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

Qinghua Ma; Zuoliang Xu; Liping Wang

doi:10.3934/jimo.2015.11.421

Article Contents

2015, Volume 11, Issue 2: 421-437. Doi: 10.3934/jimo.2015.11.421

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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
2.
Renmin University of China, Beijing 100872
3.
Hebei Normal University, Shijiazhuang 050024

Received: February 2013

Revised: April 2014

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: 65J15, 65J20, 91B28, 46N10, 47N10.

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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.