期权,期货及其衍生品第23弹.ppt

* * * * * * * Volatilities Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 * This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE Example A company’s equity is $3 million and the volatility of the equity is 80% The risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year Solving the two equations yields V0=12.40 and sv=21.23% The probability of default is N(?d2) or 12.7% Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 * The Implementation of Merton’s Model Choose time horizon Calculate cumulative obligations to time horizon. This is termed by KMV the “default point”. We denote it by D Use Merton’s model to calculate a theoretical probability of default Use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of default. Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 * Credit Risk in Derivatives Transactions (534) Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 * Three cases Contract always an asset Contract always a liability Contract can be an asset or a liability Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 General Result Assume that default probability is independent of the value of the derivative Define t1, t2,…tn: times when default can occur qi: default probability at time ti. fi: The value of the transaction at time ti R: Recovery rate * Options, Futures, and Other Derivatives, 8th Edition, Copyright ? John C. Hull 2012 General Result continued The expected loss from defaults at time ti is qi(1?R)E[max(fi,0)]. Defining ui=qi(1?R) and vi as the value of a derivative that provides a payoff of max(fi,0) at time ti, the PV of the cost of defaults is * Options, Futures, and Other Deriva