Kemna-Vorst: the geometric-average Asian closed form

An Asian option pays on the average of the underlying over its life, not the terminal spot. Two averages are used in practice: the arithmetic average (the market standard) and the geometric average. They look almost identical — and yet one has an exact price and the other does not.

Why geometric is easy and arithmetic is hard

Under Black-Scholes the spot is log-normal. The geometric average is a product of log-normals, and a product of log-normals is still log-normal — so the geometric Asian prices with a Black-style formula, just with an adjusted volatility and drift. This is the Kemna-Vorst (1990) result. The arithmetic average, by contrast, is a sum of log-normals, which is not log-normal and has no closed form — which is why the arithmetic Asian is priced by simulation.

The adjusted volatility

Averaging smooths the terminal quantity, so the geometric average carries a lower effective volatility than the spot itself — for a continuous average it works out to σ/√3, with a matching drift adjustment. Plug those into a Black formula and you have the exact price. The same smoothing is why any Asian is cheaper than the vanilla of the same strike: less dispersion in the payoff, less optionality.

The payoff: the control variate

Here is where the exact formula earns its keep. The arithmetic Asian has no closed form, so you price it by Monte Carlo — but the estimate is noisy. Because the geometric payoff is highly correlated with the arithmetic one and you know its price exactly, you can use it as a control variate: simulate both, and correct the arithmetic estimate by the geometric leg's known error, A − β·(G − E[G]). In practice this collapses the variance by an order of magnitude — the same confidence interval for a tiny fraction of the paths.