Cover’s Rebalancing Option with Discrete Hindsight Optimization

Abstract

The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover &amp; Company in their theory of <i>universal portfolios</i> (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an <i>a priori</i> view that one of the favored asset allocations (“bets”) in the set {<i>b</i>₁, …, <i>b_n</i>} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best <i>b_i</i> in hindsight, hence the point of the rock-bottom option price.