【文章內(nèi)容簡介】 of % of defaulting by the end of the first year, % by the end of the second year, and so on 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 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 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。 t h e of 2 . 4 7 % or l o se to e x p e ct w et h a t m e a n s T h i s????eOptions, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Possible Reasons for These Results ? The liquidity of corporate bonds is less than that of Treasury bonds ? Bonds traders may be factoring into their pricing depression scenarios much worse than anything seen in the last 20 years 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。 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 The RealWorld Probability of Default ? The expected cash flow from the Arated bond is % less than that from the riskfree bond ? But we still get the same price if we discount at about 38 bps per year more than the riskfree rate ? If riskfree rate is 5%, it is consistent with the beta of the Arated bond being Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Which World Should We Use? ? We should use riskneutral estimates for valuing credit derivatives and estimating the cost of default ? We should use real world estimates for calculating credit VaR and scenario analysis Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Merton’s Model (page 621 to 623) ? Merton’s model regards the equity as an option on the assets of the firm ? In a simple situation the equity value is max(VT D, 0) where VT is the value of the firm and D is the debt repayment required Options, Futures, and Other Derivatives, 5th edition 169。 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 ( ) ( )。w h e r esssOptions, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Volatilities s ?? s sE V VE EV V N d V0 0 1 0? ? ( )This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE 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 Example continued ? The probability of default is N(d2) or % ? The market value of the debt is ? The present value of the promised payment is ? The expected loss is about % ? The recovery rate is 91% Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The Implementation of Merton’s Model (. KMV Moody’s) ? Choose time horizon ? Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote