A power put option is an option with payoff (S_{T}^{alpha}left(K-S_{T} ight)^{+}), its price is denoted (operatorname{PowP}^{alpha}(x, K)).

A power put option is an option with payoff \(S_{T}^{\alpha}\left(K-S_{T}\right)^{+}\), its price is denoted \(\operatorname{PowP}^{\alpha}(x, K)\). Prove that there exists \(\gamma\) such that

\[\operatorname{DIC}^{S}(x, K, L)=\frac{1}{L^{\gamma}} \operatorname{PowP}^{\gamma-1}\left(K x, L^{2}\right)\]

From (ii) in Lemma 3.6.6.1, \(\operatorname{DIC}^{S}(x, K, L)=\frac{1}{L^{\gamma}} \mathbb{E}\left(S_{T}^{\gamma}\left(\frac{L^{2}}{S_{T}}-K\right)^{+}\right) .

Example 3.6.7.1 Prove the following relationships:
\[\begin{align*}
& D I C^{S}(x, L, L)+L \operatorname{BinDIC}^{S}(x, L, L) \\
= & \left(\frac{x}{L}\right)^{\gamma-1} e^{-\mu T}\left[P_{E}^{S}\left(x, L e^{2 \mu T}\right)-L \frac{x}{L} \operatorname{Delta} P_{E}^{S}\left(x, L e^{2 \mu T}\right)\right] \\
& =\left(\frac{x}{L}\right)^{\gamma-1} e^{\mu T} L \operatorname{BinP}^{S}\left(x, L e^{2 \mu T}\right), \\
\operatorname{DIB}(x, L) & =\operatorname{BinP}^{S}(x, L)+\left(\frac{x}{L}\right)^{\gamma-1} e^{\mu T} \operatorname{BinP}^{S}\left(x, L e^{2 \mu T}\right) \\
& \quad-\frac{1}{L} \operatorname{DIC}^{S}(x, L, L) . \tag{3.6.31}
\end{align*}\]

Lemma 3.6.6.1:

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

\[\begin{equation*}
\gamma=1-\frac{2(r-\delta)}{\sigma^{2}}, \tag{3.6.26}
\end{equation*}\]

(i) the process \(S^{\gamma}=\left(S_{t}^{\gamma}, t \geq 0\right)\) is a martingale with dynamics

\[d S_{t}^{\gamma}=S_{t}^{\gamma} \widehat{\sigma} d W_{t}\]

where \(\widehat{\sigma}=\gamma \sigma\).

(ii) for any positive Borel function \(f\)

\[\mathbb{E}_{\mathbb{Q}}\left(f\left(S_{T}\right)\right)=\mathbb{E}_{\mathbb{Q}}\left(\left(\frac{S_{T}}{x}\right)^{\gamma} f\left(\frac{x^{2}}{S_{T}}\right)\right) .\]