Partial Hedging Using Malliavin Calculus

Abstract

Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option

Duality methods for stochastic optimal control problems in finance

“…The literature on partial hedging is strongly oriented towards the theory, see e.g. Pham (2000Pham ( , 2002, Jonsson and Sircar (2002), and Nygren and Lakner (2012). We analytically solve the LPM-linked terminal wealth problem by means of the martingale technique.…”

“…In related technical studies by Jonsson and Sircar (2002), Bouchard et al (2004), and Nygren and Lakner (2012), the utility-dimension of a.o. the LPM operator is highlighted.…”