sets of asset allocations. We show how the rebalancing option can be interpreted as a certain portfolio of Margrabe-Fischer (1978) exchange options, and derive the general replicating strategy, which is a time- and state-varying convex combination of the $b_i$. We close the paper by proving that American-style rebalancing options (with general exercise price $K$) are always “worth more alive than dead” in equilibrium.

Definitions and Notation {#notation}
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We start in the Black-Scholes (1973) market with a single underlying stock whose price $S_t$ follows the geometric Brownian motion $$\frac{dS_t}{S_t}=\mu\,dt+\sigma dW_t,$$where $\mu$ is the drift, $\sigma$ is the volatility, and $W_t$ is a standard Brownian motion. There is a risk-free bond whose price $B_t:=e^{rt}$ follows $$\frac{dB_t}{B_t}=r\,dt.$$A constant rebalancing rule $b\in(-\infty,+\infty)$ is a fixed-fraction betting scheme that continuously maintains the fraction $b$ of wealth in the stock and the fraction $1-b$ of wealth in bonds. We let $V_t(b)$ denote the wealth at $t$ that accrues to a $\$1$ deposit into the rebalancing rule $b$. Thus, the trader holds $\Delta:=bV_t(b)/S_t$ shares of the stock at time $t$, and his remaining $(1-b)V_t$ dollars are invested in bonds. Maintenance of the target asset allocation generally requires continuous trading. If $0<b<1$, the trader must sell a 