Pricing Americans with finite-difference

[hiAndrewQuinn]

"Pricing American options is an open problem in the quantitative finance. It has no closed form solution similar to the Black-Scholes formula for European options."

I'm fascinated by this. Why not? Is it some kind of regulation thing?

[Sniffnoy]

I think you've misunderstood the terminology a bit. "European options" and "American options" don't mean "options in Europe" and "options in America"; they're just names for two different styles of options. I assume there is some real geographic origin to the naming convention, but I'm pretty sure both exist in both places.

Other geographic naming styles for options are definitely more arbitrary; Asian options are called that simply because they were invented in Tokyo, for instance, rather than necessarily being particularly common in Asia. Meanwhile Bermuda and Canary options are called that because they're somewhere inbetween American and European options in terms of how they work; they have no real connection to Bermuda or the Canary islands.

[leplen]

American options can be exercised at any time at the investor's discretion. This means the instrument has a maximum duration, but the actual duration is up to choice of the option holder, which you can't model with an equation.

[gituliar]

As already mentioned American / European / Bermuda / Asian options vary by features of option contracts. They don't belong to a particular regulatory regime. For some reason its names are associated to the geo locations. I'm sure there are people who can wrap that in a nice narrative. :)

[sheepscreek]

European style option contracts are exercisable/allocated only at the time of expiration.

Whereas American style option contracts can be exercised at any time up to the time of expiration.

The Black-Scholes formula is only applicable to European style option contracts.

[Beijinger]

"The Black-Scholes equation"

The Black-Scholes equation assumes a random walk/Gaussian distribution. This assumption is basically flawed, since it is a Levi flight.

[smabie]

and yet, in practice, it doesn't matter that much (most of the time)

models just have to useful, they don't have to be correct

[scott00]

This method will work but will require a large grid and consequently be quite slow. And order of magnitude or two faster than this is possible if you are clever.

[quanto]

what's a faster method than a Crank Nicolson implementation discussed in the post?

[gituliar]

Indeed, Andersen-Lake method is much faster, but is limited to vanilla problems.

The finite-difference covers wider range of problems, including stochastic volatility models, like SABR or Heston.

[mikrl]

I’ve been taking an online course in mathematical finance although it’s mostly analytic, so not much in the way of numerics and all of the options are European / fixed term.

Thanks for the article! It will be interesting to see how early exercise affects the PDE solutions.

Stochastic calculus is a few levels above undergrad physics, but it has motivated me to understand measure theory when before I couldn’t make head nor tail of it. Having a concrete end is a fantastic motivator :)

[gituliar]

You don't need fancy math to study financial modeling, like measure theory. Focus on market dynamic, like in physics try to model that with math and programming. Newton built solid models without advanced math of XX or XIX centuries. Of course, some advanced effects require advanced math, but those are built on top of simple theories, like General Relativity on top of the Newton theory.

[keithalewis]

It is encouraging to see posts by people trying to teach themselves well-known theory. It used to be college professors writing a book to do that. There is no longer a barrier to entry for randoms.

[esafak]

I'd rename "Americans" to "American options" for clarity.