ln(X) later sigma_nd_2 = sigmaDiff_2 * timeStop + i * sigma_pi_2 jumpProb = (Exp(-lambda * timeStop) * (lambda * timeStop) ^ i) _ / WorksheetFunction.Fact(i) driftPart = X_0 * Exp(mu_nd + (sigma_nd_2 / 2))


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norm_n = (Log(X_0) + (mu_nd + sigma_nd_2)) / Sqr(sigma_nd_2) norm_n_F_br = (Log((w_0 - 1) / w_1) - Log(X_0) - (mu_nd)) / Sqr(sigma_nd_2) norm_n_G = (Log(X_0) + (mu_nd)) / Sqr(sigma_nd_2) norm_n_br = (-Log((w_0 - 1) / w_1) + (Log(X_0) + (mu_nd))) / Sqr(sigma_nd_2) normProb = WorksheetFunction.NormSDist(norm_n) normProb_br = WorksheetFunction.NormSDist(norm_n_br) normProb_G = WorksheetFunction.NormSDist(norm_n_G) normProb_F_br = WorksheetFunction.NormSDist(norm_n_F_br) expValue_S = expValue_S + driftPart * jumpProb * normProb expValue_B = expValue_B + driftPart * jumpProb * (normProb_br - normProb) survProb = survProb + jumpProb * normProb_G survProbF_br = survProbF_br + jumpProb * normProb_F_br Next i stockValue = Exp(-discountRate * timeStop) * (expValue_S - survProb) bondValue = Exp(-discountRate * timeStop) * _ (survProb + (1 - w_0) * ((1 - survProb) - survProbF_br) + w_1 * expValue_B) 'Test for erronous values: If bondValue < 0 Then 'Log of neg. number MsgBox "Closed form solution produces non-real result " & vbCrLf & _ "Please decrease w_0 or increase w_1." Range("d13").Activate Exit Sub End If 'Times 10000 to get bps: bondSpread = 10000 * ((-Log(bondValue) / timeStop) - discountRate) 'Uodate findings to public vars: pubBondValue = bondValue pubBondSpread = bondSpread pubStockValue = stockValue End Sub


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C VBA Code for Changing Diffusion Volatility and Jump Volatility Public Sub runSigmaDiff_SigmaPi() 'Start timer: Dim startTime As Double startTime = Timer 'Read public variables from first sheet: If readInput = False Then Exit Sub 'Declare variables and arrays: Dim nu As Double Dim dllBVRes As Double Dim cdsPremium As Double Dim bondSpread As Double ' = credit spread Dim cdsSpread As Double Dim equity As Double 'dll more stable if we do not redim (dynamic memory allocation): Dim ArrLnX(500) As Double Dim ArrW(100000) As Double Dim ArrOut(0 To 5) As Double pubBondValue = 0 pubBondSpread = 0 pubStockValue = 0 'Start working on these graphs: Worksheets("SigmaDiff-SigmaPi").Activate ''''''''''''''''''''''''''''''''''' 'Read changing vars: ''''''''''''''''''''''''''''''''''' Dim timeStop As Double Dim stepLength As Double Dim stepNumber As Long Dim timeStopLast As Double timeStop = Range("d23").Value stepLength = Range("d25") stepNumber = Range("d27") timeStopLast = timeStop + stepLength * stepNumber Dim i As Long Dim j As Long j = 0 'Sigma_diff: Dim ArrSigmaDiff_2(0 To 2) As Double ArrSigmaDiff_2(0) = Range("g23").Value ArrSigmaDiff_2(1) = Range("g25").Value ArrSigmaDiff_2(2) = Range("g27").Value 'Update lambda and sigma_pi: Dim ArrSigma_pi_2(0 To 2) As Double Dim ArrLambda(0 To 2) As Double For i = 0 To 2 If ArrSigmaDiff_2(i) >= pubSigmaTotal_2 Then MsgBox "sigmaDiff must be less than SigmaTotal" Exit Sub End If


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ArrLambda(i) = pubLambda 'Constant for now ArrSigma_pi_2(i) = (pubSigmaTotal_2 - ArrSigmaDiff_2(i)) / ArrLambda(i) Next i ''''''''''''''''''''''''''''''''''''''''' 'Set info: ''''''''''''''''''''''''''''''''''''''''' Range("c32").Value = ArrSigmaDiff_2(0) Range("d32").Value = ArrSigmaDiff_2(1) Range("e32").Value = ArrSigmaDiff_2(2) Range("i31").Value = ArrSigmaDiff_2(0) Range("s31").Value = ArrSigmaDiff_2(1) Range("ac31").Value = ArrSigmaDiff_2(2) Range("k31").Value = ArrSigma_pi_2(0) Range("u31").Value = ArrSigma_pi_2(1) Range("ae31").Value = ArrSigma_pi_2(2) '''''''''''''''''''''''''''''''''''' 'Run model: '''''''''''''''''''''''''''''''''''' 'Clear for new run: Rows("33:126").ClearContents Range("k27").ClearContents 'Loop for every time step: Do While timeStop < timeStopLast 'Print maturity: Range("b33").Offset(j, 0).Value = timeStop 'Loop for every sigma/lambda pair: For i = 0 To 2 nu = Exp(pubMu_pi + ArrSigma_pi_2(i) / 2) - 1 'Print input: printInput pubX_0, pubDiscountRate, timeStop, pubN, pubM, pubW_0, _ pubW_1, ArrLambda(i), _ pubMu_pi, ArrSigma_pi_2(i), ArrSigmaDiff_2(i), _ pubSigmaTotal_2, pubCdsPayments, pubK 'Run dll: dllBVRes = dllPlotting(pubDiscountRate, _ ArrSigmaDiff_2(i), _ ArrLambda(i), nu, _ timeStop, pubMu_pi, _ ArrSigma_pi_2(i), _ pubX_0, pubW_0, pubW_1, pubN, pubM, pubCdsPayments, _ ArrLnX(0), ArrW(0), ArrOut(0), pubK) 'Closed form: calcClosedForm pubDiscountRate, _ ArrSigmaDiff_2(i), _ ArrLambda(i), _ timeStop, _ pubMu_pi, _ ArrSigma_pi_2(i), _ pubX_0, pubW_0, pubW_1, _ pubN, pubM, pubCdsPayments 'Print results: bondSpread = ArrOut(0) cdsPremium = ArrOut(1) cdsSpread = ArrOut(2)


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equity = ArrOut(3) Range("c33").Offset(j, i).Value = bondSpread Range("g33").Offset(j, i * 10).Value = timeStop Range("h33").Offset(j, i * 10).Value = dllBVRes Range("i33").Offset(j, i * 10).Value = bondSpread Range("j33").Offset(j, i * 10).Value = cdsPremium Range("k33").Offset(j, i * 10).Value = cdsSpread Range("l33").Offset(j, i * 10).Value = equity ' Print output from closed form: Range("m33").Offset(j, i * 10).Value = pubBondValue Range("n33").Offset(j, i * 10).Value = pubBondSpread Range("o33").Offset(j, i * 10).Value = pubStockValue Next i j = j + 1 timeStop = timeStop + stepLength Loop 'Print time spent: Range("h17").Value = Timer - startTime End Sub


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D MCMC Results

We do not link the MCMC implementation to our model in VBA and C. The reason is that there are too many assumptions implied so that it would not give correct results. All initial parameters are set inside the file gibbs.r and asset data is read from file. We put approximately 600 stock observations of Statoil in the file and estimate the parameters from these data. The algorithm could, of course, have been extended to calculate the asset value as described in chapter 4.6, but our purpose is here to show the methodology in use. The graphs on the left side show the density distributions on each of the parameters while the right side shows a trace plot of the corresponding parameter. The parameters we use can be found in the code below.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

03

Density drift diffusion

Den

sity

0 200 400 600 800 10000.

1

Trace plot drift diffusion

Iterations

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001

060

0

Density drift jump

Den

sity

0 200 400 600 800 1000

-0.0

06

Trace plot drift jump

Iterations

110 120 130 140 150 160 170

0.00

Density jump intensity

Den

sity

0 200 400 600 800 1000

40

Trace plot jump intensity

Iterations

0.006 0.008 0.010 0.012

040

0

Density drift vol

Den

sity

0 200 400 600 800 1000

0.00

5

Trace plot drift vol

Iterations

0.0020 0.0025 0.0030 0.0035

0

Density jump vol

Den

sity

0 200 400 600 800 1000

0.00

2

Trace plot jump vol

Iterations


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Variable Mean Standard deviation Diffusion drift 0.4578826 0.08269648 Jump drift -0.003356848 0.0006683849 Jump intensity 138.6237 7.971066 Diffusion volatility 0.007638973 0.0598364 Jump volatility 0.002546043 0.02818999 Figure A.1 Results from parameter estimation using MCMC The plots show that the parameters use the first iterations to approach a seemingly equilibrium before they oscilliate around this axis. The standard deviation is high for some of the parameteres, which is most likely due to a limited amount of data. We have run 1000 simulations with a burn-in period on 100, which is not counted in the calculation of the mean and standard deviation.


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E R Code for MCMC setLogReturns <- function() { size <- length(assets) tmp <- seq(1,size - 1,1) for(i in 2:size) { tmp[i-1] <- log(assets[i]/assets[i-1]) } return(tmp) } calcProb <- function() { tmp1 <- logReturn[i] - muDiff * delta tmp2 <- delta + hDiff / hJump oddsRatio <- (1 - intensity * delta) * sqrt(tmp2 ) / (intensity * delta * sqrt(delta)) * exp((tmp1 - muJump) ^ 2 / (2 * tmp2 / hDiff ) - tmp1 ^ 2 / (2 * delta / hDiff )) return(1 / (1 + oddsRatio) ) } # Initial parameters in prior muDiff <- 0.12335 muJump <- -0.00095 intensity <- 2.7 hDiff <- 80 hJump <- 500 h <- c(25,0,0,100) h <- matrix(h,2,2) bUnder <- c(muDiff, muJump) # Prior variables s2Diff <- 0.1338 vDiff <- 8 s2Jump <- 0.87 vJump <- 4.22 alfa0 <- 0.5 beta0 <- 1 # Fetch asset data from specified file and calculate log-return assets <- scan("statoil.dat") logReturn <- setLogReturns() T <- length(logReturn) delta <- 0.004 # Time between asset observations G <- 1000 # Number of simulations burnIn <- 100 # Number of burn-in periods # Run simulation with Gibbs sampler and Metropolis-Hastings xx <- matrix(0,G,5) for(g in 1:G) { # Initialize matrices Bernoulli <- c(matrix(0,T,1)) R <- matrix(0,T,1) X <- matrix(0,T,2) s2HDiff = 0 s2HJump = 0 # Calculate the posterior distribution of the jump times for(i in 1:T) { if(runif(1) <= calcProb()) { Bernoulli[i] = 1 } tmp = sqrt(delta + hDiff* Bernoulli[i] / hJump) R[i,1]= logReturn[i] / tmp X[i,1] = delta / tmp X[i,2] = Bernoulli[i] / tmp


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} # Draw jump intensity intensity = rbeta(1, alfa0 +sum(Bernoulli), beta0 + T - sum(Bernoulli)) / delta # Draw drift parameters hUpper <- hUnder + hDiff * (t(X) %*% X) bUpper <- solve(hUpper, (hUnder %*% bUnder + hDiff * (t(X) %*% R))) muDiff <- rnorm(1,bUpper[1], solve(hUpper)[1,1]) muJump <- rnorm(1,bUpper[2], solve(hUpper)[2,2]) bUnder <- c(muDiff, muJump) # Generate proposal of 1/variance of diffusion process and run Metropolis-Hastings for (j in 1:T) { tmp = logReturn[i] - muDiff * delta s2HDiff = s2HDiff + (tmp - muJump * Bernoulli[i] ) ^ 2 / (delta + hDiff / hJump * Bernoulli[i] ) } tmpV <- T + vDiff tmpS <- s2Diff + s2HDiff proposalH <- rchisq(1, tmpV ) / tmpS prob <- 1 tmp <- sqrt(1 + (hDiff - proposalH )/(hJump*delta + proposalH)) for (j in 1:T) { if (Bernoulli[i]==1) { prob <- prob * tmp } } if(runif(1) <= prob ){ hDiff = proposalH } # Generate proposal of 1/variance of jump process and run Metropolis-Hastings for (j in 1:T) { tmp = logReturn[i] - muDiff * delta s2HJump = s2HJump + Bernoulli[i] * (tmp - muJump ) ^ 2 / (delta * hJump / hDiff + 1) } tmpV <- sum(Bernoulli) + vJump tmpS <- s2Jump + s2HJump proposalH <- rchisq(1, tmpV ) / tmpS prob = ((delta * proposalH + hDiff) / (delta * hJump + hDiff)) ^ (- sum(Bernoulli) / 2) if(runif(1) <= prob ){ hJump = proposalH } xx[g,] = c(muDiff, muJump, intensity, hDiff, hJump) } # Print and plot results print("Average values: (diff drift, jump drift, intensity, diff vol, jump vol)") print(mean(xx[-(1:burnIn),1])) print(mean(xx[-(1:burnIn),2])) print(mean(xx[-(1:burnIn),3])) print(1 / mean(xx[-(1:burnIn),4])) print(1 / mean(xx[-(1:burnIn),5])) print("Standard deviations: (diff drift, jump drift, intensity, diff vol, jump vol)") print(sd(xx[-(1:burnIn),1])) print(sd(xx[-(1:burnIn),2])) print(sd(xx[-(1:burnIn),3])) print(1 / sd(xx[-(1:burnIn),4])) print(1 / sd(xx[-(1:burnIn),5])) par(mfrow = c(5,2))


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plot(density(xx[-(1:burnIn),1]), main="Density drift diffusion", xlab="") plot(xx[,1], type="l", main="Trace plot drift diffusion", xlab="Iterations", ylab="") plot(density(xx[-(1:burnIn),2]), main="Density drift jump", xlab="") plot(xx[,2], type="l", main="Trace plot drift jump", xlab="Iterations", ylab="") plot(density(xx[-(1:burnIn),3]), main="Density jump intensity", xlab="") plot(xx[,3], type="l", main="Trace plot jump intensity", xlab="Iterations", ylab="") plot(density(1/xx[-(1:burnIn),4]), main="Density drift vol", xlab="") plot(1/xx[,4], type="l", main="Trace plot drift vol", xlab="Iterations", ylab="") plot(density(1/xx[-(1:burnIn),5]), main="Density jump vol", xlab="") plot(1/xx[,5], type="l", main="Trace plot jump vol", xlab="Iterations", ylab="")

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