Lookback Pricing using FD

Pricing lookbacks using FD results in a complex PDE which has three variables. It can be solved but it's easier to reduce it to a 2D grid rather than a 3D grid. Similarity reduction as discussed earlier in Outperformance Option can be applied to lookbacks.

V(S,M,t) where M = Max(S)

Terminology: S = Spot, t = time

Payoff = Max(S,M) – alpha x k, where alpha = constant and k = strike

Let's assume S/M = y

V(S,M,t) => M V (S/M,1, t) [at time t]

=> M V (y,1,t)

which is a single variable PDE, lets choose a new symbol for this

M Vnew (y,t). At fixing times, we might have the following boundary conditional

fixing times = those times where the M = Max(S) function is applied

since V(S,M,t-) = V(S,max(S,M) ,t+) ==> This is from old PDE

Also max(S,M)/M = max(y,1)

Last equation is our variable Vnew using two variables y and t. Final price can be calculated by multiplying Vnew with S. Why S?