Dynamics of basket hedging (CreditMetrics for baskets the Black-Scholes of the Credit Derivatives market) Galin Georgiev January, 2000 Disclaime r This report represents only the personal opinions of the author and not those of J.P.Morgan, its subsidiaries or affiliates 94469_97 Summary Definition of a protection contract on an individual name and a first-todefault (FTD) protection contract on two names The CreditMetrics model for baskets: basic definitions and variations. The basket as a rainbow digital option. Greeks and dynamic hedging of baskets. Implied vs. realized correlation. 3 94469_97

Individual protection contract Suppose risk-free interest rates are zero and denote by t the present time. A A protection contract f (t , T ) maturing at time T on company A entitles the holder to receive $1 at T if A defaults prior to T (and $0 otherwise). (In the market place, this is called a zero-coupon credit swap, settled at maturity, with zero recovery). A A f ( t , T ) P (t , T ) where the latter is the risk-free probability of Note that A f (t , T ) is proportional to the credit spread default of A up to time T, i.e., of A. 4 94469_97

FTD protection contract A first-to-default (FTD) protection contract maturing at time T, on companies A and B, entitles the holder to receive $1 at T if at least one of the companies defaults prior to T (and $0 otherwise). f A B (t , T ) equals the probability The price both defaulting before T. P AB (t , T ) of A or B or 5 94469_97 The market perspective While the price of protection for individual credits is (more or less) given by the credit swaps market, there is no liquid market yet for FTD protection (or, equivalently, FTD probability). A B f

A B One needs a model to price as a rainbow derivative on and f It depends on the correlation between the underlying spreads in the nodefault state and the correlation between the corresponding default events. f 6 94469_97 The CreditMetrics formalism Assumption: A A A A P (t , T ) Probab( Z K t ,T ) N ( K t ,T ) A A Z

K where is a univariate random normal variable and t ,T is the so- called threshold (defined above; distribution). N (.) is the cumulative normal One can be more specific and define A A A P (t , T ) Probab( AT LT ) AT A A is the normally distributed firms asset level and LT is the where (fixed) firms liability level (at time T). 7 94469_97

Asset Distribution at Maturity Default Probability Liability Level Initial Asset Level 8 94469_97 Inconsistencies of the CreditMetrics model Asset Level Assets Default Liability Level 1 Liability Level 2 Assets >Liabilities => No Default Time

0 T1 T2 9 94469_97 Assuming for simplicity constant volatility (of the assets), one can rephrase the price of protection in terms of familiar option theory: A X t , T A P (t , T ) N Tt A where X t ,T is a standard Brownian motion which we call normalized A

threshold (with initial point X 0,T , depending unfortunately on T). This is A X nothing else but the price of a barrier option (digital) on the underlying t ,T struck at 0 and expiring at T. 10 94469_97 The protection contract as a barrier option A f (t , T ) as a contingent We can therefore think of the protection contract A claim (barrier option) on the underlying X t ,T A X A t ,T A A f (t , X t ,T ) P (t , T ) N Tt

Unsurprisingly, it satisfies the (normal version of) the Black-Scholes equation: f A 1 2 f A 0 A 2 t 2 ( X t ,T ) 11 94469_97 The FTD protection contract as a rainbow barrier option The FTD protection price in this context is f A B (t , T ) P A B A A B

B (t , T ) Probab( AT LT orAT LT ) which in terms of normalized thresholds means f A B (where between A B (t , X t ,T , X t ,T ) 1 N 2 ( N 2 (.) X t ,T A X t ,T A Tt ,

is the bivariate normal cumulative and B and X t ,T ). For 0 , one has f A B X t ,T B Tt , ) is the correlation f A f B f A f B 12 94469_97 Black-Scholes for the FTD protection One can easily compute the Greeks and check that our rainbow contingent claim satisfies the two-dimensional version of the (normal) Black-Scholes

equation: f A B t 1 2 f A B 1 2 f A B 2 f A B 0 A 2 B 2 A B 2 ( X ) 2 ( X ) X X 13 94469_97 Hedge ratios Since the normalized thresholds are not traded, we obviously hedge the A B (t , X A , X B ) FTD protection (the rainbow barrier option) f with A

A the two individual protection contracts (1-dim barrier options) f (t , X ) B B A B and f (t , X ). The corresponding hedge ratios , are easily computable: A B A B A B A f f f X X A / N T t 1 2

f A X A X A 14 94469_97 Convexity of the hedged portfolio The hedged FTD portfolio f A B A f A B f B is easily seen to have a negative off-diagonal convexity: 2 2 f A B A B 0 A B X X X X

and positive diagonal convexity ( 0 ): 2 A 2 2 f A B f A 0 A 2 A 2 A 2 ( X ) (X ) (X ) 15 94469_97 FTD basket protection 4.000% 3.500% 3.000% 2.500% 2.000% 1.500%

1.000% 0.500% 3 .0 0 % 0.80% 2 .6 0 % 2 .2 0 % name A prote ction 1.40% 1 .8 0 % 1 .4 0 % 1 .0 0 % 2.00% 0 .6 0 % 0 .2 0 % 0.000% 2.60% name B

prote ction 0.20% 16 94469_97 Convexity seen through the effect of individual tweaks or parallel tweaks on the hedge ratios of 5 name basket Effect of Tweaking 1 Name 90% 85% Hedge Ratio 80% Tweaked name 75% Other names 70% 65% 60% 55% 50% 0 50

100 150 200 250 Tweaked Spd (bp) 17 94469_97 Effect of Tweaking 5 Names 90% Tweaked names 85% Other names Hedge Ratio 80% 75% 70% 65% 60% 55% 50% 0

50 100 150 200 250 Tweaked Spd (bp) 18 94469_97 Implied vs. realized correlation If one buys FTD protection and continuously rehedges, the resulting P&L is T XA XB P & L n2 , , ( )d T T t where is the realized correlation and n2 (.,.,.) is the bivariate normal density. If , one is long convexity and makes money due to

rehedging (but one pays for it upfront because the money earned by selling the original hedges is less). 19 94469_97 A correlation contract ? The P&L due to continuous rehedging of the basket is clearly pathdependent. Similarly to the development of the vol contract in standard option theory, the time will come to develop a correlation contract whose payoff is path-independent and proportional to realized correlation. 20