Part II is a continuous-time counterpart to Part I. The results are somewhat richer and more delicate than those in Part I, with a greater dependence on mathematical technicalities. It is wiser to focus on the parallels than on these technicalities. Once again, the three basic forces behind the theory are arbitrage, optimality, and equilibrium.
Chapter 5 introduces the continuous-trading model and develops the Black-Scholes partial differential equation (PDE) for arbitrage-free prices of derivative securities. The Harrison-Kreps model of equivalent martingale measures is presented in Chapter 6 in parallel with the theory of state prices in continuous time. Chapter 7 presents models of the term structure of interest rates, including the Black-Derman-Toy, Vasicek, Cox-IngersollRoss, and Heath-Jarrow-Morton models, as well as extensions. Chapter 8 presents specific classes of derivative securities, such as futures, forwards, American options, and lookback options. Chapter 8 also introduces models of option pricing with stochastic volatility. Chapter 9 is a summary of optimal continuous-time portfolio choice, using both dynamic programming and an approach involving equivalent martingale measures or state prices. Chapter 10 is a summary of security pricing in an equilibrium setting. Included are such well-known models as Breeden’s consumption-based capital asset pricing model and the general equilibrium version of the Cox-Ingersoll-Ross model of the term structure of interest rates. Chapter 11 treats the valuation of equities and corporate bonds, beginning with “structural models,” based on the capital structure of the firm and incentives of equity and debt holders, then turning to “reduced-form” models, based on an assumed stochastic arrival intensity of the stopping time defining default. Chapter 12 reviews numerical methods for calculating derivative security prices in a continuous-time setting, including Monte Carlo simulation of a discrete-time approximation of security prices, and finitedifference solution of the associated PDE.