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Good evening everyone, I would like to ask a question about Monte Carlo and PDE Pricing. For an American option, which one should we use, Monte Carlo method or PDE method? The same question for an Asian option such as an Asian call? As far as I know, PDE method have a downside which is the curse of dimensionality. However, I wonder whether this should be the main reason why Monte Carlo method is the favorite one? Thanks in advance!

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For American (or any HJB problem), numerical methods are depending on the dimensionality.

Below dimension 3 (even 4), a PDE will do the job nicely, whereas above, MC methods are more appropriate.

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There are many factors to consider. But mainly, in my opinion, you may choose the method depending on the complexity of the option and the resources you have. PDE method is usually used to solve problem whose complexity level is similar to problems you may solve using trees, and that using other approaches is not suitable.

In the other hand, using Monte Carlo allows you to consider any property you may think when creating an option, as long as you can model it.

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    $\begingroup$ It's not so straightforward to value American options with Monte Carlo. You don't know the exercise boundary ex-ante -- it has to be determined at each point along the path by optimization (do you exercise or hold). The LSM method of Longstaff and Schwartz is a method for approximately solving that optimization problem, but it's not so simple to implement. $\endgroup$ – Foster Boondoggle Sep 14 '16 at 15:33
  • $\begingroup$ @FosterBoondoggle I agree. Monte Carlo can be used for anything. However, it requires LSM or something like that for early exercising. The vanilla implementation can't handle it. $\endgroup$ – SmallChess Sep 15 '16 at 1:55
  • $\begingroup$ I didn't say it is straightforward, you need to model it.But as long as you can create the model, you can value it. In my old company we created a system to valuate all kinds of options by designing their payoffs. $\endgroup$ – arodrisa Sep 15 '16 at 18:49

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