The evaluation of financial derivatives represents a central part of financial risk management. There are many types of derivatives among other path-dependent options. In this study, we aim at valuing Asian options. They are path dependent and have several benefits. For instance, their values are habitually lower than European options. Also, an Asian option on a commodity drops the risk value close to maturity. Though, the disadvantage is that they are in general difficult to value since the distribution of the payoff is usually unknown. It is agreed in the literature that a stochastic process with a jumps model for the underlying asset provides a more precise value for the option price e.g. [1]. Moreover, the volatility is not constant, and it increases during a crisis see for instance the model of [2].
This work investigates the pricing of Asian options under a modified version of the pioneer Black Scholes model [3]. It aims at suggesting an alternate model that comprises jumps (as in [1] for instance) and increased volatility (see the model of [4]).
The study will propose to model the underlying asset with a new “hybrid” stochastic differential equation with jumps and high volatility. Then, under these settings, the valuation of Asian options will be investigated based on the works of [2, 4]. Numerical techniques for finance will be used in this thesis to get a solution to the pricing problem. Several illustrations of the solution will be offered to demonstrate the efficiency of the used methods
