Asymptotic techniques and stochastic volatility in option pricing problems

Abstract

This thesis investigates the use of asymptotic techniques and stochastic volatility models in option pricing problems. For the Heston stochastic volatility model, a fast mean reverting asymptotic approach, similar to Fouque et al. (2000) is taken. The asymptotic solution derived extends on their most recent work, with the solution presented expanded out to four terms. The worthiness and robustness of the asymptotic solution is then tested by applying it to the theory of locally risk minimizing hedges. The asymptotic approach is then further developed by applying it to a real options framework, allowing for a better understanding of what the asymptotic solution actually reflects under this model, and in particular, how it affects the optimal investment threshold, a key component in real options theory. Asian options with general call type payoffs are then investigated and equivalency theorems derived linking them to Australian options under both a Black-Scholes model and a Heston stochastic volatility model. Examining Asian options from this ‘Australian’ perspective gives a new angle on how one can approach the pricing of Asian options under stochastic volatility. Advances are made in areas such as the PDE pricing equation, and Monte Carlo simulations. Finally, an asymptotic solution under a low volatility assumption in the Black-Scholes model for an Australian call option is derived. This extends the work of Dewynne and Shaw (2008), to cater for Australian options. It is argued that this can be used as an alterative to existing approximations under a low volatility regime, for both pricing general Australian call options and general Asian options through the equivalency theorems. Aside from the over arching theme of asymptotic techniques and stochastic volatility, this thesis looks at how each of the newly presented solutions and/or methods, can be of benefit to the pricing of their respective option types. In particular, focus will be placed on the usage, accuracy and computational efficiency of these techniques. In all cases, the new solutions provide a high level of accuracy compared to the true solution, and/or are much more computationally efficient than existing methodologies. The simplicity and advantages of these solutions make a valuable contribution to current option pricing techniques.