Option valuation and hedging using an asymmetric risk function: asymptotic optimality through fully nonlinear partial differential equations
by Emmanuel Gobet1, Isaque Pimentel2 and Xavier Warin3
1Centre de Mathématiques Appliquées (CMAP), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau Cedex, France
(email: emmanuel.gobet@polytechnique.edu)
2Centre de Mathématiques Appliquées (CMAP), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau Cedex, France and
Optimization, Simulation, Risques et Statistiques (OSIRIS), Electricité de France (EDF), 7 boulevard Gaspard Monge, 91120 Palaiseau, France
(email: pimentel.isaque@gmail.com)
3Optimization, Simulation, Risques et Statistiques (OSIRIS), Electricité de France (EDF), 7 boulevard Gaspard Monge, 91120 Palaiseau, France
(email: xavier.warin@edf.fr)
Abstract
Discrete-time hedging produces a residual risk, namely the tracking error. The major problem is to get valuation/hedging policies minimising this error. We evaluate the risk between trading dates through a function penalising profits and losses asymmetrically. After deriving the asymptotics from a discrete-time risk measurement for a large number of trading dates, we derive the optimal strategies minimising the asymptotic risk in the continuous-time setting. We characterise optimality through a class of fully nonlinear partial differential equations (PDEs). Numerical experiments show that the optimal strategies associated with the discrete and the asymptotic approaches coincide asymptotically.
Key words:
Hedging, Asymmetric risk, Fully nonlinear parabolic PDE, Regression Monte Carlo
JEL Classification: G13, C60
Mathematics Subject Classification (2010): 60H30, 35K55, 91G60, 91G80