Swing options valuation: a BSDE with constrained jumps approach. (English) Zbl 1247.91179

Bernhart, Marie; Pham, Huyên; Tankov, Peter; Warin, Xavier

Carmona, René A. (ed.) et al., Numerical methods in finance. Selected papers based on the presentations at the workshop, Bordeaux, France, June 2010. Berlin: Springer (ISBN 978-3-642-25745-2/hbk; 978-3-642-25746-9/ebook). Springer Proceedings in Mathematics 12, 379-400 (2012).

Summary: We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity \(\lambda \), the penalization parameter \(p > 0\) and the time step. In particular, we obtain a convergence rate of the error due to penalization of order \((\lambda p^{\alpha-\frac{1}{2}}, \forall \alpha \in (0, \frac {1}{2})\). Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.
For the entire collection see [Zbl 1238.91005].

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
91G99	Actuarial science and mathematical finance
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

backward stochastic differential equations with constrained jumps; impulse control problems; Monte Carlo methods; swing options

References:

[1]	M. Bernhart: Modelization and valuation methods of gas contracts, see Part II pp. 67-168. PhD Thesis, Université Paris VII Denis-Diderot, Preprint (2011)
[2]	Bouchard, B.; Chassagneux, J-F., Discrete-time approximation for continuously and discretely reflected BSDEs, Stochastic Processes and their Applications, 118, 12, 2269-2293 (2008) · Zbl 1158.60030
[3]	Bouchard, B.; Elie, R., Discrete-time approximation of decoupled forward-backward SDE with jumps, Stochastic Processes and their Applications, 118, 1, 53-75 (2008) · Zbl 1136.60048
[4]	A. Bensoussan and J.-L. Lions: Impulse control and quasi-variational inequalities. Gauthier-Villars (1984)
[5]	B. Bouchard and N. Touzi: Discrete-Time Approximation and Monte-Carlo Simulation of Backward Stochastic Differential Equations. Stochastic Processes and their applications, 111(2), 175-206 (2004) · Zbl 1071.60059
[6]	B. Bouchard and X. Warin: Monte-Carlo valorisation of American options: facts and new algorithms to improve existing methods. Preprint (2010) · Zbl 1247.91196
[7]	Bally, V.; Pagès, G., Error analysis of the optimal quantization algorithm for obstacle problems, Stochastic Processes and Their Applications, 106, 1, 1-40 (2003) · Zbl 1075.60523
[8]	R. Carmona and M. Ludkovski: Pricing Asset Scheduling Flexibility Using Optimal Switching. Applied Mathematical Finance, 15(6), 405-447 (2008) · Zbl 1156.91361
[9]	Carmona, R.; Touzi, N., Optimal multiple stopping and valuation of Swing options, Mathematical Finance, 18, 2, 239-268 (2008) · Zbl 1133.91499
[10]	J-F. Chassagneux, R. Elie and I. Kharroubi: Discrete-time Approximation of Multidimensional BSDEs with oblique reflections. Preprint (2010) · Zbl 1243.93128
[11]	R. Cont and P. Tankov: Financial modelling with jump processes. Chapman & Hall/CRC Pres (2003) · Zbl 1052.91043
[12]	J.-P. Chancelier, B. \( \mathcal{O}\) ksendal and A. Sulem: Combined stochastic control and optimal stopping, and application to numerical approximation of combined stochastic and impulse control. Tr. Mat. Inst. Steklova, 237, 149-172 (2000)
[13]	B. Djehiche, S. Hamadène and I. Hdhiri: Stochastic Impulse Control of Non-Markovian Process. Applied Mathematics and Optimisation, 61(1), 1-26 (2010) · Zbl 1195.93144
[14]	Djehiche, B.; Hamadène, S.; Popier, A., The Finite Horizon Optimal Multiple Switching Problem, SIAM Journal on Control and Optimisation, 48, 4, 2751-2770 (2010) · Zbl 1196.60069
[15]	R. Elie: Contrôle stochastique et méthodes numériques en finance mathématique. PhD Thesis, Université Paris Dauphine (2008) · Zbl 1185.93002
[16]	R. Elie and I. Kharroubi: Constrained BSDEs with jumps: Application to optimal switching. Preprint (2010)
[17]	Gobet, E.; Lemor, J. P.; Warin, X., A regression-based Monte Carlo method to solve backward stochastic differential equations, Annals of Applied Probability, 15, 3, 2172-2202 (2005) · Zbl 1083.60047
[18]	E. Gobet, J.P. Lemor and X. Warin: Rate of convergence of empirical regression method for solving generalized BSDE. Bernoulli, 12(5), 889-916 (2006) · Zbl 1136.60351
[19]	Hamadène, S.; Jeanblanc, M., On the Starting and Stopping Problem: Application in reversible investments, Mathematics of Operations Research, 32, 1, 182-192 (2007) · Zbl 1276.91100
[20]	Y. Hu and S. Tang: Multi-dimensional BSDE with oblique reflection and optimal switching. Probability Theory and Related Fields, 147(1-2), 89-121 (2008) · Zbl 1188.60029
[21]	Kharroubi, I.; Ma, J.; Pham, H.; Zhang, J., Backward SDEs with constrained jumps and Quasi-Variational Inequalities, Annals of Probability, 38, 2, 794-840 (2010) · Zbl 1205.60114
[22]	V. Ly Vath, M. Mnif et H. Pham: A model of portfolio selection under liquidity risk and price impact. Finance and Stochastics, 11(1), 51-90 (2007) · Zbl 1145.91025
[23]	J-P. Lemor: Approximation par projections et simulations de Monte-Carlo des équations différentielles stochastiques rétrogrades. PhD Thesis, Ecole Polytechnique (2005) · Zbl 1185.65007
[24]	M. Ludkovski: Optimal Switching with Applications to Energy Tolling Agreements. PhD Thesis, Princeton University (2005)
[25]	B. Øksendal and A. Sulem: Applied stochastic control of jump diffusions. Universitext, Springer Verlag (2006) · Zbl 1074.93009
[26]	A. Porchet: Problems of Valuation and Organization in Energy Markets. PhD Thesis, Université Paris Dauphine (2008)
[27]	R. C. Seydel: Impulse Control for Jump-Diffusions: Viscosity Solutions of Quasi-Variational Inequalities and Applications in Bank Risk Management. PhD Thesis, Leipzig Universität (2009)