5 ms·
Pricing Americans with finite-difference
- esafak 3y agoI'd rename "Americans" to "American options" for clarity.
- hiAndrewQuinn 3y ago"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 3y agoI 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.
- twic 3y ago> 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.
- Sniffnoy 3y agoHm, 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/ https://www.macroption.com/american-vs-european-options/ [2] https://www.youtube.com/watch?v=RbIzwTGN3Yc&t=11m https://www.youtube.com/watch?v=RbIzwTGN3Yc&t=11m [3] https://link.springer.com/chapter/10.1007/978-3-319-22237-0_11 https://link.springer.com/chapter/10.1007/978-3-319-22237-0_...
- deleted 3y ago[deleted]
- ravetcofx 3y agoMaybe 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.
- Sniffnoy 3y ago
- ffgjgf1 3y ago> 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).
- hiAndrewQuinn 3y agoThanks!
- leplen 3y agoAmerican 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.
- auc 3y agoThis 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.
- l33t7332273 3y agoWhy does having variable duration mean an equation cannot model this type of option?
- hfhdjdks 3y agoThe 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)
- sheepscreek 3y agoEuropean 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.
- gituliar 3y agoAs 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. :)
- scott00 3y agoThis 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 3y agowhat's a faster method than a Crank Nicolson implementation discussed in the post?
- scott00 3y agoCrank-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.
- quanto 3y agoIndeed, 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.
- scott00 3y agoYeah, 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.
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- Beijinger 3y ago"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 3y agoand 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
- Beijinger 3y ago"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 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 :-(
- luca3v 3y agoYou 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.
- smabie 3y agoYou 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
- mikrl 3y agoI’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 3y agoYou 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.
- mikrl 3y agoMaybe 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.
- keithalewis 3y agoIt 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.