【正文】 ansition matrix is necessary to value derivatives that have payoffs dependent on credit rating changes ? A riskneutral transition matrix can (in theory) be determined from bond prices Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example ? A pany’s equity is $3 million and the volatility of the equity is 80% ? The riskfree rate is 5%, the debt is $10 million and time to debt maturity is 1 year ? Solving the two equations yields V0= and sV=% Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull A Key Theoretical Reason (page 621) ? The default probabilities estimated from bond prices are riskneutral default probabilities ? The default probabilities estimated from historical data are realworld default probabilities Options, Futures, and Other Derivatives, 5th edition 169。 Samp。 20xx by John C. Hull Implied Default Probabilities Assuming That Default Can Happen on Bond Maturity Dates (Table , page 617) Time (yrs) Claim = NoDef Value Claim=Face Val+Accr Int 1 2 3 4 5 10 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Reason Why This Analysis is Simplistic ? Bonds are assumed to be zerocoupon ? The equation: Prob. of Def. (1Rec. Rate)=Exp Loss% assumes that the claim in the event of default equals the nodefault value of the bond Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example continued Oneyear riskfree bond (principal=$1) sells for Oneyear corporate bond (principal=$1) sells for or at a % discount This indicates that the holder of the corporate bond expects to lose % from defaults in the first year e ? ? ?0 05 1 0 951229. .e ? ? ?0 0525 1 0 948854. .Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Chapter 26 Credit Risk Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The RiskFree Rate ? Most analysts use the LIBOR rate as the riskfree rate ? The excess of the value of a riskfree bond over a similar corporate bond equals the present value of the cost of defaults Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Summary of Results (Table , page 612) Ma turity (ye ars ) Cu mu l. Lo ss . % Loss Du ring Y r ( %) 1 0. 24 97 0. 24 97 2 0. 99 50 0. 74 53 3 2. 07 81 1. 08 31 4 3. 34 28 1. 26 47 5 4. 63 90 1. 29 62 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull What Should We Use as the Claim Amount The best assumption seems to be that the claim amount for a bond equals the face value plus accrued interest not the nodefault value Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Theory and Practice ? In theory asset swap spreads should be slightly dependent on the bond’s coupon ? In practice it is assumed to be the same for all bonds with a particular maturity and the quoted asset swap spread is assume to apply to a bond selling for par ? This means that the spread would be quoted as bps in Example 2 and when calculating default probabilities we would assume Bj=100 and Gj= Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Bond Prices vs. Historical Default Experience yp r o b a b i l i t h i st o r i ca l 0 . 5 7 % t h e t h a n g r e a t e r m u ch is T h i s 2 . 4 7 % . is d e f a u l t ofy p r o b a b i l i t yr5 T h e r a t e .r e co v e r y z e r o A ss u m ep e r i o d . ye a r5 a o v e r v a l u e sb o n d 39。 20xx by John C. Hull Equity vs. Assets An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, sV E V N d De N ddV D r TTd d TrTVVV0 0 1 21022 12? ??? ?? ??( ) ( )ln ( ) ( )。 20xx by John C. Hull Reducing Credit Exposure (page 625) ? Collateralization ? Downgrade triggers ? Diversification ? Contract design ? Credit derivatives Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Measure 1 ? One monly used default correlation measure is the correlation between 1. A variable that equals 1 if pany A defaults between time 0 and time T and zero otherwise 2. A variable that equals 1 if pany B defaults between time 0 and time T and zero otherwise ? The value of this measure depends on T. Usually it increases at T increases. Options, Futures, and Other Derivatives, 5th edition 169。(),([)(22???r??r?Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Model Based on Credit Rating Changes (Creditmetrics) ? A