Philippe Henrotte
ITO 33 and HEC Paris
Equity-to-Credit Arbitrage
Gestion Alternative, Evry, April 2004 http://www.ito33.com

Or how to optimally hedge your credit risk exposure with equity, equity options and credit default swaps http://www.ito33.com

 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 http://www.ito33.com

http://www.ito33.com

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) http://www.ito33.com

Default is a jump with intensity p(S, t)
E
  



V
 t

1
2

2
S
2
 2
V

S
2

 dt




V
 t

1
2

2
S
2
 2
V

S
2
 pX

 dt
E
   
X
 
 r
 dt



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 http://www.ito33.com

In the risk-neutral world

V
 t

1
2

2
S
2
 2
V

S
2
 rS

V

S
 rV
 pX
X

V

V def
V def


V

S

S

S def

 max


S def
, RN




We solve the PDE opposite
X is the jump in value of the hedge portfolio


S def is the recovery value of the underlying share
V def is the recovery value of the derivative.
Example : Convertible
Bond
Game is over upon default http://www.ito33.com

Example: Convertible Bond

V
 t

1
2

2
S
2
 2
V

S
2

 r
 p
 
S

V

S
 rV
 p

V
 max


S
 
, RN
 
 We recover a fraction of face value N
 We may have the right to convert at the recovery value of the underlying share http://www.ito33.com

Example: Credit Default Swap
 Credit protection buyer pays a premium u until maturity or default event
 We model this as asset U

U
 t

1
2

2
S
2
 2
U

S
2

( r
 p

) S

U

S

( r
 p ) U
U ( S , t

)

U ( S , t

)
 u http://www.ito33.com

Example: Credit Default Swap


Credit protection seller pays a contingent amount at the time of default
We model this as asset

 V is the insured security
 
 t

1
2

2
S
2
 2 

S
2

( r
 p

) S
 

S
 r
  p (
  
)
 
100

V (
 
)
 time of default http://www.ito33.com

Example: Credit Default Swap
 R recovery rate


CDS guarantees we recover par at maturity
Simple closed forms when hazard rate is time dependent only:
U ( t , T )
 t i
T 
 t ue
 t
 t i
( r
 p ) du

( t , T )

( 1

R ) e

T
 t rdu

 1
 e

T
 t pdu




 u is such that U ( 0,T ) =

( 0,T ) at inception http://www.ito33.com

Example: Equity Options
 PDE for a Call under default risk

C
 t

1
2

2
S
2
 2
C

S
2

 r
 p
 
S

C

S
 rC
 pC
 PDE for a Put under default risk

P
 t

1
2

2
S
2
 2
P

S
2

 r
 p
 
S

P

S
 rP
 p

P

Ke
 r ( T
 t )
 http://www.ito33.com

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 http://www.ito33.com

Hedging (traditional approach)
 The hazard rate is expressed in the riskneutral 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?
http://www.ito33.com

What if there were a life after default?
(Convertible bond case)
V def

V ' ( S def
,

)

V '
 t

1
2

'
2
S
2
 2
V '

S
2
 rS

V '

S
 rV '
V ' ( S , T )

RN
V ' ( S , t )
 
S



V ' ( S ,
 t
Coupon
)

V ' ( S ,
 t
Coupon
)

RCoupon 
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 http://www.ito33.com

Switch to “default regime”
 The default regime and the no-default regime are coupled through the Poisson transition

V
 t

1
2

2
S
2
 2
V

S
2

 r
 p
 
S

V

S
 rV
 p

V

V '

S ( 1
 
), t
 
 Two coupled PDEs, with different process parameters and different initial and boundary conditions

V '
 t

1
2

'
2
S
2
 2
V '

S
2
 rS

V '

S
 rV ' http://www.ito33.com

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http://www.ito33.com

Incomplete markets


The state of no-default decomposes into subregimes of different diffusion components and different hazard rates
This replaces stochastic

( S, t ) and p ( S, t ) with
 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 http://www.ito33.com

No Default
State
Default State
Inhomogeneous p(S,t)

(S,t)
Homogeneous

1

2

2

1

2

3

3

2

3

1

1

3

1

2

3 p
2

Default p
1

Default p
3

Default
Default State http://www.ito33.com

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) http://www.ito33.com

Pricing and calibration
 If we wish to price one contingent claim relative to another, we can work in the riskneutral 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 http://www.ito33.com

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 http://www.ito33.com

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 http://www.ito33.com

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 http://www.ito33.com

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 http://www.ito33.com

Tyco Volatility Smile
Volatility
250%
200%
150%
100%
50%
0%
5
Strike
12
.5
20
30
Feb
-03
Jul-0
3
Jan
-06
Maturity http://www.ito33.com

Tyco CDS Calibration
1.6%
1.4%
1.2%
1.0%
0.8%
0.6%
1 2 3 4
Maturity
5 6 7
Market
8 9
Model
10 http://www.ito33.com

Calibrated Regime Switching Model
Brownian Volatility
Regime 1
Regime 2
49.86%
27.54%
Jumps Jump size Intensity
Regime 1 -> Regime 2 4.48%
Regime 2 -> Regime 1 -58.68%
3.34
0.169
Default Intensity
Regime 1
Regime 2
0.119
0.032
http://www.ito33.com

Tyco Convertible
 Vanilla convertible bond
 Maturing in 5 years
 Conversion ratio 4.38, corresponding to a conversion price of $22.8
http://www.ito33.com

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 http://www.ito33.com

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 http://www.ito33.com

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 http://www.ito33.com