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.
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.
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.
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.
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.
