I'm self studying and I'm having trouble with understanding the equivalent formulations of the volatility $\sigma$ of an asset $X$, as in the below problem.
In the below the problem (and the first part of the solution that I posted), what is highlighted in red implies that the statement "the volatility of a $6$-month prepaid forward on $X$ is $0.3$" is equivalent to stating "the volatility of $X$ is $0.3$.
I'm trying to convince myself of why that is true.
The volatility of an asset $X$ means the standard deviation of the return, or $\sqrt{\text{Var}(\ln X_t / X_0)} = \sqrt{\text{Var}(\ln{X_t})}$.
The standard deviation of the return on a $T - t$-month prepaid forward on $X$, $F^p_{t, T}(X)$, would be $\sqrt{\text{Var}(\ln X_T / F^p_{t, T}(X))} = \sqrt{\text{Var}(\ln{X_t})}$, since $F^p_{t, T}(X)$ is a known constant.
Hence, the two formulations for volatility are equivalent. Is this reasoning correct?