Introductory Stochastic Analysis for Finance and Insurance

CHAPTER 6

STOCHASTIC CALCULUS: ADVANCED TOPICS

In this chapter, we discuss topics that have important financial applications but are more mathematically challenging than those in Chapter 5. We begin with the Feynman-Kac formula which provides a probabilistic solution to certain types of partial differential equations (PDEs). As an application, the Black-Scholes PDE derived in Section 5.4 is solved using the Feynman-Kac formula. The Girsanov theorem is the subject of Section 6.3. The Girsanov theorem has many applications, and often plays a central role in evaluating contingent claims and derivative securities in particular. The risk-neutral valuation of contingent claims requires a probability measure equivalent to the physical probability measure, such that the present value process of underlying securities is a martingale, as seen in Section 3.5 in a discrete-time framework. When continuous-time models are considered, the Girsanov theorem is an appropriate tool to find such a probability measure. Section 6.4 deals with complex barrier hitting probabilities. We provide a detailed derivation in order to illustrate the use of the Girsanov theorem and the further use of the reflection principle. In the final section, we present results of stochastic calculus in a two-dimensional setting. These results are counterparts to those in the one-dimensional case but contain additional terms that reflect correlations between two stochastic processes. The proofs of the results are similar ...