Pricing Americans with finite-difference
tastyhedge.com
tastyhedge.com
I'm fascinated by this. Why not? Is it some kind of regulation thing?
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.
IIRC, the coiners of those terms were American, and called the simpler type European as a snub.
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_...
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).
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.
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 assumes a random walk/Gaussian distribution. This assumption is basically flawed, since it is a Levi flight.
models just have to useful, they don't have to be correct
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.
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
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 :-(
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.
( 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.
(I wish I could talk more, but yeah, legal obligations)
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...
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.
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.
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.
The finite-difference covers wider range of problems, including stochastic volatility models, like SABR or Heston.
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 :)
> 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.