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 & 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.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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