Friday, April 18, 2014

Non-linear Option Pricing

I am currently reading the book "Nonlinear Option Pricing" by J. Guyon and P. Henry-Labordère. It's quite interesting even if the first third is quite theoretical. For example they describe how to solve some not well defined non-linear parabolic PDE by relying on the parabolic envelope. They also explain why most problems lead to parabolic PDEs in finance.

The rest is a bit more practical. I stumbled upon an good remark regarding Longstaff-Schwartz: the algorithm as Longstaff and Schwarz describe it does not necessary lead to a low-biased estimate as they use future information (the paths they regress on) in the Monte-Carlo estimate. It was actually a subject of discussion with colleagues, and I analyzed the numerical impact in a simple use case in In short: it's actually more precise to include the path, even if the estimate is not purely low biased anymore, but the bias is really small in practice.

On the same subject I was a bit surprised that a recent paper on American Monte-Carlo regressed systematically on all paths instead of just a subset. One interesting part of the paper is a way to do successive regressions on different blocks of paths.

Those details are rarely discussed in papers and books. It was comforting to see that I am not alone to wonder about all this.

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