This thesis explores the theoretical foundation and practical applications of the Esscher transform in both discrete and continuous-time stochastic processes. We begin by introducing the transform for random variables and apply it to well-known probability distributions such as the Poisson, Binomial, and Normal distributions. We then extend the concept to stochastic processes, demonstrating how the Esscher transform leads to construction of an equivalent risk-neutral measure, ensuring that the discounted price process is a martingale. A key focus is its application in the Var-Gama and Black-Scholes models, where the use of Esscher transformation enables effective option pricing.
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