Option pricing for Fractal Activity Time Geometric Brownian Motion (FATGBM)

Abstract

This thesis examines option pricing for a Long Range Dependent (LRD) stochastic process with student marginal distributions called Fractal Activity Time Geometric Brownian Motion (FATGBM), introduced in Heyde (1999). We address four separate problems involving the pricing of options under FATGBM and other LRD stochastic processes. Following an introduction into the mechanics of derivative pricing, the thesis begins by addressing the problem of derivative pricing under FATGBM. We first develop the properties of FATGBM and show that the market is arbitrage-free but incomplete under this model. We then prove that there is no replicating strategy for this model except under special circumstances. We show that those special circumstances lead to the hedging of a Timer Option where interest rates are zero and we conclude by discussing the issue of completing the market by calibrating FATGBM to liquid risky assets such as European Options, as discussed in Carr et al. (2001). We then describe how to price path dependent options under FATGBM. We first propose a non-recombining tree that is used to then construct a recombining tree to price path dependent options. Further, we prove that our discrete time model converges to the continuous time one, resulting in a discrete approximation scheme for path dependent options. We then prove that the discrete approximation scheme results in an upper bound for the price of an American put. The next chapter addresses the problem of sampling from the distribution of FATGBM conditional on price history. Given that FATGBM is a LRD process, it is imperative to be able to simulate future price paths given a price path history. We propose a Markov Chain Monte Carlo (MCMC) approach to develop two algorithms for two different LRD processes, one FATGBM and one similar to FATGBM that we call FATGBM 2. We prove that the algorithms result in a Markov chain with a stationary distribution identical to the conditional distribution from which we wish to sample. We then discuss the implementation of both algorithms and compare the mixing times and features of the resultant conditional distribution. The final chapter combines the themes and results of the preceding chapters by using the MCMC algorithm in conjunction with the recombining tree developed in Chapter 2. The result is an analysis of the effect of long range dependence on option prices, the most compelling finding being that LRD has more of an impact on the option price than the impact of heavy tails alone, a phenomenon that has thus far been overlooked by the literature on option pricing. We conclude with an analysis of the implied volatility surface arising from FATGBM and discuss the implications of our research in the context of the existing literature. Show more