The Optimal Alternative to the Delta Hedge in the Black and Scholes' (1973) World
24 Pages Posted: 23 May 2010 Last revised: 27 May 2010
Date Written: May 26, 2010
The delta hedge, constituting the benchmark method when covering a position in an option, does not define the optimal investment strategy, even in the original Black and Scholes (1973)’ (BS) setting. We propose the definition of the optimal alternative to the delta neutral cover in the original BS world, using the Merton (1971)’s optimal asset allocation model. The solution derived can be seen as the optimal hedging strategy for a position in an option, or, more generally, as the optimal portfolio rule in the case of an optional payoff. Optimal portfolios composed of an option, the underlying stock and (possibly) the riskless asset are considered. It is shown that the optimal solution each time incorporates the pure delta neutral hedge, this first term being augmented by a - possibly modified - speculative fund. The results prove that, by going from the case without riskless asset to the one characterized by its presence, the general principles driving the pure delta neutral strategy do not always hold anymore.
Keywords: Delta Hedge, Optimal Hedge, Optimal Portfolio, Optional Payoff, Stochastic Dynamic Programming
JEL Classification: C61, G11, G13
Suggested Citation: Suggested Citation