When we take the arithmetic average of the underlying, A(0,T) is calculated as
Our Black-Schloes-Merton model and Black’s model rely on the assumption that the underlying price follows a lognormal distribution. And, the arithmetic average of the lognormally distributed random variables is not a lognormal.
Turnbull and Wakeman (1991) valued arithmetic average Asian options using similar formulas to those used for regular options by assuming that Save is a lognormal. He, first, calculated the first and the second moments of Save during the option life, and then assumed that the Save is lognormally distributed with the first and the second moments, and used Black’s model. Hence by setting
in the Black’s formula, we obtain price for Asian (arithmetic average) call and put options as follows
And, if the stock is paying a dividend yield at rate q per annum, we need to replace r by r−q, and the formulas for M1 and M2 will become