Journal of Applied Mathematics

Calibration of the Volatility in Option Pricing Using the Total Variation Regularization

Yu-Hua Zeng, Shou-Lei Wang, and Yu-Fei Yang

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In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.

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J. Appl. Math., Volume 2014 (2014), Article ID 510819, 9 pages.

First available in Project Euclid: 2 March 2015

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Zeng, Yu-Hua; Wang, Shou-Lei; Yang, Yu-Fei. Calibration of the Volatility in Option Pricing Using the Total Variation Regularization. J. Appl. Math. 2014 (2014), Article ID 510819, 9 pages. doi:10.1155/2014/510819.

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  • R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation,” Econometrica, vol. 50, no. 4, pp. 987–1007, 1982.
  • T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, vol. 31, no. 3, pp. 307–327, 1986.
  • D. B. Nelson, “Conditional heteroskedasticity in asset returns: a new approach,” Econometrica, vol. 59, no. 2, pp. 347–370, 1991.
  • L. R. Glosten, R. Jagannathan, and D. Runkle, “On the relation between the expected value and the volatility of the nominal excess return on stocks,” The Journal of Finance, vol. 48, pp. 1779–1801, 1993.
  • E. Sentana, “Quadratic ARCH models,” Review of Economic Studies, vol. 62, pp. 639–661, 1995.
  • J.-M. Zakoian, “Threshold heteroskedastic models,” Journal of Economic Dynamics and Control, vol. 18, no. 5, pp. 931–955, 1994.
  • R. F. Engle, D. M. Lilien, and R. P. Robins, “Estimating time-varying risk premia in the term structure: the ARCH-M model,” Econometrica, vol. 55, pp. 391–407, 1987.
  • R. F. Engle and G. G. J. Lee, “A permanent and transitory component model of stock return volatility,” in Cointegration, Causality, and Forecasting: A Festschrift in Honor of Clive W. J. Granger, R. Engle and H. White, Eds., pp. 475–497, Oxford University Press, 1999.
  • P. Christoffersen, R. Elkamhi, B. Feunou, and K. Jacobs, “Option valuation with conditional heteroskedasticity and nonnormality,” Review of Financial Studies, vol. 23, no. 5, pp. 2139–2183, 2010.
  • G. Li and C. Zhang, “On the number of state variables in options pricing,” Management Science, vol. 56, no. 11, pp. 2058–2075, 2010.
  • T. Adrian and J. Rosenberg, “Stock returns and volatility: pricing the short-run and long-run components of market risk,” Journal of Finance, vol. 63, no. 6, pp. 2997–3030, 2008.
  • S. L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, pp. 327–343, 1993.
  • J.-C. Duan, “The GARCH option pricing model,” Mathematical Finance, vol. 5, no. 1, pp. 13–32, 1995.
  • S. L. Heston and S. Nandi, “A closed-form GARCH option valuation model,” Review of Financial Studies, vol. 13, no. 3, pp. 585–625, 2000.
  • D. S. Bates, “Post-87 crash fears in the S&P 500 futures option market,” Journal of Econometrics, vol. 94, no. 1-2, pp. 181–238, 2000.
  • D. S. Bates, “Maximum likelihood estimation of latent affine processes,” Review of Financial Studies, vol. 19, no. 3, pp. 909–965, 2006.
  • J. Pan, “The jump-risk premia implicit in options: evidence from an integrated time-series study,” Journal of Financial Economics, vol. 63, no. 1, pp. 3–50, 2002.
  • J. C. Duan, “Conditionally fat-tailed distributions and the volatility smile in options,” Working Paper, Department of Finance, The Hong Kong University of Science and Technology, 1999.
  • J.-C. Duan, P. Ritchken, and Z. Sun, “Approximating GARCH-jump models, jump-diffusion processes, and option pricing,” Mathematical Finance, vol. 16, no. 1, pp. 21–52, 2006.
  • B. Eraker, “Do stock prices and volatility jump? Reconciling evidence from spot and option prices,” Journal of Finance, vol. 59, no. 3, pp. 1367–1403, 2004.
  • M. Broadie, M. Chernov, and M. Johannes, “Model specification and risk premia: evidence from futures options,” Journal of Finance, vol. 62, no. 3, pp. 1453–1490, 2007.
  • P. Christoffersen, C. Dorion, K. Jacobs, and Y. Wang, “Volatility components, affine restrictions, and nonnormal innovations,” Journal of Business & Economic Statistics, vol. 28, no. 4, pp. 483–502, 2010.
  • P. Christoffersen, S. Heston, and K. Jacobs, “Option valuation with conditional skewness,” Journal of Econometrics, vol. 131, no. 1-2, pp. 253–284, 2006.
  • P. Christoffersen, K. Jacobs, C. Ornthanalai, and Y. Wang, “Option valuation with long-run and short-run volatility components,” Journal of Financial Economics, vol. 90, no. 3, pp. 272–297, 2008.
  • S. J. Taylor and X. Xinzhong, “The incremental volatility information in one million foreign exchange quotations,” Journal of Empirical Finance, vol. 4, no. 4, pp. 317–340, 1997.
  • T. G. Andersen and T. Bollerslev, “Answering the skeptics: yes, standard volatility models do provide accurate forecasts,” International Economic Review, vol. 39, no. 4, pp. 885–905, 1998.
  • T. G. Andersen, T. Bollerslev, F. X. Diebold, and H. Ebens, “The distribution of realized stock return volatility,” Journal of Financial Economics, vol. 61, no. 1, pp. 43–76, 2001.
  • T. G. Andersen, T. Bollerslev, F. X. Diebold, and P. Labys, “The distribution of realized exchange rate volatility,” Journal of the American Statistical Association, vol. 96, no. 453, pp. 42–55, 2001.
  • T. G. Andersen, T. Bollerslev, F. X. Diebold, and P. Labys, “Modeling and forecasting realized volatility,” Econometrica, vol. 71, no. 2, pp. 579–625, 2003.
  • O. E. Barndorff-Nielsen and N. Shephard, “Econometric analysis of realized volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society. Series B. Statistical Methodology, vol. 64, no. 2, pp. 253–280, 2002.
  • O. E. Barndorff-Nielsen and N. Shephard, “Econometric analysis of realized covariation: high frequency based covariance, regression, and correlation in financial economics,” Econometrica, vol. 72, no. 3, pp. 885–925, 2004.
  • F. M. Bandi, J. R. Russell, and C. Yang, “Realized volatility forecasting in the presence of time-varying noise,” Journal of Business & Economic Statistics, vol. 31, no. 3, pp. 331–345, 2013.
  • F. Corsi, N. Fusari, and D. La Vecchia, “Realizing smiles: options pricing with realized volatility,” Journal of Financial Economics, vol. 107, pp. 284–304, 2013.
  • L. Zhang, P. A. Mykland, and Y. Aït-Sahalia, “A tale of two time scales: determining integrated volatility with noisy high-frequency data,” Journal of the American Statistical Association, vol. 100, no. 472, pp. 1394–1411, 2005.
  • P. Christoffersen, B. Feunou, K. Jacobs, and N. Meddahi, “The economic value of realized volatility: Using high-frequency returns for option valuation,” Working Paper, 2012.
  • F. Black and M. Scholes, “The pricing of options and corporate liabilities,” The Journal of Political Economy, vol. 81, pp. 637–654, 1973.
  • B. Dupire, “Pricing with a smile,” Risk, vol. 7, pp. 18–20, 1994.
  • R. Lagnado and S. Osher, “A technique for calibrating derivative security pricing models: numerical solution of an inverse problem,” Journal of Computational Finance, vol. 1, pp. 13–25, 1997.
  • C. Chiarella, M. Craddock, and N. El-Hassan, “The calibration of stock option pricing models using inverse problem methodology,” QFRQ Research Papers, University of Technology, Sydney, Sydney, Australia, 2000.
  • L. Jiang and Y. Tao, “Identifying the volatility of underlying assets from option prices,” Inverse Problems, vol. 17, no. 1, pp. 137–155, 2001.
  • S. Crépey, “Calibration of the local volatility in a trinomial tree using Tikhonov regularization,” Inverse Problems, vol. 19, no. 1, pp. 91–127, 2003.
  • V. Isakov, “The inverse problem of option pricing,” in Recent Development in Theories & Numerics: International Conference on Inverse Problems, pp. 47–55, World Scientific, Singapore, 2003.
  • H. Egger and H. W. Engl, “Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates,” Inverse Problems, vol. 21, no. 3, pp. 1027–1045, 2005.
  • P. Ngnepieba, “The adjoint method formulation for an inverse problem in the generalized Black-Scholes model,” Journal of Systemics Cybernetics and Informatics, vol. 4, pp. 69–77, 2006.
  • Z.-C. Deng, J.-N. Yu, and L. Yang, “An inverse problem of determining the implied volatility in option pricing,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 16–31, 2008.
  • J. R. Franks and E. S. Schwartz, “The stochastic behavior of market variance implied in the price of index options,” Economics Journal, vol. 101, pp. 1460–1475, 1991.
  • R. Heynen, “An empirical investigation of observed smile patterns,” Review Futures Markets, vol. 13, pp. 317–353, 1994.
  • A. N. Tikhonov, A. S. Leonov, and A. G. Yagola, Nonlinear Ill-Posed Problems, vol. 14, Chapman & Hall, London, UK, 1998.
  • H. E. Leland, “Option pricing and replication with transaction costs,” The Journal of Finance, vol. 40, pp. 1283–1301, 1985.
  • L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena, vol. 60, no. 1-4, pp. 259–268, 1992.
  • G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71, Cambridge University Press, Cambridge, UK, 1999.
  • V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Mathematics. Doklady, vol. 7, pp. 414–417, 1966. \endinput