Convertible Bond pricing and hedging

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Equity-to-Credit Problem

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

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 http://www.ito33.com

I – Traditional approach

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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?

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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

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

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II – Incomplete Markets

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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

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Tyco Convertible

 Vanilla convertible bond

 Maturing in 5 years

 Conversion ratio 4.38, corresponding to a conversion price of $22.8

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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

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