A power put option is an option with payoff ( T ( ( S T

A power put option is an option with payoff ${(}_T^{\alpha }{((-S_T)}^{+}$, its price is denoted ${}^{}(x,K)$. Prove that there exists $\gamma$ such that

$${\mathrm{DIC}}^S(x,K,L)=\frac{1}{L^{\gamma }}{\mathrm{PowP}}^{\gamma -1}(Kx,L^2)$$

From (ii) in Lemma 3.6.6.1, ${\mathrm{DIC}}^S(x,K,L)=\frac{1}{L^{\gamma }}E\left({}^{}{(\frac{{)}^2}{S_T}-K)}^{+}\right).◃$

Lemma 3.6.6.1:

Let $S$ be an underlying whose dynamics are given by (3.6.25) under the risk-neutral probability $Q$. Then, setting

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