# Asian option analytical approximation

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...

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