Author(s)

Lan Ma Nygren, Peter Lakner

Department of Management Sciences, Rider University, Lawrenceville, USA.
Department of Statistics and Operations Research, New York University, New York, USA.

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.

Partial Hedging; Malliavin Calculus; Convex Duality; Convex Risk Measure

L. Nygren and P. Lakner, "Partial Hedging Using Malliavin Calculus," Journal of Mathematical Finance, Vol. 2 No. 3, 2012, pp. 203-213. doi: 10.4236/jmf.2012.23023.