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The Asian option is cash settled, so the bank will transfer you $0.5. Delivering the shares and doing some trades is not possible. You can't buy the spot for the average price over a period, you just pay the spot price. Since you're into Asian options, I assume asian option valuation is useful for you to assess whether you're not paying too much. 2 You want to work directly with$\overline{X}$, and not some other r.v. with the same distribution, since equivalence in distribution doesn't imply that correlation remains the same. For ease of notation, I'll assume that$\mu = 0$and$\sigma = 1$. I claim that $$\text{cov}\left(\overline{X},X \right) = \frac{1}{t} \int_0^t s \ ds.$$ Note that this is ... 2 Your analysis is correct. From the risk neutral process $$dS_t = rS_tdt + \sigma S_tdW_t$$ we get $$\mathbb{E}(S_\tau|\mathbb{F}_t) = S_te^{r(\tau-t)}$$ and $$\mathbb{E}(\int_{t}^{T}S_\tau d\tau|\mathbb{F}_t) = \frac{S_t}{r}[e^{r(T-t)}-1]$$ Hence, as$S \rightarrow \infty$C(S,A,t) \sim \frac{S_te^{-r(T-t)}}{r(T-T_0)}[e^{r(T-t)}-1] = ... 2 A few tips. First note that$e^{-rt}S_t$is a martingale. So make it appear and then integrate by part to rewrite$\int S_u du$as a stochastic integral. Finally use the Ito isometry property. 1 Yes it is known in closed form. See https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian_doc/asian_doc.html section 5.1 which references an older Geman-Yor paper. 1 There is no difference in information, though the fitting algorithm may increase in complexity. First note that in practice you never have an entire curve or surface of prices$C(K,T)\$ of any kind of option. You only have a finite number of observations and even those typically have a bid and an offer. I would therefore argue that the correct picture of ...