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Looking for some help on this question.

Suppose the Black-Scholes framework holds. The payoff function of a T-year European option written on the stock is $(\ln(S^3) - K)^+$ where $K > 0$ is a constant.

Use the risk-neutral valuation or other method to find the price of the option at time $ t$. Express your result in terms of $ N(x) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}y^2} dy $.

My solution so far:

I assume that:$$dS_t=r_tS_tdt+\sigma S_t d \tilde{W}_t$$

Via Ito Lemma, I find that: $$d\ln(S_T^3)=3d\ln(S_T)=3(r-\sigma^2/2)dt+3\sigma d \tilde{W}_t$$

Now, I am a little wonky on this but I assume: $$\begin{align*} \mathbb{E}[\ln(S_T^3)-K]^+&=\mathbb{E}[3\ln(S_T)1_{S_{T}-K}]-K \mathbb{P}(3\ln(S_T)>K) \end{align*}$$ But I am not too sure how to derive the answer after that. Any help is greatly appreciated.

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    $\begingroup$ $\ln(S^3)$ has normal distribution so you can apply Bachelier formula. $\endgroup$ Dec 6, 2023 at 2:19

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