# American Institute of Mathematical Sciences

April  2015, 11(2): 421-437. doi: 10.3934/jimo.2015.11.421

## Recovery of the local volatility function using regularization and a gradient projection method

 1 College of Applied Arts and Science of Beijing Union University, Beijing 100191, China 2 Renmin University of China, Beijing 100872, China 3 Hebei Normal University, Shijiazhuang 050024, China

Received  February 2013 Revised  April 2014 Published  September 2014

This paper considers the problem of calibrating the volatility function using regularization technique and the gradient projection method from given option price data. It is an ill-posed problem because of at least one of three well-posed conditions violating. We start with the European option pricing problem. We formulate the problem by obtaining the integral equation from Dupire equation and provide a theory of identifying the local volatility function $\sigma(y,\tau)$ when the parameter $\mu\neq 0$, and then we apply regularization technique for volatility function retrieval problems. A projected gradient method is developed for recovering the volatility function. Numerical simulations are given to illustrate the feasibility of our method.
Citation: Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial and Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
##### References:
 [1] J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-659. [3] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [4] I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [5] I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization, Quantitative Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302. [6] S. Crépy, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J.Math.Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [7] Y. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47. doi: 10.1007/s00211-004-0569-y. [8] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [9] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [10] H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile and term structure in inverse option pricing, Inverse Problems, 22 (2006), 1247-1259. doi: 10.1088/0266-5611/22/4/008. [11] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8. [12] T. Hein and B. Hofmann, On the nature of ill-posedness of an inverse problem arising in option pricing, Inverse Problems, 19 (2003), 1319-1338. doi: 10.1088/0266-5611/19/6/006. [13] T. Hein, Some analysis of Tikhonov regularization of the inverse problem of option pricing in the price-dependent case, Journal for Analysis and its Applications, 24 (2005), 593-609. doi: 10.4171/ZAA/1258. [14] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. [15] B. Hofmann and R. Krämer, On maximum entropy regularization for a specific inverse problem in option pricing, J.Inv.Ill-Posed Problems, 13 (2005), 41-63. doi: 10.1515/1569394053583739. [16] J. Hull and A. White, An analysis of the bias in option pricing caused by a stochastic volatility, Advances in Futures and Options Research, 3 (1988), 29-61. [17] J. Hull, Options, Futures and Other Derivatives, Sixth Edition, People Post Press, 2010. [18] L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inverse Problems, 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311. [19] L. S. Jiang, Q. H. Chen, L. J. Wang and J. E. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. doi: 10.1088/1469-7688/3/6/304. [20] R. Krämer and M. Richter, Ill-posedness versus ill-conditioning - an example from inverse option pricing, Applicable Analysis, 87 (2008), 465-477. doi: 10.1080/00036810802032136. [21] L. Lu and L. Yi, Recovery implied volatility of underlying asset from European option price, J.Inv.Ill-Posed Problems, 17 (2009), 499-509. doi: 10.1515/JIIP.2009.031. [22] R. Merton, Option Pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [23] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, Journal of the Association for Computing Machinery, 9 (1962), 84-97. doi: 10.1145/321105.321114. [24] S. Twomey, Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions, J. Comput. Phys., 18 (1975), 188-200. [25] Y. F. Wang, Computational Methods for Inverse Problems and Their Applications, Higher Education Press, Beijing, 2007. [26] Y. F. Wang and C. C. Yang, A regularizing active set method for retrieval of atmospheric aerosol particle size distribution function, Journal of Optical Society of America A, 25 (2008), 348-356. doi: 10.1364/JOSAA.25.000348. [27] Y. F. Wang, An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function, Journal of Aerosol Science, 39 (2008), 305-322. [28] Y. F. Wang and S. Q. Ma, Projected Barzilai-Borwein methods for large scale nonnegative image restorations, Inverse Problems in Science and Engineering, 15 (2007), 559-583. doi: 10.1080/17415970600881897. [29] Y. X. Yuan, Gradient methods for large scale convex quadratic functions, Optimization and Regularization for Computational Inverse Problems & Applications (Y. F. Wang, A. Yagola and C. Yang eds.) Berlin/Beijing: Springer-Verlag/Higher Education Press, (2010), 141-155. doi: 10.1007/978-3-642-13742-6_7.

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##### References:
 [1] J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141. [2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ., 81 (1973), 637-659. [3] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problems, 13 (1997), L11-L17. doi: 10.1088/0266-5611/13/5/001. [4] I. Bouchouev and V. Isakov, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets, Inverse Problems, 15 (1999), R95-R116. doi: 10.1088/0266-5611/15/3/201. [5] I. Bouchouev, V. Isakov and N. Valdivia, Recovery of volatility coefficient by linearization, Quantitative Finance, 2 (2002), 257-263. doi: 10.1088/1469-7688/2/4/302. [6] S. Crépy, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J.Math.Anal., 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202. [7] Y. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47. doi: 10.1007/s00211-004-0569-y. [8] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [9] H. Egger and H. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [10] H. Egger, T. Hein and B. Hofmann, On decoupling of volatility smile and term structure in inverse option pricing, Inverse Problems, 22 (2006), 1247-1259. doi: 10.1088/0266-5611/22/4/008. [11] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8. [12] T. Hein and B. Hofmann, On the nature of ill-posedness of an inverse problem arising in option pricing, Inverse Problems, 19 (2003), 1319-1338. doi: 10.1088/0266-5611/19/6/006. [13] T. Hein, Some analysis of Tikhonov regularization of the inverse problem of option pricing in the price-dependent case, Journal for Analysis and its Applications, 24 (2005), 593-609. doi: 10.4171/ZAA/1258. [14] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343. [15] B. Hofmann and R. Krämer, On maximum entropy regularization for a specific inverse problem in option pricing, J.Inv.Ill-Posed Problems, 13 (2005), 41-63. doi: 10.1515/1569394053583739. [16] J. Hull and A. White, An analysis of the bias in option pricing caused by a stochastic volatility, Advances in Futures and Options Research, 3 (1988), 29-61. [17] J. Hull, Options, Futures and Other Derivatives, Sixth Edition, People Post Press, 2010. [18] L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices, Inverse Problems, 17 (2001), 137-155. doi: 10.1088/0266-5611/17/1/311. [19] L. S. Jiang, Q. H. Chen, L. J. Wang and J. E. Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. doi: 10.1088/1469-7688/3/6/304. [20] R. Krämer and M. Richter, Ill-posedness versus ill-conditioning - an example from inverse option pricing, Applicable Analysis, 87 (2008), 465-477. doi: 10.1080/00036810802032136. [21] L. Lu and L. Yi, Recovery implied volatility of underlying asset from European option price, J.Inv.Ill-Posed Problems, 17 (2009), 499-509. doi: 10.1515/JIIP.2009.031. [22] R. Merton, Option Pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [23] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, Journal of the Association for Computing Machinery, 9 (1962), 84-97. doi: 10.1145/321105.321114. [24] S. Twomey, Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions, J. Comput. Phys., 18 (1975), 188-200. [25] Y. F. Wang, Computational Methods for Inverse Problems and Their Applications, Higher Education Press, Beijing, 2007. [26] Y. F. Wang and C. C. Yang, A regularizing active set method for retrieval of atmospheric aerosol particle size distribution function, Journal of Optical Society of America A, 25 (2008), 348-356. doi: 10.1364/JOSAA.25.000348. [27] Y. F. Wang, An efficient gradient method for maximum entropy regularizing retrieval of atmospheric aerosol particle size distribution function, Journal of Aerosol Science, 39 (2008), 305-322. [28] Y. F. Wang and S. Q. Ma, Projected Barzilai-Borwein methods for large scale nonnegative image restorations, Inverse Problems in Science and Engineering, 15 (2007), 559-583. doi: 10.1080/17415970600881897. [29] Y. X. Yuan, Gradient methods for large scale convex quadratic functions, Optimization and Regularization for Computational Inverse Problems & Applications (Y. F. Wang, A. Yagola and C. Yang eds.) Berlin/Beijing: Springer-Verlag/Higher Education Press, (2010), 141-155. doi: 10.1007/978-3-642-13742-6_7.
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