This thesis focuses on the evaluation of various path-dependent options, specifically down-and-out put options, floating strike lookback options, and geometric Asian options. We consider a hybrid model with stochastic elasticity of variance and stochastic volatility as the driving factors for the underlying asset. It is well-known that obtaining closed-form solutions for these path-dependent options under stochastic volatility models is challenging.

To address this issue, we employ an asymptotic expansion approach and the Mellin transform method. By utilizing these techniques, we are able to derive explicit closed-form formulas for both the zero-order and the first-order correction terms. These formulas provide valuable insights into the pricing of the options and allow for a more comprehensive analysis.

Furthermore, we conduct a sensitivity analysis on the asymptotic terms obtained from our pricing formulas. This analysis helps us understand the impact of various factors on the option prices. Additionally, we compare the option prices calculated using our derived formulas with those obtained from Monte-Carlo simulations and the binomial tree method.

By comparing the prices derived from different models such as Black-Scholes, CEV, and SVCEV, we demonstrate the accuracy and effectiveness of our pricing formulas. The numerical comparisons highlight the strengths of our approach and emphasize the practical relevance of our findings.

In summary, this thesis contributes to the research field by providing explicit closed-form formulas for path-dependent options under a hybrid model with stochastic elasticity of variance and stochastic volatility or a fractional Brownian motion model. Through numerical analysis and comparisons with other pricing methods, we validate the accuracy and robustness of our derived formulas, thereby enhancing the understanding and applicability of option pricing in financial markets.