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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,
 spread volatility, default risk
   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
  spread risk”
 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 ?
 Answer(s): motivated by
   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
    trading in CDS)
       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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posted:7/22/2011
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