American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.   Google Scholar

[3]

M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities,, Applied Mathematical Finance, 2 (1995), 73.  doi: 10.1080/13504869500000005.  Google Scholar

[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.  doi: 10.1007/s007800200091.  Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, The Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[6]

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

[7]

T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function,, Journal of Computational Finance, 2 (1999), 77.   Google Scholar

[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.  doi: 10.1137/S0036141001400202.  Google Scholar

[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.   Google Scholar

[10]

J. Hull, Options, Futures, and Other Derivatives,, Prentice-Hall, (2005).   Google Scholar

[11]

N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces,, Journal of Computational Finance, 2 (1999), 5.   Google Scholar

[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.  doi: 10.1088/1469-7688/3/6/304.  Google Scholar

[13]

Y. K. Kwok, Mathematical Models of Financial Derivatives,, Springer, (2008).   Google Scholar

[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.   Google Scholar

[15]

R. Lagnado and S. Osher, Reconciling differences,, Risk, 10 (1997), 79.   Google Scholar

[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.  doi: 10.1007/s10957-009-9559-7.  Google Scholar

[17]

R. C. Merton, Option pricing when underlying stock return are discontinuous,, Journal of financial economics, 3 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[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.   Google Scholar

[19]

J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed,, Springer, (2006).   Google Scholar

[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.   Google Scholar

[21]

G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics,, Vol. 59, (1990).  doi: 10.1137/1.9781611970128.  Google Scholar

[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.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

[23]

P. Wilmott, Paul Wimott on Quantitave Finance,, Wiley, (2000).   Google Scholar

[24]

K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing,, Applied Numerical Mathematics, 59 (2009), 1754.  doi: 10.1016/j.apnum.2009.01.004.  Google Scholar

[25]

K. Zhang and S. Wang, Interior penalty approach to american option pricing,, Journal of Industrial and Management Optimization, 7 (2011), 435.  doi: 10.3934/jimo.2011.7.435.  Google Scholar

[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.  doi: 10.1007/s10614-013-9361-3.  Google Scholar

show all references

References:
[1]

Y. Achdou and O. Pironneau, Computational Methods For Option Pricing,, Vol. 30. SIAM, (2005).  doi: 10.1137/1.9780898717495.  Google Scholar

[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.   Google Scholar

[3]

M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities,, Applied Mathematical Finance, 2 (1995), 73.  doi: 10.1080/13504869500000005.  Google Scholar

[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.  doi: 10.1007/s007800200091.  Google Scholar

[5]

F. Black and M. Scholes, The pricing of options and corporate liabilities,, The Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[6]

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

[7]

T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function,, Journal of Computational Finance, 2 (1999), 77.   Google Scholar

[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.  doi: 10.1137/S0036141001400202.  Google Scholar

[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.   Google Scholar

[10]

J. Hull, Options, Futures, and Other Derivatives,, Prentice-Hall, (2005).   Google Scholar

[11]

N. Jackson, E. Suli and S. Howison, Computation of deterministic volatility surfaces,, Journal of Computational Finance, 2 (1999), 5.   Google Scholar

[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.  doi: 10.1088/1469-7688/3/6/304.  Google Scholar

[13]

Y. K. Kwok, Mathematical Models of Financial Derivatives,, Springer, (2008).   Google Scholar

[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.   Google Scholar

[15]

R. Lagnado and S. Osher, Reconciling differences,, Risk, 10 (1997), 79.   Google Scholar

[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.  doi: 10.1007/s10957-009-9559-7.  Google Scholar

[17]

R. C. Merton, Option pricing when underlying stock return are discontinuous,, Journal of financial economics, 3 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[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.   Google Scholar

[19]

J. Nocedal and S. Wright, Numerical Optimization Series: Springer Series in Operations Research and Financial Engineering 2nd ed,, Springer, (2006).   Google Scholar

[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.   Google Scholar

[21]

G. Wahba, Splines Models for Observational Data. Series in Applied Mathematics,, Vol. 59, (1990).  doi: 10.1137/1.9781611970128.  Google Scholar

[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.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

[23]

P. Wilmott, Paul Wimott on Quantitave Finance,, Wiley, (2000).   Google Scholar

[24]

K. Zhang and S. Wang, A computational scheme for uncertain volatility model in option pricing,, Applied Numerical Mathematics, 59 (2009), 1754.  doi: 10.1016/j.apnum.2009.01.004.  Google Scholar

[25]

K. Zhang and S. Wang, Interior penalty approach to american option pricing,, Journal of Industrial and Management Optimization, 7 (2011), 435.  doi: 10.3934/jimo.2011.7.435.  Google Scholar

[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.  doi: 10.1007/s10614-013-9361-3.  Google Scholar

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