The Asian option pricing engine uses Monte Carlo Simulation to estimate the value of the Asian option. Here, we assume the process of asset pricing applies to Black-Scholes process.
Asian Option is kind of exotic option. The payoff is path dependent and it is dependent on the average price of underlying asset over the settled period of time \(T\).
The payoff of Asian options is determined by the arithmetic or geometric average underlying price over some pre-set period of time. This is different from the case of usual European Option and American Option, where the payoff of the option depends on the price of the underlying at exercise. One advantage of Asian option is the relative cost of Asian option compared to American options. Because of the averaging feature, Asian options are typically cheaper than American options.
The average price of underlying asset could be used as strike price or the underlying settlement price at the expiry time. When the average price of underlying asset is used as the underlying settlement price at the expiry time, the payoff is calculated as follows:
payoff of put option = \(\max(0, K - A(T))\)
payoff of call option = \(\max(0, A(T) - K)\)
When the average price of underlying asset is used as the strike price at the expiry time, the payoff is calculated as follows:
payoff of put option = \(\max(0, A(T) - S_T)\)
payoff of call option = \(\max(0, S_T - A(T))\)
Where \(T\) is the time of maturity, \(A(T)\) is the average price of asset during time \(T\), \(K\) is the fixed strike, \(S_T\) is the price of underlying asset at maturity.
The average could be arithmetic or geometric, which is configurable. \(N\) is the number of discrete steps from \(0\) to \(T\).
Arithmetic average: \(A(T) = \frac{\sum_{i=0}^N S_i}{N}\)
Geometric average: \(A(T) = \sqrt[n]{\prod_{i=0}^N S_i}\)