Closed form pricing method for Asian options

Closed form pricing method for Asian options

Pay off of Asian option is depending on the average price of underling assets in some period of the life. For that reason the payoff must be according to the historical data. Comparing to European options Asian option does not have closed form pricing method to arithmetic average like a Black-Scholes, because of lognormally distribution of assets collapsed. However, in 1990 Kemna and Vorst introduced closed form pricing method for geometrical averaging options by altering volatility and cost of carry term.  Geometric average can be priced by closed form method because of the underling prices are lognormally distributed. The closed form geometrical method formulas for Asian calls and puts options are: Call

 

and put

 

 

which can be simplified as

      

and adjusted volatility and dividend are given by

                                               

 

where s  is the observed volatility, r is the risk free rate and q is the dividend yield.

As there was no close form pricing method for arithmetic average, because of not lognormal distribution of underling asset, in 1991 Turnbull and Wakeman made an assumption that distribution is approximately lognormal and introduced put forward first and second moment of average in order to price the option.

The formulas for call and put under (Turnbull and Wakeman) TW are:

 

 

Where

 

where T₂ is the time to maturity from time zero and X is the strike price.

Because of the TW approximation is based on moments of time, first moment be M₁ and the second moment be M₂, for q≠r.

 

Adjusted volatility is

 

By assuming that the average asset price is lognormal, we can value the Asian options like an option on a futures contract with

 

If average period is already began we must adjust strike price

Where T is original time t maturity, T₂   is the remaining time to maturity, X is the original strike price and S avg is the average asset price.

After Turnbull and Wakeman in 1991 introduced their close form pricing method, Levy put forward another analytical approximation and it gives more accurate results that Turnbull and Wakeman approximation.

Approximation to call

For puts

Where

And

Here the variables are the same as in Turnbul and Wakeman formula.

Here you can see that the Levy and Turnbull and Wakeman approach are the same and it give approximately same strike price results.