# zbMATH — the first resource for mathematics

Recovery of the local volatility function using regularization and a gradient projection method. (English) Zbl 1305.91241
Summary: 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.
##### MSC:
 91G60 Numerical methods (including Monte Carlo methods) 91G20 Derivative securities (option pricing, hedging, etc.) 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
Full Text:
##### References:
 [1] J. Barzilai, Two-point step size gradient methods,, IMA Journal of Numerical Analysis, 8, 141, (1988) · Zbl 0638.65055 [2] F. Black, The pricing of options and corporate liabilities,, J. Political Econ., 81, 637, (1973) · Zbl 1092.91524 [3] I. Bouchouev, The inverse problem of option pricing,, Inverse Problems, 13, (1997) · Zbl 0894.90014 [4] I. Bouchouev, Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets,, Inverse Problems, 15, (1999) · Zbl 0938.35190 [5] I. Bouchouev, Recovery of volatility coefficient by linearization,, Quantitative Finance, 2, 257, (2002) [6] S. Crépy, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization,, SIAM J.Math.Anal., 34, 1183, (2003) · Zbl 1126.35373 [7] Y. Dai, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming,, Numerische Mathematik, 100, 21, (2005) · Zbl 1068.65073 [8] B. Dupire, Pricing with a smile,, Risk, 7, 18, (1994) [9] H. Egger, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates,, Inverse Problems, 21, 1027, (2005) · Zbl 1205.65194 [10] H. Egger, On decoupling of volatility smile and term structure in inverse option pricing,, Inverse Problems, 22, 1247, (2006) · Zbl 1112.91032 [11] H. W. Engl, Regularization of Inverse Problems,, Mathematics and its Applications, (1996) · Zbl 0859.65054 [12] T. Hein, On the nature of ill-posedness of an inverse problem arising in option pricing,, Inverse Problems, 19, 1319, (2003) · Zbl 1086.91028 [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, 593, (2005) · Zbl 1109.35120 [14] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options,, Review of Financial Studies, 6, 327, (1993) · Zbl 1384.35131 [15] B. Hofmann, On maximum entropy regularization for a specific inverse problem in option pricing,, J.Inv.Ill-Posed Problems, 13, 41, (2005) · Zbl 1086.91029 [16] J. Hull, An analysis of the bias in option pricing caused by a stochastic volatility,, Advances in Futures and Options Research, 3, 29, (1988) [17] J. Hull, Options, Futures and Other Derivatives,, Sixth Edition, (2010) · Zbl 1087.91025 [18] L. S. Jiang, Identifying the volatility of underlying assets from option prices,, Inverse Problems, 17, 137, (2001) · Zbl 0997.91024 [19] L. S. Jiang, A new well-posed algorithm to recover implied local volatility,, Quantitative Finance, 3, 451, (2003) [20] R. Krämer, Ill-posedness versus ill-conditioning - an example from inverse option pricing,, Applicable Analysis, 87, 465, (2008) · Zbl 1152.91527 [21] L. Lu, Recovery implied volatility of underlying asset from European option price,, J.Inv.Ill-Posed Problems, 17, 499, (2009) · Zbl 1167.91372 [22] R. Merton, Option Pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3, 125, (1976) · Zbl 1131.91344 [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, 84, (1962) · Zbl 0108.29902 [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, 188, (1975) [25] Y. F. Wang, Computational Methods for Inverse Problems and Their Applications,, Higher Education Press, (2007) [26] Y. F. Wang, A regularizing active set method for retrieval of atmospheric aerosol particle size distribution function,, Journal of Optical Society of America A, 25, 348, (2008) [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, 305, (2008) [28] Y. F. Wang, Projected Barzilai-Borwein methods for large scale nonnegative image restorations,, Inverse Problems in Science and Engineering, 15, 559, (2007) · Zbl 1202.94077 [29] Y. X. Yuan, Gradient methods for large scale convex quadratic functions,, Optimization and Regularization for Computational Inverse Problems & Applications (Y. F. Wang, 141, (2010) · Zbl 1277.90089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.