black scholes - Expectation of $frac {S_{T_2}} {S_{T_1}}$ at $T_0$

Is my below computation correct (assuming flat volatlity Black Scholes model, flat interest rate curve):

$\mathbb{E}(\frac {S_{T_2}} {S_{T_1}}| \mathcal{F}_{T_0})$

$ = \mathbb{E}{\frac{S_{T_0}e^{(r-\frac{\sigma^2}{2})T_2+\sigma W_{T_2}}}{S_{T_0}e^{(r-\frac{\sigma^2}{2})T_1+\sigma W_{T_1}}}}$

$=\mathbb{E}(e^{r(T_2-T_1)-\frac{1}{2}\sigma^2(T_2-T_1)+\sigma(W_{T_2}-W_{T_1})})$

$=e^{r(T_2-T_1)-\frac{1}{2}\sigma^2(T_2-T_1)+\frac{1}{2}\sigma^2(T_2-T_1)}$

$ = e^{r(T_2-T_1)}$

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EDIT: Can anyone please re-confirm one of the steps above? $\mathbb{E}(e^{r(T_2-T_1)-\frac{1}{2}\sigma^2(T_2-T_1)+\sigma(W_{T_2}-W_{T_1})})$ $=e^{Mean(.) + \frac{1}{2}Variance(.)}$ $Mean(.) = r(T_2-T_1)-\frac{1}{2}\sigma^2(T_2-T_1)$ $Variance(.) = \mathbb{E}[\{\sigma(W_{T_2}-W_{T_1})\}^2]=\mathbb{E}[\sigma^2\{(W_{T_2})^2 +(W_{T_1})^2 -2W_{T_1}W_{T_2}\}]=\sigma^2(T_2+T_1-2T_1) = \sigma^2(T_2-T_1)$

I think I got it all correct, now! :-)

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Related Question - Do we have an analytical formula (under standard Black Scholes) for -

$\mathbb{E}((\frac {S_{T_2}} {S_{T_1}}-K)^+| \mathcal{F}_{T_0})$ paid at $T_2$

My attempt .. basically using the Black Scholes pricing formula for call option -

$\mathbb{E}((\frac {S_{T_2}} {S_{T_1}}-K)^+| \mathcal{F}_{T_0}) = e^{r(T_2-T_1)}N(d_1)-KN(d2)$

where $d_1= \frac{\ln(\frac{e^{r(T_2-T_1})}{K})+\frac {\sigma^2(T_2-T_1)}{2})}{\sigma \sqrt(T_2-T_1)}$

$d_2= \frac{\ln(\frac{e^{r(T_2-T_1})}{K})-\frac {\sigma^2(T_2-T_1)}{2})}{\sigma \sqrt(T_2-T_1)}$

I would multiple with the discounting factor $e^{-r (T_2-T_0)}$ to the above formula to get the price at $T_0$.