Sinopsis
This chapter discusses the most important concepts in derivatives models, including risk-neutral pricing and no-arbitrage pricing. We derive the renowned Black- Scholes formula using these concepts. We also discuss fundamental formulas and techniques used for pricing derivatives in general, as well as those needed for the remainder of this book. In section 1.1, we discuss forward contracts, the most basic and fundamental derivative contract. In section 1.2, we derive the Black-Scholes partial differential equation (PDE). In section 1.3, we discuss the concept of riskneutral pricing and derive Black-Scholes equations for European calls and puts using risk-neutral pricing. In section 1.4, we provide a simple implementation for pricing these European calls and puts. In section 1.5, we discuss the pricing of American options. In section 1.6, we discuss fundamental pricing formulas for derivatives in general. In section 1.7, we discuss the important change of numeraire technique—useful for changing asset dynamics and changing drifts. In section 1.8, Girsanov’s theorem and the Radon-Nikodym derivative are discussed for changing probability measures to equivalent martingale measures. In section 1.9, we discuss the T-forward measure, a useful measure for pricing many derivatives; and finally, in section 1.10, we discuss considerations for choosing a numeraire in pricing. (A probability review is provided in Appendix A at the back of the book and a stochastic calculus review is provided in Appendix B.)
Content
- Black-Scholes and Pricing Fundamentals
- Monte Carlo Simulation
- Binomial Trees
- Trinomial Trees
- Finite-Difference Methods
- Exotic Options
- Stochastic Volatility
- Statistical Models
- Stochastic Multifactor Models
- Single-Factor Interest Rate Models
- Tree-Building Procedures
- Two-Factor Models and the Heath-Jarrow-Morton Model
- LIBOR Market Models
- Bermudan and Exotic Interest Rate Derivatives
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