mfe230vb 2011 section all

7/29/2019 MFE230VB 2011 Section All

1/77

MFE 230VB GSI Section 1

Frank Fung

09/09/2011


2/77

Project Think:

Is there any hidden information in credit data?

Can we/how can we exploit such information?

Two parts:

1. What signal(s)?

2. How do you execute your strategies according to thosesignals?

Examples of signals: P/E ratio, LIBOR-OIS spread,

Examples of strategies: Trend, Contrarian, Always good to have a story behind signals/strategies. tojustify that you are not merely datamining

Frank Fung MFE230VB GSI


3/77

Project (cont.) Things to consider:

Signals can be exogenous, e.g. Moodys rating

Signals can also be from your own model, e.g. PD You would have to calibrate your model to market prices. How much

time are you ready to spend on that?

Especially in credit finance, what is a good measure of risk? Standarddeviation? VaR? Max. drawdown?

There are so many rates (LIBOR, Treasury, Fed Fund) and so

many spreads (TED, LIBOR-OIS, CDS) in the market.Understand their differences and pick the right ones.

Practical concerns: bid-ask, transaction cost, margin



4/77

Project (cont.) More things to consider:

If your arbitrage strategy is betting on two quantities to

converge, does the convergence take too long? An alternative strategy can also serve as the benchmark

If you want to do this, pick a zero-alpha strategy as benchmark

. . -

level, long-short ratio), can you optimize the strategy w.r.t.

these parameters?

Can you identify the source of your PnL ? (performance

attribution) If so, can you hedge away the unwanted ones?



5/77

Project (cont.) Seemingly trivial points:

Label all the axes and graphs on the plots

Properly label your equations/sections, especially if you want torefer to them later

Citation (authors, paper title, year)

st stan ar errors an or test stat st cs ; t e mean va uealone does not say much

Put the exhaustive, detailed data and numerical results in your

appendices; save the main body for a concise table that

contains the most interesting results



6/77

Lecture Review - Overview Structural Models in a few lines:

Assume full knowledge of the firm

Default Firm asset dynamics Focus on whydefaults occur

Examples:

Merton Black-Cox

KMV

Give good PD prediction



7/77

Lecture Review - Overview(cont.) Reduced Form Models in a few lines:

Do not assume full knowledge of the firm

Default

Hazard rate or alike(hidden variable) Focus on whendefaults occur

Examples: arrow and Turnbull 1992

Duffie and Singleton [1999] Usually fit market prices better

Very nice survey article: Bohn [2000], A Survey of Contingent-Claim Approaches to Risky

Debt Valuation



8/77

Lecture Review Merton Model Advantage:

Similar to B-S. People like B-S.

PD and DD connected by simple, analytic relation Limitations:

Asset dynamics not observable

Default can only occur at maturity Short maturity spread too small (overnight spread is actually

zero)

Can be improved by introducing jump



9/77

Lecture Review Improving Merton Black-Cox

Allows for default before maturity

Whenever asset value hits default barrier Default barrier can be time dependent

KMV

In Merton, PD = N(-DD); in KMV, PD = f(DD) where thefunction f() is fitted to historical data

Historical data is likely to be skewed and fat-tailed compared to anormal distribution

Inhomogeneous capital structure (short- vs. long-term debts,common vs. preferred stocks)



10/77

Lecture Review Short Maturities Under Merton:

Problem:

Short maturity risk structure very sensitive to quasi-debt-to-firm ratio Cause:

Default is not recognized before maturity, even though asset leveldrops below barrier

Problem: Short maturity spread too small

Cause: Asset dynamics, modeled as a GBM, is continuous. It takes a finite

amount of time for it to diffuse. Remedies: Adding jump, using Imperfect Information

Models



11/77

Digression Jump Process SDE

where dN is a Poisson counting process

Note:

Can still do one-step MC in the presence of jumps

Ref: Paul Wilmott on Quantitative FinanceCh. 57



12/77



13/77


Frank Fung

09/11/2011


14/77

Agenda Lecture review

Merton model walkthrough

A reduced form credit model that looks like short ratemodel: Duffie and Singleton [1999]



15/77

Lecture Review Probability Measure A few words on probability measure

In credit derivative texts people usually omit the nuances ofP-

vs. Q-measure. Be cautious about probability measure, especially when dealing

with structural models:

,

measure

You then calculate the DDP, and hence the PDP

What about if you want to find the price of a bond using PDP?

Have to translate it into risk-neutral measure, PDQ

See Bohn, Active Credit Portfolio Management pp.177



16/77

Digression Volatility So many kinds of it:

Actual

Not directly observable

Kalman filter

Historical/Rolling

A backward-looking proxy to the actual vol.

ow ong o we oo ac or ro ng ca cu a on eware o reg me

change Implied

Model dependent

Just an alternative way to quote the price. Much like bond price vs. YTM

Local/Forward Time- and price-dependent function thats consistent with the market

E.g. Dupire formula



17/77

Di-digression Whats so forwardabout

forward volatility? There is an analogy between bond pricing and option

pricing:

YTM Implied volatility Both are ways to quote price by giving the value of a parameter

Instantaneous forward rate Forward volatility

Both require walking through a realized path



18/77

Lecture Review FI Instruments Spreads i.e. Gimme a single number that reflects the relative

values of two securities. There are dumb and smart waysto churn out such a number: Yield Spread Simple subtraction

Z-S read Lets account for the fact that the

instrument has CFs not only at maturity

OAS Lets account for the fact that there isembedded option in the instrument

Defined on a tree



19/77

Lecture Review Survival Analysis Following are all related but different concepts (you will

see these again in the ABS class):

F(t) Survivor function Alive up to t

f(t) Survival density

s e pro a y o a ure w n e per o ,

(t) Hazard function

(t)dt is the conditional probability of failure within the period (t, t+dt)

Ref: Kalbfleish The Statistical Analysis of Failure Time Data



20/77

Lecture Review Survival Analysis (cont.) For your quick reference (Tbeing the failure time):



21/77

Merton Model Walkthrough After taking two credit classes, you should know

(KNOW-KNOW, not sorta know) how to implement

Merton model, arguably the simplest non-trivial structuralmodel

Whats the best way to assure that?

Lets walk it through together!



22/77

Merton Model Walkthrough (cont.) Not going to show you the picture with bell-shaped curve

One-year historical stock price time series

Et=[0.3500, 0.3652, 0.4187, 0.3565, 0.3808, 0.3910, 0.3568, 0.3468, 0.3568,0.4644, 0.5699]

I know that

Why? Because I cheated, these are data simulated using

those parameters

In practice, extract the drift and volatility ofE by Rolling calculation

MLE and Kalman filter



23/77

Merton Model Walkthrough (cont.) Other parameters:

X= 0.24 default barrier (total book liability)

T= 2 maturity, or horizon

r = 0.05 risk-free rate

Our objective is to deduce

. From HW1,

Equating drift

Equating diffusion

Under the Merton model framework the stock, or equity E, isa call option to the firm asset:



24/77

Merton Model Walkthrough (cont.) From previous slide:

From HW1,

Equating drift

Equating diffusion

era on s requ re ecause e quan es we are so v ng

for are deeply embedded within the ugly partialderivatives

Now Maturity

Historical stock data spans this region



25/77

Merton Model Walkthrough (cont.) The iteration: (need initial guess of [ ]0 )

1. Back out vectorA from vector E, using [ ]i as input (

find the underlying stock price given the option price) 2. Use vectorA to calculate the realized [ ]i+1

. ee t s set o ers s gn cant y rom t e

previous iteration

4. If the difference is smaller than some threshold, call it a day;

else, keep iterating

MFE230VB GSI Frank Fung


26/77

Merton Model Walkthrough (cont.) When the iteration converges: (red = Asset, blue = Equity)

The spread betweenA and E

is almost a constant. Why? Because

0.55

0.6

0.65

0.7

0.75

0.8

. ,

Check: exp(-0.05*3)*0.24 = 0.21

= (0.0376,0.2743) Ref: Jovan [2009], The Merton Structural Model and IRB

Compliance

0 1 2 3 4 5 6 7 8 9 10

0.35

0.4

0.45

0.5



27/77

Merton Model Walkthrough (cont.) Summary

Have equity data (stock price series) up to now

Want for the calculation of DD and hence PD Asset process is latent, i.e. not observable

But we know(under Merton framework, anyway) that equity is

a van a ca to rm asset

This allows us to back out that are consistent with

the equity process



28/77

Reduced Form Model Revisited Many flavours of reduced form models

Duffie and Singleton [1999]

Cathcart and El-Jahel [1998] Vaillant [2001]

Jarrow and Turnbull [1995]

They all try to deduce default probabilities from marketprices of credit derivatives (risky bonds are creditderivatives too!)

Well look at two of them in greater details



29/77

Case Study Duffie and Singleton [1999] Modeling Term Structures of Defaultable Bonds

Very similar to short rate model

Recap: Vasicek model Risk-free pure discount bond price is (model independent)

Short rate dynamics is Turns out that for Vasicek model (and other affine models) the

expectation value can be evaluated analytically

Fit to risk-free yield curve to find the three short rate dynamicsparameters

Use the fitted short rate dynamics to price other derivatives



30/77

Case Study D-S [1999] (cont.) D-S proposes that we can play a similar trick for risky

instruments

A risky pure discount bond price is

ris the risk-free short rate, hdt is the conditional defaultprobability, L is the fractional LGD

Default manifests itself as a deeper discounting

Suppose both rand hL follow square-root processes (i=1for rand i= 2 for hL):



31/77

Case Study D-S [1999] (cont.) This is essentially a 2-factor CIR model

Closed-form solution exists for bond price (see for

example Chen and Scott [2002])

Two ste s: 0.063

Fit rto the Treasury curve Fir hL to the spread

0 5 10 15 20 25 300.056

0.057

0.058

0.059

0.06

0.061

0.062



32/77

Case Study D-S [1999] (cont.) Summary

Objective:

Fit risk-free rate dynamics to risk-free yield curve Fit credit spread dynamics to risky yield curve

Use the fitted parameters to price other derivatives

Can choose different short rate and default processes

Usually would use multifactor models for both rand hL

D-S also discusses a defaultable HJM framework

An LMM with default risk is studied Schonbucher [2000] These all resemble what youve seen in the fixed income class



33/77



34/77


Frank Fung

09/20/2011


35/77

Agenda

HW1

Section 2 Review

Reduced form model: Jarrow LandoTurnbull



36/77

Last Time

Reduced form model by Duffie and Singleton

is intensity (hazardous rate) based

is identical in principle to short rate models calibrating of which amounts to fitting the parameters of the

SDE


o ay we ta a out a re uce orm mo e t at

is rating based

Is analogous to binomial tree, just that there are more

branches

calibrating of which amounts to solving for the risk-neutralprobabilities


37/77

Case Study Jarrow Lando Turnbull [1997]

A Markov Model for the Term Structure of Credit Risk Spreads

JLT [1997] takes a Markov chain approach

Credit migration/transition matrix

Lets consider a simple example, with 3 credit classes:

As shown above is the transition matrix describing anInvestment-Junk-Bankruptcy economy, assuming thatbankruptcy (=/= default) is an absorbing state

In practice you can have N-by-N matrix covering all ratings This is in the P-measure, and can be estimated using rating

agency data



38/77

Case Study JLT [1997] (cont.)

The transition matrix in Q-measure

is what is needed for pricing

-

This matrix in general can be time-dependent. In discretetime setting, we have a unique matrix at each timestep i

The n-step transition probability is simply the matrix

product



39/77


Whats the plan?

Ingredient 1: P-measure probability from rating agency

Ingredient 2: R-N derivative connecting P- and Q-measure

Ingredient 3: Risky bond price Q-measure probability

Important Assumption



40/77


As an example (3 credit ratings, 4 years): Step 1

Obtain the P-measure transition matrix from rating agency

Step 2 Get the risk-free and risky bond market prices

p = [99.2, 98.7, 96.5, 93.8] for 1-, 2-, 3- and 4-year maturitiesvinv = [98.1, 97.3, 95.4, 91.6] for 1-, 2-, 3- and 4-year maturities

vjunk = [97.0, 90.1, 82.3, 78.9] for 1-, 2-, 3- and 4-year maturities

vban = [96.8, 85.1, 79.4, 66.3] for 1-, 2-, 3- and 4-year maturities

Step 3 Estimate delta, the proportional recovery when default occurs



41/77


Step 4 (see Matlab code)

Calculate

Step 5 (see Matlab code)

Calculate iteratively for t>1

1 1

12

31

23

0

.

12

31

23

0

.

12

31

23

0

0.5

1

12

31

23

0

0.5

1



42/77


How to price derivative using the matrices?

For example, to price a call option on a junk bond maturing in 2 yrs

expiring in 1 yr

with strike K


option? 6! Replicating portfolio consists of p(0,2), vinv(0,2), vban(0,2),

vjunk(0,1), vjunk(0,2), B(0)

The junk can end up being (I-d), (I-n), (J-d), (J-n), (B-d) or (B-n)

Ref: Jarrow [1995], Pricing Derivatives on SecuritiesSubject to Credit Risk


43/77


Summary Objective:

Fit the Q-measure transition matrix to the bond prices

Use the transition matrix to price other derivatives

Final remarks: One advantage is that the framework considers different ratings

explicitly

The assumption that credit quality is independent of short rate reasonable for investment grade, doubtful for junk bonds

I considered discrete time case in the example

Both of the above can be relaxed

Beware of negative probability

Ref: Schoenbucher, Credit Derivative ModelsCh.8



44/77



45/77


Frank Fung

09/27/2011


46/77

Agenda

CDO pricing, as simple as it can get

Sensitivity of different tranches to correlation

How to create AAA tranches Caveats



47/77

Last Time

A reduced form model

By Jarrow Lando Turnbull

That takes a Markov chain approach Which gives the risk-neutral probabilities

And explicitly take ratings into account



48/77

CDO Pricing

If we are to choose the most simplistic model

In equity option pricing:

Black-Scholes

In FX option pricing:

Garman-Kohlhagen


(turns out that you can have a model named after you just by re-labeling the variables)

In fixed income pricing:

Short rate models such as Vasicek, CIR

In CDO pricing:

???


49/77

CDO Pricing Vasicek Model

Vasicek gets the last laugh again!

Lots of assumptions to simplify things:

Large asset pool Homogeneous assets within the pool

Single factor to describe default


Gaussian copula

Constant correlation across assets and time

Disclaimer: Vasicek Model is a pricing model, not a

credit fitting model (i.e. Merton, Black-Cox, Duffie andSingleton)

Vasicekrequires PD calculated elsewhere


50/77

CDO Pricing Vasicek Model (cont.)

In other words, given the PD:

CDS spread

CDO tranche coupon rate


Zti is a stochastic variable that describes the accumulated losssuffered by the tranche during coupon period i, as apercentage of notional


51/77


Setup and Notations

Normalize everything: dividing by total portfolio value

Tranches are defined by attachment points [KU,KL]

0 10.05 0.20


In this case the equity tranche takes the first 5% loss, the Mezzanine

tranche takes the next 15%, the senior tranche takes the rest

Cumulative loss of whole portfolio, Ztotal, up to a moment

and Cumulative loss of tranche j, Z, up to a moment are

related through

Z = min(Ztotal,KU) min(Ztotal,KL)


52/77


Derivation (briefly):

LHP assumption infinitely many assets

Pairwise correlation is a single constant, i.e. Time independent

Uniform across all assets


s ng e ran om quan y i,t e erm nes asse e au a .

Xi,t has systematic and idiosyncratic components:

The unconditional default probability of all firms is pt, hence

homogeneous


53/77


Gis the unconditional probability that fractional portfolio loss

is below , 0 < < 1

This is the so called Gaussian Copula


,

portfolio probability How do we make use ofG?

Recall that the tranche spread depends on E[Z],Zthe loss

suffered by the tranche


54/77


0 10.05 0.20

= 0.09


is the loss given default KU/L is the upper/lower attachment points, 0 < KU/L < 1

e.g. a Mezzanine tranche might have KL = 5%, KU= 20%

Stop for a moment and make sense of the above Now that we know how to calculate E[Z], we can carry out

numerical integration and find the tranche coupon rate


55/77


0.01

0.015

0.02

0.025

0.03Yield Curve

0.02

0.03

0.04

0.05

0.06Unconditional PD


maturity = 2yr payment = 3m = 0.54

tranching = [0% 3% 6% 20% 100%] (4 tranches)

We look at two correlation scenarios

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01


56/77

Discussion: Correlation Sensitivity

High correlation

hurts senior tranche

benefits equity tranche

Just as we expected!

We have so far assumed -8-6

-4

-2

0

2Log of coupon rate vs. rho


that is constant acrossall tranches

This is all very well

until the market slaps you in the face and tells you yourewrong

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-16

-14

-12

-10

Equity

Senior


57/77

Discussion: Correlation Smile

Whenever you see a smile

Not a good sign

Model is too simple

In stock option pricing we have seen the volatility smile

Take the market option prices


Calculate the impliedvolatilities

See a smile (or smirk) across different strikes

In CDO pricing we might see a correlation smile

Take the market tranche coupons

Calculate the impliedcorrelations

See a smile across different tranches

B C d C l i


58/77

Base vs. Compound Correlation

There are two types of implied tranche correlations: Compound correlation:

Use market prices for [0% 5%], [5% 20%], [20% 100%] Back out the implied correlations

Base correlation:


For the following equity tranches ([0% 5%], [0% 20%], [0% 100%]):

Back out the implied correlations

Why consider base correlation? It looks not as intuitive Base correlation smile is usually smoother than compound correlation

smile

Di i CDO D Id ?


59/77

Discussion: CDO, a Dangerous Idea?

Market value of coupon rate for AAA tranche ~ 2%

Using our very simple model:

Start with junk asset with PD = 4% (S&P rating B) If we form a CDO with 2 tranches, how much equity tranche

would be sufficient to produce a AAA senior tranche?


Equivalently, how much equity tranche would make the coupon

rate for the senior tranche < 2%?

Equity

Tranche

Senior

Tranche

Senior

coupon

[0% 1%] [1% 100%] 2.36%

[0% 3%] [3% 100%] 2.04%

[0% 5%] [5% 100%] 1.83%

Di i CDO D Id ? ( )


60/77

Discussion: CDO, a Dangerous Idea? (cont.)

Questions:

CDO issuers are usually required to retain the equity tranche.

Does it help protecting the investors?

CDO pricing is hard. Why do the issuers still include so many

asset in the pool? Wouldnt it make more sense to have fewer?


Di i CDO D Id ? ( t )


61/77


Excerpt from Coval [2007]:

Althoughthe equity tranche isvery riskyit is exposed

primarily to diversifiablelosses.

as the value of N (# of assets) becomes largertranches bear

progressively more systematic risk.


,

while the highest credit rating is AAA, the supplierslever up thesesecurities to match more closely the default probabilities of other

AAA-rated securities.

senior CDO tranches have significantly different systematic risk

exposures than their credit rating matched, single-name

counterparts

Di i CDO D Id ? ( t )


62/77


Going back to our two questions: CDO issuers are usually required to retain the equity tranche.

Does it help protecting the investors?

CDO pricing is hard. Why do the issuers still include so manyasset in the pool? Wouldnt it make more sense to have fewer?

If Coval 2007 is ri ht


Vasicek CDO pricing model way too simplistic

Senior tranche is riskier than most people would like to think

Senior (equity) tranche is usually over-(under-)priced

A CDO tranche with AAA rating and a bond with AAA rating

mean very different things Coval [2007], Economic Catastrophe Bonds

Summary


63/77

Summary

Lots of assumptions required to make CDO pricing tractable:

Large asset pool

Homogeneous assets within the pool

Single factor to describe default

Gaussian copula


Just as the Black formula in the IR market, Vasicek is anindustry standardin the sense that its convenient to quote the

implied correlation

Ref: Elizalde [2005], Understanding and Pricing CDOs

Ref: Hull[2004], Valuation of a CDO and an nth to Default CDS


64/77



65/77


Frank Fung

10/04/2011

Agenda


66/77

Agenda

HW2 Problem 4

Credit derivative market

Final thoughts/ideas on project


Last Time


67/77

Last Time

Pricing CDO with Vasicek model

Assuming large, homogeneous asset pool

Gaussian copula

Default of individual assets are correlated with correlation

Produce the correct qualitative results tranche values depend


on

But being a constant is too strict an assumption

Risk of CDO

A very thin equity tranche is enough to produce AAA senior

bonds Coval [2007]: risk of senior and equity tranches are

qualitatively different

HW2 Problem 4


68/77

HW2 Problem 4

Volatilities

0.024

0.026

0.028

0.03

30-day rolling

90-day rollingGARCH(1,1)


0 50 100 150 200 250 300 350 400 4500.01

0.012

0.014

0.016

0.018

0.02

0.022

HW2 Problem 4 (cont )


69/77

HW2 Problem 4 (cont.)

Default probabilities

0.03

0.035

0.04

0.045

Normal

t-3

t-30


0 50 100 150 200 250 300 350 400 450 5000

0.005

0.01

0.015

0.02

0.025

Credit Derivatives


70/77

Credit Derivatives

CDO

CDS

nth

to default Indexes: CDX, ABX


Credit default swaption

Credit Derivatives (cont.)


71/77


CDO

By now you are familiar with:

Cash CDO with pool of assets

Synthetic CDO with vanilla CDSs

Theres a third way to create a CDO


nth to default CDS

Unlike vanilla CDS, nth to default CDS has to be priced in a

portfolio context

Hence it has correlation exposures, just like a CDO tranche

does (a vanilla CDS does not) Makes it a convenient tool to bet on correlation



72/77


Credit indexes

A recent development (


73/77

j

Try not to use simple moving average for mean and

volatility. Alternatives include:

Kalman filter

GARCH

Exponential moving average


Strategy not profitable?

Apply your techniques to a different firm/index

Once you have the infrastructures coded up it should be easy

to change

Project (cont.)


74/77

j ( )

Dont know what to discuss?

With the abundance of tools and functionalities on Matlab you

can achieve a lot without too much sweat

Some examples:

tcdf(), trnd(), tpdf() instead of normcdf(), normrnd() and


portvrisk() for portfolio VaR Plotting and graphing

Project (cont.)


75/77

j ( )

A point on backtesting:

You fit your model to the in-sample portion of the data. It

looks good.

Now you try it on the out-of-sample portion. The

performance is disappointing.


- ,

the out-of-sample performance looks good. In this case the out-of-sample data is not truly out-of-sample

Alternative: divide the dataset into 3 portions:

The largest part is in-sample (~60-70%)

Use a small set for the the tweaking

Leave enough (>20%) for truly out-of-sample test

References


76/77

Personal favorite

Wilmott, Paul Wilmott on Quantitative Finance

All-purpose reference

Hull, Options, Futures and Other Derivatives Comprehensive coverage on credit modeling

Bohn, Active Credit Portfolio Management in Practice

ot or t e a nt- earte

Brigo, Interest Rate Models Credit models overview

Bohn [2000], A Survey of Contingent-Claim Approaches to Risky Debt Valuation

Good luck with your projects!

Feel free to email me questions concerning project/fixedincome and credit in general


Cheatsheet


77/77

Merton Model vol.s

Merton Model PD-DD mapping

CDS rate

IRS rate


mfe230vb 2011 section all

Documents