Corrections to Fourier Transform Method for Nonparametric Estimation of Volatility with Applications in Risk Management

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研究生：	陳姿穎 Chen, Tzu-Ying
論文名稱：	Corrections to Fourier Transform Method for Nonparametric Estimation of Volatility with Applications in Risk Management
指導教授：	韓傳祥 Han, Chuan-Hsiang
口試委員:
學位類別：	碩士 Master
系所名稱：	科技管理學院 - 計量財務金融學系 Department of Quantitative Finance
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 volatility, Fourier transform method, importance sampling, Value-at-Risk, backtesting
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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
Introduction and Literature Review 1
A Non-Parametric Estimation for Volatility Process: Fourier Transform Method 4
1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 First-Power Fourier Coefficients . . . . . . . . . . . . . . . . . . . 6
2.2 Second-Power Fourier Coefficients . . . . . . . . . . . . . . . . . . 7
3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Evaluation of First-Power Fourier Coefficients . . . . . . . . . . . 8
3.2 Evaluation of Second-Power Fourier Coefficients . . . . . . . . . . 8
3.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 An Example: Local Volatility Model . . . . . . . . . . . . . . . . . . . . 10
Corrected Fourier Transform Method 12
1 Model-Free Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Model-Dependent Approach . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1 Vasicek Model (Ornstein-Uhlenbeck Process) . . . . . . . . . . . . 16
2.2 Heston Model (Cox-Ingersoll-Ross Process) . . . . . . . . . . . . . . . . . . . . . 18
3 Estimation of Stochastic Volatility Model Parameters . . . . . . . . . . . 20
3.1 Vasicek Model (Ornstein-Uhlenbeck Process) . . . . . . . . . . . . 21
3.2 Heston Model (Cox-Ingersoll-Ross Process) . . . . . . . . . . . . . . . . . . . . . 22
4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Application: VaR/CVaR Estimation in Risk Management 26
1 Definition of VaR/CVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Calculation of VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Basic Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . 28
2.2 Variance Reduction: Importance Sampling . . . . . . . . . . . . . 28
2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Calculation of Conditional VaR . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Basic Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . 31
3.2 Variance Reduction: Importance Sampling . . . . . . . . . . . . . 32
3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Two Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 A Comparison of the Three Methods . . . . . . . . . . . . . . . . 35
4.3 Tests of VaR Accuracy: Backtesting . . . . . . . . . . . . . . . . . 36
4.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Conclusion 52

                                

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Springer

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