PRICING CREDIT DERIVATIVES AND CREDIT RISK

Ed Watson

A t hesis s u bmit ted in conformity wit h the requirements for the degree of Master of Science

Graduate Department of Mathematics University of Toronto

@Copyright by Ed Watson 2000

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Pricing Credit Derivatives and Creclit Risk

Abstract

CVe assume the unort hodox premise that every financial transaction is born from a disagreement about its value. The focus is on credit derivatives, which involve the transfer of credit risk from one party to another. We discuss the pricing of a credit default swap, first from the risk-neutral perspective of the market maker. Then. turning to the risk/reward perspective of the investor on the other side of the trade, a n approach is put forward based on utility functions. which does not require the Caiissian assumption behind Modern Portfolio Theory We end with a discussion of derivative counterparty risk. and how to price the exotic default swap that hedges it.

'1 would üke to thank my supervisor Prof. Luis Seco at the University of Toronto. and his research team, including Dave Saunders. Acknowledgements &O go to my former colleagues at Greenwich Nat West for their rnotivating infiuence - .Uan Thomson. Aaita 5liilar, Dave Palmer and Nick Palmer. I would especidy like to thank Xikos Manolis for many interesthg discussions and iUejandro de los Santos for proof-reading the paper. Aii remaining errors are of course rny own.

Contents

1 Summary 3

2 A 'vanilla' Credit Default Swap 6 2.1 A market-maker's perspective . . . . . . . . . . . . . . . . . . 6

3.i.i WLat is a L o d ? . . . . . . . . . . . . . . . . . . . . . G - 2.1.2 Pricing a risk-free bond . . . . . . . . . . . . . . . . . (

2.1.3 Pricing a risky bond (the easy way) . . . . . . . . . . S 2.1.4 O v e ~ e w of credit pricing literature . . . . . . . . . . 9 '1.1.5 Review of reduced-form models . . . . . . . . . . . . . 15 3.1.6 What is a credit default swap? . . . . . . . . . . . . . 19 2.1 . 7 Hedging a credit default swap . . . . . . . . . . . . . . 20 2.1.8 Hedging Nit h bonds of different rnaturities . . . . . . . 25 2.1.9 An illustraton of a defauIt swap hedge . . . . . . . . . 26

2.2 The investor's perspective . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Modern Portfolio Theory . . . . . . . . . . . . . . . . 30 2.2.2 Investor preference in a non-Gaussian world . . . . . . 32 '2.2.3 Pricing investor preferences . . . . . . . . . . . . . . . 38 '2.2.4 CreditVaR models . . . . . . . . . . . . . . . . . . . . 42

3 An 'exotic' Credit Default Swap 48 3.1 Counterparty ri& modeling . . . . . . . . . . . . . . . . . . . 48

3.1.1 What is an interest rate swap? . . . . . . . . . . . . . 49 3.1.2 Credit exposure on a swap . . . . . . . . . . . . . . . . 50 3.1.3 Review of counterparty risli modeling . . . . . . . . . 51 3.1.4 Cornpu ting expect ed exposures . . . . . . . . . . . . . 52 - . 3.2 Pricing and hedging counterparty risk . . . . . . . . . . . . . aa 3.2.1 A well-known result . . . . . . . . . . . . . . . . . . . 55 3.2.2 Setting up a hedge . . . . . . . . . . . . . . . . . . . . 55 3.2.3 Re-balancing the hedge . . . . . . . . . . . . . . . . . 57 3.2.4 Default occurs and the position is closed . . . . . . . . 58 3.2.5 The total cost of the hedge . . . . . . . . . . . . . . . 59 3.2.6 The net cost in the limit . . . . . . . . . . . . . . . . . 60 3.2.7 A note on some assumptions made . . . . . . . . . . . 63

3.3 Counterparty ri& and reward . concluding rernarks . . . . . . 64

1 Summary

Every financial contract exists precisely because its counterparties disagree

as to its d u e . Each party must have a motimtion for making the tradeL.

Whiie this is a generai truth about aii economic transactions. in this paper

we will concentrate on credit as the traded product . We imagine a sirnplified

world of market-makers (banh) and end-users (investors. on the whole) and

look at how each might rationally price their side of a credit derivative.

The bank in our transaction is concerned with the cost of the hedge. and

thus risk neutral pricing, while the investor seeks to maximize some 'return-

versus-ri&' tradeoff. This vague concept of return-versus-risk we shall make

more precise, via the use of profit-and-los utility functions and avoiding the

Gaussian assumptions of Modern Portfolio Theory.

Specifically. and as the implied motif behind cliapter 2. consider the case

where a market-rnaker sells a simple credit default swap CO an investor.

We first look at this transaction from the market-malier's point of view.

.ksiirning complete markets and that he intends to run a flat book. we would

like to find a hedge for the trade and t hus derive its risk-neut rai price. Before

we do this. we give a brief introduction to fixed income securities (bonds).

and review the literature on credit pricing, both the so-called *structural*

models which developed from the theory of Robert Merton. and the newer

reduced-jonn rnodels which rely more on hedging and l e s on the estimation

of unknown parameters. Using the reduced-form approach, but in a very

generai sense, we go on to show that being 'long' both the (coupon-bearing)

bond and the credit default swap (of equivalent rnaturity) is a good but

irnperfect hedge. In fact, perhaps surprisingly, t here is no perfect hedge to

the simple credit default swap derïmble from bond positions at dl. -

'The orthodox view is somewhat different: that counterparties do agree on a fair price, and that one's 'motivation* for making a trade (and m y risk/reward analpis) is left out of the direct valuation process.

Our investor on the other side of the trade has a different perspective. In

one sense he is less pedantic, because he considers buying protection (from

a risk-free bank) as equivalent to seUing the loan. However, unlike the bank

that simply hedges its position, he is more worried about the implications of

ronc~ntra tinn in h i s portfnlio. HP wishes to determine on which of his loans

he should buy protection. To this end we need to consider the dynamics

of credit in a way we did not need to for the market rnaker. We look at

some currently fashionable CreditVâR' models that allow us to specify a

dependence (correlation) between changes in the various credits wit hin the

investor's portfolio in order to measure its risk. The somewhat contrasting

notions of risk and retum enjoy a separate e-xistence within Modern Port-

folio Theory. which we review before proposing an alternative formulation

which unifies the two concepts. Our approach, based on specifving a utility

function and the whole profit-and-10% distribution (not just a VaR mea-

sure). is compatible nrith the fat-tailed distribution of returns typical in the

credit market.

This credit market encompasses not only the fixed incorne world but aIso

the counterparty risk embedded in the the huge over-the-counter derivatives

market. So as the motif behind chapter 3, consider the case where an end-

user2 wishes to buy protection against the default of one of its derivatives

counterparties. In our example the underlying derivative is an interest rate

swap, the workings of which we explain in a non-technical way. We review

some of the methods used to measure this 'swap credit riskg (also known

as 'counterparty risk'), and find an expression for the expected losses it

generates. The sirnilarity between this expression and an option pricing

formula lead us to suppose that it can be hedged. By experimentation pie

find that a dynamic hedge c m be spthesized from a strip of credit default

swaps and a strip of (credit-risk-free) cal1 options defined on the interest

this tirne, less lîkely an investor and more Likdy another bank 'Le. a series of defauit swaps with diiferent maturities

rate swap.

The ability to hedge the exotic default swap means that. when pricing coun-

terparty risk. not only must the default probabilities be market-implied but

the expected exposures on the underlying transaction rnust be computed us-

ing the risk-neutral measure too. Aithough this is perhaps counter-intuitive.

it certainly fits into the historical pattern where-by the ri&-neutral measure

has progressively pervaded financial t heory4. We argue t ha t . like the fised-

income investor in the previous chapter, this end-user should mark his book

on a risk-neutrd basis, and only then 'pricdn' the effect of its variability

over a ri&-horizon. Interestingly. this risk-horizon is comparable not to the

Length of the underlying swap but to the short time needed to re-hedge the

credit risk.

'For e-xample, before Black and Scholes (19'73) it was generally believed that the risk- neutral measure should be used when pricing forward contracts but not options.

2 A 'vanilla' Credit Default Swap

The title of the chapter reflects the underlying motif. where we suppose that

a market-maker seiIs a simple ('vanilla') credit default swap to an investor. T T l u e t d r cuista dt di, iii u u r tliuuglit e:ipcrimerii, becaise ùur cûunier-

parties have different implicit principals of vaiuation. We consider first the

point of view of the market-maker.

2.1 A market-maker's perspective

Our market-maker's motivation is the fee (or bid/offer spread) that he

charges the client for the trade. and in order to protect this steady in-

corne stream. he must try to hedge the default swap. Default swaps are

derived from more primitive fixed income instruments. such as bonds. so

before pricing a default swap we must be able to price the underlying bond.

2.1.1 What is a bond?

A bond is a promise to pay cash in the future. Cash in the future has a

value today? so companies (and goverments) issue bonds as a way of raising

capital. Most bonds are coupon bearing. that is to Say they promise to pay

a rate of interest (usually fixed) for a period of time before redeeming the

principal the investor originaily paid. Non-coupon beanng bonds are much

l e s prevalent in the market place, but because of their illustrative potential

are popular in the Literature. where th- are always refered to as 'zere

coupon bonds'. The cashflows from a coupon- bearing bond are illustrated

in Figure 1.

principal redeemed ut maturity

coupom

time

principal

Figure 1: The cashflows for a coupon-bearing bond

2.1.2 Pricing a risk-free bond

The term ~rislr-free bond' is used to mean a bond which bears no default rkk.

so that ail its promised cashflows will be honoured with certainty. Risk-free

bonds may not r e d y esist. but they are still instructive c o n c e ~ t s . ~ Bonds

can exchange bands a t any time. and even the price of 'risk-free bonds'

luctuates, as interest rates change. For exampie. an existent bond that

promises a 6% coupon would not be so attractive if 7% bonds have just

been issued. If we suppose that an interest rate exists for every .maturity'

(future point in time), we can determine the pnce of a bond by summing

d l the promised cashflows, each first discounted by the appropriate interest

rate for its corresponding maturity. So if we let rt denote the annualiied

risk-free interest rate for maturity t , then a zero-coupon bond promising a

single dollar cashflow a t time t has a pnce B, given by

In the derivatives üterature interest rates used are dmost always contin-

uowly compounded. not annually compounded for instance, and we s h d

5Sometimes people d e r to US goverment bonds as being 'risk-free'.

T

foUow this example. With continuously compounded rates. the formula be-

In the case of bonds bearing coupons of c; a t times ti ( i = 1.2. .... n ) and

re&ruiiiig vue J d a r uf priiicipd d t h é t , ive sinipiy è:<icrtd th& fûiiiiüki

to give

rUternatively (and by convention) we might wish to postdate the esistence

of a single interest rate (usually c d e d the yield) with which we can discount

every cashflow to arrive at the same bond price, B. Then there is a direct

one-to-one correspondence between each bond pnce B, and its yield y,,,

defined bÿ

2.1.3 Pricing a risky bond (the easy way)

When the promised cashflows are no longer certain. we can posit a series of

pmbabzlities that the bond's issuer will default a t different points during the

life of the asset, and partial recovery rates in the event of default. We might

even postdate on how t hese probabilities could evolve in time (if necessarp) . PVe WU Save this discussion for a later section. However. an alternative

formulation simply decrees t hat cashflows need to be discounted by a higher

interest rate to compensate investors for the risk that the promised cashflows

will not materialize. To this end we might as weU Say we are adding a credit

spread, st ont0 the risk-free rate rt. In general there is no reason why the

'simply by the monotone relationship between bond pnces and interest rates

8

spread should not vary with tirne, just as the risk-free rate does. So' with

spreads a function of tirne, our formula for the zero-coupon bond becomes

In the above example we added a spread to each zero rate rti but alterna-

tively we could have added a single spread rate ont0 the yield yt, as in the

previous su bsection.

Notice that we haven't really provided much insight into the pricing of risky

debt. To say that a risky bond can be priced by adding a spread ont0 the

risir-free rates used for discounting is almost a tautology.

Plenty of modeIs have been proposed. homever, that do try to provide some

insight into the default process that causes this credit spread to exist. Mer-

ton (1974) postulated a firm-value process that would give rise to default if it

became too low. Other academics developed his ideas further. Then came a

new 'reduced form' approach based on an exogenously defined Poisson-type

default process. In the following sections we review some of these develop

ments.

2.1.4 Overview of credit pricing literat ure

The first important mode1 for d u i n g defaultable bonds was developed by

Robert Merton (19i4), using the principles of option pricing developed by

Black and Scholes (1973) and Merton (1973). He argued that a cashflow

promised to the holder of a corporate bond would be honoured if and only

if the d u e of the corporate's assets were sufficient. At the maturity of the

debt, the bond-holder would receive the smaller of two quantities: the value

of the firm's assets and the face value of the bond. Interestingly. this payoff

closely resembles the payoff from an option. Specificdy. it is equal to the

face value of the bond minus the payoff from a put option. the right (but

not. th^ ohtigation) tn sel1 the va.lite of the firm for an amount pqita1 to 6 h ~

bond's principle.

With the benefits of the previous year's insights into option pricing, Merton

ivas able to solve explicitly for the pnce of the risky zero-coupon bond. -111

the Black-Scholes assumptions are made, namely risk free interest rates, r,

are constant, and Geometric Brownian Motion is assumed for the d u e of

the firm's assets, Ft . In other words

for a standard Wiener process (Brownian motion) and some constant

volatility a. Let u s say the debt has face \due D and matures at time T at

which point the bond-holder recieves the pay-off

min(&. D) = D - max(D - Fr, O).

Figure 2 illustrates a sample path for the firm's asset price process. In this

case. the asset price a t rnaturity is below the face value of the debt and

default occurs.

Working through the usual Black-Scholes analysisi for a European put o p

tion gives that the value of the bond at time t is

where p( ...) is the Black-Scholes pricing formula for a European put option.

namely

p(S , X. o,r. T - t ) = ~ e - ' ( ~ - ' ) N ( - d ~ ) - SN(-d i )

' ~ h e standard derivat ion uses [to's formula and the no-arbitrage condition to denve a partiai differential equation for the option price, which is then solved with the option pay-off as boundary condition.

Figure 2: In Merton's model default occurs if F(T) < D

where d l = ( I n ( S / S ) + ( r + a2/2)(T - t ) ) / o JT-tT-t and d2 = dl - a m .

By varying T. we can obtain zero-coupon bond p r i m for different ma-

turities. or equivalently a tenn structure of credit spreah. analagous to

the term structure of risk-free interest rates obtained from risk-free bond

prices of varying maturities. CVe can observe from Merton's model that

different spreads are obtained for different values of the firm's leuerage.

1 = D ~ - ' ( ~ - ' ) / F ~ . and volatility G of the firmes value Ft.

Interestingiy. there is a pattern to the sorts of term-structures of spread o b

tainable. High-quality issuers tend to have upward-sloping term structures.

low-quality issuers have downward sloping term structures and in the mid-

dle nre have humped-shape term structures, as shown in Figure 3. This is

consistent with historical default rates for issuers of different credit ratings

as reported by the ratings agencies, and, in terrns of credit spreads observed

in the market? Sarig and Warga (1989) and Fons (1994) found some em-

pirical consistency with Merton6 mode!. On the other hand' Heiwege and

Turner (1999) found that when comparing the term structures for individual

issuers, credit spreads for low rated issuers were often, in fact, upward s lop

credit spread

I>-* high-gmde

Figure 3: Merton's model produces particular term-structure shapes

ing. Further. K m . Ramaswarmy and Sundaresan (1993) rather crucially

showed that using realistic values of leverage and volatility of the firm's

value do not produce yield spreads close to those of the market.

Merton's model has an elegant simplicity. but a few further drawbacks should

be noted. Fundamentally. the d u e of the firm is an unobservable quantity.

so that pricing the cornpany's debt amounts to guessing the value of the

firm's assets (and volatility there-of). -4 l e s commonly noted dran-back.

but in some sense equally important, is the fact that even if the firm value

were somehow observable, the risk-neutral valuation assumptions behind the

Black-Scholes analysis rely on the ability of the option writer to hedge by

taking a position in the underlying. Here the underlying is the firm's assets.

and to Say the least it is hard to imagine a way of continuously trading in

the firm's assets. . G o to note is that the model assumes that al1 debt is due

a t a single point in time, and that for different senàorities (classes of debt),

one would need to know them ail in order to adapt the model to prke any

of t h m .

Several models have been developed based on Merton's idea and they are

coilec tively refered to as k t ructural' or 'firm value' models. T hey generally

address some of the rninor drawbacb of Merton (1974) but not the major

drawbacb: concerning the inability to observe the value of or to trade in the

firm's assets. We do not provide an exhaustive review of the literature but

mention some of them here to e v e a flavour.

Gesky (1977) extendeci Merton's model to accomodate coupon payrnents as

well as the redernption of principal. In so doing, it became a problem of

pricing a cornpound option, which Gesicy addressed.

Black and COX (1976) aiiowed for early default. mhere-as in &Ierton's rnodel

default could onty occur at the debt's maturity. Default is assumed to occur

whenever the value of the firm reaches a certain lower threshold. which they

model as an exponentially affine function of tirne. The d u e of the firm

follows not lognormal diffusion as in Slerton's model. but instead a square

root diffusion process.

Longstaff and Schwartz (1995) introduced stochastic interest rates. which

they correlate with the firm value process. The former is modeled according

to the Vasicek (1971) mean-reverting short rate process. and the Iatter is

standard Geometric Brownian motion. Explicitly we have

dr = (a - $)dt + a2(p&V1 + dl - p2&V2)

where F is the firm value? r is the risk-free short-term interest rate, IVl and

tV2 are independent Brownian motions and p, al, a. d. and p all constants.

Foilowing Black and Cox. they specify a lower t hreshold for the value of the

firm. If this barrier is hit, default occurs immediately. nith recoveq rates

exogenously defined.

Although there are continuing developments dong the iines of 'clIerton's

model, much recent development on the pricing of risky debt has taken a

completely different , and some would Say more promising, direction, Where-

as structurai models al1 assume a firm \Aue process. which is unforcunately

unobservable, another approach is to model default as a Poisson-type pro-

ces . A good deal of research has now been done based on these models.

whirh are refered to in the literatirre as .reduced form' or *hazard rate' mod-

els. Two major differences become apparent between the structural and the

reduced form models. First, in a reduced form model. default can be a sud-

den surprise. In the structural models default was rnuch more 'predictable'

in the sense that a stochastic process tiad to cross a threshold for default to

occur (for example, the firm's assets had to be exhausted). Secondly, and

related to this. is that a reduccd form model specifies default probabilities

exogenously, or rather as fundamental inputs as opposed to probabilities to

be computed from some other 'more fundamental' process. Once a process

is postulated for the hazard rate, its parameters are backed out from observ-

able market variables only, for example bond prices. This met hod provides a

consistent framework for the risk-neut rai pricing and hedging of derivatives

b a s 4 on credit risk or risky debt.

Ive propose the following analogy. An equity trader might use fundamental

analysis to indentiS. mispriced stocks, w hile an equity deriuatiues trader

would quite rightly prefer to use a model that did agree with observed

stock prices in order to be able to hedge in the underlying. So similady.

a corporate bond trader might use structural models where appropriate in

order to find trading opportunities in the bond markets. At the same time.

a credit derivatives trader would generally assume that bonds were correctly

priced in order to be able to use the information contained therein to price

and hedge more sophisticated derivatives using a reduced form model. In

the following section we review some of the developments in reduced form

modeling .

2.1.5 Review of reduced-form models

reduced-form models are based on the specification of a Poisson-type

default process. The most straight forward to consider is perhaps that of

Jrrror.. ax! TkrlibiiI (1999). Tho rPciwry rate ic defi-!t k a constant. and

the default time is exponentiaily distnbuted with parameter X (mean = l / X ) .

Equivalently, we might Say that default is given by the first jump of a Poisson

process with intensity A. Risk-free interest rates are taken to be independent

of the default process and are modeled as a single factor diffusion process.

They implement the mode1 as a discrete time tree Nith four branches at each

node, corresponding to default or no-default and whether short-term interest

rates go up or down. Specifically. the (risky) forward rates are modeled (in

the continuous time version) as

df (t. T ) = (a(t . T) - B ( t . T)X)dt + o(t. T)dWl+ d ( t . T)dXt

where f ( t . T) are the (ris-) forward rates. X'' is the Poisson process with

intensity A. Wi is standard Brownian motion and a. O , O are (non-stochastic)

functions of t and T. The idea behind this representation is to rnake use

of a weil-known resuit from foreign currency option pricing, by considering

the pnce of an issuer's promised dollar to be the price of a certain dollar

mutilplied by an 'exchange rate' to incorporate the probability of default

and recovery in default.

Jarrow, Lando and Turnbull (1997) simiiarly kept the default process inde-

pendent from risk-free interest rates. However, they formulated an interest-

ing default process, based on the idea of credit mtings and the probabilities

of *migratingt form one rating to anot her. The idea of credit ratings cornes

from the credit rating agencies. such as Moodys or Standard and Poors.

The agencies traditionally assign their own ratings to corporate debt. the

implication being that each rating defines a probability of default, or rather

a term structure of default probabilities. In addition to t heir publishing h i s

torical levels of default for their different ratings. they also provide historicd

transition rates. that is to Say probabilities of moving from one rating to an-

other. Jarrow, Lando and Turnbull decided to mode1 the default process in

this way. Specificdy, they defined a -space' of credàt rnting states {1.Z ..... K}

and in t h e rontiniiniis timr wttins a CIarknv rhain [on t h e ratine s j a r ~ ) .

defined by its generator rnatrix

where

To obtain the matrix of transition probabilities over a finite time period t

ive sirnply compute the matrix exponential

P( t ) = exp (At).

They started with historical migration rates. and then applied a transforma-

tion (mulitplication by a diagonal matnc) in order to make their cumulative

default rates (last column of the P ( t ) matrices) consistent nrith bond prices.

This is equivalent to changing from the red-world to the risk-neu tral mea-

sure! by adding a rislc premium to each transition probability. However. t his

risk premium is fised, irrespective of rating, which is perhaps a questionable

assumption.

Duffie and Singleton (1999) (and earlier versions) dowed for a dependency

between riskfree interest rates and the default process. They first con-

sider the discrete-time setting. At each time step, the bond either defaults

or it does not. So. under the risk neutral measure. the price of a risky

bond (promising a dollar at time T say) must be equal to the discounted.

probability-weighted sum of the recovery rate a t the next tirne step (should

the bond have defaulted by then) and the expected price a t the next step

~ i v e n no defa~ilt. Or. formally U

where Bt k the market value of the risky bond a t time t. 0, is the recovery

in the event of defauk a t time S. h, is the probabiiity of defaulting between

times s and s+ 1 given no default by S. r, is the default-free short rate. The

notation E?( ...) rneans EQ( IF^). the conditional espectation ni th respect

to the naturai filtration { F t } .

Solving Equat ion 1 recursively we ob t ah. explici tly

Intuitively. the price of the bond at time t is the expected discounted value of

recovery times the probability of defaulting *at some time' (but not before).

plus the erpected discounted payoff in the event of no default.

At this point Duffie and Singleton make their key assumption. They s u p

pose that the recovery level is given as some fraction of the market value

immediately prior to default. In other words they define

E P [ ~ S + I I = (1 - LS)EP[&+II

for some adapted process L,. Substituting this result into Equation 1. and

t hen solving recursively, gives

= EF (- Ri)]

As the time steps shrink in length,

There are further technical conditions that need to be satisified in the con-

tinuous time setting, which they detail. What they had in mind for the

continuous time version is clearly

This neat result allows the default probabilities and recovery rates to be

bundled up together into a credit spread, which can then be manipulated

dong with the risk-free rate with one of the many e-xisting interest rate

models. Its key assurnption that makes this possible is the-recovery of

market value'. where-by the bond-holders receive a fraction of the market

value of t heir debt immediateiy prior to default. However, this is a c t u d y

at odds with standard bancruptcy practice, where recovery is a specified

fraction of the bond's principal.s

In summary, the great advantage of the Duffee and Singleton (1999) a p

proach and of reduced-form models over structural models in general is that

t hey malie no assumptions regarding unobservable processes. Default inten-

sities (sometimes inextricably intertwined with recovery rates) are inferred

from tradable prices only. This lends this type of mode1 very well to the

pricing of derivatives whose value depends on these defaultabIe bond p ~ c e s ,

for example a Credit Default Swap.

' ~ u f f i e (1998 b) addresses this by extending the mode1 to incorporate Fractional Re- covery of Par.

Up until default

periodic premium m

Figure 4: The cashflows in a Default Swap

2.1.6 What is a credit default swap?

-1 credit default swap is a contractual agreement between two parties. Party

A promises to pay an amount to party B contingent upon the defadt of a

reference asset. In exchange. B pays h a fixed fee. which is usuaily payable

periodically and up until the reference asset defaults (or the expiry of the

default swap). Figure 4 illustrates the payments that occur up until the

default of the reference asset and at the time of default.

This representation is of a default swap in the popular form whereby upon

default ehere is a cash settlement equal to the difference between the de-

faulted value of the bond (the recovery rate) and the par value. .hother

popular scheme involves the physicai delivery of the defaulted bond in ex-

change for par. -1 further, but less cornmon, scheme involves a fixed payment

upon default .

In the following section we attempt to price and hedge such a derivative.

2.1.7 Hedging a credit default swap

In this section we show that. as expected, a risky zero coupon bond and its

credit default swap are a perfect hedge (except for the remaining exposure to

risic-fie 12tps, Of ~91~r5e). WP I.EP fnrmal notation to ma kr this i d ~ n p r ~ r i s ~ :

in fact making very few assumptions about the interest rate and default

processes. We go on to show that the same is not true for coupon-bearing

bonds.

Consider the price at time t of a risky zerecoupon bond B(t. T) redeem-

ing, for ease of exposition. one dollar of principal a t cime T. The risk-free

short rate r ( t ) and the analagous hazard rate h ( t ) are stochastic processes.

both adapted to the natural filtration Ft, and possibly having some degree

of mittual dependence. Regardless of when default occurs. the payment is

assumed to take place at time T. and R is the recovery rate, not necessar-

ily a constant but at least adapted to the filtration (exogenously derived.

perhaps) .

By The Fundamentai Theorem of Finance, the bond pnce is the expected

discounted payoff under the ri&-neutral rneasure Q, i.e.

where

are random variables and E?[ ...] means E ~ [ ... 1 Ft].

NOW, we assume that a credit default swap exists, with upfront premium

already paid. which in the event of default pays out (1 - R) at time S. Shen

its pnce is similarly given by its expected discounted payoff under Q. namelp

t hat

C(t, T) = E~[ZH(I - R)].

So together, they have value

which is the price of the riskless bond. Encouraged by this. we might rea-

sonably hope that a similar resuit would hold for coupon-bearing bonds and

their credit default swaps. We shall show how wrong this is. even with

strong restrictions on the processes.

Suppose in fact that the processes r ( t ) and h(t) and the variable R are dl

mutually independent. So that in the above example the formulae would

become

where

~ = E ? [ Z ] and h = E~[H].

'Yow suppose there is a series of bonds (same issuer and seniority) which is

-dense' in the sense that there is a bond that matures on every coupon date

of the longest bond.

Suppose that the recovery rate R is esogenously derived and independent of

maturity. Without loss of generality our maturities are t l . t2, .... t,,. with ri&-

free discount factors d l , dz, ..., dn, cumulative default probabilities h l . l a 2 , .... h,

and risky bond prices B I , B2, ..., B,. Suppose for each bond Bi the coupon is

a fixed ci. which can vary between bonds of different maturities but is fixed

for any one issue. So the bond with the shortest maturity pays a coupon

cl and one dollar of face value, bot h a t tirne t 1. In default we receive re-

covery on the one dollar of principal only. The bond's price is given by its

discounted expected payoff, namely

Rearranging gives us the risk-neutral default probability in terrns of the

bond price. Thus Bl/dl - 1 - c l

hl = R - 1 - c l

So we can -bacb: out' the probability of default for the first time period from

the risky bond price. aithough notice thar it is no t uniqueiy ciererrnineci

without specifying a recovery rate too. Later ive shall explore the difficulties

that this gives us. In the meantime. however. we can proceed to pnce the

second bond in a similar way, to obtain

which we can rearrange to give

In general. for the bond maturing a t tn. we have

which we can re-group bÿ hi term to give

In this way we can buiid a l the default probabilities hi recursively from the

bond prices' -bootstrapping' style' beginning with h l , then h2 and so on up

to hn, in a sirnilar way to yield curve construction. We need to assume a

recovery rate R to do this, though.

Now let's use the default probabilities to price a credit default swap on the

bond maturing a t time tn. A default swap is generally constructed with its

premium a t a level such that the transaction has zero market value on trade

date.g -At any future point in time, however. the swap wili generdy be

off-market of course. and its value is determined by t akng the present value

of a l l expected cashflows, as one would for any derivative. So we assume

t h ~ r a is 22 ey&ten_t tracle, w h q ~ th@ pr~rn iurn is fiserf a t p and paya h b a t

times t 1. tZ . .... fn- 1 or until the default of the reference asset. (It would be

unusual to expect a premium to be paid at the bond's maturity t , or in

the event of default. just as you wouldn't need to pay a home-and-contents

insurance premium after your house has just burnt down.)

The p i ce of the default swap is made up of two legs. the expected payout

due to potential default D$ and the prernium payable D;. The payout will

be (1 - R) in the event of default, so we have

and the premium p is payable up until time t,,l so Iong as there is no

default . Hence

giving the total (net) pnce of the default swap as

Now let us see how good a hedge it is if we are long the bond and have

default swap protection. Combining Equations 3 and 4 give t hat our two-

trade portfolio has value

'This is easily accomplished. Set equation 4 equaI to zero and solve for the premium P.

Certainly. the terrns don't seem to have cancelled as we might have hoped

for. However. to determine how good (or bad) a hedge this is. we need to

consider the terms that are multiplied by the default probabilities h i . The

third line should not concern us because it simply contains fixed terms c,

and p discounted by different risk free rates. .As such we have only sensitivity

to ri&-free rates, which can easiiy be hedged by interest rate derivatives. By

inspection. we notice that the largest of the other terms is the last term on

the second line, -hndncn. This is not only because h, > hnWl > . . . > hi. but aiso because the multiplicands of the other hi are smaller. For a default

swap entered on a par bond. for example. (c, - p ) is approximately the

ri&-free rate. which is itself roughly equal to ( d i - d i + i ) especially for s m d

i. Even for a default swap entered on a fairly distressed bond. (c , - p)

NilI still be positive (and small), so it d l help to partly offset (di - d i + L ) .

On the other hand dncn is equal or greater to the risk-free rate. in general.

This makes sense intuitively. A bond that is about to mature will redeern

principal plus final coupon. but if it defaults during the period immediately

p k r to maturity Nill redeem only recovery R. The default swap will pay

out (1 - R) making a total of 1 but the finai coupon payment is lost. So.

informally. this is the greatest risk in holding a coupon-bearing bond and

its credit default swap. The final coupon may or may not materialise.

What about the case where protection is bought on a bond that is very

distressed (one that has been 'downgraded' by the market). In this case

the premium is very high, perhaps even larger that the coupon itself, and

each of the multiplicands becomes signifiant. Again, intuitively t his makes

sense. If we are paying a very high premium. we would prefer that the

bond defaulted sooner rather than later, so as to extracate ourselves from

the high premium cashstream. For the sake of cornpletenes, we might also

consider the case where protection is purchased on a risky bond that has

recently appreciated significantly, alt hough t his scenario is of course rnuch

less tikely. In that case the multipiicands could actually be negative (with

en >> d l - d2 and p small) so that we would prefer default to occur later (if

nt all! so as to pick up the attractive coupon pavments dong the way. .Us0

worthy of note is the fact that. although the recovery R does not appear

at dl in the d u e of the hedged portfolio, indirectly it affects the quality

of the hedge in the sense that we have Our hi specified by bond prices only

by assuming a value of R. -4 lower assumed value for R gives higher hi and

vice versa.

In summary. our hedge of 'default swap plus bond' is pretty good. especially

when protection aas bought at a time when the bond was trading near to

par. However it is not a perfect hedge and is more sensitive to the default

process dynamics (change in default probabilities) if protection had beeo

bought when the bond were in a distressed state. 1s there a better hedge

possible. though, by matching the hi terms in the default swap price by

bonds with different rnaturities? We shdl now explore this idea.

2.1.8 Hedging with bonds of different maturities

If we look at Equation 3 and Equation 4, we notice that, when added, they

give a residual term in h,. Suppose we purchased, instead of one bond, an

amount equal to the ratio of the h, terms. namely

1 - R

so that our portfolio becomes

In this way we have cancelled the h, term. Suppose then we choose an

amount of the bond B,,I to cancel the h,, term and so on, recursively

until we have cancelled every hi term. The formulae get messy very quickly.

but no matter - we can hedge out ail terms with default probabilities in them.

But t o do this we have assumed a fixed recovery value, and in practice we

cannot determine this value wit hout knowing the default probabilities. We

are caught in a circular argument and forced to admit defeat. A default

swap can be hedged fairly well with the bond of the same maturit- or a

str ip of bonds. but no perfect hedge is possibleL0. In the following section

we give a worked example to dernonstrate this.

2.1.9 An illustraton of a default s w a p hedge

Let us go through a worked example to illustrate the points in the previous

3 subsections. Suppose we have an A-rated 5 year corporate bond promising

an annuai coupon of 7.00%. With the implied default probabilities and ri&-

free zero-coupon yields in Figure 5 and an implied recovery rate of 50%. the

bond trades a t $107.41 per $100 dollars of notional.

The price of a 5-year default swap can also be ~ o r n ~ u t e d ' ~ from this table.

using Equation 4, and in our example we get a net price closetc+the-money,

a t $0.34.

'ODuffie (1998 a) shows how a defauitable Floating Rate Note (FEIN) can be used to construct a perfect hedge to a defauit swap if the payout is defined to include accrued interest on the FR!'. As Duffie mentions, this wouid be an atypical speciîication for a credit default swap. " W e assume a premium of 20 basis points per annum due for the next 4 years.

Figure 5: Risk-free rates and default probabilities

time 1 3 -

1 3 4 5

Increasing the default probabilities12. we can re-cornpute prices for the bond

and for the default swap. We find t hat the bond depreciates while the default

swap appmciates. but the changes do not exactly offset, as iliustrated in

risk-free rate 5% 5% 5% 5% *5 %

Figure 6. Loosely speaking, we might Say that the hedge is about 85%

effective.

risk-free discount factor

0.952 0.907 0.864 0.833 0.784

r [ Using default rates 1 Default rates up 1 Change in

default probability (cumulative)

O. 1% t

0.5% 1.0% 1.6% 2.5%

1 1 1

1 total 1 $107.75 1 $100.71 1 - $0 .O4

5 year bond

Figure 6: Assessing the hedge under a change in default rates

Suppose, instead, we kept the five year default swap but took positions in

each of 5 bonds. maturing respectively in 1, 3. 3, 4 and 5 years. We can

choose our positions in such a way that our exposure to default rates is

zero, using the method outlined in Equation 5 . rUthough our portfolio is

100% hedged against a change in default probabilities. we find t hat it is now

sensitive to the implied recovery rate. See the totals in Figure 7.

in Figure .5 $107.41

This is aii of interest to the market-rnaker whose job may be to run as

fiut a book as possible (in other words. to avoid taking N k positions). On

12by 1% per year, so that, for example. the one year default probability increases to 1.1% and the five year cumulative rate increases by 5 x 1%.

byO.l%perannum $101.15

d u e -30.26

CDS

portfolio description 1 1 position

Iyr bond

MTM value rates 1 def. rates 1 Rec. up from

2yr bond 3yr bond 4yr bond 5yr bond

total

Figure 7: Hedging with a strip of bonds

price $0.34

the other hand, an investor accepts talring on risk as the necessary price for

expecting a given level of return on his investment . Only if he were happy to

earn the risk-free rate would he put ail his cash in US Treasuries or similar

size 100%

(vi r tudy) riskless securities. The investor wants to know which risks to

take on and which to hedge. We will attempt to value his book in such a

unchanged $0.34

way that this question is naturally answered. This is the topic of the nest

section.

up 1% $2.53

2.2 The investor's perspective

50% to 75% -$O.Il

Our investor has a particular portfolio of fixed incorne instruments (loans or

bonds) and wishes to know on which of t hem s hould he buy credit protection . But the challenge of invest ment policy is normally framed slightly differently

from this and as follows: for a given level of expected return demanded, how

can we construct a portfolio with the smallest possible ri&?

The classical solution to this challenge was given by Markowitz. Tobin and

Sharpe. Markowitz (1959) (and earlier papers) em phasized the importance

of diuersijtcation within a portfolio of stocks. He showed how an individual

stock's value in a portfolio should depend not just on its expected return but

also its covariance with the rest of the portfolio. Gaussian assumptions need

to be made about the return on a stockt so that a portfolio's expected return

and variance there-of uniquely describe it. -4 set of mean-kariance efficient

portfolios is computed, whereby for each level of expected return demanded.

t h e r ~ is no 0 t h pnssihl~ portfolin of stnrks with a. srnall~r m-riance. Tnhin

(1958) showed chat only one of these *efficient * portfolios was really efficient

and then, depending on the investor's degree of risk-aversion. more or less of

this optimal portfolio should be bought, with the rest invested in the ri&-

free asset. William Sharpe (1964) then showed that. in *equilibrium' ( d l

stocks fairly priced). Tobin's efficient stock portfolio is in fact the market

portfolio of al1 t radable stocks.

Our investor. on the other hand. has a portfolio of jùed income instruments

(loans and bonds) and not stock. X Gaussian mode1 may be just about

passable for a stock portfolio, but it is well knomn chat for a portfolio con-

taining credit risk it is quite inaccurate. This is due to the nature of credit.

where 'most of the time' we expect the promised cashflows to materidise

but with a srnall probability we will Iose our shirt. This gives rise to very

fat left tails in the probabiiity distribution of returns. We would also Iike to

relax the assumption t hat Our investor holds some sort of 'market portfolio'.

For esample, in reality he might be a commercial bank Nith a compara-

tive advantage lending to a particular geographicd or industrial sector. He

doesn't so much wish to know how he could hold the ideal 'market portfolio'

as to whether he can irnprove his risk-return profile by off-loading some of

his existent credit risk (by purchasing credit protection. for example). In

addition, the capital structure of his business (whatever it is) may result in

a particular risbaversion profile. Modern portfolio t heory does no t require

this ri& profile to be specified: a lower variance of return is always preferable

to a higher variance (for given expected return), regardless of risk-profile.

But with these Gaussian assumptions gone, we now need to bring back this

concept.

We begin with a review of Modern Portfolio Theory. Then ive change tack.

dropping the Gaussian assumptions and notional of optimal portfolios and

proceed with a different approach. This is based on the (fat-tailed) distri-

bution of the investor's profit-and-los (PSIL) and a utility function that

describes our investork relative preference to different P9;L values.

2.2.1 Modern Portfolio Theory

Harry Markowitz is credited with 'inventing' the concept of diversification.

He realised t hat investors should care about both the expected return and

the risk of their investments. Otherwise the rational investment strategy

would simply be to invest al1 your capital in the stock which promised the

highest expected return. He formalised the adage that "one shouldn't put al1

oneWs eggs in the same basketFL3 and set about finding out which portfolios

could be deemed eficent, in the sense that for a given level of expected

return. there nras no other portfolio with smaller 'risli'. Any given stock

should be vdued not only on the b a i s of its expected return. but also on

the basis of the CO-dependency of its return with that of every other stock.

Consider a world consisting of a number of stocks. each Nith Gaussian re-

turnsL4 and a degree of dependence between thern, uniquely defined by a

covariance matrk because of our Gaussian assumption. Different stocks

promise different expected returns, so the same will hold for different com-

binations there-of (portfoGos). For aII the portfolios promising a given level

of expected return, some r d have a larger variance of return that oth-

ers. hlarkowitz posed the problem of minimizing this variance subject to a

which, rather surprisingly, was not even the received wisdom of the day, judging by the criticisrn his theory attracted peynes (1939), in Varian (1993)l.

14We couid think of 'returns' in terms of the present value of future dividends, say.

constraint on the expected return demanded by the investor. Considering

the set of al1 possible portfolios and plotting their standard deviation of re-

turn against their ezpected return gives a bounded set (with the boundary

a hyperbola). This is illustrated in Figure S.

, q e r ted return

standard %uiation

Figure 8: Combinations of expected return and standard deviation

Tobin (1958) introduced into the mode1 a riskless asset. He brilliantly

showed that by investing in this risliless asset one could not only extend

the combinations of risk acheivable for a given level ofexpected return. but

also that there was a unique 'rnagical' stock portfolio that should be a p

propriate for all investors. shown in Figure 9 as the point (o. p) . By partly

investing also in the riskless asset or by borrowing and investing euen more

in this magical portfolio ('leveraging') an investor could find his optimal

investment strategy for a given level of ri&-aversion. The addition of the

straight line y = (p/a)x + r ont0 the diagram illustrates this fact. that al1

mean-variance efficient portfolios can be acheived nrith the same magical

portfolio of stock.

William Sharpe provided the insight that, in equilibrium, this 'magical'

portfolio is the market portfolio. For, if not, then a given stock would be

relatively ovenmlued (by consideration of its cova.riance with the rest of the

market) and lack of demand would drive its price d o m (and hence expected

return up) until equilibrium were obtained.

standard deviat ion

Figure 9: Introducing a riskless asset identifies an 'optimum' stock portfolio

Discussion of optimal portfolios aside, it is worth noting t hat any two portfo-

lios offering the same expected return and the same variance of return are in

fact offering an identical distribution of return. This is because a Gaussian

distribution is parameterized by its mean and voriance. So. regardless of an

investor's particular risk appetite. there is never a situation where he would

prefer a portfolio with a higher variance (for a given expected return)15.

This happily leaves risk-aversion out of the analysis in Modern Portfolio

Theory. We are interested in risk-aversion oniy to determine how to divide

our capital between the market portfolio and the riskless asset.

Returns on risky debt are highly non-Gaussian. and investors do not often

hold the market portfolio. For this reason, we propose the foliowing mode1

to incorporate these facts.

2.2.2 Investor preference in a nomGaussian world

.U investors are ri&-averse, Given the choice between two investments

promising the same expected return. an investor dl always choose the least

risky. But what do we mean by 'risky ' ? Can risk be characterized by a single - -

15- intuitive, almost obvious, idea is formaliseci in the next section after we have introduced utility hctions.

number? Both the variance of the portfolio (in Nodern Portfolio Theory)

and ualue-ut-risk (VaR), as currently used in risk management, are measures

that attempt to quant@ risk with a single number. Value-at-risk measures

a percentile from the tail (the 95th for example) and as sttch is deemed

by corne t r? h~ a 'het t~r ' meapitrP than varianre herariw it iisiially picks

up fat taiis to a greater extent. and it is the left tail of the distribution of

returns that concerns the risk manager. Neither a percentile rneasure or a

moment can tell the whole story. however - we can easily generate counter-

examples showing two distributions with an identical moment or percentile

rneasure where one is 'clearly' more risky than the other. This is illustrated

in Figure 10.

probabiiity density

VaR

Figure 10: Percentile measures can be misleading

Left of the vertical line, the area under the graph is the same for both

the Cauchy distribution and the triangle. so they have the same VaRL6.

although the Cauchy distribution is 'clearly' more risky. This is al1 rather

vague, though, and begs the question: what is this concept of risk that

these measures only seem to be partly addressing? Looking a t the diagram

above, it seems as if the ieft tail is the key. But how fur into the tail should

"The 'VaR' percentile shown in Figure 10 is approlamately a 70th percentile. not a 95th or 99th, for the sake of visuai clarity.

we be interested, and by how much are we interested? We need a scheme

by which we can answer these questions, and it is to utility functions that

we naturally look. A utility function d l give different 'weights' to different

parts of the P&L distribution where required. and by t a k n g the expectation

by which we can evaiuate every P&L distribution on an equivalent basis.

a(.)

Figure 11: A utility for every PkL value defines a concave function

Consider a concave utility function, strictly increasing, and wit h a large neg-

ative second derivative for large negative values, as shown in Figure 11. We

tnight evaluate two PSrL distributions for example, an uncertain PkL dis-

tribution (Gaussian, say) with mean value $6m. and a certain cash amount

of $5mG. and find that, with our utility function in Figure 11. both mea-

sures give the same expected utility. Our investor would t hen be indifferent

with respect to chosing between these two scenarios. This is illustrated in

Figure 12.

Alternatively we might nrish to compute the expectation of this utilitÿ func-

tion under each of the probability distributi~ns in Figure 10. to conciude

that the triangular distribution has a higher utfity than our Cauchy distri-

"i.e. a L~ed PSrL of S5m achieved with probability 1

34

utility

under

Figure 12: A utility function defines an equivalence between distributions

bution. Equivalently we might Say i t has a lower risk. but as 'ri&' has a

certain established meaning nre shall avoid the term and stick with utility

which of course wraps up the concepts of risk and expected return into a

single measure.

We ought to test our mode1 against some of our common-sense notions in

order to verify its reasonableness. To this effect. suppose we compared the

expected utility under two different Gaussian distributions -4 and B. which

have the same mean but where -4 has a lower mriance than B. A quick

glance is almost enough to verify that the espectation under A is higher

than the expectation under B. This is a consequence of our utility function

being concave. We shall try to formalise this.

Let u(x) be any concave utility function, and XI, -Y2 be two Gaussian

distributions. with

Xi - Y ( p y q)

and

Proof: let fi(x) and f2(x) be the respective probability density functions.

So, explicitly

e -(x-pj2;2tr: e-ix-pj? and fi(4 =

JZiraq

Let

so t h a t

(This is easily verified.)

Partition the state space of possible P&L values (-x. ,cc) into 3 disjoint

subsets. narnely

This is illustrated in

Now, as the integral

Rearranging, Ne can P

s- = ( -cap-< i l

S = ( p - d . p S - d )

S+ = b+d.m)

Figure 13.

of every density function is unity, we have

probabilz'ty density

Figure 13: Partitioning sample space to compare two Gaussian distributions

Therefore. because u ( . ) is concave tve have

y the symrnetry of f (S-) with f (S+) and Jensen's Inequality. Rearranging

again we obtain

which proves the result. When returns are Gaussian our theory is consistent

wit h Modern Portfolio T heory.

This is reassuring. The other trivial result to check is that if our investor

is completely risk neutml, then E ~ [ U ( S ~ ) ] = E~[U(&)] . This is true. A

risk-neutral investor has a straight line utility function u(x) = as mithout

loss of generdity. So, trividy

As an aside, it is worth noting that various alternative measures of risk have

been suggested in the literature. Dembo (1991) [in Saunders (1997)] has

suggested a generaiization of the notion of risk. referred to as regret which

measures the risk of a PSiL distribution relative to a benchmark distribution.

-1 function p,(rr) : ;VI -t R from a subset of nice random variables .LI to the

real numbers R qualifies as a regret firnctional if and onlÿ if the folIowing

a x i n m ~ are ca ticficicl.

Xotice that if our portfolio T is -re-cenrred' so that its espected d u e is zero.

t hen

- E'[u(T)]

satisfies t hese axioms for any concave u (.) , wit h T the trivial O 'dist ribu-

tion'. l8

2.2.3 Pricing investor preferences

This utility approach ailows us to compare any two risk-distributions on an

equivalent basis. and can determine which is more attractive to our investor.

given a specified utility function. However. so far it only gives us relative

value analysis between distributions - we don't have any idea what our units

are. We would like to convert our utility back into a cash amount for val-

uation purposes. To this end, we simply apply the inverse of the utility

function, to arrive at a cash value that our investor would be indifferent to

receiving (with certainty) compared Nith taking on a particular risk distri-

bution. (See Figure 13 for esample.) So, if S is our probability distribution

of PkL, and u(x) is our utility function, then we are interested in the value - -

"1 have slightly abused the notation here, with uf.) alternatively mapping either .CI + R or R + R but the meaning shodd be dear.

where u - l ( . ) is the inverse of our utility function. which we said was strictly

monotone (and therefore invertible) .

Now assume that we are looking a t some risk horizon a finite tirne from

today, Say t . so strictly speaking we should discount our value back to today

for pricing purposes. Let V be the value today of holding our risk position

over a time t with corresponding ri&-free rate r. Then

The application of this equation is illustrated in Figure 14. We assume here

that the vdues our randorn variable X takes are risk-neutral prices. the idea

being t hat our investor holds a position unhedged over the time period (O. t )

because the ri&-neutml value today (value when hedged) is smaller than

the value Y of keeping the risk unhedged until time t. But it is likely in

practice that our investor is not going to hedge his position a t time t either.

We would hope that the reason investments are held, in the real world. for

some period of time is a rational one - there is *more in it' for the investor

remaining unhedged.

So now consider the scenario where me want to compute the d u e today of

our investment, given that r e will re-hedge a t time t l if and only if the ri&-

neutral value (at time t l ) is greater than the value (a t time t [ ) of rernaining

unhedged until time t2 . At time t2 assume we close out our position anyway.

Then we have a recursive vaiuation formula, with the value at time O [today)

1 price today

risk-neutml portfolio valu

at time t

discounted real- world espected 4-1 ) pro bability utility demit y

Figure 14: Evaluating the utility of the unhedged portfolio

with Vi, ri is the ri&-free forward rate. from time t;,i to time t ; , as seen

from today. The assumptions are that our utility function u ( x ) is constant

through time. and that riskfree interest rates are independent of our value

process Vt (although this could be relaved if necessary). -4s usual. our

notation E![ ...] means E ~ [ ... 1 Ft] where Ft is the natural filtration.

In theory. we would define a Vo. Cf,. K, ... and so on until the maturity of

the portfolio. In practice this would be difficult. Furthermore, the naturd

question to ask at each trading time ti is not whether to hedge all or nothing,

but whether to hedge something and what. In this case. for the sake of

practicality we would consider just one time period. If our portfolio were

made up of n positions. so that at time t our risk-neutral book value is

then ive would define Vo by

= rnmau~' sçx

where we write S -Y to mean that S is a sub-portfolio of the larger

portfolio S = {-Y l , -Y2 . . . , -Yn) and

We ieave the positions in the subset S unhedged. and we hedge the rest.

For the sake of completeness, we could extend this formulation recursively,

so that at each -trading' time ti we get the choice of hedging any combination

of our existing ( u n hedged) positions. Then we would get. in the multi-period

model,

where for any Si C S we define

and, in a recursive fashion,

and so on.

Our restriction that Sk+L Sk means that at each stage, we can hedge

further (i.e. sel1 positions) but we cannot unhedge positions (buy new ones).

This might seem restrictive from a theoretical standpoint. but once we look

at the financial practicalities we re&e it is essential. For: othemke. we

would be vaiuing a hedged position by taking into account the possible future

benefit of unhedging it a t some later time. This is equivaient to valuing our

book on the basis of positions that we do not even have. a rather dubious

valuation rnethodology which might not receive general appro td amongst

accounting standards su t horities and regulators!

We emphasize that this recursive valuation method is not really practical

from a processing point of view. In practice we would define our portfolio's

\ d u e by Equation 6 and Equation 7, This would mean running a risk model

(on- once) ta compute the real-world distribution of our positions* future

(risk-neutrd) values s at tirne t. .As yet we have made no mention of

this risk model. In the next section we dkcuss how WC might model this

distribution of portfolio value a t the risk horizon.

2.2.4 CreditVaR models

Suppose we have a portfolio of risky debt (loans, say) and wish to compute its

price distribution a t some future point in time, t Say. Traditionally this risk

horizon is taken to be much larger (1 year. for example) than the risk horizon

for computing market risk (say 10 days) due to the difficulty in extracating

oneself from a credit risk position. Where-as a market risk hedge could be

adjusted daily. it might be much harder to close out an Illiquid corporate

loan position for example. Credit derivatives are addressing this disparity.

by allowing an investor to re-balance his credit risk in a much shorter time

frame. Default protection on a small corporate may be less liquid than the

interest rate swaps market, but still there is a good chance of being able to

a find a protection seller in the relatively short term. Certainly the days of

having to hold debt positions until maturity are gone.

So it rnight seem reasonable to look a t a risk horizon that is somewhat

longer than that n o r m d y used for market risk. but considerably shorter

t h m the maturity of the portfolio itself. We might consider a one month

or six month risk horizon for example. We wish to know the distribution

(under the reai-world measure P) of value a t the horizon. Shere are severaI

competing methods currently used to do this. We s h d outiine two of the

most common methods in use.

Credit Suisse First Boston (1997) (CSFB') developed ~ r e d i t ~ i s k + ~ ~ , a

method that sirnulates default only. Suppose nre have Y counterparties

where the probability of default by each before time t is pi. Assuming inde-

pendence between defaults. we have that the expected nurnber of defaults

and with the additionai assumption that pi are dl smail. me have that the

total number of defaults is Poisson. In other words, the probability of n

defaults is

If we then combine this with an exogenously derived distribution of loss

given default, we have a very quick way of calculating a loss distribution.

The problem is our independence assumption. In the real world defaults are

certainly not m u t u d y independene.

CSFB iooked at this problem of dependence in the following way. They

noted that there is a statist icdy significant variation in default rates from

year to p a r . One might Say that the default probabilities pi depended on the

general level of the rnacro-economy (or sorne other hidden variables). What-

ever the reason. t his could be modeled by assiiming a probability distribution

for the default rates themse1ves1? By sampling from this distribution. then

using the short-cut provided by Equation 8 (together with a distribution of

l o s in default), a complete PSrL distribution could be produced. A corre-

lation between default rates is implicit by using this method, and helps to

''In their modei the proportional changes in default probability are assumed to be the same for each 'categoryT (country or industry for example)

produce the *fat tailed' distribution typical for credit losses, as depicted in

Figure 15. pro babilit y densit y

Figure 15: P&L due to credit events typically has a fat left tail

Traditionally, the user would read off a certain percentile from this distribu-

tion, say the 99th. This percentile is usudly refered to as the CreditVaR'

and can be used to set a level of Econornic Capitd to hold against these

losses. One motivation for setting aside this Economic Capital is to provide

a cushion against bancruptcy that worlrs a (high) proportion x of the tirne.

where (1 - x) is the histoncal probability of default for a .LA rated cornpany

( say) , in the hope that the Rating Agencies will notice this and d o w them

to keep their rating! Another motivation is that the Regulatory Rules for

Capital Adequacy (for banks) seem to be moving in this direction.

At this stage we are not concerned with this CreditVaR nurnber. or any

particular percentile level. We Nish to use the whole distribution in order

to cornpute our ezpected utility.

One drawback of CSFB's model is that it only respects oittright default.

-4 general deterioration in credit worthiness is not picked up by the model.

Perhaps t hi is not important, t hough - traditionally loans were orîginated

then held to maturity. Without marliing them to market. the only thing

that did matter was whether the borromer defaulted or not. But here cornes

the rub. If we are going to restrict ourselves to a ri& horizon considerably

shorter than the portfolio's rnaturity. irnpücit in this is the assumption that

we can close out a deterioratine position at (or beforel the risk horizon.

To close out a position, ive rnust pay the prevailing market rate for the

distressed debt. but this model rites nothing off against distressed debt a t

the horizon. We see from this that there is a trade-off. By considering a

shorter risk horizon. we then need to be able to mark our positions more

accurately to the market at the horizon. J.P.Morgan (1997) attempted to

provide such a *mark-to-market' CreditVaR model with ~redi t l~ le t r ics~" .

Recall how Jarrom. Lando and Turnbull (97) ('JLT') made use of a transition

matriu to describe the probabilities of a firm's credit rating changing rhrough

time. Credithletrics uses a similar approach to this. although the goal of

JLT was pricing not risk management per se. Jarrow. Lando and Turnbuil

make a transformation to the matrk of historical transition probabilities

to convert them into risk-neutral probabilities. Credit'vlet rics has no need

to do this. for it is the real-world distribution of portfolio value at the risk

horizon that we are after. On the other hand, ive have an additional problem.

Using the transition rnatrix approach for CreditVaR means we do need to

find a way of introducing a dependence between the migrations of different

borrowers. J.P.Morgan's approach is to define correlated Gaussian random

variables, and map each random variable into credit rating space. (So for

example, a very high value would map a BBB to a .LL4 - the highest rating

- where-as a very low value maps BBB into default.)

A correlation structure between Gaussian random variables is uniquely and

easily realised (by C holesky decomposition, for example). Determining the

correlations is an altoget her harder problem. J.P.Morgan appeal to the

Merton (1974) model of corporate debt and show how a correlation between

bond prices can be determined by the correlation between their respective

equity prices and by a few other variables such as the (elusive) firm-%due.

Then. with a sample from a multimriate Gaussian distribution. each uni-

variate (i.e. marginal) is mapped into rating space (for its respective issuer)

Figure 16: Credithletrics mapping a Gaussian into a row of the transition matrk

In this way, by Monte Carlo simulation we can obtain a sample of possible

combinations of rating States for our portfolio. Then. given a firm's credit

rating, our respective bond-holding is valued using the risk-neut raI pricing

of the market, so that a BBB bond gets priced according to the market's

risieneutrd default probabilities for a BBB of appropriate maturity (Le. not

from historical default rates).

Credithletrics is a superior mode1 for considering the impact of credit changes

over the medium term (1 to 6 months, say) because it incorporates the

mark-temarket effect of ratings migration as well as outright default. -As

the credit derivatives market becomes more liquid and credit risk becomes

easier to hedge. the standard risk-horizon used is likely to become shorter.

When this happens, it d become more important to find a CreditVaR

model that additiondy allows for credit spread changes urithin rating, as

the change in portfolio value due to spreads changing within rating Nil1

become proportionally larger.

In the meantime. we have aU the necessarily tools to price our investor's port-

folio on a risk/reward basis. The CreditVaR model gives us the distribution

of ri&-neutral value at the risk horizon. Our utiiity function then evaluates

this distribution and returns an *economic vaiue' of holding the portfolio up

until the risk horizon. Shen by considering the effect of rernoving different

subsets of the portfolio, we can identify uneconomic concentrations of credit

risk and decide exactly what if anything to hedge.

Thus far we have assumed that our end-user's credit risk is derived from

a portfoIio of fixed income instruments. But credit risk dso exists on an?

derivative contract where one or other party may default on its contractual

obligations. Since the explosive growt h in over- t he-counter derimtives in the

19SOs, dealing with the credit risk embedded in a bank's derivatives book

has become a significant activity for both the industry and its regulators. In

the following chapter we consider the credit risk that arises from a derivative

transaction and attempt to derive its price.

3 An 'exotic' Credit Default Swap

In this chapter we look at the pricing of counterparty rL9k This is the credit

ri& that a firm takes on when entering into a derivatives transaction with

a UefiiiikaLUk t û ~ ~ t c i . ~ a ~ t y .

We introduce the topic of counterparty risk using the example of an interest

rate swap to illustrate a derivative with two-sided potential credit esposure.

We review some of the measures used for risk management. and discuss

how to compute expected credit exposures over the life of the trade. In

computing expectations we omit any mention of measure at this stage.

Then. proceeding in a more formal fashion. we discuss how this risk can be

hedged (at least in theory) using a combination of default swaps and cal1

options defined on the underlying transaction. By demonstrating the corn-

pleteness of the hedge. we find that we have *inadvertently0 priced the risk.

This fact establishes that . amongst other things. our expected exposures

in fact need to be cornputed using the ri&-neutral measure, and not the

real-world measure.

3.1 Counterparty risk modeling

Most derivâtive transactions involve either or bot h parties taking on an

eiement of credit risk. X firm may need to consider the possibility that

its counterparty to a transaction wül not honour its cashflow obligations.

In some over-the-counter'O agreements. there is credit risk from only one

partyk perspective. A good exampie is an option, where the premium is

paid upfront, and the trade cannot subsequently have positive value to the

200ver-the-counter agreements are bespoke contracts between two parties wit hout the intermediation of an organised exchange

option writer. On the ot her hand, in a swap contract, the trade can generally

have positive value to either party. This makes swaps more interesting from

a counterparty risk perspective, and in some sense they are the most generul

derivatives to consider. In the following section we explain the workings of

an interest rate swap. then proceed to discuss how to measure the credit

risk it generates.

3.1.1 What is an interest rate swap?

-4 swap is an agreement between two parties to exchange cashflows in the

future. In an interest rate swap, party X agrees to pay party B fixed interest

on a notional amount at pre-agreed dates in the future. in eschange for

interest payments at the floating rate. In the typical (vanilla) contract the

notional amounts. currencies and payment dates are the same for both legs

of the trade. This is illustrated in Figure 17. (The cashflows are shown from

the point of view of party A.)

floutirtg interest payments

jùed interest payments

Figure 17: The cashflows from an interest rate swap

The fixed rate of interest is usually determined on trade date so that the

contract has zero (risk-neutrai) net value to each party. Depending on the

movement of interest rates after trade date, however, the swap can become

a liability to one party and therefore an asset to the other.*' In addition.

an interest rate swap generally has a life of several years, during which time

there rnay be a reasonable probability of one or other party defaulting. The

fact that the trade can develop a positive value to either partv over this

time makes it particularly interesting to model from the point of view of

counterparty exposure. We now examine some of the techniques typically

used to model this exposure.

3.1.2 Credit exposure on a swap

An interest rate swap entered 'at-the-money' (so neither party is required to

make an upfront payment) will have, a t any future tinie. a positive value to

one of the parties, depending on the movement of interest rates. If interest

rates rise (more than 'expected'). the swap d l become more valuable to the

party which is receiving the floating rate and paying the fixed rate. Con-

versely. with M i n g rates, the swap becornes more vduable to the receiver

of fixed payments. Of course. over the life of the trade the swap can take

on positive value first to one party, then the other, in accordance with the

ups and downs of the market.

If we look a t the trade from party A's perspective. his rislr2* is that party B

will default a t some time in the future when the trade has positive value to

.LZ3 So the modeling challenge to party A is to determine the distribution

of potential future (positive) exposure to B conditional on B defaulting at

some future point during the life of the swap.

negative mark-temarket (MTM) to one, positive &[TM to the other 12aside h m any unhedged market ris& that the position may be contributing ta '='CVe assume that party B fin& a third party wiUing to cake on his side of the trade, in

the event that B is in-the-money on the swap and defadts.

3.1.3 Review of counterparty risk modeling

The typical question asked is: what is the potential exposure to A should

party B default at some future time t. In other words. what is the distribu-

tien nf the conditional ranclom n r i a h b d ~ f i n ~ r t hy

where -Yt is the exposure at time t, r is the time of default and F(r = t ) is

the natural o-field generated by { r = t ) .

The following simplifying assumption is often made. We suppose that the

potentid exposure at time t is independent of the default process for party

B. In other words we take

-Y; = -Yt.

This 'independence' assumption has been made explicitly in the lirerature

severai tirne~.~"Ve shall return to this issue in due course. In the meantirne,

nre wish to know the distribution of -Yt. In particular. we might wish to know

the gworst' -Yt can be (wit h a given degree of confidence). So if our confidence

level were 95%, we would want to know

for each time t . Sirnulating the future mark-to-market (SITM) of the swap

through time2' allows this profile

to be computed. For an interest rate swap, this profile might look something

Iike the one shown in Figure 18.

2 4 ~ a r example, EuiI and White (1995), Huli and White (1992) and Jarmw and Turnbuii (1995)- "or by using alternative d y t i c d methods

credit exposure

Figure 1s: 'Worst' case esposure on an interest rate swap

This kind of rnodeling gives us an idea of what we mzght lose. with a given

degree of confidence. if our counterparty defaults at some point during the

life of the swap. This sort of information might arguably be usefuI for s bank

set ting ezposure limits against particular coun terparty names. but it does

not directly help to p k e this credit risk. More reasonably we might antici-

pate that it is the ezpected future exposures that wouId be more important

here.

3.1-4 Comput ing expected exposures

In fact it is true in a very precise sense that it is the expected exposures

that should concern us, from a pricing point of view. We s h d elaborate

on this in a future secton. For now, though, we presume the reason we are

interested in the trade's e ~ ~ e c t e d * ~ future positive MTM is t hat. toget her

with the probabüity of our counterparty defaulting (and the recoven; rate)?

this d reveal our total q e c t e d loss.

We recall chat if the trade has a negative MTM when our counterparty

"under some appropriate prubability measure, which we omit to mention for now

defaults, we lose nothing. 'iegative exposure means zero exposure, as illus-

trated in Figure 19.

Figure 19: Credit exposure as a function of contract d u e

This is sure to introduce a max(0, -Yt) term into the expression for the es-

pected exposure at default. Our totaI expected loss is then given as the

expected default probability t imes recoveq times eqected czposure. in te-

grated (or summed) over the Me of the swap." Consider the discrete time

setting, where t is today. T is the maturity of the swap. and time has n (T - t )

equalIy spaced jumps from t to T. Then. esplicitly our total expected 1 0 s

is

wit h

where *Ys is the MTM at tirne S. r ( s ) is ri&-free short rate at time S. h, is

the probability of defaulting between times s and s + 1 given no default by

" ~ e e section 2.1.5 on Reduced Form Models and in particdar D&e and SingIeton ( 1999).

s, and L,+ is the loss given default (between times s and s + 1). -1s always?

ail variables are presumed to be processes adapted to the natural filtration.

With the independence a ~ s u r n ~ t i o n * ~ back in place, this formula becomes

where kjln is defined as in Equation 10. The first erpectation in the summa-

tion l o o h Like it could be the price of some sort of forward starting default

swap. Cnlike a typicai pricing formula. however, it has no discounting in-

terest rate term. -4s such it is Ioob more like the price of a default swap on

some strange notional whose value is expressed as a present d u e dollar.

The second espectation in the summation look very much like an option

pricing formula. In particular it is the price2%t time t of a swaptzon agree-

ment. which is the right to enter into (what's left of) the swap agreement a t

time j/n. We note that this 'optionality' arises from the fact that negative

exposure counts as zero exposure. rather like an option espiring out-of-

the-money has zero value, and is consistent with the hockey-stick shape in

Figure 19 which is reminiscent of a cd1 option payoff.

In the follonring section we see if we can use the resemblance between deriva-

tives prices and these expectation terms in order to find a hedge for this risk.

"Sec the previous section. "if our elpectatiaru were under risk-neutral measure

3.2 Pricing and hedging counterparty risk

3.2.1 A weU-known result

In the nert subsection we are coing to make use of the following well-known

practical result from derivatives pricing.

Suppose two assets -4 and B follow diffusion processes. Then a derivative

that promises to redeem the product

at a future time T is hedged by taking positions in both asset -4 and asset

B. adjusting the hedge dynamically through time.30

3.2.2 Setting up a hedge

We attempt to define a derivative hedge whose price replicates the terms in

Equation 11. To this end. consider the discrete time setting of the previous

section. We have LV = n(T - t ) discrete and equally spaced time-jumps from

t to T.

First define H;( t ) to be the pnce at time t of a default swap which pays out

one dollar less the recovery rate, after the j-th step in the event of default

during that step. By The Fundamentai Theorem of Finance31 it equals the

discounted espected payout under the risk neutral measure Q. Hence

' * ~ t this stage we leave this r d t vague and unproven because we WU need it in a slightly more gened form.

"We ignore the question of whether these default swaps can themseives be hedged by more primitive instnunents, and we treat tbem as primary instruments in their own right.

using the same notation as in the previous section.

Similarly define Sj(t) to be the price at time t of an option to enter into the

underlying (interest rate) swap at time t + j / n . So that, similarly.

Discounting complicates the following formulae unnecessari- so for ease of

exposition we set al1 risk-free rates to zero. If t his seems incompatible wit h

t here being an interest rate swap in the first place. consider an alternative

trade. So in the foiiowing section. we will work with the simplified formulae

and

Now define as our hedge a portfolio of default swaps and interest rate swap

tions, as follows.

. Purchase H,(t) swaptions s ~ . ~ ~ Purchase S j ( t ) default swaps H,.

The idea behind this hedge is to use the result from section 3.2.1. In our case.

Sj (t) follows a diffusion process but Hj ( t ) is a jump-diffusion. Nevert heless.

we intend to demonstrate that the hedge still worlrs.

Precisely. a t time t Our hedge portfolio has value

.. number number lV A n

2'"' ( t ) = C HI ( t ) Sj ( t ) + Sj (t) Hj(t). 32 That is to Say, for each j, observe the value of Hj ( t ) and buy that many swaptions.

56

ive need to be mindful that those numbers (under the braces) are just

'arnounts' of something eise, so they are not functions of time. In other

words, a t some time s > t , our hedge has value

:v 2'"' ( s ) = 1 H j ( t )Sj (s) + Sj ( t) H, (s)

j= 1

noting the positions of the s and the t.

Consider the first time step. from t to t + l l n . Either we have default or

we do not. If not. then assuming a diffusion process for the hazard rates.

we have a neiv set of 'diffused' default swap prices. And assurning another

(independent) diffusion process for the variables underlying the interest rate

swap. we have a new set of *diffused7 swaption prices. So. unless default

occurs, the new d u e of our portfolio is

expired swaption ., active hedge

The near-term default swap Hl (t ) expires wort hless so it is omit ted from the

formula. -4s for the near-term swaption (SI ( t t ) ) . it has fulfilled its hedging

function and is noiv redundant. For this reason. we need to specify that

the swaptions are cash-settled so as to avoid inadvertently entering into the

underlying swap as the swaption expires. .Assuming no default tooli place.

dl remaining swaptions and default swaps need to be rebaianced.

3.2.3 Re-balancing the hedge

This means, at time t l , making a purchase of

AR: = (Hj( t l ) - H,(to)) swaptions Sj

AS: = (S j ( t l ) - Sj ( to) ) default swaps Elj

for each j = 2,3? 4, ..., .M.

After adjusting our portfolio, it has value

Then after the second time-step, assuming no default occurs. we have

already settled just expired ..

Mter a second adjustment of AH; swaptions and AS: default swaps.

and so on. re-balancing the portfolio at every time step. .\ssuming no default

occurs in the first k - I time steps. have

already settled

3.2.4 Default occurs and the position is closed

We have assumed diffusion for the swaption and default swap prices thus

far, but what happens a t default (the 'jump' part of the 'jumpdiffusion'

hazard-rate process)? Well, suppose chat default occurs during the Ir-th

time-stept i.e.

t + (k - i ) / n < r < t + k/n. Then the k-th default swap expires in-the-money (H&) = Lk) and ali the

rest becorne worthless. This gives us a portfolio value of

expired swaptions active swap tions A r L

% 0 \

Notice t hat the middle term Sk(tk-,)Lk is very similar to the loss on coun-

terparty default with regard to our underlying interest rate swap, which is

Sk(tk)Lk.

FVe now know the value of our portfolio after default. but mhat about the

total cost of the hedgeS? This is just the initial cost of the replicating portfolio

plus the cost of readjusting it at every time step.

3.2.5 The total cost of the hedge

Our position has cost us the initial hedge plus the re-hedging. Esplicitly

this is

1 r e hedging

Leaving h d f of the first summation aione (for example. the one labeiled

snraptions). and distributing the other haif (labeiled default swaps) so as to

group terms together with a commoo j suffix, gives us

Rearranging again we O bt ain

where we wili show that the double summation tends to zero in the Iimiting

case. First we relate these costs to the d u e of the portfolio at default.

3.2.6 The net cost in the Iimit

To recap, ive started ntith a counterparty risk Y ( t ) at the initiai time t . We

then set up a replicating portfoüo of total cost c("). and at default we incur

a l o s of L ( t r ) by which time our portfolio has a value of z(") (tk). Hence

our counterparty risk Y ( t ) has incurred a net cost at time t k of

Many of the terms in c(") and -2(")(tk) partly cancel? leaving

We want to show that the first four terms on the right hand side of the

Equation 14 tend to zero JF the tirne-step &ri& to zero' !ei&g juct the

fifth term. To this end. notice that H, - l / n because it is default protection

for a single time step. 'low the default probabiiities follow some (unspecified)

diffusion process (except at default). so that each AH, 5 ( l / n ) The

swaption prices follow a diffusion too. so t har AS, - m. The means that

the first term

k

C H,(~,-~)AS: k ( l / n ) f i fi.

Similarly the second term

The double sumrnation term is the interesting one. The outer (1)-summation

behaves like n because it is summing the changes from different contracts.

We cannot clairn any strong diversification here - for example in a single fac-

tor diffusion mode1 the contracts with different rnaturities would al1 move

together. However, the inner (2)-summation sums consecutive changes in

the same contract. T hese consecutive changes have zero covariance because

our swaption and default swap prices are martingaies. This gives us -diversi-

fication' benefit through the Central Limit Theorem. Siightly more formally.

Let

k-i j

j=l l=l

for standardized Gaussian random variables 4, and some constants a:. bf

and 4. Then. because we assume independence between 4 and c:. we have

t hat

E[R,] = O

and the variance is given by

for some constants di and standard Gaussian & and which are al1 mu t u d y

covariant-zero". Using this independence. we have that ~ [ & e ~ ] = O and

E[{'<~] = O for al1 1 # m. Therefore

for some constant d. So E[R:] tends to zero as n tends to infinity. The

mean is also zero, therefore Rn + O in probability, as n + m.

.As for the fourth term in Equation 14, rather straight-fomardly. A s ~ L ~ - m*

''The c and C terms are mutudy independent by our assumption of independence between our swaption and default swap prices, where-as between Meren t ci terms and between ciSemnt C' terms we have zero covariance by the Martingale property of traded prices,

In the lirnit as n tends to infinity, al1 these terms tend to zero" in probability,

so we have IV

~ ( " ' ( t ~ ) -t C Hj(to)Sj( to) gf V ( t k ) .

Now, crucially. this expression does not depend on the time of default or

even whether default occun. Our counterparty risk V ( t ) was exchanged

for a payout with a certain present value of Fr(tk). So by appealing to the

principal of no arbitrage,

This is exactly the same as Equation 11 but with the rather vague Et[ ...] Q now replaced by Et [...], the expectatioa with respect to the risk-neutml

meosure Q. Thus. we have shown that it is right to think of the cost of

counterparty risk as the erpected loss, but importantiy we take expectations

not under the real-world measure but under the risk-neutral measure. even

to compute erpected future exposures.

3.2.7 A note on some assumptions made

To ease the working in the previous sections we ignored the effect of dis-

counting and set the risk-free interest rate to zero. With suitable notation,

discounting could be re-introduced into the formulae. with little effect on the

result unless we suppose stochastic interest rates that are actually correlated

with the default process. Independence was &O assurned between the coun-

terparty default process and our underlying trade. If we were to relax this

assumption and posit an instantaneous correlation between the swaption

and defauit swap prices, an alternative tractable formulation could be re-

alised for the price of counterparty default. With this scheme it is suggested

34Here, zero means the distribution that is a constant zero.

63

that the assumption of an exogenous Poisson jump at default is maintained

nevert heless.

In this chapter we have considered the counterparty risk on a single under-

lying trade. In practice, however, several existing trades may exist with a

counterparty. with protection required against the total loss in the event of

default. With netting enforceable betwen different contracts at default. we

are concerned only with the net positive exposure. Our expression for the

expec ted positive exposure becomes slighrly more complicated. .Ut houg h

it retains its resemblance to an option pricing formula. it involves several

underlyings. In this way it could be interpretted as a basket option pricing

formula, where the 'option' still has a strike price of zero but is now on the

basket of underlying trades.

3.3 Counterparty risk and reward - concluding remarks

Traditionally, counterparty risk modeling is based on worst case analusis.

Nith potential future esposure measured with some confidence level. Then a

combination of qualitative and quantitative rnethods are used to determine

credit line availabiiity. At the same time. the pncing of t his credit risk is

kept away from the question of whet her t here is appetite to transact furt her

business with a counterparty. Further. there is no real notion of a 'risk-

horizon' in the market-risk sense: the nearest t hing there is to a risk horizon

is perhaps the maturity of the derivative.

However, in section 3.2 we show how counterparty risk can be hedged and

therefore priced on a risk-neutral basis. This puts it on an 'equal footing'

with the credit risk on loans, where hedging can be effected nrithin the short

tirne frame required to malce a credit derivative transaction. With this in

mind. we need to consider the variabity of this pnce only up until a future

tmding-time horizon, a t which point we hedge the risk if it is economic to

do so. based on a risk/reward (PkL utility) evaluation (see section 2.2.3).

Interestingly this variability is affected not only by changes in credit spreads

but also by changes in the mark-temarket of the underlying tramaction.

$urprking!y. hwviwr, ~ ~ T S ~ - F I ~ S P OxpnstirPs rhrolighniir th^ l i f ~ nf th^ rrnrb

turn out not to overly concern us. In the case of a swap, the increase in

credit risk due to a significant rise in its mark-temarket is offset by the

'swaption' part of the exotic hedge.

In this way, counterparty risk ends up being framed in pricing terms. with

the discovery of an unlihly sounding hedge forcing us to price this coun-

terparty risk using the riskneutral measure. This risk-neutral cost of credit

stili has an associated *market risk'. however (where by market ' we mean

the credit market). Then this residual market risk is also reduced to a

pricing problem. but this time via the use of risli/reward a n a l ~ i s and P9;L

utility functions.

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