= Portfolio with constant stock value in bond units
{title2=$V_t/B_t=w_0/B_0+\theta\log(S_t/S_0)+\theta(\sigma^2/2-\rho)t$}
In the <Black-Scholes model>, take $B_t=e^{-\rho(T-t)}$ as a <zero-coupon bond> <numéraire>. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The <self-financing conditions for smooth stock and bond holdings> give
$$
h_t=\frac{w_0}{B_0}-\theta+\theta\log(S_t/S_0)+\theta(\sigma^2/2-\rho)t
$$
units of the bond. Since $V_t=g_tS_t+h_tB_t$, this yields the displayed wealth. Under the <risk-neutral measure>, $V_t/B_t=w_0/B_0+\theta\sigma W_t^Q$: the discounted gain is a constant multiple of <Brownian motion>.
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