I'm trying to approximate the price of an Asian option via the Black-Scholes formula by considering the discrete arithmetic average as a log-normal distribution.

$$ A_{T}(n):=\frac{1}{n} \sum_{i=1}^{n} S_{t_{i}} $$

To that end, the first step would be to compute the first two moments of $A_{T}(n)$

$$ \begin{aligned} \mathbb{E}\left[A_{T}(n)\right] &=f_{1}\left(S_{0}, r, \sigma, n\right), \\ \operatorname{Var}\left[A_{T}(n)\right] &=f_{2}\left(S_{0}, r, \sigma, n\right), \end{aligned} $$

then selecting the log-normal parameters corresponding to this mean and variance by moment-matching. In other words, considering the approximation $\ln A_{T}(n) \approx \mathcal{N}\left(\mu_{A_{T}(n)}, \sigma_{A_{T}(n)}\right)$ and calibrate $\mu_{A_{T}(n)}$ and $\sigma_{A_{T}(n)}$ by solving the system of two equations

$$ \begin{aligned} \mathbb{E}\left[A_{T}(n)\right] &=g_{1}\left(\mu_{A_{T}(n)}, \sigma_{A_{T}(n)}\right), \\ \operatorname{Var}\left[A_{T}(n)\right] &=g_{2}\left(\mu_{A_{T}(n)}, \sigma_{A_{T}(n)}\right), \end{aligned} $$

where $g_{1}(\mu, \sigma)$ and $g_{2}(\mu, \sigma)$ are functions that give the mean and variance of a random variable having a log-normal distribution of parameters $\mu$ and $\sigma$.

Any help would be appreciated...

  • $\begingroup$ See Shreve (Stochastic Calculus for Finance II, chapter 7.5 (Asian Options) for more details). $\endgroup$ Jan 7, 2022 at 12:45


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