# Asian Options-Change of Numeraire

Assume the risk-free bond $$B_t$$ and the stock $$S_t$$ follow the dynamics of the Black & Scholes model

without dividends (with interest rate r, stock drift $$\mu$$ and volatility $$\sigma$$). Show that $$S_{u;T} := \frac{S_{u}}{S_T}$$ under the measure $$Q^S$$ (with the stock as a numeraire) can be written as $$exp\{(-r-\frac{\sigma^2}{2})(T-u)+\sigma\hat{W}_{T-u}\}$$ where $$\hat{W}_t$$ for $$t\in[0,T]$$ has the same law of a Wiener process under the $$Q^S$$ measure.

I'm stuck on how to solve this question. Would really appreciate the help.

Let $$\mathbb{Q}$$ be the risk-neutral probability measure which uses the risk-free bank account $$(B_t)$$ as numeraire. In general, $$\mathrm{d}B_t=r_tB_t\mathrm{d}t$$. In the Black-Scholes setting, $$r_t\equiv r$$, we have $$B_t=e^{rt}$$.
The stock measure $$\mathbb{Q}_S$$ uses the compounded stock price $$S_te^{qt}$$ as numeraire and is defined via the Radon Nikodym derivative \begin{align*} \frac{\mathrm{d} \mathbb{Q}_S}{\mathrm{d}\mathbb{Q}}(t) &= \frac{B_0}{B_t}\frac{S_te^{qt}}{S_0} \\ &= \frac{1}{e^{rt}}\exp\left(\left(r-q-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right)e^{qt} \\ &=\exp\left(-\frac{1}{2}\sigma^2t+\sigma W_t\right) \\ &= \mathcal{E}\left(\sigma W_t\right), \end{align*}
using that $$\mathrm{d}S_t=(r-q)S_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$$. Recall that $$(W_t)$$ is a standard Brownian motion under $$\mathbb{Q}$$. Using Girsanov's theorem, we know that $$\mathbb{Q}_S\sim\mathbb{Q}$$ and that the process \begin{align*} \hat{W}_t &=W_t-\sigma t \end{align*} is a standard Brownian motion under $$\mathbb{Q}_S$$.
So, we conclude with \begin{align*} S_{u,T} &= \frac{S_u}{S_T} \\ &= \exp\left(\left(r-q-\frac{1}{2}\sigma^2\right)u+\sigma W_u -\left(\left(r-q-\frac{1}{2}\sigma^2\right)T+\sigma W_T\right)\right) \\ &= \exp\left(\left(r-q-\frac{1}{2}\sigma^2\right)(u-T)-\sigma (W_T-W_u)\right) \\ &= \exp\left(\left(r-q-\frac{1}{2}\sigma^2\right)(u-T)-\sigma \left(\hat{W}_T +\sigma T -\left(\hat{W}_u+\sigma u\right)\right)\right) \\ &\overset{d}{=} \exp\left(-\left(r-q-\frac{1}{2}\sigma^2\right)(T-u)-\sigma \big(\hat{W}_{T-u}+\sigma(T-u)\big)\right) \\ &= \exp\left(-\left(r-q+\frac{1}{2}\sigma^2\right)(T-u)-\sigma \hat{W}_{T-u}\right) \\ &\overset{d}{=} \exp\left(-\left(r-q+\frac{1}{2}\sigma^2\right)(T-u)+\sigma \hat{W}_{T-u}\right). \end{align*}