Let's say we write a standard call option on $S_t$ which pays $Max[0, S_t-K] \,\forall \, t \in T $. Given that $\frac{dS}{S} = \mu \,dt + \sigma \,dW_t$, and, $V_T = (S_T -K)_+$, we can solve this under the Black-Scholes framework as:
$$V_t[S_t,K,\sigma_S,r,t] = S_t \varPhi[d_1] - K e^{-r (T-t)} \varPhi[d_1 - \sigma \sqrt{T-t}]$$
where: $\varPhi[x]$ is a cumulative distrbution function; and,
$$d_1 = \frac{\ln\left(\frac{S_t}{K}\right)+{(r+\sigma^2/2)(T-t)} }{\sigma \sqrt{T-t}}$$
What is the expected variance of this option's returns, $\sigma_V$? I.e., how does the process $E \left[ V_t \right]$ evolve wrt time?
Intuitively, the logarithmic variance should be defined if we constrain that the option must take non-zero, positive values.
I ask because I am trying to assess what might be called a compound option in which the parameters are adapted for $V_t$.