本文在Christoffersen et al. (2009)的框架下推導出多因子隨機波動率模型選擇權定價公式的封閉解。探討將該模型應用在信用，選擇權和風險管理上。我們主要研究隨機波動率對信用價差收益率曲線隨時間變化的影響。我們認為多因子的隨機波動率模型可以更好地擬合信用價差，因此，模型校準結果顯示出兩個隨機波動率因子更靈活，可以捕捉信用價差期限結構隨時間變化的特性，其中一個因子代表長期波動率，另外一個代表短期波動率。在對隱含波動率曲面的模型校準上，對比Han（2011）的兩階段蒙地卡羅校準方法，我們的模型有更小的加權平均MSE和更快的計算速度。最後，該模型還可以計算風險中性下的違約概率和VaR。

This article develops a closed-form solution of multifactor stochastic volatility option pricing model which is developed from Christoffersen et al. (2009) framework. We apply this model on Credit Spreads, Equity Options, and Risk Management. In particular, we look at the effect of having stochastic volatility in the structural approach and study the effects of time scales on the credit spread yield curves for the stochastic volatility. We argue that this model with multifactor stochastic volatility can produce more realistic credit spreads. Thus, the calibration reveal how the introduction of two volatility factors can generate a wide range of combinations associated with short-term and long-term patterns corresponding to credit spreads. In Second application, we solve the calibration problem of implied volatility surfaces. The results reveal that our model has relatively lowest total MSE and faster computing speed. Finally, our model also can calculate risk-neutral default probability.

Abstract ........................................................................................................................... i
Acknowledgements ...................................................................................................... iii
Table of Contents ......................................................................................................... iii
Chapter 1 Introduction and Literature Review .............................................................. 1
Chapter 2 Modeling Option Price with Two-factor Stochastic Volatility Model .......... 4
2.1 A stylized model with stochastic volatility .......................................................... 4
2.2 Calculate call option price ................................................................................... 5
2.3 Calculate characteristic function of the log-price ΨlnST(ω) ................................ 8
Chapter 3 ApplicationModeling Credit Spreads ...................................................... 9
3.1 Pricing credit spreads ........................................................................................... 9
3.2 Numerical Illustration ........................................................................................ 10
3.3 Calibration to market credit spreads .................................................................. 14
3.3.1 Calibration Procedure ................................................................................. 14
3.3.2 Calibration Data and Results ...................................................................... 15
Chapter 4 Application: Calibration to Implied Volatility ........................................ 21
4.1 Model Calibration to Implied Volatility Surface ............................................... 21
4.2 Comparison ........................................................................................................ 23
Chapter 5 ApplicationⅢ: Calculate Default Probability and VaR .............................. 25
5.1 Calculate Risk-Neutral Default Probability ....................................................... 25
5.2 Calculate VaR by Fourier Transform method ................................................... 25
Chapter 6 Conclusion ................................................................................................... 27
Appendix ...................................................................................................................... 28
References .................................................................................................................... 30

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