I am stuck at a assignment problem where I have to compute the price of an exotic option.
I am given the values the prices of option $C(X;k) = E[max(0,X_T - k)]$ for different strike prices $k$ and I have to compute the exotic price
$D(X;k) = E[I(X_T > k)]$ for the same set of $k$s.
I did this using difference of sum and got
$D(X;k) = C(X;k) - C(X;k+1)$
Now using these I have to compute another exotic price:
$P(X;k) = E[max(X_T - k,0)^2]$
I tried using the same method to get :
$P(X;k) - P(X;k-1) = 2C(X;k) - D(X;k+1)$
but to numerically calculate $P(X;k)$ at some k, I need the value of $P(X;k+1)$ and I have no initial value for the pricing $P$.
Any help would be great