Second Generation Exotics
The products introduced above in Chapter 8, while heavily traded in FX, all have a value at expiry VT that depends on some nontrivial combination of either the asset price at expiry ST or the continuously monitored minimum mT or maximum MT over the interval [0, T].
More correctly, we should say that we can handle an indicator function 1 on mT and/or MT by imposition of a suitable Dirichlet boundary condition on a backward time-stepping finite difference PDE scheme or a suitable extinguishing condition for a forward time-stepping Monte Carlo scheme. This means that we can price barriers, such as a double knock-out, with the value at expiry given by . The use of Dirichlet boundary conditions to handle KO barriers, which we may call weakly path-dependent products after the fashion of Wilmott (1998), extends to partial, forward and windowed barriers, as well as barriers with discretely observed fixings.
However, if we are asked to price up a fixed/floating strike lookback call or put, with payoff given by for ω = ±1, then the methods from Chapter 8 are inapplicable. We’ve basically reached the limits of how far we can go with a simple PDE with appropriately ...