Valuation (Cont)
▪ Onecalculates the first two moments of the probability distribution of the
arithmetic average in a risk-neutral world exactly and then fit a lognormal
distribution to the moments.
▪ Consider a newly issued Asian option that provides a payoff at time T
based on the arithmetic average between time zero and time T. The first
moment, M 1 and the second moment, M2, of the average in a risk-neutral
world can be shown to be
푀 1 =푒 푟−푞 푇− 1
푟−푞 푇
푆 0푀 2 =
2 푒^2 푟−푞+푒(^2) 푇
푆 02
(푟−푞+휎^2 )(2푟−2푞+휎^2 )푇^2
- 2 푆 02
(푟−푞)푇^2
1
2 푟−푞 +휎^2
−
푒 푟−푞푇
푟−푞+휎^2
where r is the interest rate and q is the devidend yield and 푞 ≠푟.