OJS  Vol.5 No.6 , October 2015
Pricing American Options Using Transition Probabilities: A Dynamical Systems Approach
Abstract: We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.
Cite this paper: Elizondo, R. , Padilla, P. and Bladt, M. (2015) Pricing American Options Using Transition Probabilities: A Dynamical Systems Approach. Open Journal of Statistics, 5, 525-542. doi: 10.4236/ojs.2015.56056.

[1]   Bank, P. and Föllmer, H. (2002) American Options, Multi-Armed Bandits, and Optimal Consumption Plans: A Unifying View. Paris-Princeton Lectures on Mathematical Finance, 1814, 1-42.

[2]   Bladt, M. and Rydberg, T.H. (1998) An Actuarial Approach to Option Pricing under the Physical Measure and without Market Assumptions. Insurance: Mathematics and Economics, 22, 65-73.

[3]   Bally, P., Pages, G. and Printems, J. (2005) A Quantization Tree Method for Pricing and Hedging Multidimensional American Options. Mathematical Finance, 15, 119-168.

[4]   Broadie, M. and Cao, M.H. (2008) Improved Lower and Upper Bound Algorithms for Pricing American Options by Simulations. Quantitative Finance, 8, 845-861.

[5]   Carr, P., Jarrow, R. and Myneni, R. (2006) Alternative Characterizations of American Put Options. Mathematical Finance, 2, 87-106.

[6]   Geske, R. and Johnson, H.E. (1984) The American Put Option Valued Analytically. The Journal of Finance, 39, 1511-1524.

[7]   Han, H.D. and Wu, X.N. (2004) A Fast Numerical Method for the Black-Scholes Equation of American Options. SIAM Journal on Numerical Analysis, 41, 2081-2095.

[8]   Ikonen, S. and Toivanen, J. (2008) Efficient Numerical Methods for Pricing American Options Under Stochastic Volatility. Numerical Methods for Partial Differential Equations, 24, 104-126.

[9]   Merton, R.C., Foreword by Samuelson, P.A. (1995) Continuous-Time Finance. Blackwell, Massachusetts.

[10]   Musiela, M. and Rutkowski, M. (1998) Martingale Methods in Financial Modelling. Springer-Verlag, Berlin.

[11]   Tangman, D.Y., Gopaul, A. and Bhuruth, M. (2008) A Fast High-Order Finite Difference Algotithm for Pricing American Options. Journal of Computational and Applied Mathematics, 222, 17-29.

[12]   Jia, Q. (2009) Pricing American Options using Monte Carlo Methods. Department of Mathematics Uppsala University, U.U.D.M. Project Report.

[13]   Uys, N. (2005) Optimal Stopping Problems and American Option. Master of Science Dissertation Submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg.

[14]   Wilmott, P., Dewynne, J. and Howison, S. (1993) Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford.

[15]   Levendorskii, S. (2004) The American Put and European Options Near Expiry, Under Lévy Processes. Department of Mathematics, University of Leicester, Leicester, 1-30.

[16]   Hyungsok, A. and Wilmott, P. On Trading American Options. OCIAM, Oxford University, Oxford.

[17]   Luenberger, D.G. (1998) Investment Science. Oxford University Press, Oxford.

[18]   Méndez, R.E. (2007) Correlación Temporal en la Valuación de Derivados. Tesis de Maestra en Ciencias Matemáticas, UNAM, Mexico City.

[19]   Risken, H. (1989) The Fokker-Planck Equation: Methods of Solution and Applications. Springer-Verlag, Berlin.

[20]   Werner, H. and Lefever, R. (1984) Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology. Springer-Verlag, Berlin.

[21]   Carr, P. and Hirsa, A. (2002) Why Be Backward? Forward Equations for American Options. Morgan Stanley/Courant Institute, NYU, New York, 1-25.

[22]   Evans, L. (1998) Partial Differential Equations. American Mathematical Society, Providence.

[23]   Lyuu, L. (2000) Financial Engineering and Computation: Principles, Mathematics and Algorithms. Cambridge University Press, New York.

[24]   Pontryagin, L., Andronov, A. and Vitt, A. (1989) Appendix: On the Statistical Treatment of Dynamical Systems. Springer-Verlag, New York, 329-348.

[25]   Zhang, W.-B. (1991) Synergetic Economics: Time and Change in Nonlinear Economics. Springer-Verlag, Berlin.

[26]   Llenera-Garcés, F. (2000) Una Nota Sobre Valoración de Opciones Americanas y Arbitraje. Investigaciones Económicas, XXIV, 207-218.

[27]   Mikosch, T. (1999) Elementary Stochastic Calculus with Finance in View. World Scientific Publishing, Singapore.

[28]   Steele, J.M. (2001) Stochastic Calculus and Financial Applications. Springer-Verlag, New York.

[29]   Padilla, P. and Bladt, M. (2001) Nonlinear Financial Models: Finite Markov Modulation and Its Limits. In: Avellaneda, M., Ed., Quantitative Analysis in Financial Markets, Collected Papers of the New York University Mathematical Finance Seminar, Vol. III, World Scientific Publishing, Singapore, 159-171.

[30]   Elizondo, R. (2009) Incorporación de Factores Macroeconómicos en los Modelos de Valuación de Productos Derivados. Thesis de Doctorado en Ciencias, IIMAS, UNAM, Mexico City.

[31]   Elizondo, R. and Padilla, P. (2008) An Analytical Approach to Merton’s Rational Option Pricing Theory. Analysis and Application, 6, 169-182.

[32]   Jarrow, R.A. (1998) Preferences, Continuity and the Arbitrage Pricing Theory. The Review of Financial Studies, 1, 159-172.

[33]   Odegaard, B.A. (2007) Financial Numerical Recipes in C++.

[34]   Numerical Implementation Website.

[35]   Hull, J. (2000) Options, Futures, and Other Derivatives. Prentice Hall, Upper Saddle River.

[36]   Broadie, M. and Detemple, J. (1996) American Option Valuation: New Bounds, Approximations and a Comparison of Existing Methods. The Review of Financial Studies, 9, 1211-1250.

[37]   Chesney, M. and Jeanblanc, M. (2003) Pricing American Currency Options in a Jump Diffusion Model. 1-19.

[38]   Christ Churh College (2004) Nonlinear Black Scholes Modelling: FDM vs FEM. A Thesis Submitted in Partial Fulfilment of the Requirements for the MSc in Mathematical Finance, Oxford University, Oxford.

[39]   Leung, L.T. and Po-Shing, W.S. (2002) Valuation of American Options via Basis Functions. Department of Statistics, Stanford University, Technical Report No. 2002-28, 1-29.

[40]   Longstaff, F. and Schwartz, E. (2001) Valuing American Options by Simulation: A Simple Least-Squares Approach. This Paper Is Posted at the Scholarship Repository, University of California, Oakland.

[41]   Matache, A.M., Nitsche, P.A. and Schwab, C. (2003) Wavelet Galerkin Pricing of American Options on Lévy Driven Assets. Research Report No. 2003-06, Zürich, 1-26.