Exotic equity derivatives: a comparison of pricing models and methods with both stochastic volatility and interest rates
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The traditional Black Scholes methodology for exotic equity option pricing fails to capture the features of latent stochastic volatility and observed stochastic interest rate factors exhibited in financial markets today. The detailed study presented here shows how these shortcomings of the Black Scholes methodology have been addressed in literature by examining some of the developments of stochastic volatility models with constant and stochastic interest rates. A subset of these models, notably with models developed within the last two years, are then compared in a simulated study design against a complex Market Model. Each of the select models were chosen as “best” representatives of their respective model class. The Market Model, which is specified through a system of Stochastic Differential Equations, is taken as a proxy for real world market dynamics. All of the select models are calibrated against the Market Model using a technique known as Differential Evolution, which is a globally convergent stochastic optimiser, and then used to price exotic equity options. The end results show that the Heston-Hull-CIR Model (H2CIR) outperforms the alternative Double Heston and 4/2 Models respectively in producing exotic equity option prices closest to the Market Model. Various other commentaries are also given to assess each of the select models with respect to parameter stability, computational run times and robustness in implementation, with the final conclusions supporting the H2CIR Model in preference over the other models. Additionally a second research question is also investigated that relates to Monte Carlo pricing methods. Here the Monte Carlo pricing schemes used under the Black Scholes and other pricing methodologies is extended to present a semi-exact simulation scheme built on the results from literature. This new scheme is termed the Brownian Motion Reconstruction scheme and is shown to outperform the Euler scheme when pricing exotic equity derivatives with relatively few monitoring or option exercise dates. Finally, a minor result in this study involves a new alternative numerical method to recover transition density functions from their respective characteristic functions and is shown to be competitive against the popular Fast Fourier Transform method. It is hoped that the results in this thesis will assist investment and banking practitioners to obtain better clarity when assessing and vetting different models for use in the industry, and extend the current range of techniques that are used to price options.