3.8 Equivalent Martingale Measures... 38 3.9 Girsanov and Market Prices of Risk.·..........·.· 39 3.10 Black-Scholes Again 43 3.11 Complete Markets .. 44 3.l2 Optimal Trading and Consumption.....·,...····. 46 3.13 Martingale Solution to Merton's Problem............ 50 4 Term-Structure Models 54 4.1 One-Factor Models ... 55 4.2 Term-Structure Derivatives..... 60 4.3 Fundamental Solution ..................... 63 4.4 Multifactor Term-Structure Models....····.· 64 4.5 Affine Models...·.....·....··········· 66 4.6 The HJM Model of Forward Rates..·.···..····.· 69 5 Derivative Pricing 73 5.1 Forward and Futures Prices.... 73 5.2 Options and Stochastic Volatility 76 5.3 Option Valuation by Transform Analysis 80 6 Corporate Securities 84 6.1 Endogenous Default Timing........·.....···.. 85 6.2 Example:Brownian Dividend Growth.... 87 6.3 Taxes,Bankruptcy Costs,.Capital Structure·.....···. 91 6.4 Intensity-Based Modeling of Default.·.......····.· 93 6.5 Zero-Recovery Bond Pricing................... 96 6.6 Pricing with Recovery at Default ....... 98 6.7 Default-Adjusted Short Rate............... 99 23.8 Equivalent Martingale Measures . . . . . . . . . . . . . . . . . 38 3.9 Girsanov and Market Prices of Risk . . . . . . . . . . . . . . . 39 3.10 Black-Scholes Again . . . . . . . . . . . . . . . . . . . . . . . 43 3.11 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . 44 3.12 Optimal Trading and Consumption . . . . . . . . . . . . . . . 46 3.13 Martingale Solution to Merton’s Problem . . . . . . . . . . . . 50 4 Term-Structure Models 54 4.1 One-Factor Models . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Term-Structure Derivatives . . . . . . . . . . . . . . . . . . . . 60 4.3 Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Multifactor Term-Structure Models . . . . . . . . . . . . . . . 64 4.5 Affine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 The HJM Model of Forward Rates . . . . . . . . . . . . . . . 69 5 Derivative Pricing 73 5.1 Forward and Futures Prices . . . . . . . . . . . . . . . . . . . 73 5.2 Options and Stochastic Volatility . . . . . . . . . . . . . . . . 76 5.3 Option Valuation by Transform Analysis . . . . . . . . . . . . 80 6 Corporate Securities 84 6.1 Endogenous Default Timing . . . . . . . . . . . . . . . . . . . 85 6.2 Example: Brownian Dividend Growth . . . . . . . . . . . . . . 87 6.3 Taxes, Bankruptcy Costs, Capital Structure . . . . . . . . . . 91 6.4 Intensity-Based Modeling of Default . . . . . . . . . . . . . . . 93 6.5 Zero-Recovery Bond Pricing . . . . . . . . . . . . . . . . . . . 96 6.6 Pricing with Recovery at Default . . . . . . . . . . . . . . . . 98 6.7 Default-Adjusted Short Rate . . . . . . . . . . . . . . . . . . . 99 2