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研究生:陳姿穎
研究生(外文):Chen, Tzu-Ying
論文名稱:CorrectionstoFourierTransformMethodforNonparametricEstimationofVolatilitywithApplicationsinRiskManagement
指導教授:韓傳祥韓傳祥引用關係
指導教授(外文):Han, Chuan-Hsiang
學位類別:碩士
校院名稱:國立清華大學
系所名稱:計量財務金融學系
學門:商業及管理學門
學類:財務金融學類
論文種類:學術論文
論文出版年:2010
畢業學年度:98
語文別:英文
論文頁數:55
中文關鍵詞:隨機波動度傅立葉轉換方法重點抽樣法風險值回溯測試
外文關鍵詞:stochastic volatilityFourier transform methodimportance samplingValue-at-Riskbacktesting
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This thesis consists of two parts. In the first part, we aim to estimate parameters arising from stochastic volatility models by means of the nonparametric Fourier transform method (Malliavin and Mancino, 2002, 2009). Under the assumption that data satisfy the continuous semimartingale property, this Fourier transform method is based on integration of the time series rather than on their differentiation. Due to some boundary deficiency in numerical approximation (Reno, 2008), we propose some correction methods including model-free and model-dependent approaches to the Fourier estimation.

In the second part, the Fourier transform method is applied to VaR (Value at Risk) and CVaR (Conditional Value at Risk) estimation under stochastic volatility models. Through Monte Carlo simulations with importance sampling, we test the performance of VaR with our corrected Fourier transform method using some foreign exchange and the S&P 500 index data. We find that our corrected Fourier transform method under stochastic volatility models outperforms other VaR measurements from historical simulation, RiskMetrics, and GARCH(1,1) model.
Contents
1 Introduction and Literature Review 1
2 A Non-Parametric Estimation for Volatility Process: Fourier Transform Method 4
2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 First-Power Fourier Coefficients . . . . . . . . . . . . . . . . . . . 6
2.2.2 Second-Power Fourier Coefficients . . . . . . . . . . . . . . . . . . 7
2.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 Evaluation of First-Power Fourier Coefficients . . . . . . . . . . . 8
2.3.2 Evaluation of Second-Power Fourier Coefficients . . . . . . . . . . 8
2.3.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 An Example: Local Volatility Model . . . . . . . . . . . . . . . . . . . . 10
3 Corrected Fourier Transform Method 12
3.1 Model-Free Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Model-Dependent Approach . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Vasicek Model (Ornstein-Uhlenbeck Process) . . . . . . . . . . . . 16
3.2.2 Heston Model (Cox-Ingersoll-Ross Process) . . . . . . . . . . . . . . . . . . . . . 18
3.3 Estimation of Stochastic Volatility Model Parameters . . . . . . . . . . . 20
3.3.1 Vasicek Model (Ornstein-Uhlenbeck Process) . . . . . . . . . . . . 21
3.3.2 Heston Model (Cox-Ingersoll-Ross Process) . . . . . . . . . . . . . . . . . . . . . 22
3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Application: VaR/CVaR Estimation in Risk Management 26
4.1 Definition of VaR/CVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Calculation of VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.1 Basic Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . 28
4.2.2 Variance Reduction: Importance Sampling . . . . . . . . . . . . . 28
4.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Calculation of Conditional VaR . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Basic Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . 31
4.3.2 Variance Reduction: Importance Sampling . . . . . . . . . . . . . 32
4.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.1 Two Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.2 A Comparison of the Three Methods . . . . . . . . . . . . . . . . 35
4.4.3 Tests of VaR Accuracy: Backtesting . . . . . . . . . . . . . . . . . 36
4.4.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Conclusion 52
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surement". Handbook of Financial Econometrics, ed. by Y. Ait-Sahalia and L. P.
Hansen. Amsterdam: North-Holland Press, forthcoming.
[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., (1999). "Coherent Measures of
Risk". Mathematical Finance, 9, 203-28.
[3] Barucci, E., Malliavin, P. and Mancino, M. E. (2005). "Harmonic Analysis Methods
for Nonparametric Estimation of Volatility: Theory and Applications: Theory and
Applications". Stochastic Process and Applications to Mathematical Finance, 1-34
[4] Barucci, E. and Reno, R. (2002). "On Measuring Volatility of Diffusion Process with
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casting Performance". International Financial Markets, Institutions and Money, 12,
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metric Measurement by Volatility Feedback". C. R. Acad. Sci. Paris, 334, 505-508
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HJM Geometry: An Application of Ito Calculus to Financial Statistics". Japanese
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[15] Shreve, S. (2004), "Stochastic Calculus for Finance II: Continuous-Time Models".
Springer
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