Abstract
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 λ, the penalization parameter p > 0 and the time step. In particular, we obtain a convergence rate of the error due to penalization of order \({(\lambda p)}^{\alpha -\frac{1} {2} },\forall \alpha \in \left (0, \frac{1} {2}\right )\). 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.
MSC2010: 91G20, 60H10
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
M. Bernhart: Modelization and valuation methods of gas contracts, see Part II pp. 67–168. PhD Thesis, Université Paris VII Denis-Diderot, Preprint (2011)
B. Bouchard and J-F. Chassagneux: Discrete-time approximation for continuously and discretely reflected BSDEs. Stochastic Processes and their Applications, 118(12), 2269–2293 (2008)
B. Bouchard and R. Elie: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Processes and their Applications, 118(1), 53–75 (2008)
A. Bensoussan and J.-L. Lions: Impulse control and quasi-variational inequalities. Gauthier-Villars (1984)
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)
B. Bouchard and X. Warin: Monte-Carlo valorisation of American options: facts and new algorithms to improve existing methods. Preprint (2010)
V. Bally and G. Pagès: Error analysis of the optimal quantization algorithm for obstacle problems. Stochastic Processes and Their Applications, 106(1), 1–40 (2003)
R. Carmona and M. Ludkovski: Pricing Asset Scheduling Flexibility Using Optimal Switching. Applied Mathematical Finance, 15(6), 405–447 (2008)
R. Carmona and N. Touzi: Optimal multiple stopping and valuation of Swing options. Mathematical Finance, 18(2), 239–268 (2008)
J-F. Chassagneux, R. Elie and I. Kharroubi: Discrete-time Approximation of Multidimensional BSDEs with oblique reflections. Preprint (2010)
R. Cont and P. Tankov: Financial modelling with jump processes. Chapman & Hall/CRC Pres (2003)
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)
B. Djehiche, S. Hamadène and I. Hdhiri: Stochastic Impulse Control of Non-Markovian Process. Applied Mathematics and Optimisation, 61(1), 1–26 (2010)
B. Djehiche, S. Hamadène and A. Popier: The Finite Horizon Optimal Multiple Switching Problem. SIAM Journal on Control and Optimisation, 48(4), 2751–2770 (2010)
R. Elie: Contrôle stochastique et méthodes numériques en finance mathématique. PhD Thesis, Université Paris Dauphine (2008)
R. Elie and I. Kharroubi: Constrained BSDEs with jumps: Application to optimal switching. Preprint (2010)
E. Gobet, J.P. Lemor, X. Warin: A regression-based Monte Carlo method to solve backward stochastic differential equations. Annals of Applied Probability, 15(3), 2172–2202 (2005)
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)
S. Hamadène and M. Jeanblanc: On the Starting and Stopping Problem: Application in reversible investments. Mathematics of Operations Research, 32(1), 182–192 (2007)
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)
I. Kharroubi, J. Ma, H. Pham and J. Zhang: Backward SDEs with constrained jumps and Quasi-Variational Inequalities. Annals of Probability, 38(2), 794–840 (2010)
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)
J-P. Lemor: Approximation par projections et simulations de Monte-Carlo des équations différentielles stochastiques rétrogrades. PhD Thesis, Ecole Polytechnique (2005)
M. Ludkovski: Optimal Switching with Applications to Energy Tolling Agreements. PhD Thesis, Princeton University (2005)
B. Øksendal and A. Sulem: Applied stochastic control of jump diffusions. Universitext, Springer Verlag (2006)
A. Porchet: Problems of Valuation and Organization in Energy Markets. PhD Thesis, Université Paris Dauphine (2008)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bernhart, M., Pham, H., Tankov, P., Warin, X. (2012). Swing Options Valuation: A BSDE with Constrained Jumps Approach. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds) Numerical Methods in Finance. Springer Proceedings in Mathematics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25746-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-25746-9_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25745-2
Online ISBN: 978-3-642-25746-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)