Exact Replication of the Best Rebalancing Rule in Hindsight

Abstract

This article prices and replicates the financial derivative whose payoff at T is the wealth that would have accrued to a $1 deposit into the best continuously-rebalanced portfolio (or fixed-fraction betting scheme) determined in hindsight. For the single-stock Black–Scholes market, Ordentlich and Cover (1998) only priced this derivative at time-0, giving . Of course, the general time-t price is not equal to . The author completes the Ordentlich–Cover (1998) analysis by deriving the price at any time t. By contrast, the author also studies the more natural case of the best-levered rebalancing rule in hindsight. This yields , where b(S, t) is the best rebalancing rule in hindsight over the observed history [0, t].

The author shows that the replicating strategy amounts to betting the fraction b(S, t) of wealth on the stock over the interval [t, t + dt]. This fact holds for the general market with n correlated stocks in geometric Brownian motion: C(S, t) = (T/t)^n/2 exp(rt + b′Σb·t/2), where Σ is the covariance of instantaneous returns per unit time. This result matches the O(T^n/2) “cost of universality” derived by Cover in his “universal portfolio theory” (1986, 1991, 1996, 1998), which super-replicates the same derivative in discrete-time. The replicating strategy compounds its money at the same asymptotic rate as the best-levered rebalancing rule in hindsight, thereby beating the market asymptotically. Naturally enough, the American-style version of Cover’s Derivative is never exercised early in equilibrium.

TOPICS: Derivatives, portfolio construction, performance measurement, statistical methods