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Pricing a Call Option on the Cube of a Black–Scholes Stock

Article Quant Q&A · Author: Anon
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Summary

The answer frames the claim as a European payoff based on the cube of the stock price at maturity. It invokes risk-neutral discounted expected payoff, then applies Itô’s lemma to the cubed stock process. This produces a process with a transformed drift and volatility, after which the response proposes treating the cube as a new underlying and substituting its parameters into the standard Black–Scholes call formula.

The response does not work through the resulting formula or explain how the normal distribution terms change. Its transformation also needs care: under risk-neutral pricing, the stock drift should be the risk-free rate, and the drift and diffusion coefficients for the cube must be derived consistently. Therefore, the proposed substitution is a starting idea, not a fully checked pricing derivation; the underlying is a nonlinear transformation of the original stock rather than an independently traded asset.

Key ideas

  • Price a contingent claim as the discounted risk-neutral expectation of its payoff.
  • Itô’s lemma can derive the dynamics of a power of a Black–Scholes stock.
  • A cubed stock has transformed drift and volatility that affect the option formula inputs.
  • The answer omits a completed formula and requires careful risk-neutral derivation.

Tags

OptionsDerivatives pricingStatistics
Full text
# Calculate the price at time t=0


# Calculate the price at time t=0












Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model (with interest rate r, stock drift $\mu$ and volatility $\sigma$).

Calculate the price at time $t = 0$ of a derivative with maturity T and payoff $(S^3_t-K)^+$. I know I need to use the Black Scholes formula for price of a call to find the price of the derivative but the formula also contains $N(d_1)$ and $N(d_2)$ so how would this get affected?

## Answer by ab94 (score 3, accepted)

https://quant.stackexchange.com/a/48967

I don't understand the question but I can try. I think the problem is to find the price of a contingent claim that has payoff $(S_T^3 - K)^+$. The well-known pricing formula is: \begin{equation} \pi(t)=\mathbb{E}^\mathbb{Q}[e^{-r(T-t)}(S_T^3 - K)^+|\mathcal{F}_t] \end{equation} Now put $Y=S^3$, by using Ito's Lemma \begin{equation} dY(t)=dS^3(t)=3S^2(t)dS(t) + \frac126S(t)\sigma^2S^2(t)dt \end{equation} In Black-Scholes model \begin{equation} dS(t)=\mu S(t) dt + \sigma S(t) dW(t) \end{equation} So we have: \begin{equation} dY(t)=3\mu S^3dt + 3\sigma^2S^3dt + 3\sigma S^3dW=(3\mu + 3\sigma^2)Ydt + 3\sigma YdW \end{equation} Now we define \begin{align} \tilde{\mu}&=3\mu + 3\sigma^2 \\ \tilde{\sigma}&=3\sigma \end{align} Now suppose $Y$ is a new stock with drift $\tilde{\mu}$ and volatility $\tilde{\sigma}$ and just substitute in the Black-Scholes formula for an european option with underlying $Y$ and strike $K$.

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