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Equity-to-Credit Problem

Philippe Henrotte

ITO 33 and HEC Paris

Equity-to-Credit Arbitrage

Gestion Alternative, Evry, April 2004


Or how to optimally hedge your credit risk exposure with equity, equity options and credit default swaps


Agenda

Traditional approach: diffusion + jump to default The notion of hazard rate Inhomogeneous model (local vol & hazard rate) Calibration and hedging problems

More robust approach: jump-diffusion + stochastic volatility Incomplete markets Homogeneous model Optimal hedging


I – Traditional approach


The equity price is the sole state variable Structural models of the firm: Default is

triggered by a bankruptcy threshold (certain or uncertain: Merton, KMV, CreditGrades)

Reduced-form model: Default is triggered by a Poisson process of given intensity, a.k.a. “hazard rate”

Synthesis: making the hazard rate a function of the underlying equity value (and time)


Default is a jump with intensity p(S, t)

dtS

VS

t

V

2

222

2

1

X

Given no default before t: With probability (1 – pdt):

no default With probability pdt:

default Taking expectations (in

the risk-neutral probability)

Risk-free growth of the hedged portfolio

dtpXS

VS

t

VE

2

222

2

1

dtrE


In the risk-neutral world

We solve the PDE opposite

X is the jump in value of the hedge portfolio Sdef is the recovery value

of the underlying share Vdef is the recovery value

of the derivative. Example : Convertible Bond

Game is over upon default

pXrVS

VrS

S

VS

t

V

2

222

2

1

defdef SSS

VVVX

RNSV defdef ,max


Example: Convertible Bond

We recover a fraction of face value N We may have the right to convert at the

recovery value of the underlying share

RNSVprVS

VSpr

S

VS

t

V,max

2

12

222


Example: Credit Default Swap

Credit protection buyer pays a premium u until maturity or default event

We model this as asset U

utSUtSU ),(),(

UprS

USpr

S

US

t

U)()(

2

12

222



Credit protection seller pays a contingent amount at the time of default

We model this as asset V is the insured security

)()(2

12

222

prS

SprS

St

defaultoftimeV )(100



R recovery rate CDS guarantees we recover par at maturity Simple closed forms when hazard rate is

time dependent only:

T

tt

dupr

i

it

tueTtU)(

),(

T

t

T

t

pdurdu

eeRTt 1)1(),(

u is such that U(0,T) = (0,T) at inception


Example: Equity Options

)(2

222

2

1 tTrKePprPS

PSpr

S

PS

t

P

pCrCS

CSpr

S

CS

t

C

2

222

2

1

PDE for a Call under default risk

PDE for a Put under default risk


Example: Equity Options

The jump to default generates an implied volatility skew

Problem of the joint calibration to implied volatility data and credit spread data

Calibrate (S, t) and p(S, t)? In practice, we use parametric forms and

p as S 0


Hedging (traditional approach)

The hazard rate is expressed in the risk-neutral world (calibrated from market data)

Collapse of the bond floor (negative gamma)

The delta-hedge presupposes that credit risk has been hedged with a CDS (or a put, …)

Volatility hedge?


What if there were a life after default?(Convertible bond case) Share does not jump to

zero Issuer reschedules the

debt Holder retains

conversion rights It may not be optimal to

convert a the time of default

),(' defdef SVV

'''

'2

1'2

222 rV

S

VrS

S

VS

t

V

RNTSV ),('

RCoupontSVtSV CouponCoupon ),('),('

StSV ),('


Switch to “default regime” The default regime and the no-default regime

are coupled through the Poisson transition

'''

'2

1'2

222 rV

S

VrS

S

VS

t

V

tSVVprVS

VSpr

S

VS

t

V),1('

2

12

222

Two coupled PDEs, with different process parameters and different initial and boundary conditions


Conclusion: the status of default/no default is the second state variable


II – Incomplete Markets


Incomplete markets

The state of no-default decomposes into sub-regimes of different diffusion components and different hazard rates

This replaces (S, t) and p(S, t) with stochastic and stochastic p

It turns the model into a homogenous model Markov transition matrix between regimes Stock jumps between regimes yield the

needed correlations with vol and default


Inhomogeneous

Default State

No Default State

p(S,t)

(S,t)

Homogeneous

1

2

3

Default State

21

23 32

p1Default

31

p3Default

12

p2Default

13


Incomplete markets In a Black-Scholes world without hedging, you

can use the BS formula with any implied volatility value

Perfect replication in the BS world imposes uniqueness: the implied volatility had better be the volatility of the underlying

Under a general process (jump-diffusion, stochastic volatility, default process, etc.), perfect replication is not possible…

…and many non arbitrage pricing systems are possible (risk neutral probabilities)


Pricing and calibration If we wish to price one contingent claim

relative to another, we can work in the risk-neutral probability. This is called “calibration”: Reverse engineer the prices of the Arrow-

Debreu securities from the market prices of a given set of contingent claims

Use the AD prices, or risk-neutral probability measure, to price a new contingent claim

Whenever we wish to price a contingent claim “against the underlying” (by expressing the optimal hedging strategy), we have to work in the real probability


Pricing through optimal hedge The “fair value” of a contingent claim is the

initial cost of its optimal dynamic replication strategy (for some optimality measure)

This requires the knowledge of the historic or real probability measure…

…while calibration only recovers a risk neutral probability

We need to know the drift or the Sharpe ratio of the underlying

The drift of the underlying drops out of the Black-Scholes pricing formula, not of the Black-Scholes world


Calibration is just a pricing shortcut(It has nothing to say about hedging) Examples:

Calibration of the risk-neutral default intensity function p(S, t) from the market prices of vanilla CDSs, or risky bonds

Calibration of the risk-neutral jump-diffusion stochastic volatility process from the market prices of vanilla options

To express the hedge, we have to transform back the risk-neutral probability into the real probability


Hedging credit risk

Using the underlying only The notion of HERO Correlation between regimes and stock price

Reducing the HERO Using the CDS to hedge credit risk and an

option to hedge volatility risk (typically, hedging the CB)

Using an out-of-the-money Put to hedge default risk (typically, hedging the CDS)

Completing the market


Tyco

Tyco, 3 February 2003 Stock price $16 Sharpe ratio 0.3 Joint calibration of options and CDS Option prices fitted with a maximum

error of 4 cents CDS up to 10 years


Tyco Volatility Smile

Feb-03

Jul-03

Jan-06

512.5

2030

0%

50%

100%

150%

200%

250%

Volatility

Maturity Strike


Tyco CDS Calibration

0.6%

0.8%

1.0%

1.2%

1.4%

1.6%

1 2 3 4 5 6 7 8 9 10

Maturity

Qu

arte

rly

Pre

miu

m

Market Model


Calibrated Regime Switching Model

Regime 1 49.86%Regime 2 27.54%

Brownian Volatility

Jumps Jump size IntensityRegime 1 -> Regime 2 4.48% 3.34Regime 2 -> Regime 1 -58.68% 0.169

Regime 1 0.119Regime 2 0.032

Default Intensity


Tyco Convertible

Vanilla convertible bond Maturing in 5 years Conversion ratio 4.38,

corresponding to a conversion price of $22.8


Optimal Dynamic Hedge

With the underlying alone HERO is $9.8 If one uses the CDS with a maturity of 5

years on top of the underlying, the HERO falls to $5

If we add the Call with the same maturity and strike price $22.5, the HERO falls down to a few cents and an almost exact replication is achieved


Optimal Dynamic Hedge

As a result, the convertible bond has been dynamically decomposed into an equity call option and a pure credit instrument

This is the essence of the Equity to Credit paradigm


References E. Ayache, P. Forsyth, and K. Vetzal: Valuation of

Convertible Bonds with Credit Risk. The Journal of Derivatives, Fall 2003

E. Ayache, P. Forsyth, and K. Vetzal: Next Generation Models for Convertible Bonds with Credit Risk. Wilmott, December 2002

E. Ayache, P. Henrotte, S. Nassar, and X. Wang: Can Anyone Solve the Smile Problem?. Wilmott magazine, January 2004

P. Henrotte: Pricing and Hedging in the Equity to Credit Paradigm. FOW, January 2004

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