The Pricing of Asian Options in High Volatile Markets: A PDE Approach
Financial derivatives are very important tools in risk management since they decrease uncertainty. Moreover, if used effectively, they can grow the income and save the cost. There are many types of financial derivatives, for instance: futures/forwards, options, and swaps. The present thesis deals with the pricing problem for Asian options.
The main aim of the thesis is to generalize the Asian option pricing Partial Differential Equation (PDE) in order to handle post-crash markets where the volatility is high. In other words, we seek to extend the work on the Asian option pricing PDE under the well-known Black-Scholes model to a high volatility model. To this end, we first set up a model that accounts for high volatile situations and we solve the Stochastic Differential Equation (SDE) of the underlying asset price. Our illustrations confirm the high volatile behavior of the model. We then derive the Asian option PDE for the suggested model. The resulting PDE is reduced from two-dimensional space to one-dimensional space using a change of variable. Moreover, we derive a relationship between the Asian options prices of the Black-Scholes model and our high volatility model where the increase in volatility is a deterministic function of the interest rate.