Construction of risk-neutral measure in a brownian motion with exotic option

Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing problems. In contrast to the Black-Scholes and other complex option-pricing models that require solutions to stochastic differential equations, the binomial option-pricing model is mathematically simple. We are trying to show how to price a derivative security by determining the initial capital which requires hedging a short position in the derivative security. The overriding objective of this research paper is to discover a clever way to solve the partial differential equation using risk-neutral probability measure. The main goal of this study is fourfold: (1) to derive the joint density for a Brownian motion with drift and its maximum to data, (2) to introduce the change of the probability measure with risk-neutral approach in the pricing of equity derivatives from real-world to risk-neutral by using Binomial structure model, (3) to extend this approach to treat the price the special type of option called a barrier option, and (4) to compute the risk-neutral price at time zero of the up-and-out call.