Moment Matching for a Lognormal Asian Option Approximation

Summary

The document outlines an analytical approximation for pricing an Asian option whose payoff depends on a discrete arithmetic average of the underlying asset at several observation times. It proposes treating that average as approximately lognormally distributed, then matching the approximation's mean and variance to the true first two moments of the average under a pricing model. Once the corresponding lognormal parameters are obtained, the approximation could be used with a Black–Scholes-style calculation.

The approach is presented as a proposed method rather than a completed derivation: the functions for the average's moments and the matching lognormal moments are left unspecified, and no option price, comparison, or numerical evidence is given. Its central simplification is replacing the distribution of an arithmetic average with a lognormal distribution. The document does not assess the accuracy of that assumption or explain how errors may vary with monitoring frequency, volatility, or maturity, so those limits would need separate analysis before relying on the approximation.

Key ideas

• The payoff's discrete arithmetic average is approximated as a lognormal random variable.
• The proposed approximation matches the average's mean and variance to determine lognormal parameters.
• The resulting distribution is intended to support a Black–Scholes-style Asian option valuation.
• The document gives the method outline but does not derive the moments or report pricing accuracy.

Tags

# Asian option analytical approximation


# 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...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.