EUROPEAN AVERAGE STRIKE OPTION

Question
Briefly describe the main features of a EUROPEAN AVERAGE STRIKE option, explaining in particular how average strike options differ from European
average rate and European vanilla options.
YOU MAY ASSUME that it the payoff of a path-dependent option is dependent on the quantities S(T) and
Z T
0
f(S(τ ), τ ) dτ
then the independent variable
I =
Z t
0
f(S(τ ), τ ) dτ
satisfies the stochastic differential equation
dI = f(S,t)dt.
Use this fact and Ito’s lemma (which you may use without proof) to show
that the value V = V (S, I,t) of such an option satisfies the partial differential
equation
Vt + f(S,t)VI +
1
2
σ
2S
2VSS + rSVS − rV = 0
where, as usual, V , S, t, σ and r denote respectively the option value, the
asset price, time, the volatility and the interest rate.
Show that in the case of the continuously sampled arithmetic average strike
the governing differential equation may be reduced to one involving only two
independent variables by setting
V (S, R,t) = IW(R,t)
where
R =
S
R t
0 S(τ ) dτ
and
R =
S
I
.