Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 91G60 , 91G20; Secondary: 90C33.

 Citation:

•  [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.