快速傅立葉轉換法在風險管理下的應用｜國立清華大學博碩士論文庫

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研究生：	蔡崇湋 Tsai, Chung-Wei
論文名稱：	快速傅立葉轉換法在風險管理下的應用 Fast Fourier Transform Method in Risk Management
指導教授：	韓傳祥 Han, Chuan-Hsiang 徐南蓉 Hsu, Nan-Jung
口試委員:	吳慶堂牛繼聖
學位類別：	碩士 Master
系所名稱：	理學院 - 統計學研究所 Institute of Statistics
論文出版年：	2012
畢業學年度：	100
語文別：	英文
論文頁數：	44
中文關鍵詞：	快速傅立葉轉換、風險值、隨機波動率模型、離散誤差、截斷誤差
外文關鍵詞：	Fast Fourier Transform, value at risk, Stochastic Volatility Model, sampling error, trancation error
相關次數：	點閱：3 下載：0
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在這篇論文中，我們提出了使用快速傅立葉轉換(FFT)來計算風險值(VaR)的方法，一旦我們知道了標的資產價格的自然對數的特徵函數時，我們就能夠使用快速傅立葉轉換方法來計算風險值。這樣的發法是很有用的，因為在許多情況下標的資產價格的自然對數的特徵函數很容易計算。而使用快速傅立葉轉換計算風險值時我們能夠在確保準確度的情況下減少計算的時間。為了確保準確度，我們也分析了真實值與數值方法之間的誤差上界。

In this thesis, we propose Fast Fourier Transform (FFT) method to compute VaR. Once we know the characteristic function of natural logarithm of the underlying asset price, we are able to compute VaR by FFT method. This is very powerful because the characteristic function can be computed easily in a lot of models. Using FFT method to compute VaR can efficiently reduce the computational time without losing the accuracy. In order to check the accuracy, we would like to analyze the bound of error between the true value and the numerical computation.

Table of Contents    1
Chapter 1   Introduction    3
Chapter 2  Modified Fourier Transform Method for Volatility Estimation    5
2.1  Fourier Series    5
2.2  Fourier Transform Method    6
2.3  Modified Fourier Transform Method – Beas Reduction    7
Chapter 3  Stochastic Volatility Model    9
3.1  Heston Model ( CIR Process )    9
3.2   Parameter Estimation    10
Chapter 4  Apply Fast Fourier Transform method to VaR Estimation    12
4.1   Introduction of Fast Fourier Transform    12
4.2   From Characteristic Functions to VaR/CVaR Estimation    13
4.3   Evaluation of VaR by FFT Method    17
Chapter 5  Bounds for Sampling and Truncation Errors    19
5.1  Truncation Error    20
5.2  Sampling Error    20
5.3  Bound of Total Error    23
Chapter 6 Empirical Studies    25
6.1  Dataset with different time period    25
6.2  Introduction of the two methods    25
6.3  Introduction of Backtesting    26
6.4  Empirical Results    28
Chapter 7  Conclusion    37
Appendix    38
Reference    44

                                

[1] C.-H. Han, W.-H. Liu, and T.-Y. Chen. (2011). An Improved Procedure for VaR/CVaR Estimation under Stochastic Volatility Models. Submitted.
[2] C. Broto and E. Ruiz. (2004). Estimation methods for stochastic volatility models: A survey. Journal of Economic Surveys 18(5), 613-649.
[3] J. C. Hull. (2008). Options, Futures, and Other Derivatives. Seventh Edition. Prentice Hall.
[4] J. C. Hull. (2010). Risk Management and Financial Institutions. 2nd Edition. Prentice Hall.
[5] P. Malliavin and M. E. Mancino. (2002). Fourier series method for measurement of multivariate volatilities. Finance and Stochastics. 6, 49-61.
[6] P. Carr and D. Madan. (1999). Option Valuation Using the Fast Fourier Transform. Journal of Computational Finance.
[7] P. Malliavin and M. E. Mancino. (2009). A Fourier Transform Method for Nonparametric Estimation of Multivariate Volatilities, The Annals of Statistics, 37, 1983-2010.
[8] R. Reno. (2008). Nonparametric estimation of the diffusion coefficient of stochastic volatility models. Econometric Theory. 24(5), 1174-1206
[9] U. Cherubini, G. Della Lunga, S. Mulinacci, and P. Rossi. (2010). Fourier Transform Methods in Finance. Wiley Finance.

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