5 - Exponential martingales, change of measure and financial applications

Summary

Summary We begin this chapter by studying two different types of ‘exponential’ of a Lévy-type stochastic integral Y. The first of these is the stochastic exponential, dZ(t) = Z(t–)dY(t), and the second is the process e^Y. We are particularly interested in identifying conditions under which e^Y is a martingale. It can then be used to implement a change to an equivalent measure. This leads to Girsanov's theorem, and an important special case of this is the Cameron–Martin–Maruyama theorem, which underlies analysis in Wiener space. In Section 5.3, we prove the martingale representation theorem and this is then applied to obtain the chaos decomposition for multiple Wiener–Lévy integrals. We then give a brief introduction to Malliavin calculus in the Brownian case. The final section of this chapter surveys some applications to option pricing. We discuss the search for equivalent risk-neutral measures within a general ‘geometric ‘Lévy process’ stock price model. In the Brownian case, we derive the Black–Scholes pricing formula for a European option. In the general case, where the market is incomplete, we discuss the Föllmer–Schweitzer minimal measure and Esscher transform approaches. The case where the market is driven by a hyperbolic Lévy process is discussed in some detail.

In this chapter, we will explore further important properties of stochastic integrals, particularly the implications of Itô's formula.