Asian Options

Companies whose sales or purchases arise over a period use Asian options to hedge price moves. The selling financial institution builds the valuation and hedge for a payoff set by an average.

Payoff set by the average of observed prices

An Asian option is a path-dependent option whose payoff depends not only on the price at expiry but also on prices during the period. Because it depends on the average price of the underlying asset over the period, it is also called an average-price option¹. The classification of exotic options and the general points on valuation and hedging are covered in Overview of Exotic Options.

The average is determined by the observation dates and the type of average. The contract sets the observation dates. Continuous observation is treated in theory, but in practice averages over discrete observation dates at regular intervals are the norm². The observation dates are written $t_1, t_2, \dots, t_n$, the underlying price on each date is written $S(t_i)$, and the last observation date $t_n$ is the expiry. The period from the first observation date to the last is called the averaging period³.

There are two kinds of average, arithmetic and geometric. The arithmetic average adds the observed prices and divides by their number. The geometric average multiplies them and takes the n-th root of that product (n is the number of prices). The arithmetic average is the common one, and the geometric average is sometimes used².

$$A = \frac{1}{n}\sum_{i=1}^{n} S(t_i)$$ $$G = \left(\prod_{i=1}^{n} S(t_i)\right)^{1/n}$$

$A$ is the arithmetic average and $G$ is the geometric average. Below, "the average" on its own means the arithmetic average.

The payoff is set in one of two ways. In the type with a fixed strike price $K$, called the average-price type, the average is compared with the strike. In the type that uses the average in place of the strike, called the average-strike type, the price at expiry is compared with the average².

In each case the payoff is 0 when the difference is negative. The payoff of an average-price call is, in a formula³:

$$\max(A - K,\ 0)$$

The average-price type has the form of a vanilla call option with the price at expiry replaced by the average. The average-strike type pays the amount by which the price at expiry is above or below the average over the period. Settlement is generally by cash settlement, with no delivery of the underlying asset¹.

Payoff and a numerical example

Consider a hypothetical example in which the price of the underlying asset is observed six times, at the end of each month, and the sixth observation is the expiry. Company A buys one unit of an average-price call with a strike price of 200 yen. For comparison, the table also shows the payoffs of a vanilla call with a strike price of 200 yen (a European option, set by the price at expiry only) and of an average-strike call. The payoff columns show the amounts before the premium is paid.

On path P, the price at expiry of 230 yen is 30 yen above the strike price of 200 yen, so the vanilla call pays 30 yen. The average-price call pays 25 yen, the amount by which the average of 225 yen exceeds the strike price⁴. On path Q, the price rises to 260 yen along the way and returns to 190 yen at expiry. The vanilla call pays 0 yen, but because the average is 220 yen, the average-price call pays 20 yen. On path R, the price jumps close to expiry and the price at expiry is 270 yen. The vanilla call pays 70 yen, but the average is exactly 200 yen, so the average-price call pays 0 yen. On the same path the average-strike call pays 70 yen, because the price at expiry is 70 yen above the average.

The buyer gains on some paths and loses on others. On path Q, where the price rises and then comes back, the average-price type pays more than the vanilla. On path R, where the price rises just before expiry, the vanilla pays more. On path S, where the price keeps falling, every type pays 0 and the premium paid is not returned.

Figure description

Payoff at expiry of buying one unit of an average-price call with a strike price of 200 yen. The horizontal axis is the arithmetic average of observed prices, not the price at expiry. The premium is not included. The payoff is 0 yen when the average is 200 yen or less, and rises by 1 yen for each 1 yen the average rises above 200 yen. The shape is the same as that of a vanilla call; the difference is that the horizontal axis is the average, not the price at expiry.

Why averaging lowers the premium

The payoff of the average-price type is set by the average, which moves less than the price at a single point in time. For this reason, Asian options tend to trade at a lower premium than vanilla options¹. That an average is less volatile than a single price is an advantage of this type⁵.

Across the four paths in the previous section, the price at expiry ranges from 140 yen to 270 yen, a spread of 130 yen. The averages on the same paths range from 165 yen to 225 yen, a spread of 60 yen⁴. The smaller dispersion of the average across paths is a factor behind the lower premium.

A lower premium does not mean the payoff is always smaller than the vanilla's, however. On path Q the average-price call pays 20 yen and the vanilla pays 0 yen, so the average-price call is higher. For the buyer, the lower premium and the smaller payoff when the price rises just before expiry are two sides of the same mechanism.

Geometric averages have a formula; arithmetic averages need approximation or numerics

The valuation starts from the assumption of the Black-Scholes model that the price of the underlying asset follows a lognormal distribution. Under this assumption the geometric average is lognormally distributed at expiry, so a price can be found with a Black-Scholes-type formula⁵. An arithmetic average of lognormally distributed prices is not lognormal. It is a sum of correlated lognormal variables whose distribution is generally unknown, so the price cannot be written in closed form⁵.

There are several ways to value an option on an arithmetic average⁵.

• Solving a partial differential equation numerically
• The Monte Carlo method, in which the closed-form formula for an option on a geometric average can be used as a variance-reduction tool (a control variate)
• An approximation that replaces the distribution of the average with a lognormal distribution that matches its first two moments (mean and variance), known as the Wilkinson approximation

For an arithmetic average observed continuously, an analytical formula has been published². For the arithmetic average over discrete observation dates used in practice, valuation relies on approximations or numerical methods. When valuing partway through the averaging period, it is enough to know the average of the prices observed up to the valuation date and the current price to value the remaining period⁶.

For the same observed prices, the arithmetic average is greater than or equal to the geometric average. For example, if the two observed prices are 100 yen and 400 yen, the arithmetic average is 250 yen and the geometric average is 200 yen⁴. So for average-price calls with the same strike price and observation dates, the payoff of the arithmetic-average call is at least that of the geometric-average call, and its price is not below it either⁴. If the observed prices are all the same on every path, the two averages are equal, and so are the payoffs and the prices.

With any of these methods, the value depends on the assumptions of the model used (model risk). The idea of the Monte Carlo method is covered in Monte Carlo Simulation.

Sensitivity of the payoff falls as expiry approaches

An observed price does not change afterwards. Only the prices not yet observed can move the average.

Consider the point where four of the six observations are done. If both of the remaining two prices rise by 1 yen, the average rises by only 2/6, about 0.33 yen⁴. When the average is above the strike price, the payoff of an average-price call moves by as much as the average. So the size of the payoff's response to the price of the underlying asset (the counterpart of delta) is at most 0.33 yen. With one observation left, this upper limit becomes 1/6, and once all observations are done the payoff is fixed. For a vanilla call above the strike price, a 1 yen move in the price at expiry moves the payoff by 1 yen.

In other words, the upper limit on how much the payoff responds to the underlying price falls as observations proceed. This applies only to the average-price type, however. The payoff of the average-strike type is "price at expiry minus average", and if the price on the last observation rises by 1 yen, the payoff moves by 5/6 yen⁴. The upper limit on sensitivity does not fall as observations proceed, so the same view does not apply.

When a seller hedges with the underlying asset or vanilla options, the cost of rebalancing and the hedging error remain, as with other exotic options.

Fit with a company's revenues and costs

For a company that pays or receives money for sales or purchases over a period, an Asian option set by the average over the period can match the hedged item better than a vanilla option set by the price at a single point in time². Another advantage given is that it is hard to influence the payoff by moving the price at the single point of expiry⁵.

Take the example from the previous section. Company A buys the same quantity of a raw material at the market price at the end of each month for six months. The raw material price moves in the same way as the underlying price in the earlier table. The purchase cost per unit is the average of the six purchase prices. Compare the case where Company A buys an average-price call with a strike price of 200 yen with the case where it buys a vanilla call with a strike price of 200 yen. The figures in the table are per unit and exclude the premium.

With the average-price call, on paths P, Q and R, where the price rose, the purchase cost after the payoff is the strike price of 200 yen⁴. This is because the purchase price moves with the average, so the payoff matches it exactly. With the vanilla call, on a path like Q where the price rises along the way and returns by expiry, the rise in purchase cost is not covered by the payoff, and the cost stays at 220 yen. On a path like R where the price rises only at expiry, the payoff exceeds the rise in purchase cost and the cost falls to 130 yen.

On path S, where the price fell, the payoff is 0 yen in both cases and the purchase cost falls to 165 yen. Company A has paid the premium in this case as well, however, so its cost is higher than if it had not bought the option, by the amount of the premium paid.

This example rests on the assumptions that the quantity purchased is the same every month and that the purchase price moves in the same way as the price of the option's underlying asset. If the purchase dates or quantities do not match the observation dates, or the purchase price and the underlying price diverge, the difference remains.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (February 1998), Section 4330.1 "Options", "Exotic Options" (Page 2). It supports the statements on the dependence on the average, the tendency toward a lower premium, and cash settlement. ↩ ↩² ↩³

2. Mark Broadie and Jérôme Detemple, "Option Pricing: Valuation Models and Applications" (Management Science 50(9), 2004), Appendix A.2.3 "Asian Options" (p. 1171). It supports the statements on the payoff definitions, the types of average, discrete observation, corporate hedging, and the analytical formula for a continuously observed arithmetic average. ↩ ↩² ↩³ ↩⁴ ↩⁵

3. Steven Kou, "Discrete Barrier and Lookback Options" (Handbooks in Operations Research and Management Science Vol. 15, 2008), Section 1 "Introduction" (p. 343). It supports the statement that a discretely observed average-price call pays the difference between the average and the strike at expiry. ↩ ↩²

4. Calculations for the text. Each average is the sum ÷ 6, and the spreads are 270 − 140 = 130 and 225 − 165 = 60. The sensitivities of the average-price call to the observed prices are 2/6 and 1/6, and for the average-strike type 1 − 1/6 = 5/6. The arithmetic average of 100 and 400 is 250 and the geometric average is √(100 × 400) = 200, and the ordering follows from the inequality between the arithmetic and geometric means. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷

5. Manuel Moreno and Javier F. Navas, "Australian Asian Options" (Universitat Pompeu Fabra, Working Paper 680, version of 25 February 2003), Section 1 "Introduction" (pp. 1–3; PDF pages 3–5). It supports the statements on the advantages of averaging, the distributions of geometric and arithmetic averages, and the methods for valuing an arithmetic average. ↩ ↩² ↩³ ↩⁴ ↩⁵

6. Broadie and Detemple, same paper, Section 5.1.1 "Path-Dependent Options" (pp. 1165–1166). It supports the statement that an Asian option can be valued by tracking only the current price and the average to date. ↩