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The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is $$ V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right] $$ where the maximum is taken over all stopping times (exercise strategies $\tau>0$ permissible in the contract). With a PDE operator such as you have, the instantaneous ...

I would start with explaining random walk (this should be fairly simple) and then making a connection to heat equation in discrete time. This paper is doing exactly this and by leaving out technicalities you should make this pretty intuitive for students. Basically the intuition is as follows: At each integer time unit, the heat at each point is spread ...

I am not sure any of the other answers mentioned this but the main reason you should not use an option model to buy/sell the underlying (BS or other) is that the option models are more about market-making in options and hedging using the underlying rather than forecasting the underlying. The layman way to understand this is that: using an option model, you ...