Mathematics of Financial Markets presents the mathematics that underpins pricing models for derivative securities in modern financial markets, such as options, futures and swaps. This new edition adds substantial material from current areas of active research, such as coherent risk measures with applications to hedging, the arbitrage interval for incomplete discrete-time markets, and risk and return and sensitivity analysis for the Black-Scholes model.
This work is aimed at an audience with a sound mathematical background wishing to learn about the rapidly expanding ﬁeld of mathematical ﬁnance. Its content is suitable particularly for graduate students in mathematics who have a background in measure theory and probability.
The emphasis throughout is on developing the mathematical concepts re-quired for the theory within the context of their application. No attempt is made to cover the bewildering variety of novel (or ‘exotic’) ﬁnancial instru-ments that now appear on the derivatives markets; the focus throughout remains on a rigorous development of the more basic options that lie at the heart of the remarkable range of current applications of martingale theory to ﬁnancial markets.
The ﬁrst ﬁve chapters present the theory in a discrete-time framework. Stochastic calculus is not required, and this material should be accessible to anyone familiar with elementary probability theory and linear algebra. The basic idea of pricing by arbitrage (or, rather, by nonarbitrage) is presented in Chapter 1. The unique price for a European option in a single-period binomial model is given and then extended to multi-period binomial models.
Chapter 2 introduces the idea of a martingale measure for price pro-cesses. Following a discussion of the use of self-ﬁnancing trading strategies to hedge against trading risk, it is shown how options can be priced using an equivalent measure for which the discounted price process is a mar-tingale. This is illustrated for the simple binomial Cox–Ross–Rubinstein pricing models, and the Black–Scholes formula is derived as the limit of the prices obtained for such models.
Chapter 3 gives the ‘fundamental the-orem of asset pricing’, which states that if the market does not contain arbitrage opportunities there is an equivalent martingale measure. Explicit constructions of such measures are given in the setting of ﬁnite market models. Completeness of markets is investigated in Chapter 4; in a com-plete market, every contingent claim can be generated by an admissible self-ﬁnancing strategy (and the martingale measure is unique). Stopping times, martingale convergence results, and American options are discussed in a discrete-time framework in Chapter 5.
The second ﬁve chapters of the book give the theory in continuous time. This begins in Chapter 6 with a review of the stochastic calculus. Stopping times, Brownian motion, stochastic integrals, and the Itˆo diﬀerentiation rule are all deﬁned and discussed, and properties of stochastic diﬀerential equations developed.
The continuous-time pricing of European options is developed in Chap-ter 7. Girsanov’s theorem and martingale representation results are de-veloped, and the Black–Scholes formula derived. Optimal stopping results are applied in Chapter 8 to a thorough study of the pricing of American options, particularly the American put option. Chapter 9 considers selected results on term structure models, forward and future prices, and change of num´eraire, while Chapter 10 presents the basic framework for the study of investment and consumption problems.
- Pricing by Arbitrage
- Martingale Measures
- The Fundamental Theorem of Asset Pricing
- Complete Markets and Martingale Representation
- Stopping Times and American Options
- A Review of Continuous-Time Stochastic Calculus
- European Options in Continuous Time
- The American Option
- Bonds and Term Structure
- Consumption-Investment Strategies