Mathematical Techniques in Finance

CHAPTER 6Option Pricing

The risk‐neutral valuation framework of Chapter 5 provides the mechanism for pricing any contingent claims. The celebrated Black‐Scholes‐Merton option pricing formula can be derived by following the steps of risk‐neutral valuation: positing that an asset's returns follows the limiting form of a random walk, ensuring risk‐neutrality by equating forward prices and expected prices, and computing the expected discounted value of the option payoff.

6.1 RANDOM WALK AND BROWNIAN MOTION

A symmetric random walk and its continuous time limit, a Brownian Motion, are typically used to model the evolution of the underlying asset or more typically the return of the underlying asset, leading to lognormal dynamics for the underlying asset.

6.1.1 Random Walk

A symmetric random walk is an example of a discrete time random process: a collection of random variables indexed by time. Different realizations of the random variables as functions of time are called sample paths and denoted by the generic symbol , . We will generally suppress the second argument unless necessary.

The symmetric random walk starts at the origin, and at each time‐step increases or decreases by with ...