Pricing Americans with finite-difference
tastyhedge.com"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 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.
> 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_...
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).
This is really well researched and I enjoyed following along, thank you.
> 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).
Thanks!
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.
This is subtly incorrect.
An American option should be priced assuming that the option is optimally exercised, otherwise this would create a soft arbitrage opportunity. The difficulty is determining when the option is optimally exercised because it depends on several potentially unknown and difficult to model factors.
Why does having variable duration mean an equation cannot model this type of option?
The point is not whether it can be modeled (it can), but whether you can get a closed form solution for it (you can't...or we don't know how for now)
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. :)
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.
"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
You can see that the market deviates from the model and accounts for the fact that the Brownian motion model of Black-Scholes underestimates the probability of big moves: typically, options for the same security and the same expiration date have different IV, with options ATM having a lower IV and options deep ITM or deem OTM having larger IV.
If the market believed in the model, options for the same security and the same expiration date would all have the same IV, which would be whatever volatility the market thinks the security is going to have.
You can just input diff IVs for BS for diff strikes, you don't have to actually ditch BS.
After all vol is the free parameter for BS
That being said, not an expert on non-d1 products so I could be wrong about how this is dealt with in practice
"and yet, in practice, it doesn't matter that much (most of the time)"
It does not matter? https://en.wikipedia.org/wiki/Long-Term_Capital_Management
Options far out of the market are underpriced. Mandelbrot, investing on the stockmarket is riskier than you think. But then, the opposite must also be true. It can be more lucrative than expected. I currently hold some far out of the money options. Unfortunately, the underlying stock goes against me :-(
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.
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)
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...
I go through academic literature on a regular basis, hoping that some kind of really major improvement might magically appear. Usually the ideas are great, but they don’t survive real life equities markets ( from dividends to non convex payoffs, local vol etc )
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.
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.
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.
You can train a small NN on a bunch of synthetic data and the inference will be pretty fast.
Curious to see some references to people doing this. Have any ?
was thinking exactly this... and even better, why not on real data, especially since real, clean, data in this field is not at all hard to find.
Real historical data might not cover current market state.
For example, option price among other params depends on the interest rate. For the last decade interest rate was around 0% in Europe and slightly higher in US. If you train on that data only, there is no chance to "learn" option prices in the high-interest-rate environment which we saw for the last few years. Hence, you need synthetic data to learn that region of the market space.
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.
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 :)
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.
Maybe it’s the course I’m taking but a lot of the results are surprising to me.
> such and such is a martingale so this term goes to zero
This is why I dug into stochastic calculus and from there to measure theory, because it seems even the rigorous treatment of Brownian Motion springs out of Kolmogorov’s extension theorem… and every section I’ve read on optimal stopping is over my head rn.
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.
I'd rename "Americans" to "American options" for clarity.