In this thesis I consider the problems associated with Monte-Carlo pricing and hedging contingent claims on financial securities, where the underlying asset appears as a supervolatile stochastic process. The original motivation for these considerations was to price options for the Queensland electricity spot market.
The presence of supervolatility, which is sometimes represented in the form of a jump-diffusion process or Levy process, presents many challenges for any numerical pricing method including Monte-Carlo pricing. In this thesis I have addressed some of these difficulties to fairly price, using Monte-Carlo pricing, a series of Asian style options on the Queensland electricity spot price.
Chapter 1 opens with some introductory comments about the Queensland electricity market along with a brief description of the difficulties faced when pricing contingent claims on any supervolatile security. Background reading in mathematical and financial theory required to understand these issues appears in Chapter 2. In Chapter 3 I introduce a simulation based method that utilises a genetic algorithm to estimate the parameters of a supervolatile process.
An analysis of a recently developed stable numerical scheme for solving Stochastic Ordinary Differential Equations (SODEs) is provided in Chapter 4. This analysis allows for the development of a class of higher order, stable numerical schemes for the solution of SODEs, a matter which is explored in Chapter 5. Chapter 5 also includes a thorough numerical analysis of the error and stability properties of one member of this higher order class of methods and extends this method to produce numerical schemes for the solution of SODEs using alternate stochastic processes, such as Levy and Cox processes.
Chapter 6 introduces, and empirically analyses the error of, a non-parametric method, called Canonical valuation, for calculating the risk-neutral price of any European contingent claim where the underlying follows an unknown ergodic, Markov chain. I utilise this method to construct, and analyse, a replicating portfolio for the same class of processes. These methods of pricing and hedging are then employed to construct a risk-neutral Monte-Carlo method for pricing and hedging contingent claims.
Chapter 7 explicates the relevance and necessity of all these advances to price a small number of Asian style options for the Queensland electricity spot price market. A summary of the work contained in this thesis is presented in the Conclusion, along with some thoughts on future research directions.