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How can I compute the derivative of the payoff function for an American put option?

In the paper "Smoking adjoints: fast Monte Carlo Greeks" by Giles and Glasserman (2006) they compare two methods to calculate pathwise derivatives:

  1. Forward method
  2. Adjoint method

Both of these methods requires the derivative of the payoff function wrt. the parameter. E.g. to approximate the delta, one needs to compute

$$\frac{\partial g}{\partial X(0)}$$

where $g$ is the payoff function and $X(0)$ is the (spot) value of the underlying at time 0. However, they do not specify how this is done for American options. I am concerned that it very well might depend on the optimal stopping time $\tau^*$.

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    $\begingroup$ I am not sure their method works if there is an optimal stopping time. Perhaps Least Squares Monte Carlo might be more suitable here. I think you might be able do a bump and revalue (using the same random seed) for Americans in LSMC. The paper on LSMC for American options: people.math.ethz.ch/~hjfurrer/teaching/… $\endgroup$
    – Frido
    May 4 at 16:35
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    $\begingroup$ Thanks for suggestion Frido. I've implemented a bump-and reval approach using LSMC and also a finite difference solution. These solutions work just fine but the pathwise approaches seem to have some advantages on their own and may even be applied in an AAD-setup. Thus having massive improvements in both speed, memory and accuracy. $\endgroup$
    – Landscape
    May 4 at 20:43
  • $\begingroup$ Ah, sorry I couldn't be of more help. Wasn't aware you'd already tried LSMC. Then I suppose you've also already tried the available analytical approximations which eases Greeks calculations. I cannot see how the Giles and Glasserman can work with stopping times. Perhaps others may have some ideas. $\endgroup$
    – Frido
    May 4 at 21:29
  • $\begingroup$ I am not sure which "analytical approximations" you are referring to. Do you have any specific ones in mind? You might be on to something as the authors mention something about that more complicated payoff may be approximated by e.g. MC. To clarify the two pathwise approaches mentioned in the paper are already approximations as they use the Euler discretization. $\endgroup$
    – Landscape
    May 5 at 5:23
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    $\begingroup$ By analytical approx I meant for instance the Barone-Adesi-Whaley approx. But I'm not sure how this can be of use in a pathwise simulation context. Besides, the assumptions of BAW are quite stringent (constant vol etc). $\endgroup$
    – Frido
    May 5 at 8:30

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