I'm doing some work on pricing basic derivatives against a simple market in continuous time - I believe I'm pricing the power option correctly, but when I try to price its' inverse, I'm not nearly as certain: the result $\textit{feels}$ off.
Anyway to be more concrete, I'm working with a market (($\Omega$, $\mathscr{F}$, $\mathbb{F}$, $\mathbb{P}$), $X$) with $d$ = 1 and $X = X^{1}$ (i.e. a single risky asset), where --
$dX_{t} = \sigma X_{t}dW_{t}, X_{0} = 1$
-- for a Brownian motion process $W$. This is - I believe - the same representation of a stochastic differential given by Girsanov's theorem.
I'm trying to price $H = X_{T}^{2}$, the power option, and its' inverse $\frac{1}{H}$, by working out their value under a unique ELMM $\mathbb{P}^{*}$, with a Radon-Nikodym derivative of the form $\frac{d\mathbb{P}^{*}}{d\mathbb{P}} = e^{-\frac{\mu}{\sigma}W_{T} - \frac{1}{2}(\frac{\mu}{\sigma})^{2}T}$.
Writing out my steps in full for pricing $H$, I get:
$\mathbb{E}$($H$)
= $\mathbb{E}$($X_{T}^{2}$)
= $\mathbb{E}((X_{0} e^{-\frac{1}{2}\sigma^{2}T + \sigma W_{T}})^{2})$
= $X_{0}^{2}~\mathbb{E}(e^{-\sigma^{2}T + 2\sigma W_{T}})$
= $X_{0}^{2}~e^{-\sigma^{2}T}~\mathbb{E}(e^{2\sigma W_{T}})$
= $X_{0}^{2}~e^{-\sigma^{2}T}~e^{\frac{1}{2}(2\sigma)^{2}T}$
= $X_{0}^{2}~e^{\sigma^{2}T}$
= $1^{2}~e^{\sigma^{2}T}$
= $e^{\sigma^{2}T}$
Giving me a non-discounted price for $H$ at time 0 of $e^{\sigma^{2}T}$.
Now if I try to do the same with the inverse power option, I end up sitting with a very similar result...
$\mathbb{E}(\frac{1}{H})$
= $\mathbb{E}(X_{T}^{-2})$
= $\mathbb{E}((X_{0}e^{-\frac{1}{2}\sigma^{2}T + \sigma W_{T}})^{-2})$
= $X_{0}^{2}~\mathbb{E}(e^{\sigma^{2}T - 2\sigma W_{T}})$
= $X_{0}^{2}~e^{\sigma^{2}T}~\mathbb{E}(e^{-2\sigma W_{T}})$
= $X_{0}^{2}~e^{\sigma^{2}T}~e^{\frac{1}{2}(-2\sigma)^{2}T}$
= $X_{0}^{2}~e^{3\sigma^{2}T}$
= $1^{2}~e^{3\sigma^{2}T}$
= $e^{3\sigma^{2}T}$
Am I going mad, or does that make sense? I suppose I'm somewhat rubber-ducking here, but additional insights would be greatly appreciated! If I'm eliding anything in terms of 'why I did step X' please let me know: I can't quite see the forest for the trees anymore.
Cheers!
Just updating this to say that these calculations ended up being correct.