Abstract
It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not preserve the convexity or monotonicity properties. In this paper, we propose a method of approximation for American options which can preserve both convexity and monotonicity. The resulting values can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds. Other application of the algorithm include finding optimal hedging strategies.
MSC code: 91G60 91G20.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andersen, L. and Broadie, M. (2004). Primal-dual simulation algorithm for pricing multidimensional American options. Management Science, 50:1222–1234.
Bally, V., Pagès, G., and Printems, J. (2005). A quantization tree method for pricing and hedging multidimensional American options. Math. Finance, 15(1):119–168.
Ben-Ameur, H., Breton, M., and Martinez, J.-M. (2009). A dynamic programming approach for pricing derivatives in the GARCH model. Management Science, 55(2):252–266.
Boyle, P. (1977). Options: A Monte Carlo approach. Journal of Financial Economics, 4:323–338.
Brennan, M. and Schwartz, E. (1977). The valuation of American put options. Journal of Finance, XXXII:449–462.
Broadie, M. and Detemple, J. (1997). The valuation of American options on multiple assets. Mathematical Finance, 7:241–286.
Broadie, M., Detemple, J., Ghysels, E., and Torrès, O. (2000). Nonparametric estimation of American options’ exercise boundaries and call prices. Journal of Economic Dynamics and Control, 24:1829–1857.
Broadie, M. and Glasserman, P. (1997). Pricing American-style securities using simulation. Journal of Economic Dynamics and Control, 21:1323–1352.
Broadie, M. and Glasserman, P. (2004). A stochastic mesh method for pricing high-dimensional American options. Journal of Computational Finance, 7:35–72.
Carriere, J. (1996). Valuation of the early-exercise price for options using simulations and nonparametric regression. Insurance: Mathematics and Economics, 19:19–30.
Clément, E., Lamberton, D., and Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance Stoch., 6:449 – 471.
Cox, J., Ross, S., and Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7:229–263.
Duan, J.-C., Gauthier, G., Sasseville, C., and Simonato, J.-G. (2003). Approximating American option prices in the GARCH framework. The Journal of Futures Markets, 23:915–929.
Duan, J.-C., Gauthier, G., and Simonato, J.-G. (2004). Numerical pricing of contingent claims on multiple assets and/or factor - A low-discrepancy Markov chain approach derivatives in the GARCH model. Technical Report 80, GERAD.
Duan, J.-C. and Simonato, J.-G. (2001). American option pricing under GARCH by a Markov chain approximation. J. Econom. Dynam. Control, 25(11):1689–1718.
Fu, M., Laprise, S., Madan, D., Su, Y., and Wu, R. (2001). Pricing American options: A comparison of Monte Carlo simulation approaches. Journal of Computational Finance, 4(3):39–88.
Haugh, M. B. and Kogan, L. (2004). Pricing american options: A duality approach. Operations Research, 52(2):pp. 258–270.
Hocquard, A., Papageorgiou, N., and Rémillard, B. (2007). Optimal hedging strategies with an application to hedge fund replication. Wilmott Magazine, (Jan-Feb):62–66.
Kargin, V. (2005). Lattice option pricing by multidimensional interpolation. Math. Finance, 15(4):635–647.
Laprise, S., Fu, M., Marcus, S., Lim, A., and Zhang, H. (2006). Pricing American-style derivatives with European call options. Management Science, (1):95–110.
Longstaff, F. and Schwartz, E. (2001). Valuing American options by simulation: A simple least-square approach. The Review of Financial Studies, 14:113–147.
Neveu, J. (1975). Discrete-parameter Martingales. North Holland, Amsterdam.
Papageorgiou, N., Rémillard, B., and Hocquard, A. (2008). Replicating the properties of hedge fund returns. Journal of Alternative Invesments, 11:8–38.
Rémillard, B. and Rubenthaler, S. (2009). Optimal hedging in discrete and continuous time. Technical Report G-2009-77, Gerad.
Rogers, C. (2002). Monte Carlo valuation of American options. Mathematical Finance, 12:271–286.
Schweizer, M. (1995). Variance-optimal hedging in discrete time. Math. Oper. Res., 20(1):1–32.
Tilley, J. (1993). Valuing American options in a path simulation model. Transactions of the Society of Actuaries, 45:83–104.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Del Moral, P., Rémillard, B., Rubenthaler, S. (2012). Monte Carlo Approximations of American Options that Preserve Monotonicity and Convexity. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds) Numerical Methods in Finance. Springer Proceedings in Mathematics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25746-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-25746-9_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25745-2
Online ISBN: 978-3-642-25746-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)