Abstract
We investigate and compare two dual formulations of the American option pricing problem based on two decompositions of supermartingales: the additive dual of Haugh and Kogan (Oper. Res. 52:258–270, 2004) and Rogers (Math. Finance 12:271–286, 2002) and the multiplicative dual of Jamshidian (Minimax optimality of Bermudan and American claims and their Monte- Carlo upper bound approximation. NIB Capital, The Hague, 2003). Both provide upper bounds on American option prices; we show how to improve these bounds iteratively and use this to show that any multiplicative dual can be improved by an additive dual and vice versa. This iterative improvement converges to the optimal value function. We also compare bias and variance under the two dual formulations as the time horizon grows; either method may have smaller bias, but the variance of the multiplicative method typically grows much faster than that of the additive method. We show that in the case of a discrete state space, the additive dual coincides with the dual of the optimal stopping problem in the sense of linear programming duality and the multiplicative method arises through a nonlinear duality.
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References
Andersen L. (2000). A simple approach to the pricing of Bermudan swaptions in the multi-factor Libor market model. J. Comput. Financ. 3: 5–32
Andersen L. and Broadie M. (2004). A primal–dual simulation algorithm for pricing multi-dimensional American options. Manage. Sci. 50: 1222–1234
Bolia, N., Glasserman, P., Juneja, S.: Function-approximation-based importance sampling for pricing American options. In: Proceedings of the 2004 Winter Simulation Conference, pp. 604–611 (2004)
Broadie M. and Glasserman P. (1997). Pricing American-style securities by simulation. J. Econ. Dyn. Control 21: 1323–1352
Broadie M. and Glasserman P. (2004). A stochastic mesh method for pricing high-dimensional American options. J. Comput. Finance 7: 35–72
Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury Press, Inc., North Scituate (1995)
Glasserman P. (2004). Monte Carlo Methods in Financial Engineering. Springer, Berlin
Haugh M. and Kogan L. (2004). Pricing American options: a dual approach. Oper. Res. 52: 258–270
Jamshidian, F.: Minimax optimality of Bermudan and American claims and their Monte-Carlo upper bound approximation. NIB Capital, The Hague. http://www.finance-research.net/enaa203t3_jamshidian.html (2003)
Karatzas I. and Shreve S.E. (1991). Brownian Motion and Stochastic Calculus 2nd edn. Springer, New York
Karlin S. and Taylor H.M. (1975). A First Course in Stochastic Processes 2nd edn. Academic Press, San Diego
Kolodko A. and Schoenmakers J. (2006). Iterative construction of the optimal Bermudan stopping time. Finance Stoch. 10: 27–49
Longstaff F.A. and Schwartz E.S. (2001). Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14: 113–147
Nocedal J. and Wright S.J. (1999). Numerical Optimization. Springer, Berlin
Rogers L.C.G. (2002). Monte Carlo valuation of American options. Math. Finance 12: 271–286
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Chen, N., Glasserman, P. Additive and multiplicative duals for American option pricing. Finance Stoch 11, 153–179 (2007). https://doi.org/10.1007/s00780-006-0031-3
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DOI: https://doi.org/10.1007/s00780-006-0031-3