Exact Replication of the Best Rebalancing Rule in Hindsight

TL;DR

This paper derives the exact pricing and replication strategies for a derivative based on the best hindsight rebalancing rule in continuous and multi-asset markets, showing it outperforms the market asymptotically.

Contribution

It provides a closed-form solution for the derivative's price at any time and extends the analysis to multiple correlated stocks, connecting to universal portfolio theory.

Findings

Exact time-dependent pricing formulas derived.

Replication strategy involves betting the optimal fraction of wealth.

Strategy asymptotically outperforms the market.

Abstract

This paper 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 $C_0=1+\sigma\sqrt{T/(2\pi)}$. Of course, the general time-$t$ price is not equal to $1+\sigma\sqrt{(T-t)/(2\pi)}$. I complete the Ordentlich-Cover (1998) analysis by deriving the price at any time $t$. By contrast, I also study the more natural case of the best levered rebalancing rule in hindsight. This yields $C(S,t)=\sqrt{T/t}\cdot\,\exp\{rt+\sigma^2b(S,t)^2\cdot t/2\}$, where $b(S,t)$ is the best rebalancing rule in hindsight over the observed history $[0,t]$. I show 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: we get $C(S,t)=(T/t)^{n/2}\exp(rt+b'\Sigma b\cdot t/2)$, where $\Sigma$ is the covariance of instantaneous returns per unit time. This result matches the $\mathcal{O}(T^{n/2})$ "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, we find that the American-style version of Cover's Derivative is never exercised early in equilibrium.