Li, Minqiang (2009): A Quasianalytical Interpolation Method for Pricing American Options under General Multidimensional Diffusion Processes.

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Abstract
We present a quasianalytical method for pricing multidimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the BlackScholes model, our method achieves the same e±ciency as BaroneAdesi and Whaley's (1987) quadratic approximation with our method being generally more accurate for outofthemoney and longmaturity options. When applied to Heston's stochastic volatility model, our method is shown to be extremely e±cient and fairly accurate.
Item Type:  MPRA Paper 

Original Title:  A Quasianalytical Interpolation Method for Pricing American Options under General Multidimensional Diffusion Processes 
Language:  English 
Keywords:  American option; Interpolation method; Quasianalytical approximation; Critical bound ary; Heston's Stochastic volatility model 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods G  Financial Economics > G1  General Financial Markets > G13  Contingent Pricing ; Futures Pricing 
Item ID:  17348 
Depositing User:  Minqiang Li 
Date Deposited:  02. Oct 2009 10:14 
Last Modified:  11. Feb 2013 21:09 
References:  Amin, K., Khanna, A. (1994). Convergence of American option values from discrete to continuous time financial models. Mathematical Finance 4, 289304. BaroneAdesi, G. (2005). The saga of the American put. Journal of Banking & Finance 29, 29092918. BaroneAdesi, G., Whaley, R. (1987). E±cient analytical approximation of American option values. Journal of Finance 42, 301320. Bergman, Y., Grundy, B., Wiener, Z. (1996). General properties of option prices. Journal of Finance 51, 15731610. Bensoussan, A. (1984). On the theory of option pricing. Acta Applicandae Mathematicae 2, 139158. Black, F., Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637654. Boyle, P.P. (1977). Options: A Monte Carlo approach. Journal of Financial Economics 4, 323338. Breen, R. (1991). The accelerated binomial option pricing model. Journal of Financial and Quantitative Analysis 26, 153164. Brennan, M., Schwartz, E. (1977). The valuation of American put options. Journal of Finance 32, 449462. Brennan, M., Schwartz, E. (1978). Finite di®erence methods and jump processes arising in the pricing of contingent claims: A synthesis. Journal of Financial and Quantitative Analysis 13, 461474. Broadie, M., Detemple, J.B. (1996). American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211{1250. Broadie, M., Glasserman, P. (1997). Pricing Americanstyle securities using simulation. Journal of Economic Dynamics and Control 21, 13231352. Bunch, D., Johnson, H.E. (1992). A simple and numerically e±cient valuation method for American puts using a modi¯ed GeskeJohnson approach. Journal of Finance 47(2), 809816. Bunch, D., Johnson, H.E. (2000). The American put option and its critical stock price. Journal of Finance 55(5), 23332356. Carr, P. (1998). Randomization and the American put. Review of Financial Studies 11, 597626. Carr, P., Faguet, D. (1995). Fast accurate valuation of American options. Working paper, Cornell University. Carr, P., Jarrow, R., Myneni, R. (1992). Alternative characterizations of American put options. Mathematical Finance 2, 87106. Chung, SL. (2000). American option valuation under stochastic interest rates. Review od Derivatives Research 3(3), 283307. Clement, E., Lamberton, D., Protter, P. (2001). An analysis of a least squares regression algorithm for American option pricing. Finance and Stochastic 6, 449471. Cox, J.C., Ross, S.A., Rubinstein, M. (1979). Option pricing: A simpli¯ed approach. Journal of Financial Economics 7, 229264. Curran, M. (1995). Accelerating American option pricing in lattices. Journal of Derivatives 3,817. Detemple, J.B. (2006). Americanstyle Derivatives: Valuation and Computation. CRC Press, Taylor and Francis Group, London. Detemple, J.B., Feng, S., Tian, W. (2003). The valuation of American call options on the minimum of two dividendpaying assets. Annals of Applied Probability 13, 953983. Detemple, J.B., Tian, W. (2002). The valuation of American options for a class of di®usion processes. Management Science 48, 917937. Finucane, T.J., Tomas, M.J. (1996). American stochastic volatility call option pricing: A lattice based approach. Review of Derivatives Research 1(2), 183201. Figlewski, S., Gao, B. (1999). The adaptive mesh model: A new approach to e±cient option pricing. Journal of Financial Economics 53, 313351. Geske, R. (1979). The valuation of compound options. Journal of Financial Economics 7, 6381. Geske, R., Johnson, H.E. (1984). The American put option valued analytically. Journal of Finance 39, 15111524. Heston, S.L. (1993). A closedform solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327343. Huang, J., Subrahmanyam, M., Yu, G. (1996). Pricing and hedging American options: A recursive integration method. Review of Financial Studies 9, 277330. Hull, J., White, A. (1990). Valuing derivative securities using the explicit ¯nite di®erence method. Journal of Financial and Quantitative Analysis 25, 87{100. Ito K., Toivanen, J. (2006). Lagrange multiplier approach with optimized ¯nite di®erence stencils for pricing American options under stochastic volatility. Report B 6/2006, Department of Mathematical Information Technology, University of JyvÄaskulÄa. Jacka, S.D. (1991). Optimal stopping and the American put. Mathematical Finance 1, 114. Jaillet, P., Lamberton, D., Lapeyre, B. (1990). Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 21, 263289. Johnson, H.E. (1983). An analytic approximation for the American put price. Journal of Financial and Quantitative Analysis 18(1), 141148. Ju, N. (1998). Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Review of Financial Studies 11, 627646. Ju, N., Zhong, R. (1999). An approximate formula for pricing American options. Journal of Derivatives 7, 3140. Kamrad, B., Ritchken, P. (1991). Multinomial approximating models for options with k state variables. Management Science 37, 16401652. Karatzas, L. (1988). On the pricing of American options. Applied Mathematics and Optimization 17, 3760. Khaliqa, A.Q.M., Vossb, D.A., Kazmic, S.H.K. (2006). A linearly implicit predictorcorrector scheme for pricing American options using a penalty method approach, Journal of Banking & Finance 30(2), 489502. Kim, I.J. (1990). The analytic valuation of American options. Review of Financial Studies 3, 547572. Kim, I.J. (1994). Analytical approximations of the option exercise boundaries for American futures option. Journal of Futures Markets 14, 124. Kim, I.J., Yu, G. (1996). An alternative approach to the valuation of American options and applications. Review of Derivatives Research 1, 6186. Kim I.J., Jang, BG. (2008). An alternative numerical approach for valuation of American options: A simple iteration method. Working paper, Yonsei University. Lamberton, D. (1993). Convergence of the critical price in the approximation of American options. Mathematical Finance 3, 179190. Lamberton, D. (1998). Error estimates for the binomial approximation of American put options. Annals of Applied Probability 8, 206233. Lewis, A.L. (2000). Option Valuation under Stochastic Volatility with Mathematica Code. Finance Press. Newport Beach, California. Li, M. (2008). Approximate inversion of the BlackScholes formula using rational functions. European Journal of Operational Research 185(2), 743759. Li, M. (2009). Analytical approximations for the critical stock prices of American options: a performance comparison. Review of Derivatives Research, forthcoming, DOI:10.1007/s1114700990443. Li, M., Lee, K. (2009). An adaptive successive overrelaxation method for computing the BlackScholes implied volatility. Quantitative Finance, forthcoming. Li, M., Pearson, N.D. (2009). A horse race among competing option pricing models using S&P 500 index options. Working paper, Georgia Institute of Technology. Longsta®, F.A., Schwartz, E.A. (2001). Valuing American options by simulations: A simple leastsquares approach. Review of Financial Studies 14, 113147. McDonald, R.D., Schroder, M.D. (1998). A parity result for American options. Journal of Computational Finance 1, 513. McKean, H.P. Jr. (1965). Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Management Review 6, 3239. MacMillan, L.W. (1986). An analytic approximation for the American put prices. Advances in Futures and Options Research 1, 119139. Medvedev, A., Scaillet, O. (2009). Pricing American options under stochastic volatility and stochastic interest rates. Journal of Financial Economics, forthcoming. Merton, R.C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141183. Moreno, M., Navas, J. (2003). On the robustness of least squares Monte Carlo (LSM) for pricing American derivatives. Review of Derivative Research 6, 107{128. Myneni, R. (1992). The pricing of American option. Annals of Applied Probability 2(1), 123. Omberg, E. (1987). The valuation of American put options with exponential exercise policies. Advances in Futures and Options Research 2, 117142. Parkinson, M. (1977). Option pricing: the American Put. Journal of Business 50, 2136. Peskir, G. (2005). On the American option problem. Mathematical Finance 15(1), 169181. Rendleman, R., Bartter, B. (1979). TwoState option pricing. Journal of Finance 34, 10931110. Rogers, C. (2002). Monte Carlo valuation of American options. Mathematical Finance 12, 271286. Rubinstein, M., J. Cox. (1982). Options Markets. Prentice Hall, Englewood Cliffs, NJ. Samuelson, P.A. (1967). Rational theory of warrant pricing. Industrial Management Review 6, 1331. Schroder, M. (1999). Changes of numeraire for pricing futures, forwards and options. Review of Financial Studies 12, 11431163. Stentoft, L. (2004). Assessing the least squares MonteCarlo approach to American option valuation. Review of Derivatives Research 7, 129168. Sullivan, M.A. (2000). Valuing American put options using Gaussian quadrature. Review of Financial Studies 13(1), 7594. Van Moerbeke, P. (1976). On optimal stopping and free boundary problems. Archive for Rational Mechanics and Analysis 60, 101148. Zhu, SP. (2006). An exact and explicit solution for the valuation of American put options. Quantitative Finance 6(3), 229242. Zhu, SP., He, ZW. (2007). Calculating the early exercise boundary Of American put options with an approximation formula. International Journal of Theoretical and Applied Finance 10, 12031227. Zhylyevskyy, O. (2009). A fast Fourier transform technique for pricing American options under stochastic volatility. Review of Derivatives Research, forthcoming. DOI:10.1007/s1114700990416. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/17348 