Pricing a European Power Call on a Squared Stock Price

Summary

The document poses a derivatives-pricing problem: valuing a European payoff based on the square of an underlying stock price when the stock follows a geometric Brownian motion. It applies Itô's lemma to the squared price and observes that the resulting process has volatility twice that of the original stock, along with a drift that includes an additional volatility term.

The author reports that directly treating this transformed process as a standard Black–Scholes stock, with its derived drift interpreted as the risk-free rate, gives an incorrect option value. A referenced alternative parameterization uses the original risk-free rate and an adjusted dividend yield. The question asks why this adjustment is appropriate and what fails in the initial approach. The document provides the setup and identifies the modeling issue, but it does not include an answer or derivation, so it is not a complete pricing method. Its value is in highlighting that a transformed stochastic process's drift cannot automatically be used as a tradable asset's financing rate.

Key ideas

• Squaring a geometric Brownian stock price changes both its drift and volatility under Itô's lemma.
• The squared process has volatility twice that of the original stock.
• The drift of a transformed process cannot automatically be treated as the risk-free rate in Black–Scholes.
• The question highlights an adjusted dividend-yield representation but does not derive or validate it.
• A complete valuation requires careful use of risk-neutral pricing assumptions.

Tags

# Pricing of European Power Call Option via Black-Scholes formula: reasoning?


# Pricing of European Power Call Option via Black-Scholes formula: reasoning?












I want to price a European Power Call option (without dividend yield) with payoff $\max\{S_T^2-X, 0\}$, where $T$ is the maturity and $X$ the strike. Let $(S_t)_{t\ge 0}$ be the price process of an underlying with dynamics

$dS_t=S_t(r dt + \sigma dW_t)$.

So first I observed that $dS_t^2=2S_tdS_t+d\langle S\rangle_t=S_t^2\left((2r+\sigma^2)dt+2\sigma dW_t\right)$

So $\tilde{S}_t$ is a BS-stock with volatility $\tilde{\sigma}=2\sigma$ and interest rate $\tilde{r}=2r+\sigma^2$. Then I applied the BS-formula for option pricing which gives a call price.

But this leads to the wrong result. As i looked it up the right way is to choose $\tilde{\sigma}=2\sigma$, $\tilde r = r$ and a dividend $q=-(\sigma^2+r)$.

Now my question is: What is the reasoning behind this choice? Why is the price of a European Power Call option without dividend yield derived by using Black-Scholes formula with dividends? That doesn't make sense to me. And why did my straightforward approach lead to a wrong result?

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