Pricing a Squared-Stock Payoff with a Black-Style Formula

Summary

The document poses a derivatives-pricing question about a payoff based on the square of a stock price when the stock follows geometric Brownian motion. It describes a proposed shortcut: express the squared terminal stock value using a forward-like quantity, with an adjusted initial level and volatility, then apply a Black-style call formula. The central conceptual issue is whether that use of the formula requires an actually tradable asset with the adjusted volatility.

No answer or derivation is included, so the document does not establish when the transformation is valid or provide a price. It is useful as a statement of a modeling question, but not as a self-contained pricing method. In particular, it leaves the risk-neutral measure, discounting, payoff mapping, and assumptions needed to justify the forward representation unresolved; readers should not treat the proposed shortcut as proven by this text.

Key ideas

• The question considers a call-like payoff on the square of a stock price under geometric Brownian motion.
• It presents a proposed forward representation with adjusted level and volatility for use in a Black-style formula.
• The author asks whether a tradable underlying with that adjusted volatility is required.
• The document supplies no answer or derivation, leaving the pricing justification unresolved.

Tags

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


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.