Pricing Timer Options under Jump-Diffusion Processes

Sammanfattning: Timer options are relatively new exotic options with the feature that they expire as soon as the accumulated realized variance exceeds a predefined level. This construction leads to a random time to maturity instead of having a fixed exercise day. As shown by Bernard an Cui [7], Timer options can be priced by solving a partial differential equation or by time-changing the stock price process and then using Monte-Carlo methods when assuming a diffusion process for the stock price and the variance. The purpose of this thesis is to show the results of [7] and then to extend their pricing techniques to jump-diffusion processes for the stock price. The jumps are assumed to follow a compound Cox process with independent and identically distributed jumps. Due to the jumps, the partial differential equation extends to a partial integro-differential equation. Furthermore, one can time-change the stock price process like in [7] and then use an adapted Monte-Carlo method with control variates to efficiently simulate the price of a Timer option. As an example, results for Timer Calls are shown when using Monte-Carlo methods. Finally, the pricing error for Timer Calls is studied when assuming a stock price process with continuous paths although it jumps.