# Black's formula for a call option on a non-tradable underlying

I am looking for an explanation of the following fact, which seems to be rather simple yet I am missing something. Say that $S_t$ is a stock following GBM $$dS_t = r S_td_t + \sigma S_t dW_t,$$ and I want to price a derivative with payoff $max(S_T^2-K,0)$. I know how to do this using the risk-neutral expectation, but it's rather long and apparently unnecessary.

I read in a book (Mark Joshi's Concepts...) another solution which uses the forward price. If $F_T(t)$ is the forward price of $S$ at time $t$, then one can write

$$F_T(T)^2 = (F_T(0)^2 e^{\sigma^2 T})e^{-\frac{\sigma'^2}{2} T + \sigma'\sqrt{T}N(0,1)},$$ with $\sigma'=2\sigma$, so then one can use Black's formula with forward equal to $F_T(0)^2 e^{\sigma^2T}$ and volatility $\sigma'$.

But how is Black's formula justified here exactly? I assume that the formula can be applied for a call option on the forward price of an asset, but how do we know there exists a tradable asset with volatility $\sigma'$, for instance? Or is this even necessary?

• Hi @dbluedesk. A similar question was asked here: quant.stackexchange.com/questions/26240/…. Gordon proposes a direct approach and I suggest another one directly inspired by Mark Joshi's work – Quantuple Feb 21 '17 at 14:45
• Hello @Quantuple, thank you! I think my question is: when you say in the linked answer that $N_{t,T}$ is an asset, how do you get this? I always thought you could use any self-financing portfolio as a numeraire, but I don't see how you can reproduce your $N_{t,T}$ – dbluesk Feb 21 '17 at 15:03
• A numéraire is essentially a (tradable) asset whose price is always positive. $N(t,T)$ is merely the $t$-value of a contingent claim with payoff $S_T^2$. – Quantuple Feb 21 '17 at 15:41