# zbMATH — the first resource for mathematics

Dual pricing of multi-exercise options under volume constraints. (English) Zbl 1303.91167
Summary: In this paper, we study the pricing problem of multi-exercise options under volume constraints. The volume constraint is modelled by an adapted process with values in the positive integers, which describes the maximal number of rights to be exercised at a given time. We derive a representation of the marginal value of an additional $$n$$th right as a standard single stopping problem with a modified cash-flow process. This representation then leads to a dual pricing formula, which generalizes a result by N. Meinshausen and B. M. Hambly [Math. Finance 14, No. 4, 557–583 (2004; Zbl 1169.91372)] from the standard multi-exercise option (with at most one right per time step) to general constraints. We also state an explicit Monte Carlo algorithm for computing confidence intervals for the price of multi-exercise options under volume constraints and present numerical results for the pricing of a swing contract in an electricity market.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60G40 Stopping times; optimal stopping problems; gambling theory 91G60 Numerical methods (including Monte Carlo methods) 65C05 Monte Carlo methods
Full Text:
##### References:
 [1] Aleksandrov, N., Hambly, B.: A dual approach to multiple exercise option problems under constraints. Math. Meth. in Oper. Res. 71, 503–533 (2010) · Zbl 1200.60034 [2] Andersen, L., Broadie, M.: A primal-dual simulation algorithm for pricing multidimensional American options. Manag. Sci. 50, 1222–1234 (2004) [3] Bardou, O., Bouthemy, S., Pagès, G.: When are swing options bang-bang and how to use it? Preprint (2007). http://arxiv.org/pdf/0705.0466 · Zbl 1233.91255 [4] Belomestny, D., Bender, C., Schoenmakers, J.: True upper bounds for Bermudan products via non-nested Monte Carlo. Math. Finance 19, 53–71 (2009) · Zbl 1155.91376 [5] Bender, C., Schoenmakers, J.: An iterative method for multiple stopping: convergence and stability. Adv. Appl. Probab. 38, 729–749 (2006) · Zbl 1114.60033 [6] Benth, F.E., Benth, J.S., Koekebakker, S.: Stochastic Modelling of Electricity and Related Markets. World Scientific, Singapore (2008) · Zbl 1143.91002 [7] Benth, F.E., Kallsen, J., Meyer-Brandis, T.: A non-Gaussian Ornstein–Uhlenbeck process for electricity spot price modeling and derivatives pricing. Appl. Math. Finance 14, 153–169 (2007) · Zbl 1160.91337 [8] Bouchard, B., Ekeland, I., Touzi, N.: On the Malliavin approach to Monte Carlo approximation of conditional expectations. Finance Stoch. 8, 45–71 (2004) · Zbl 1051.60061 [9] Carmona, R., Dayanik, S.: Optimal multiple stopping of linear diffusions. Math. Oper. Res. 33, 446–460 (2008) · Zbl 1221.60061 [10] Carmona, R., Touzi, N.: Optimal multiple stopping and the valuation of swing options. Math. Finance 18, 239–268 (2008) · Zbl 1133.91499 [11] Carrière, J.F.: Valuation of the early-exercise price for options using simulations and nonparametric regression. Insur. Math. Econ. 19, 19–30 (1996) · Zbl 0894.62109 [12] Davis, M.H.A., Karatzas, I.: A deterministic approach to optimal stopping, with applications. In: Kelly, F.P. (ed.) Probability, Statistics and Optimization: A Tribute to Peter Whittle, pp. 455–466. Wiley, Chichester (1994) · Zbl 0855.60041 [13] Haggstrom, G.: Optimal sequential procedures when more than one stop is required. Ann. Math. Stat. 38, 1618–1626 (1967) · Zbl 0189.18301 [14] Hambly, B., Howison, S., Kluge, T.: Modeling spikes and pricing swing options in electricity markets. Quant. Finance 9, 937–949 (2009) · Zbl 1182.91176 [15] Haugh, M., Kogan, L.: Pricing American options: a duality approach. Oper. Res. 52, 258–270 (2004) · Zbl 1165.91401 [16] Ibáñez, A.: Valuation by simulation of contingent claims with multiple early exercise opportunities. Math. Finance 14, 223–248 (2004) · Zbl 1090.91051 [17] Jaillet, P., Ronn, E.I., Tompaidis, S.: Valuation of commodity based swing options. Manag. Sci. 50, 909–921 (2004) · Zbl 1232.90340 [18] Lions, P.L., Regnier, H.: Calcul du prix et des sensibilités d’une option américaine par une méthode de Monte Carlo. Techn. Report, Univ. Dauphine Paris (2001) [19] Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14, 113–147 (2001) · Zbl 1386.91144 [20] Lucia, J., Schwartz, E.: Electricity prices and power derivatives: evidence from the Nordic power exchange. Rev. Deriv. Res. 5, 5–50 (2002) · Zbl 1064.91508 [21] Meinshausen, N., Hambly, B.M.: Monte Carlo methods for the valuation of multiple-exercise options. Math. Finance 14, 557–583 (2004) · Zbl 1169.91372 [22] Rogers, L.C.G.: Monte Carlo valuation of American options. Math. Finance 12, 271–286 (2002) · Zbl 1029.91036 [23] Tsitsiklis, J.N., Van Roy, B.: Regression methods for pricing complex American-style options. IEEE Trans. Neural Netw. 12, 694–703 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.