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Exact Replication of the Best Rebalancing Rule in Hindsight

Alex Garivaltis
The Journal of Derivatives Summer 2019, 26 (4) 35-53; DOI: https://doi.org/10.3905/jod.2019.26.4.035
Alex Garivaltis
is an assistant professor of economics at Northern Illinois University in DeKalb, IL
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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 Embedded Image. Of course, the general time-t price is not equal to Embedded Image. 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 Embedded Image, 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(Tn/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

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The Journal of Derivatives: 26 (4)
The Journal of Derivatives
Vol. 26, Issue 4
Summer 2019
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Exact Replication of the Best Rebalancing Rule in Hindsight
Alex Garivaltis
The Journal of Derivatives May 2019, 26 (4) 35-53; DOI: 10.3905/jod.2019.26.4.035

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Exact Replication of the Best Rebalancing Rule in Hindsight
Alex Garivaltis
The Journal of Derivatives May 2019, 26 (4) 35-53; DOI: 10.3905/jod.2019.26.4.035
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