Asian Options

A plain call or put only looks at the price on one single day, the expiry. An Asian option looks at the average price over a stretch of time instead. The name has nothing to do with geography; the first ones were priced in a Tokyo office, and the label stuck.

Why would anyone want the average? Picture an airline that buys jet fuel every single day of the year. It doesn't care about the price of oil on December 31st; it cares about the average price it paid all year. An option written on that average hedges its real exposure far better than one written on a single day's close. Same story for a company converting a steady stream of foreign revenue, or a fund that dollar-cost-averages into a position. Their risk is spread over many days, so their hedge should be too.

The payoff

For an average-price call, the payoff at expiry is

$$\text{payoff} = \max(\bar{S} - K,\; 0), \qquad \bar{S} = \frac{1}{n}\sum_{i=1}^{n} S_{t_i}.$$

Read it in words: $\bar{S}$ is the average of the underlying's price sampled at $n$ fixed dates $t_1, \dots, t_n$ (the "fixings"), and $K$ is the strike you locked in. If the average beat the strike, you collect the difference; if not, you collect nothing. A put just flips it to $\max(K - \bar{S}, 0)$.

Notice what the picture shows: the average line barely moves compared with the wild path around it. That is the whole point, and the whole consequence.

Why it's cheaper

Averaging a bumpy series kills volatility. The average of many prices swings far less than any single price, and low volatility means a cheap option. As a rough guide, a geometric-average Asian behaves like a vanilla whose volatility is only about $\sigma / \sqrt{3}$, roughly 58% of the original. Less wobble, less to pay.

Worked example

You hold an average-price call, strike $K = 100$, that samples the price at the end of each of four quarters. The four fixings come in at 96, 108, 104, and 112.

• The average is $\bar{S} = (96 + 108 + 104 + 112)/4 = 420/4 = 105$.
• Payoff $= \max(105 - 100,\; 0) = 5$.

Compare that with a vanilla call on the same strike. If the price on the final day happened to be 112, the vanilla would have paid $\max(112 - 100, 0) = 12$, more than double. But if that final day had instead gapped down to 92 after a strong year, the vanilla would pay nothing, while the Asian still pays 5 off the healthy average. The Asian trades away the lucky (and unlucky) extremes of a single day for something steadier.

Two design choices, and the pricing catch

There are two knobs. First, what you average: an average-price Asian averages the underlying (as above); an average-strike Asian instead sets the strike equal to the average, so it pays $\max(S_T - \bar{S}, 0)$. Second, how you average: an arithmetic mean (the ordinary sum-over-n) or a geometric mean (the n-th root of the product).

That second choice hides a real headache. The geometric average of lognormal prices is itself lognormal, so it has a clean, closed-form price much like Black-Scholes. The arithmetic average, the one everybody actually trades, is a sum of correlated lognormals, which is not lognormal and has no closed form. Desks price it by Monte Carlo or by a moment-matching approximation, and they cut the Monte Carlo noise by using the geometric Asian's exact price as a control variate.

Where it trips people up

• The average locks in as you go. By the last fixing, most of $\bar{S}$ is already decided, so a late move barely helps. An Asian's sensitivity to the spot (its delta and gamma) fades toward expiry, the opposite of a vanilla, whose gamma spikes at the end.
• Fixings matter enormously. Daily averaging behaves very differently from monthly. Always check the schedule; a "missed" or holiday fixing changes the payoff.
• Cheaper is not free. You gave up the fat upside of a single lucky close. If you actually wanted exposure to the terminal price, an Asian is the wrong tool.

For the wider family of path-dependent payoffs, the closed forms that survive, and the ones that force numerical methods, see Exotic Options and Barrier Options.