A penalty-based method from reconstructing smooth local volatility surface from American options

Previous Article
Neural network smoothing approximation method for stochastic variational inequality problems
JIMO Home
This Issue
Next Article
A family of extragradient methods for solving equilibrium problems

April 2015, 11(2): 631-644. doi: 10.3934/jimo.2015.11.631

A penalty-based method from reconstructing smooth local volatility surface from American options

Kai Zhang ^1, and Kok Lay Teo ^2,

1.	China Center for Special Economic Zone Research, Shenzhen University, 3688 Nanhai Ave., Shenzhen, 518060, China
2.	Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845, Australia

Received November 2013 Revised May 2014 Published September 2014

Abstract
Related Papers

This paper is devoted to develop a robust penalty-based method of reconstructing smooth local volatility surface from the observed American option prices. This reconstruction problem is posed as an inverse problem: given a finite set of observed American option prices, find a local volatility function such that the theoretical option prices matches the observed ones optimally with respect to a prescribed performance criterion. The theoretical American option prices are governed by a set of partial differential complementarity problems (PDCP). We propose a penalty-based numerical method for the solution of the PDCP. Typically, the reconstruction problem is ill-posed and a bicubic spline regularization technique is thus proposed to overcome this difficulty. We apply a gradient-based optimization algorithm to solve this nonlinear optimization problem, where the Jacobian of the cost function is computed via finite difference approximation. Two numerical experiments: a synthetic American put option example and a real market American put option example, are performed to show the robustness and effectiveness of the proposed method to reconstructing the unknown volatility surface.

Keywords: penalty method, option pricing., Complementarity problem, volatility surface, inverse problem.

Mathematics Subject Classification: Primary: 91G60 , 91G20; Secondary: 90C3.

Citation: Kai Zhang, Kok Lay Teo. A penalty-based method from reconstructing smooth local volatility surface from American options. Journal of Industrial and Management Optimization, 2015, 11 (2) : 631-644. doi: 10.3934/jimo.2015.11.631

References:

[1]	Y. Achdou and O. Pironneau, Computational Methods For Option Pricing, Vol. 30. SIAM, 2005. doi: 10.1137/1.9780898717495.
[2]	A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262.
[3]	M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, 2 (1995), 73-88. doi: 10.1080/13504869500000005.
[4]	F. E. Benth, K. H.Karlsen and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance and Stochastics, 7 (2003), 277-298. doi: 10.1007/s007800200091.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[6]	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.
[7]	T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100.
[8]	S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM Journal of Mathematical Analysis, 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202.
[9]	J. Huang and J. S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56.
[10]	J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005.
[11]	N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces, Journal of Computational Finance, 2 (1999), 5-32.
[12]	L. Jiang, Q. Chen, L. Wang and J. 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.
[13]	Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, Berlin, 2008.
[14]	R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of the inverse problem, Journal of Computational Finance, 1 (1997), 13-25.
[15]	R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83.
[16]	W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293. doi: 10.1007/s10957-009-9559-7.
[17]	R. C. Merton, Option pricing when underlying stock return are discontinuous, Journal of financial economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.
[18]	B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, Journal of Computational Finance, 5 (2002), 69-97.
[19]	J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed, Springer, Berlin, 2006.
[20]	S. Stojanovic, Implied volatility for American options via optimal control and fast numerical solutions of obstacle problems, Differential Equations and Control Theory, 225 (2002), 277-294 .
[21]	G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611970128.
[22]	S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3.
[23]	P. Wilmott, Paul Wimott on Quantitave Finance, Wiley, New York, 2000.
[24]	K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Applied Numerical Mathematics, 59 (2009), 1754-1767. doi: 10.1016/j.apnum.2009.01.004.
[25]	K. Zhang and S. Wang, Interior penalty approach to american option pricing, Journal of Industrial and Management Optimization, 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435.
[26]	K. Zhang, K. L. Teo and M. Swartz, A robust numerical scheme for pricing American options under regime switching based on penalty method, Computational Economics, 43 (2014), 463-483. doi: 10.1007/s10614-013-9361-3.

show all references

References:

[1]	Y. Achdou and O. Pironneau, Computational Methods For Option Pricing, Vol. 30. SIAM, 2005. doi: 10.1137/1.9780898717495.
[2]	A. Anderson and J. Andresen, Jump diffusion process: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262.
[3]	M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Applied Mathematical Finance, 2 (1995), 73-88. doi: 10.1080/13504869500000005.
[4]	F. E. Benth, K. H.Karlsen and K. Reikvam, A semilinear Black and Scholes partial differential equation for valuing American options, Finance and Stochastics, 7 (2003), 277-298. doi: 10.1007/s007800200091.
[5]	F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062.
[6]	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.
[7]	T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100.
[8]	S. Crepey, Calibration of the local volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM Journal of Mathematical Analysis, 34 (2003), 1183-1206. doi: 10.1137/S0036141001400202.
[9]	J. Huang and J. S. Pang, A mathematical programming with equilibrium constraints approach to the implied volatility surface of American options, Journal of Computational Finance, 4 (2000), 21-56.
[10]	J. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, Englewood Cliffs, 2005.
[11]	N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces, Journal of Computational Finance, 2 (1999), 5-32.
[12]	L. Jiang, Q. Chen, L. Wang and J. 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.
[13]	Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer, Berlin, 2008.
[14]	R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of the inverse problem, Journal of Computational Finance, 1 (1997), 13-25.
[15]	R. Lagnado and S. Osher, Reconciling differences, Risk, 10 (1997), 79-83.
[16]	W. Li and S. Wang, Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293. doi: 10.1007/s10957-009-9559-7.
[17]	R. C. Merton, Option pricing when underlying stock return are discontinuous, Journal of financial economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.
[18]	B. F. Nielsen, O. Skavhaug and A. Tveito, Penalty and front-fixing methods for the numerical solution of American option problems, Journal of Computational Finance, 5 (2002), 69-97.
[19]	J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed, Springer, Berlin, 2006.
[20]	S. Stojanovic, Implied volatility for American options via optimal control and fast numerical solutions of obstacle problems, Differential Equations and Control Theory, 225 (2002), 277-294 .
[21]	G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611970128.
[22]	S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254. doi: 10.1007/s10957-006-9062-3.
[23]	P. Wilmott, Paul Wimott on Quantitave Finance, Wiley, New York, 2000.
[24]	K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing, Applied Numerical Mathematics, 59 (2009), 1754-1767. doi: 10.1016/j.apnum.2009.01.004.
[25]	K. Zhang and S. Wang, Interior penalty approach to american option pricing, Journal of Industrial and Management Optimization, 7 (2011), 435-447. doi: 10.3934/jimo.2011.7.435.
[26]	K. Zhang, K. L. Teo and M. Swartz, A robust numerical scheme for pricing American options under regime switching based on penalty method, Computational Economics, 43 (2014), 463-483. doi: 10.1007/s10614-013-9361-3.

[1]	Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235
[2]	Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[3]	Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial and Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[4]	Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial and Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783
[5]	Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[6]	Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021077
[7]	Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial and Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259
[8]	Ming Chen, Chongchao Huang. A power penalty method for the general traffic assignment problem with elastic demand. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1019-1030. doi: 10.3934/jimo.2014.10.1019
[9]	Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022017
[10]	Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2703-2714. doi: 10.3934/jimo.2020090
[11]	Ming-Zheng Wang, M. Montaz Ali. Penalty-based SAA method of stochastic nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 241-257. doi: 10.3934/jimo.2010.6.241
[12]	Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005
[13]	Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205
[14]	Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[15]	Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
[16]	Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102
[17]	Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485
[18]	Kim Knudsen, Matti Lassas, Jennifer L. Mueller, Samuli Siltanen. Regularized D-bar method for the inverse conductivity problem. Inverse Problems and Imaging, 2009, 3 (4) : 599-624. doi: 10.3934/ipi.2009.3.599
[19]	Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[20]	Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

2020 Impact Factor: 1.801

Tools

Metrics

Other articles
by authors

on AIMS
Kai Zhang Kok Lay Teo
on Google Scholar
Kai Zhang Kok Lay Teo

[Back to Top]