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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, China |
2. | Renmin University of China, Beijing 100872, China |
3. | Hebei Normal University, Shijiazhuang 050024, China |
References:
[1] |
J. Barzilai and J. Borwein, Two-point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.
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. Google Scholar |
[3] |
I. Bouchouev and V. Isakov, The inverse problem of option pricing,, Inverse Problems, 13 (1997).
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).
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.
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.
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.
doi: 10.1007/s00211-004-0569-y. |
[8] |
B. Dupire, Pricing with a smile,, Risk, 7 (1994), 18. Google Scholar |
[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.
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.
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, (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.
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.
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. Google Scholar |
[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.
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. Google Scholar |
[17] |
J. Hull, Options, Futures and Other Derivatives,, Sixth Edition, (2010). Google Scholar |
[18] |
L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices,, Inverse Problems, 17 (2001), 137.
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.
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.
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.
doi: 10.1515/JIIP.2009.031. |
[22] |
R. Merton, Option Pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125. Google Scholar |
[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.
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. Google Scholar |
[25] |
Y. F. Wang, Computational Methods for Inverse Problems and Their Applications,, Higher Education Press, (2007). Google Scholar |
[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.
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. Google Scholar |
[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.
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, (2010), 141.
doi: 10.1007/978-3-642-13742-6_7. |
show all references
References:
[1] |
J. Barzilai and J. Borwein, Two-point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.
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. Google Scholar |
[3] |
I. Bouchouev and V. Isakov, The inverse problem of option pricing,, Inverse Problems, 13 (1997).
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).
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.
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.
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.
doi: 10.1007/s00211-004-0569-y. |
[8] |
B. Dupire, Pricing with a smile,, Risk, 7 (1994), 18. Google Scholar |
[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.
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.
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, (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.
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.
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. Google Scholar |
[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.
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. Google Scholar |
[17] |
J. Hull, Options, Futures and Other Derivatives,, Sixth Edition, (2010). Google Scholar |
[18] |
L. S. Jiang and Y. S. Tao, Identifying the volatility of underlying assets from option prices,, Inverse Problems, 17 (2001), 137.
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.
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.
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.
doi: 10.1515/JIIP.2009.031. |
[22] |
R. Merton, Option Pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125. Google Scholar |
[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.
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. Google Scholar |
[25] |
Y. F. Wang, Computational Methods for Inverse Problems and Their Applications,, Higher Education Press, (2007). Google Scholar |
[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.
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. Google Scholar |
[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.
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, (2010), 141.
doi: 10.1007/978-3-642-13742-6_7. |
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