# Quant Strategy Credit

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```					CREDIT CONTINGENT
OPTIONS
Frédéric Abergel
Ecole Centrale de paris
Introduction

Credit Contingent option: a payout is
delivered conditionally to a credit event not
occuring before the maturity, nor any
coupon date, of the option
Credit Contingent Option
Some examples:
 Option with a counterparty which may default
(go bankrupt)
 Convertible Bond
 Extinguishable swap that terminates when the
counterparty fails to fulfill one of its obligations
 A note paying equity-based coupons (say) as
long as a given bond does not default
MODELLING BY
REPLICATION
Single asset, single credit to begin with
Replicating strategy based on delta-
hedging the asset and dynamically trading
CDS
Three sources of risk: asset volatility,
CREDIT MODELLING
Book by Duffie and Singleton “Credit Risk”

Numerous papers by Schönbucher on
Credit Derivatives

A few references (Rutkowsky, Jeanblanc-
Pique) on modelling by replication
MODELLING BY
REPLICATION

 Modelling assumptions
 A jump-diffusion for the asset
 A diffusion for the intensity process governing both
the default process and the jump of the asset
 The asset may or may not jump, depending on
whether there is a default correlation or not
 The same Cox-Poisson process drives the
credit event and the potential jump of the asset
 The asset and spread driving factors are
correlated
dSt   t St dWt  t St dt   J  1 St d  t
1

d t  at dWt  bt dt
2

d W1 , W2   t
  dt
MODELLING BY
REPLICATION
Hedge portfolio approach:
 consider a “credit contingent” contingent claim
 use the asset and (in theory) two risky
instruments (CDS, bonds with different
maturities) to eliminate the risk factors
 Determine the risk premia via calibration
 Get the pricing equation and the hedges
HEDGE PORTFOLIO SET
UP

1 unit of contract Vt
- t units of one risky bond
-  t units of another risky bond
- t units of asset
HEDGE PORTFOLIO SET
UP

V c V
dPt      dSt      d t  V   V   d  t 
S       
 V 1 2 2  2V 1 2  2V                       2V 
       S          a            a S           dt 
 t 2         S 2
2      2
S 
 t dBt1c   t dBt2 c   t  Bt1  Bt1  d  t 
 t  Bt2  Bt2  d  t   t dStc   t  J  1 St d  t
RISK FREE PORTFOLIO

V
 t
S
V   V     t  J  1 S  t  Bt1  Bt1   t  Bt2  Bt2 
V      B1      B 2
 t      t
             

  V 1 2 2  2V 1 2  2V                      2V  
         S           a           a S         
  t 2         S  2
2      2
S  
dPt                                                         dt
                                      
  t  B1  a  B1    t  B2  a  B2 
2 2                     2  2

             2                    2             
  t                        t
2                        2

RISK FREE PORTFOLIO

V
V  V   S  J  1 S   t  Bt1  Bt1   t  Bt2  Bt2 
    

V       B1      B 2
 t      t
              
 V 1 2 2  2V 1 2  2V                      2V          B1 a 2  2 B1 
      S           a            a S             t                
 t 2         S 2
2    2
S            t     2  2 
 B2 a 2  2 B2 
 t                
  t   2  2 
RISK FREE PORTFOLIO
Three equations in two unknowns: there
exists a zero linear combination of the
rows of the matrix with non zero weights
(basic algebra)
Hence there exist two functions k,q of the
independent variables (the market prices
of risk) such that the following set of
equations holds:
B1 a 2  2 B1                       B1
            k  Bt  Bt   q
1   1

t   2    2

B2 a 2  2 B2                        B2
            k  Bt  Bt   q
2   2

t   2     2


V 1 2 2  2V 1 2  2V                    2V
  S          a            a S       
t 2         S 2
2      2
S
           V                  V
k  V  V  

 J  1 S   q
             S                 
PRICING EQUATION

V 1 2 2  V 1 2  V
2          2             2
V
  S          a            a S      
t 2         S 2
2      2
S
           V                  V
k  V  V  

 J  1 S   q
             S                 
MARKET PRICE OF
RISK(S)
 k is the “market price
of jump risk, q is the
“market price of
 Once they are fixed,
the bond price obtains                                         
 T                   
formally as              E  exp    k  u , u  du  t   
                                    
under the risk-neutral             t                           
diffusion for the
hazard rate              d t  adWt  qdt   2
MARKET PRICE OF
RISK(S)
 Question: how to choose and estimate the risk
premia ?
 Model tractability
 Market information
 Example: affine functions of the hazard rate
 Leads to simple formula for the risky bond
 CIR process so as to ensure nonnegativity of the
hazard rate
MODELLING BY
REPLICATION
This approach is interesting because:
 Gives a price associated with a hedge
 Helps one determine which parameters can
be calibrated, which one should be estimated
from historical parameters.
 Separates between the spread risk and the
default risk (relevant since there is dynamic
SOME EXAMPLES
 Credit Contingent Stock:
the contract is to deliver 1
unit of a given stock
subject to no credit event.

 Payoff
ST 1 T
 Going back to the pricing
equation, one can easily
see that variables can be
separated
CREDIT CONTINGENT
STOCK
V  Q  , t  S
 Particular solution

where Q is a “risky
bond” with correlation   Q 1 2  2Q          Q
 a          a
and jump corrections,    t 2      2

solution to the 1D          Q
q      JQ  0
equation                    
CREDIT CONTINGENT
STOCK
 In the deterministic case, one recovers the
simple pricing formula:
stock * survival probability

 The contract is not identically delta-1: as
maturity approaches and no credit event occurs,
it will tend to 1. The continuous rebalancing in
stock is financed by being structurally seller of
CDS.
SOME EXAMPLES
Extinguishable (XCCY) swap: a type of
deal where we pay (receive) coupons in
one currency and the counteparty receives
(pays) coupons in another, with full
notional exchange at the beginning and at
the end.

Equivalent to a series of risky FX forwards
as in the previous example
XCCY SWAP
 There are potentially
two credit curves for   B F a 2  2 B F                    B F
               a  k2 
the counterparty, one    t      2     2

in each currency.         rF  Jk3  B F  0
Shifting from one to
the other involves a       B D a 2  2 B D         B D
             k2
compo-bond type of          t      2     2

correction:                  rD  Jk3  B D  0
SOME EXAMPLES
Convertible and Exchangeable Bond: a
bond that can be converted at any time
during a predefined conversion period into
shares of the company (convertible) or of
another company (exchangeable)
Typically, a 2 factor, american-style option
CONCLUDING REMARKS
 First steps towards joint modelling of credit- and
asset-based derivatives
 Single asset, single credit well advanced
 Future developments:
 Make our generic derivatives credit contingent by
incorporating more sophisticated dynamics and
calibrators for the asset
 Simulate in a consistent and dynamically tractable
correlation of default times

```
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 views: 89 posted: 7/22/2011 language: French pages: 25
Description: Quant Strategy Credit document sample