【正文】 and Other Derivatives, 5th edition 169。(),([)(])。 20xx by John C. Hull Netting (page 624) Netting clauses state that is a pany defaults on one contract it has with a financial institution it must default on all such contracts ????????????NiiNiiVRVR110,m a x)1()0,m a x ()1( to f r o m l o ss t h e ch a n g e s T h i sOptions, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Bond Prices vs. Historical Default Experience ? The estimates of the probability of default calculated from bond prices are much higher than those from historical data ? Consider for example a 5 year Arated zerocoupon bond ? This typically yields at least 50 bps more than the riskfree rate Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull ? The analysis can be extended to allow defaults at any time ? It is important to distinguish between the default probability density and the hazard rate ? The default probability density, q(t) is defined so that q(t)dt as the probability of default between times t and t+dt as seen at time zero ? The hazard rate is the probability of default between times t and t+dt conditional on no earlier default ???tdhethtq 0)()()(ttExtending the Analysis to Allow Defaults at Any Time Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Typical Pattern (See Figure , page 611) Spread over Treasuries Maturity Baa/BBB A/A Aa/AA Aaa/AAA Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example continued ? Similarly the holder of the corporate bond expects to lose or % in the first two years ? Between years one and two the expected loss is % e ee? ? ? ?? ?? ?0 05 2 0 0550 20 05 2 0 009950. .. .Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Value Additivity ? If claim amount equals nodefault value, value of a coupon bond is sum of values of constituent zerocoupon bonds ? The same is not true when claim amount equals face value plus accrued interest Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull RiskNeutral Probabilities The analysis based on bond prices assumes that ? The expected cash flow from the Arated bond is % less than that from the riskfree bond ? The discount rates for the two bonds are the same This is correct only in a riskneutral world Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example Cumulative probability of default 1 2 3 4 5 A % % % % % B % % % % % C % % % % % Suppose there are three rating categories and riskneutral default probabilities extracted from bond prices are: Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Pros and Cons ? Reduced form approach can be calibrated to known default probabilities. It leads to low default correlations. ? Structural model approach allows correlations to be as high as desired, but cannot be calibrated to known default probabilities. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Measure 1 vs Measure 2 n o r m a l tem u l t i v a r i a be to a ss u m e d be ca n t i m e s su r v i v a l dt r a n sf o r m e b e ca u se co n si d e r e d a r e co m p a n i e sm a n y w h e nu se to e a si e r m u ch is It 1. M e a su r e t h a n h i g h e rt l y si g n i f i ca nu su a l l y is 2 M e a su r ef u n ct i o n . ond i st r i b u t iy p r o b a b i l i t n o r m a l b i v a r i a t e cu m u l a t i v e t h e is w h e r ea n d:v e r sa v i ce a n d 2 M e a su r e f r o m ca l cu l a t e d be ca n 1 M e a su r eMTQTQTQTQTQTQTuTuMTTuTuMTPBBAABAABBAABABBAAB])()(][)()([)()(])。 20xx by John C. Hull The Loss Given Default (LGD) ? For derivatives we need to distinguish between a) those that are always assets, b) those that are always liabilities, and c) those that can be assets or liabilities ? What is the loss in each case? Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Do Default Probabilities Increase with Time? ? For a pany that starts with a good credit rating default probabilities tend to increase with time ? For a pany that starts with a poor credit rating default probabilities tend to decrease with time Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Relaxing Assumptions ? This analysis assumes constant interest rates, and known recovery rates and claim amounts ? If default events, riskfree rates, and recovery rates are independent, results hold for stochastic interest rates, and uncertain