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Numerical methods for the pricing of swing options: a stochastic control approach. (English) Zbl 1142.91502
Summary: In the natural gas market, many derivative contracts have a large degree of flexibility. These are known as Swing or Take-Or-Pay options. They allow their owner to purchase gas daily, at a fixed price and according to a volume of their choice. Daily, monthly and/or annual constraints on the purchased volume are usually incorporated. Thus, the valuation of such contracts is related to a stochastic control problem, which we solve in this paper using new numerical methods. Firstly, we extend the Longstaff-Schwarz methodology (originally used for Bermuda options) to our case. Secondly, we propose two efficient parameterizations of the gas consumption, one is based on neural networks and the other on finite elements. It allows us to derive a local optimal consumption law using a stochastic gradient ascent. Numerical experiments illustrate the efficiency of these approaches. Furthermore, we show that the optimal purchase is of bang-bang type.

91G60 Numerical methods (including Monte Carlo methods)
93E03 Stochastic systems in control theory (general)
Full Text: DOI
[1] A. Benveniste, M. Metivier, and P. Priouret, Adaptive Algorithms and Stochastic Approximations, Springer: Berlin Heidelberg New York, 1990. · Zbl 0639.93002
[2] D. P. Bertsekas, Dynamic Programming and Optimal Control, vols. 1 and 2, Athena Scientific: Belmont, Massachusetts, 1995. · Zbl 0904.90170
[3] D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case. vol. 139 of Mathematics in Science and Engineering, Academic: New York, [Harcourt Brace Jovanovich Publishers], 1978. · Zbl 0471.93002
[4] L. Clewlow, C. Strickland, and V. Kaminski, ”Valuation of Swing contracts,” Technical report, Energy Power Risk Management, Risk Waters Group, July 2001.
[5] E. Gobet and R. Munos, ”Sensitivity analysis using Itô-Malliavin calculus and martingales. Application to stochastic control problem,” SIAM Journal on Control and Optimization vol. 43(5) pp. 1676–1713, 2005. · Zbl 1116.60033
[6] G. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. The Johns Hopkins University Press: Baltimore, MD. xxvii, p. 694, 1996. · Zbl 0865.65009
[7] S. Haykin, Neural Networks: A Conprehensive Foundation, McMillan: New York, 1994. · Zbl 0828.68103
[8] P. Jaillet, E. I. Ronn, and S. Tompaidis, ”Valuation of commodity-based Swing options,” Management Science vol. 50 pp. 909–921, 2004. · Zbl 1232.90340
[9] I. Karatzas, ”On the pricing of American options,” Applied Mathematics and Optimization vol. 17(1) pp. 37–60, 1988. · Zbl 0699.90010
[10] H. J. Kushner, and J. Yang, ”A Monte Carlo method for sensitivity analysis and parametric optimization of nonlinear stochastic systems,” SIAM Journal on Control and Optimization vol. 29(5) pp. 1216–1249, 1991. · Zbl 0745.93088
[11] H. J. Kushner, and G. Yin, Stochastic Approximation Algorithms and Applications. Springer: Berlin Heidelberg New York, 1997. · Zbl 0914.60006
[12] A. Lari-Lavassani, M. Simchi, and A. Ware, ”A discrete valuation of Swing options,” Canadian Applied Mathematics Quarterly vol. 9(1) pp. 35–74, 2001. · Zbl 1073.91035
[13] F. Longstaff, and E. S. Schwartz, ”Valuing american options by simulation: a simple least squares approach,” The Review of Financial Studies vol. 14 pp. 113–147, 2001. · Zbl 1386.91144
[14] S. Raikar, and M. Ilić, ”Interruptible physical transmission contracts for congestion management,” Power Engineering Society Summer Meeting, 2001, IEEE vol. 3 pp. 1639–1646, 2001.
[15] A. C. Thompson, ”Valuation of path-dependent contingent claims with multiple exercise decisions over time: the case of Take-or-Pay,” Journal of Financial and Quantitative Analysis vol. 30 pp. 271–293, 1995.
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