Financial markets comprise of the spot market and its derivative market. The spot market contains information that is backward-looking. On the other hand, the derivative market uses a forward-looking concept. Model calibration, which is the focus of this paper, is a crucial tool to analyze forward information, which is frequently utilize when regarding pricing, hedging, and risk management.

A model calibration problem is mainly about solving a nonlinear optimization problem to find the best fit of implied volatility surface. This kind of problem can be solved by many different methods, including Fourier transform, perturbation methods, numerical PDEs, etc. The above methods are based on deterministic approaches, which are restricted to simple models; these lack flexibility and are inapplicable in high-dimensional models. Hence Monte Carlo simulation is employ to solve complex high-dimensional models.

We propose a multi-factor Stochastic Volatility Model (SVM) that takes different frequency data into account. Literatures find the dynamic of implied volatilities is better fitted under multi-factor SVM. The accuracy of the Monte Carlo simulation can be improved using a variance reduction method, known as, the Martingale Control Variate (MCV). The advantage of better fitting comes with the problem of massive computation. We involve parallel computing on the Graphics Processing Unit (GPU) to help solve this problem. GPUs are first used for computer graphing, which turn increasingly programmable and computationally powerful especially on calculations that are carried out simultaneously. Combining GPU parallel computing with the Martingale Control Variate method will result in a model calibration of multi-factor SVM that is even more precise and more feasible to analyze option data in real time.

After developing the model, the time dependent volatility model is used to calculate the Volatility Index (VIX) with the use of the term structure of VIX published by the Chicago Board Options Exchange (CBOE) we can solve the identification problem that we faced in multi-factor SVM.

Abstract i
Acknowledgements ii
Table of Contents iii
Chapter 1 Introduction and Literature Review 1
Chapter 2 Methodology: Two-Stage Monte Carlo Calibration with GPU Computing 3
2.1 Fourier Transform Method with Correction 3
2.2 Monte Carlo Simulation with GPU Computing 7
2.3 Optimization 10
2.4 Two-Stage Monte Carlo Calibration 12
Chapter 3 Single-Factor Stochastic Volatility Model 13
3.1 The Single-Factor Stochastic Volatility Model 13
3.2 Two-Stage Monte Carlo Calibration 14
3.3 Comparison of Monte Carlo Calibration Methods 18
Chapter 4 Multi-Factor Stochastic Volatility Model 18
4.1 The Two-Factor Stochastic Volatility Model 19
4.2 Two-Stage Monte Carlo Calibration 20
4.3 Consistent Test 25
4.4 Other Multi-Factor SVM Calibration 27
Chapter 5 Comparison 29
Chapter 6 Conclusion and Future Work 34
Reference 36
Appendix 38

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