Barriers already showed that a path feature can kill or activate a vanilla payoff; this topic takes the next step. Commodity desks, structured-equity books, and FX exotic traders routinely write contracts whose payoff depends on an average, an extremum, or a deferred choice — not just on whether a level was crossed. Those contracts still price as discounted risk-neutral expectations, but the state space grows: the average (or running max) becomes a second Markov factor, the PDE dimension rises, and Monte Carlo bias from discrete monitoring becomes a first-class modelling choice. Closing this topic fills the Asian treatment promised in the binomial / barrier overview and hands the smile / multi-asset frontier to the next topic and to advanced derivatives.
By the end of this topic you should be able to:
- distinguish arithmetic vs geometric Asians, and continuous vs discrete monitoring, and explain why geometric Asians admit a closed form under GBM while arithmetic ones generally do not;
- write the Asian pricing PDE with an average state variable and follow the similarity reduction that collapses it to one spatial dimension under GBM;
- classify fixed and floating lookbacks, state the reflection-principle pricing intuition under GBM, and name the discrete-max bias that appears in trees and Monte Carlo;
- price compound, chooser, and forward-start options by reduction to vanillas (or a small tree of vanillas) where the reduction applies;
- choose among PDE, tree, and Monte Carlo for a given path-dependent contract.
In this lesson
- Contents
- The big picture
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