The prolonged boom in the US and European stock markets has led to increased interest in the mathematics of security markets, most notably in the theory of stochastic integration. This text gives a rigorous development of the theory of stochastic integration as it applies to the valuation of derivative securities. It includes all the tools necessary for readers to understand how the stochastic integral is constructed with respect to a general continuous martingale.
The author develops the stochastic calculus from first principles, but at a relaxed pace that includes proofs that are detailed, but streamlined to applications to finance. The treatment requires minimal prerequisites-a basic knowledge of measure theoretic probability and Hilbert space theory-and devotes an entire chapter to application in finances, including the Black Scholes market, pricing contingent claims, the general market model, pricing of random payoffs, and interest rate derivatives.
Continuous Stochastic Calculus with Application to Finance is your first opportunity to explore stochastic integration at a reasonable and practical mathematical level. It offers a treatment well balanced between aesthetic appeal, degree of generality, depth, and ease of reading.
Table of Contents
Covergence of Random Variables
Optional Sampling of Closed Submartingale Sequences
Maximal Inequalities for Submartingale Sequences
Continuous Time Martingales
The Covariation Process
One Dimensional Brownian Motion
Measurability Properties of Stochastic Processes
Stochastic Integration with Respect to Continuous Semimartingales
Change of Measure
Representation of Continuous Local Martingales
APPLICATION TO FINANCE
The Simple Black Scholes Market
Pricing of Contingent Claims
The General Market Model
Pricing of Random Payoffs at Fixed Future Dates
Interest Rate Derivatives
Separation of Convex Sets
The Basic Extension Procedure
Positive Semidefinite Matrices
Kolmogoroff Existence Theorem
"As a reference or a second book on stochastic calculus, Meyer is outstanding. In a formal, highly rigorous manner, he develops stochastic calculus, all the while focusing on topics of primary interest to financial engineers. … I highly recommend Meyer. It is an excellent introduction and reference on stochastic calculus."
- Glyn A. Holton of Contingency Analysis