Corrections to Dynamic Volatility Matrix Estimation by Fourier Transform Method with Applications

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研究生：	張藝馨 Chang, Yi-Hsin
論文名稱：	Corrections to Dynamic Volatility Matrix Estimation by Fourier Transform Method with Applications
指導教授：	韓傳祥 Han, Chuan-Hsiang
口試委員:	吳慶堂牛繼聖顏如儀韓傳祥
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：	2011
畢業學年度：	99
語文別：	英文
論文頁數：	51
中文關鍵詞：	變異數共變異數矩陣、(修正後)傅立葉轉換方法、瞬時波動率、投資組合違約機率
外文關鍵詞：	Volatility Matrix, (Corrected) Fourier Transform Method, Instantaneous Volatility, Portfolio Default Probability
相關次數：	點閱：3 下載：0
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We provide some simple and effective correction schemes to reduce the bias arising from an estimation of volatility matrix by the nonparametric Fourier transform method, proposed by Malliavin and Mancino (2002, 2009). The dynamic volatility matrix is defined in a multivariate diffusion process. Correction schemes include a linear method and an affine method. Simulation studies demonstrate effectiveness of these correction methods.
For applications, we apply the corrected Fourier transform method to three empirical studies. We find multiple time scales of volatility given different frequencies of data. The linearity between VIX square and instantaneous variance is confirmed. Moreover, we estimate default probabilities for a portfolio of S&P 500 index and CDX through the basic Monte Carlo method and an importance sampling scheme. We find that the importance sampling scheme performs better in this application.

1 Introduction and Literature Review                       1
2 Introduction to Fourier Transform Method                 4
  2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . 4
  2.2 Fourier Transform Method . . . . . . . . . . . . . . 5
    2.2.1 First-Power Fourier Coefficients . . . . . . . . 5
    2.2.2 Second-Power Fourier Coefficients . . . . . . .  6
    2.2.3 Smoothing . . . . . . . . . . . . . . . . . . .  6
  2.3 Numerical Implementation . . . . . . . . . . . . . . 7
    2.3.1 Evaluation of First-Power Fourier Coefficients . 7
    2.3.2 Evaluation of Second-Power Fourier Coefficients  8
    2.3.3 Examples for Fourier Transform Method . . . . .  8
3 Corrections to Fourier Transform Method                 10
  3.1 MLE Correction to Volatility Estimation . . . . . . 10
  3.2 Quadratic Variation Corrections to Correlation
      Estimation . . . . . . . . . . .. . . . . .. . . . .11
    3.2.1 Linear Method . . . . . . . . . . . . . . . . . 12
    3.2.2 Affine Method . . . . . . . . . . . . . . . . . 12
  3.3 Estimation of Stochastic Volatility and Dynamic
      Correlation Model Parameters . . . . . . . . . . .  13
    3.3.1 Vasicek Model . . . . . . . . . . . . . . . . . 14
    3.3.2 Jacobi Process . . . . . . . . . . . . . . . .  15
  3.4 Simulation Studies . . . . . . . . . . . . . . . .  16
    3.4.1 Numerical Examples for Stochastic Volatility .  16
    3.4.2 Numerical Examples for Correlation . . . . . .  17
4 Applications of Volatility Matrix Estimations           25
  4.1 Instantaneous Volatility for Different-Frequency
      Data . . . . . . . .  . . . . . . . . .  . . . . .  25
    4.1.1 Deseasonalized . . . . . . . . . . . . . . . .  25
    4.1.2 Parameters of Vasicek Model under Different-
          Frequency Data . . . . . . . . . . . . . . . . .27
  4.2 Test for Linearity between VIX^2 and Instantaneous
      Variance . . . . . . . . . . . . . . . . . . . .. . 32
    4.2.1 Dataset . . . . . . . . . . . . . . . . . . . . 32
    4.2.2 Test for Linearity: General Linear Teat
          Approach . . . . . . . . . . . .. . . . . .. .  32
    4.2.3 Empirical Results . . . . . . . . . . . . . . . 34
  4.3 Default Probability for a Portfolio of S&P 500 Index
      and CDX . . .  . . . . . . . . . . . . . . . . . .  36
    4.3.1 Dataset . . . . . . . . . . . . . . . . . . . . 36
    4.3.2 Empirical Results . . . . . . . . . . . . . . . 36
5 Conclusion                                              42
A                                                         44
  A.1 The Basic Monte Carlo Method and An Importance
      Sampling Scheme for Estimating Portfolio Default
      Probability . . . . . . . . . . . . . . . . . . . . 44
    A.1.1 Definition of Portfolio Default Probability . . 44
    A.1.2 Calculation of Portfolio Default Probability .  45
    A.1.3 Numerical Examples . . . . . . . . . . . . . .  47
Reference . . . . . . . . . . . . . . . . . . . . . . . . 50

                                

[1] Chen, T.Y. (2010), "Correction to Fourier Transform Method for Nonparametric Estimation for Volatility," Master Thesis, National Tsing Hua University.
[2] Engle, R. (2009), "Anticipating Correlations," Princeton University Press.
[3] Gourierouxa, C. and Jasiak, J. (2006), "Multivariate Jacobi process with application to smooth transitions," Journal of Econometrics, 131, 475-505.
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[6] Malliavin, P. and Mancino, M. E. (2002), "Fourier Series Method for Measurement of Multivariate Volatilities," Finance and Stochastics, 6, 49-61.
[7] Malliavin, P. and Mancino, M. E. (2009), "A Fourier Transform Method for Nonparametric Estimation of Multivariate Volatility," The Annals of Statistics, 37, 1983-2010.
[8] Mattiussi, V. and Iori, G. (2007), "A Nonparametric Approach to Estimate Volatility and Correlation Dynamics," Working Paper, Department of Economics, City University.
[9] Priestlet, M. (1979), "Spectral Time Series Analysis," Wiley.
[10] Reno, R. (2008), "Nonparametric Estimation of the Diffusion Coefficient of Stochastic Volatility Models". Econometric Theory, 24, 1174-1206.
[11] Shreve, S. (2004), "Stochastic Calculus for Finance II: Continuous-Time Models," Springer.
[12] Wild, C.J. and Seber, G.A.F. (1999), "Chance Encounters: A First Course in Data Analysis and Inference," John Wiley and Sons Ltd.
[13] Zhang, J.E., and Zhu, Y. (2006), "VIX Futures," The Journal of Futures Markets, 26(6), 521-531.

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