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Pricing Americans with finite-difference

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Re: Pricing Americans with finite-difference

#21

Post by fakesson »

scott00 wrote: post #20
fakesson wrote: post #19

Earlier quoted context omitted.

"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?

I googled and found slides there Leif extends the method to discrete dividends: https://www.math.cmu.edu/CCF/CCFevents/shreve/abstracts/L.An...

( I'm a fixed income quant, so I didn't look for it until now.) For a more advance model than Black-Scholes, e.g. local vol I don't expect it can be extended, and one would then need use some PDE based method.

Re: Pricing Americans with finite-difference

#22

Post by quanto »

scott00 wrote: post #18
quanto wrote: post #13

Earlier quoted context omitted.

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 s…

Indeed, a transformation (of some kind) is fairly standard, including the derivation for the standard analytic solution for European options.

AFAIK, discontinuous first derivative per se may act as a seed to an oscillation due to its high frequency content that are not captured by any finite resolution algorithm (n.b. Gibbs phenomenon). But it is Crank-Nicolson that characteristically creates these oscillatory problems -- in other words, there are algorithms that can gracefully handle the discontinuity without creating oscillation.

Re: Pricing Americans with finite-difference

#23

Post by smabie »

Beijinger wrote: post #12

"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.

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

Re: Pricing Americans with finite-difference

#24

Post by quanto »

fakesson wrote: post #21
scott00 wrote: post #20

Earlier quoted context omitted.

Is that method extensible to discrete dividends?

I googled and found slides there Leif extends the method to discrete dividends: https://www.math.cmu.edu/CCF/CCFevents/shreve/abstracts/L.An... ( I'm a fixed income quant, so I didn't look for it until now.) For a more advance model than Black-Scholes, e.g. local vol I don't expect it can be extended, and one would then need use some PDE based method.

Your intuition is quite correct. These methods (Leif et al) do not extend well to different boundary or intermediate conditions that are quite necessary in real life scenarios. AFAIK, there are a few teams on the Street that do fairly advanced numerical analysis, but most resort to Monte Carlo or some statistically-informed perturbation theory.

(I wish I could talk more, but yeah, legal obligations)

Re: Pricing Americans with finite-difference

#25

Post by scott00 »

quanto wrote: post #22
scott00 wrote: post #18

Earlier quoted context omitted.

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 s…

Indeed, a transformation (of some kind) is fairly standard, including the derivation for the standard analytic solution for European options. AFAIK, discontinuous first derivative per se may act as a seed to an oscillation due to its high frequency content that are not captured by any finite resolution algorithm (n.b. Gibbs phenomenon). But it is Crank-Nicolson that characteristically creates these oscillatory proble…

Yeah, the discretization interacts with the oscillation for sure. Full implicit is better than CN with regards to oscillation for instance, but I don't think would be a net win. Running a few implicit steps before switching to CN might help, though I've never tried it.

Re: Pricing Americans with finite-difference

#26

Post by fakesson »

quanto wrote: post #24
fakesson wrote: post #21

Earlier quoted context omitted.

I googled and found slides there Leif extends the method to discrete dividends: https://www.math.cmu.edu/CCF/CCFevents/shreve/abstracts/L.An... ( I'm a fixed income quant, so I didn't look for it until now.) For a more advance model than Black-Scholes, e.g. local vol I don't expect it can be extended, and one would then need use some PDE based method.

Your intuition is quite correct. These methods (Leif et al) do not extend well to different boundary or intermediate conditions that are quite necessary in real life scenarios. AFAIK, there are a few teams on the Street that do fairly advanced numerical analysis, but most resort to Monte Carlo or some statistically-informed perturbation theory. (I wish I could talk more, but yeah, legal obligations)

Monte Carlo might be ok to OTC derivatives, however for automatic market making of exchange traded option, which are mostly American, it would be too slow.

After a bit more googling, I found these more recent slides by Jesper Andersen, where he believes that the Leif et al method could be extended for local vol (see page 25): https://www.cqfinstitute.org/sites/default/files/4%20-%20Jes...

Re: Pricing Americans with finite-difference

#27

Post by logicchains »

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?

You can train a small NN on a bunch of synthetic data and the inference will be pretty fast.

Re: Pricing Americans with finite-difference

#28

Post by ravetcofx »

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…

Maybe Think of American options like concert tickets that you can use anytime before the show. Unlike tickets that only work on the day of the event (like European options), these tickets (American options) can be used whenever you want. Figuring out their price is tricky because you need to consider when it's best to use them, making it harder to have a simple formula like we do for tickets with fixed dates.

Re: Pricing Americans with finite-difference

#29

Post by Sniffnoy »

ravetcofx wrote: post #28
Sniffnoy wrote: post #11

Earlier quoted context omitted.

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…

Maybe Think of American options like concert tickets that you can use anytime before the show. Unlike tickets that only work on the day of the event (like European options), these tickets (American options) can be used whenever you want. Figuring out their price is tricky because you need to consider when it's best to use them, making it harder to have a simple formula like we do for tickets with fixed dates.

Sorry, did you mean to reply to a different comment? This doesn't appear related to what I was discussing (the etymology of the terms; I'm aware of what they mean).

Re: Pricing Americans with finite-difference

#30

Post by ffgjgf1 »

Sniffnoy wrote: post #4
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?

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…

> but I'm pretty sure both exist in both places.

Many big European companies have both American and European options available for them. I haven’t really seen any in the US (for individual stocks).

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