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Pricing Americans with finite-difference

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Re: Pricing Americans with finite-difference

#11

Post by Sniffnoy »

twic wrote: post #7
Sniffnoy wrote: post #4

Earlier quoted context omitted.

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 ca…

> I assume there is some real geographic origin to the naming convention, but I'm pretty sure both exist in both places. IIRC, the coiners of those terms were American, and called the simpler type European as a snub.

Hm, looking into it, I'm having trouble finding an easy answer. A quick search turned up this article [1] which says that yes Paul Samuelson named it that way for that reason, and links to this video [2] where he claims exactly that. But it also links to a source suggesting that the terms might predate Samuelson and be geographic in origin after all.

Well, if nothing else, it should at least be easy to check that (even if they existed elsewhere earlier) Samuelson thought he was independently inventing these terms, right? Except, in the paper where he supposedly invented these terms [3], he introduces them as follows:

> However, the simple integral (24) does give a solution under all cases to the simpler case of a warrant that can be exercised only at the end of the period T. We might call this a "European warrant" by analogy with the "European call," which, unlike the American call that is exercisable at any time from now to T, is exercisable only at a specified terminal date.

Note that nowhere previously does the paper make use of the "American" or "European" terminology, so it sounds like "European call" and "American call" here are references to pre-existing terminology -- suggesting that he didn't invent it after all! Huh.

Well, that got murkier than I expected. Don't really want to investigate further right now, but sounds like he didn't actually invent the terminology after all...?

[1] https://www.macroption.com/american-vs-european-options/

[2] https://www.youtube.com/watch?v=RbIzwTGN3Yc&t=11m

[3] https://link.springer.com/chapter/10.1007/978-3-319-22237-0_...

Re: Pricing Americans with finite-difference

#14

Post by 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 :)

Re: Pricing Americans with finite-difference

#15

Post by [deleted] »

Sniffnoy wrote: post #11
twic wrote: post #7

Earlier quoted context omitted.

> I assume there is some real geographic origin to the naming convention, but I'm pretty sure both exist in both places. IIRC, the coiners of those terms were American, and called the simpler type European as a snub.

Hm, looking into it, I'm having trouble finding an easy answer. A quick search turned up this article [1] which says that yes Paul Samuelson named it that way for that reason, and links to this video [2] where he claims exactly that. But it also links to a source suggesting that the terms might predate Samuelson and be geographic in origin after all. Well, if nothing else, it should at least be easy to check that (ev…

[deleted]

Re: Pricing Americans with finite-difference

#16

Post by sheepscreek »

hiAndrewQuinn wrote: post #3

"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?

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.

Re: Pricing Americans with finite-difference

#18

Post by scott00 »

quanto wrote: post #13
scott00 wrote: post #9

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.

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

Crank-Nicolson is probably the least objectionable part of the method, but I prefer ADE.

There are two numerically painful parts of the problem: the advection term and the oscillation inducing terminal condition (because it has a discontiuous derivative). I like to deal with advection by transforming the equation to an advection free equation. I'm under NDA on the best solution to the oscillatory terminal condition so I can't give that one away unfortunately.

Re: Pricing Americans with finite-difference

#19

Post by fakesson »

quanto wrote: post #13
scott00 wrote: post #9

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.

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

"High Performance American Option Pricing" by Leif Andersen et al is many orders of magnitude faster than any finite difference method or other PDE / tree method. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2547027

Given the exercise boundary, the American Option Price can be written exactly as a one-dimensional integral. That is the key insight to this superior method.

Re: Pricing Americans with finite-difference

#20

Post by scott00 »

fakesson wrote: post #19
quanto wrote: post #13

Earlier quoted context omitted.

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

"High Performance American Option Pricing" by Leif Andersen et al is many orders of magnitude faster than any finite difference method or other PDE / tree method. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2547027 Given the exercise boundary, the American Option Price can be written exactly as a one-dimensional integral. That is the key insight to this superior method.

Is that method extensible to discrete dividends?
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