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SAMSI Credit Risk Working Group Presentation

Model for Unified Valuation of Credit and Equity Derivatives

Apoorv Mathur(Ongoing work)

Joint Work with Prof. Jean-Pierre Fouque

November 16, 2005North Carolina State University

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Outline for this talk

• Objective• Background

– Valuation of Credit and Equity Derivatives– Credit Risk: Structural and Intensity Based Approaches– Structural: Two factor Stochastic Volatility Model– Intensity Based: Cox Process (Doubly Stochastic Process)

• Intuition• Model• Results

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Objective

A Model where Default may occur by• Hitting a Barrier• Jump to Default

Valuation of Credit, Equity Derivatives• Implied Volatility

– European Call Options

• Credit Spreads– Defaultable Zero coupon Bonds with no recovery

Explain/Fit Data, Economic Intuition, Tractability

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Background

• Structural Approach– Diffusion to Default– Merton Model, Black Cox Model

• Intensity Based Approach– Jump to Default– Poisson Process

• Hybrid Approach– Default may occur through diffusion or a jump

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• Two factor Stochastic Volatility Model– “Stochastic Volatility Effects on Defaultable Bonds” by Jean-Pierre

Fouque, Ronnie Sircar, Knut Solna (2004)– It is a Structural model, so Default Probability/ Credit Spreads for very

short maturities go to zero

• Cox Process (Doubly Stochastic Process)– “Credit Risk” SAMSI Course Notes by Ronnie Sircar (2005)– Need explanation for assuming intensity as stochastic process

• Unified Valuation of Corporate Liabilities, Credit Derivatives (Short Maturities), and Equity Derivatives– “A Jump to Default Extended CEV Model” by Vadim Linetsky, Peter

Carr (2005)– CEV assumes perfectly negative correlation between Stock price and

the Volatility and therefore it is unrealistic

Background

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Intuition behind the Model• Stock Price and Value of the Firm are modeled using a single framework

• Default can occur through jumps that kill the underlying stock price process (send it to a cemetery state) or by hitting a prescribed barrier

• Default is an absorbing state

• Two common underlying economic factors are driving the volatility and the jump intensity

– Mathematically they are just slow and fast mean reverting stochastic processes

• The stochastic volatility factors are negatively correlated with the stock price (leverage effect)

– Choose Negative correlation between Stock Price & Volatility– Implied Volatility Skew

• The default indicators (and jump intensities) are positively related to the historical volatilities

– Choose the jump intensity to vary proportionately to Variance– Credit Spreads

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Basic Notation

Value of the Underlying:

Volatility Driving Factors:

Volat

{ Pre-Default,0 Post-Defaul

ility:

t}

,

( , )

(

Jump Intensity: , )

t t

t t

t t

t t

S S

Y Z

Y Z

Y Z

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The Model

Y Y( 1 )

1

( 2 )

Z 2 Z

*

(0)

1 2 2 ( ) ( , ) ,

( ) 2 ( , ) 2 ,

( ( , ))

Under the risk neutral measure P ,

( , ) ,

( ,

t Y t t t t

t Z t t t t

t t t t t t t t t t

t t

dY m Y Y Z dt dW

dZ m Z Y Z dt dW

dS r q Y Z S dt Y Z S dW

Y Z

( ) ( )1

21

2

0

,

1

1

2

( , ) ( ) ( ) ( , ),

inf{ | ( , ) } where is Exp(1) r.v., Intens

where inf{ }, , , > 0 and > 0

a

) ( ) ,

inf{ | }, Structural Mo

ity Mod

del

el

t t

i j

t t t t

t

s s

Y

t i

Z

t

t j

t e

t S D

Y Z b t t Y

dW dW

Z

t Y Z e

dt

ds e

1 2

re small parameters, corresponding to fast and slow mean-reversion,

respectively. And ( , ), ( , ) are the market prices of risk.t t t tY Z Y Z

Pre-default Stock Price

(Post Default, St = 0)

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The Model

T { > T}

European Call Option

Defaultabl zero coupon

( )

= 1

Bond

T TC S K

P

T

( )

Implied Volatility

Y

1Y = log( )

ield

BSC I Co

Po rT

*

*

Call Option

Defaultable zero coupon Bo

( )

( )

nd

rTT

rTT

Co E e C

Po E e P

Payoff Price of Claim

Computations

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Special Cases

1, 2

1 1

1

2

0

2

1

For our Model, we have,

where

inf{ | },

Structural Mod

inf{ }

D>0, (

inf{ | ( , ) } where

) 0, ( ) 0,

D=0, ( ) 0, ( )

is Exp(1) r.v.,

Intensity Mode

0

el

,

l

t

t

s st Y Z ds e e

t S D

b t t

b t t

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Monte Carlo Simulations

• n = 1000 or 5000, dt = 0.001 or 0.0005• So = 50, D = 0 or 35, Yo = 0, Zo = 0, m = 0;

• r = 0.02, q = 0, σ1 = 0.2;

• ρ01 = -0.1, ρ02 = -0.1, ρ12 = 0;

• ε = 0.1, δ = 0.05;

• ν1 = 0.5 or 1, ν2 = 0.5;

• b = 0, 0.1 or 0.2, λ1 = 0 or 2;

• K = [42,45,48,50,52,53,55,57];• T = [1/8,1/4,1/2,3/4,1];

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Results

• Implied Volatility Skews for a Call Option

• Term Structure (short maturities) for Credit Spreads of a Zero coupon Bond with no recovery

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Implied Volatility1 No Defaults (v1,v2 = 0.5) 2 Black Cox Model (v1,v2=0.5)

3 Intensity Model (b=0.2, λ1=0) 4 Intensity Model (b=0.2,λ1=2)

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Yield Spreads1 Structural Black Cox Model (v1,v2 = 0.5) 2 Structural Black Cox Model (v1=1,v2=0.5)

3 Intensity Model (b=0.2, λ1=0) 4 Intensity Model (b=0.2,λ1=2)

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Yield Spreads

5 Hybrid Model (D=35, v1,v2 = 0.5, b=0.1,λ1=0) 6 Hybrid Model (D=35, v1=1,v2=0.5, b=0.1,λ1=0)

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Thank You