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BSDEs, càdlàg martingale problems and orthogonalisation under basis risk

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  • Ismail Laachir

    (UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris, Lab-STICC_UBS_CACS_IAS - Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance - UEB - Université européenne de Bretagne - European University of Brittany - ENIB - École Nationale d'Ingénieurs de Brest - UBS - Université de Bretagne Sud - UBO - Université de Brest - Télécom Bretagne - IBNM - Institut Brestois du Numérique et des Mathématiques - UBO - Université de Brest - ENSTA Bretagne - École Nationale Supérieure de Techniques Avancées Bretagne - IMT - Institut Mines-Télécom [Paris] - CNRS - Centre National de la Recherche Scientifique)

  • Francesco Russo

    (UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris, OC - Optimisation et commande - UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - IP Paris - Institut Polytechnique de Paris)

Abstract

The aim of this paper is to introduce a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type. A significant application concerns the hedging problem under basis risk of a contingent claim $g(X_T,S_T)$, where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes $(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes.

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  • Ismail Laachir & Francesco Russo, 2016. "BSDEs, càdlàg martingale problems and orthogonalisation under basis risk," Post-Print hal-01086227, HAL.
    • Handle: RePEc:hal:journl:hal-01086227
    DOI: 10.1137/140996239
    Note: View the original document on HAL open archive server: https://inria.hal.science/hal-01086227v2
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    References listed on IDEAS

    as
    1. Hardy Hulley & Thomas A. McWalter, 2015. "Quadratic Hedging of Basis Risk," JRFM, MDPI, vol. 8(1), pages 1-20, February.
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    4. repec:dau:papers:123456789/11525 is not listed on IDEAS
    5. Stefan Ankirchner & Gregor Heyne, 2012. "Cross hedging with stochastic correlation," Finance and Stochastics, Springer, vol. 16(1), pages 17-43, January.
    6. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2010. "Analysis of Fourier Transform Valuation Formulas and Applications," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(3), pages 211-240.
    7. Michael Monoyios, 2004. "Performance of utility-based strategies for hedging basis risk," Quantitative Finance, Taylor & Francis Journals, vol. 4(3), pages 245-255.
    Full references (including those not matched with items on IDEAS)

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