A backward dual representation for the quantile hedging of Bermudan options. (English) Zbl 1339.91114
Summary: Within a Markovian complete financial market, we consider the problem of hedging a Bermudan option with a given probability. Using stochastic target and duality arguments, we derive a backward algorithm for the Fenchel transform of the pricing function. This algorithm is similar to the usual American backward induction, except that it requires two additional Fenchel transformations at each exercise date. We provide numerical illustrations.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 91G80 | Financial applications of other theories |
| 49L20 | Dynamic programming in optimal control and differential games |
| 60J60 | Diffusion processes |
References:
| [1] | V. Bally and G. Pagès, {\it Error analysis of the optimal quantization algorithm for obstacle problems}, Stochastic Process. Appl., 106 (2003), pp. 1-40. · Zbl 1075.60523 |
| [2] | V. Bally, G. Pagès, and J. Printems, {\it A quantization tree method for pricing and hedging multidimensional American options}, Math. Finance, 15 (2005), pp. 119-168. · Zbl 1127.91023 |
| [3] | G. Barles and P. E. Souganidis, {\it Convergence of approximation schemes for fully nonlinear second order equations}, Asymptotic Anal., 4 (1991), pp. 271-283. · Zbl 0729.65077 |
| [4] | D. P. Bertsekas and S. E. Shreve, {\it Stochastic Optimal Control: The Discrete Time Case}, Math. Sci. Engrg. 139, Academic Press, New York, London, 1978. · Zbl 0471.93002 |
| [5] | B. Bouchard, R. Elie, and A. Réveillac, {\it BSDEs with weak terminal condition}, Ann. Probab., 43 (2015), pp. 572-604. · Zbl 1321.60123 |
| [6] | B. Bouchard, R. Elie, and N. Touzi, {\it Stochastic target problems with controlled loss}, SIAM J. Control Optim., 48 (2009), pp. 3123-3150. · Zbl 1202.49028 |
| [7] | B. Bouchard and T. Nam Vu, {\it A stochastic target approach for P&L matching problems}, Math. Oper. Res., 37 (2012), pp. 526-558. · Zbl 1297.91150 |
| [8] | B. Bouchard and T. N. Vu, {\it The obstacle version of the geometric dynamic programming principle: Application to the pricing of American options under constraints}, Appl. Math. Optim., 61 (2010), pp. 235-265. · Zbl 1185.49029 |
| [9] | B. Bouchard and X. Warin, {\it Monte-Carlo valuation of american options: Facts and new algorithms to improve existing methods}, in Numerical Methods in Finance, Springer, Heidelberg, 2012, pp. 215-255. · Zbl 1247.91196 |
| [10] | M. G. Crandall, H. Ishii, and P.-L. Lions, {\it User’s guide to viscosity solutions of second order partial differential equations}, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67. · Zbl 0755.35015 |
| [11] | I. Ekeland and R. Temam, {\it Convex Analysis and Variational Problems}, Vol. 1, North-Holland, Amsterdam, Oxford; American Elsevier, New York, 1976. · Zbl 0322.90046 |
| [12] | H. Föllmer and P. Leukert, {\it Quantile hedging}, Finance Stoch., 3 (1999), pp. 251-273. · Zbl 0977.91019 |
| [13] | H. Föllmer and P. Leukert, {\it Efficient hedging: Cost versus shortfall risk}, Finance Stoch., 4 (2000), pp. 117-146. · Zbl 0956.60074 |
| [14] | Y. Jiao, O. Klopfenstein, and P. Tankov, {\it Hedging under Multiple Risk Constraints}, preprint, http://arxiv.org/abs/1309.5094 arXiv:1309.5094v1 [q-fin.RM], 2013. · Zbl 1360.91136 |
| [15] | R. T. Rockafellar, {\it Convex Analysis}, Princeton University Press, Princeton, NJ, 1997. · Zbl 0932.90001 |
| [16] | M. Schweizer, {\it On Bermudan options}, in Advances in Finance and Stochastics, Springer, Berlin, 2002, pp. 257-270. · Zbl 1011.91050 |
| [17] | H. M. Soner and N. Touzi, {\it Stochastic target problems, dynamic programming, and viscosity solutions}, SIAM J. Control Optim., 41 (2002), pp. 404-424. · Zbl 1011.49019 |
| [18] | H. M. Soner and N. Touzi, {\it A stochastic representation for mean curvature type geometric flows}, Ann. Probab., 31 (2003), pp. 1145-1165. · Zbl 1080.60076 |
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