Asian options

Asian options are path-dependent options, with payoffs that depend on the average price of the underlying asset or the average exercise price. There are two categories or types of Asian options: average rate options (also known as average price options) and average strike options. The payoffs depend on the average price of the underlying asset over a predetermined time period. An average is less volatile than the underlying asset, therefore making Asian options less expensive than standard European options. Asian options are commonly used in currency and commodity markets. Asian options are of interest in markets with thinly traded assets. Due to the little effect it will have on the option’s value, options based on an average, such as Asian options, have a reduced incentive to manipulate the underlying price at expiration.

The payoffs at expiration for an average price Asian option is:

max(f(Savg - X), 0)

where

Savg is the average at the expiration date

X is the strike price

f is 1 for a call or -1 for a put

The payoffs at expiration for an average strike Asian option is:

max(f(ST - Savg), 0)

where

ST is the underlying asset price at expiration

Savg is the average at the expiration date

f is 1 for a call or -1 for a put

Geometric vs. Arithmetic price averaging

The Geometric average is the nth root of the product of the n sample points. The Arithmetic average is the sum of the stock values divided by the number of sampling points. Although Geometric Asian options are not commonly used in practice, they are often used as a good initial guess for the price of arithmetic Asian options. This technique is used to improve the convergence rate of the Monte Carlo model when pricing arithmetic Asian options.

The AveragePrice function returns the theoretical value, sensitivities, the implied volatility, and the implied strike value of a European arithmetic or geometric average price option using either the Levy model for arithmetic or the Rubinstein model for geometric average price options. The Geometric model supports both continuous and discrete time intervals.

The AverageStrike function returns the theoretical value, sensitivities, the implied volatility, and the implied strike value of a European arithmetic or geometric average strike option using either the Levy model for arithmetic or the Rubinstein model for geometric average price options. The Geometric model supports both continuous and discrete time intervals.

The AsianMC function returns the theoretical value of a European Average Price Asian Option using either an Antithetic or Control Variate Monte Carlo technique.

The AsianSpreadMC function returns the theoretical value of a European Two Asset Asian Spread Option using a Monte Carlo technique.