Asian Option

An Asian option, also called an average option, is an option whose payoff is based on the average price of the underlying asset over a stated period. A standard option looks only at the price on the day it is exercised. An Asian option replaces that single reading with a mean of readings taken daily, weekly or monthly during the life of the contract.

Asian options are among the most widely used exotic options. They are traded mainly over the counter and are common in commodity, energy and currency markets, where companies buy or sell steadily through the year and care about the average price they receive. Some exchanges also list average-price contracts on commodities. The name is generally traced to bankers working in Tokyo in the late 1980s and does not describe where the option trades.

Asian options are classified in three ways.

• Average price or average strike. In an average price option, the average replaces the final price and is compared with a fixed strike. In an average strike option, the average becomes the strike and is compared with the final price.
• Arithmetic or geometric. The arithmetic average is the ordinary mean and is what almost all traded contracts use. The geometric average is the n-th root of the product of n prices. It is always at or below the arithmetic average and is important mainly because it is easier to value.
• Call or put. As with any option, a call option gains when prices are high and a put option gains when prices are low.

Contracts also specify the observation dates and the price source, such as an exchange settlement price or a published benchmark. Exercise is normally at the end only, as in a European option.

The payoff of an average price call at expiry is:

and the arithmetic average is:

• $A$ is the average price over the observation dates.
• $K$ is the strike price.
• $S_i$ is the price of the underlying on observation date $i$, and $n$ is the number of dates.

An airline buys a five-month average price call on oil with a strike of $77 a barrel. The monthly observations are $78, $82, $85, $79 and $76. Their sum is 400 and the average is $80. The payoff is 80 − 77 = $3 a barrel. A standard call with the same strike would have paid nothing, because the final price of $76 is below the strike. The put works in reverse, paying $\max(K - A, 0)$.

Averaging matches the exposure of many hedgers. An importer that converts currency every month, or a refinery that buys crude every day, is affected by the average rate over the period, and an Asian option offsets that exposure more closely than a string of separate contracts.

Averaging also lowers cost. The average of a series of prices varies less than any single price, so the effective volatility is lower and the option premium is usually smaller than for a comparable standard option. A third benefit is resistance to price distortion. In a thinly traded market, a single closing price can be pushed by a large order or by deliberate market manipulation. An average over many days is much harder to move.

The Black-Scholes model assumes the underlying price follows a lognormal distribution. The geometric average of lognormal prices is itself lognormal, so a geometric Asian option has an exact formula of the same form. The arithmetic average is not lognormal, and no exact closed formula exists for the contracts that actually trade.

Arithmetic Asian options are therefore priced numerically. Monte Carlo simulation generates many price paths, computes the average and payoff on each, and discounts the mean payoff. The geometric price is often used as a control to make the simulation converge faster. Analytical approximations that match the first moments of the average to a lognormal distribution are common where speed matters.

As the averaging period passes, more of the average becomes known and the option grows less sensitive to the underlying. Its delta shrinks toward zero near the end, whereas a standard at-the-money option becomes more sensitive as expiry approaches. This makes late-stage hedging easier for the dealer.

The buyer accepts a trade-off. An Asian option does not capture a sharp price move late in its life, since one extreme reading is diluted by the others. A buyer exposed to the price on one specific date is better served by a standard option. Because the contracts are bespoke, buyers also bear counterparty credit risk and may find them hard to sell before expiry.