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Pricing Americans with finite-difference

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Re: Pricing Americans with finite-difference

#31

Post by Beijinger »

smabie wrote: post #23
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

"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 :-(

Re: Pricing Americans with finite-difference

#32

Post by hiAndrewQuinn »

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…

Thanks!

Re: Pricing Americans with finite-difference

#33

Post by gituliar »

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?

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. :)

Re: Pricing Americans with finite-difference

#34

Post by gituliar »

mikrl wrote: post #14

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…

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.

Re: Pricing Americans with finite-difference

#35

Post by gituliar »

logicchains wrote: post #27
quanto wrote: post #13

Earlier quoted context omitted.

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.

Curious to see some references to people doing this. Have any ?

Re: Pricing Americans with finite-difference

#36

Post by gituliar »

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.

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.

Re: Pricing Americans with finite-difference

#37

Post by luca3v »

smabie wrote: post #23
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

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.

Re: Pricing Americans with finite-difference

#38

Post by Reviving1514 »

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…

This is really well researched and I enjoyed following along, thank you.

Re: Pricing Americans with finite-difference

#39

Post by gully_boy »

logicchains wrote: post #27
quanto wrote: post #13

Earlier quoted context omitted.

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.

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.

Re: Pricing Americans with finite-difference

#40

Post by gituliar »

gully_boy wrote: post #39
logicchains wrote: post #27

Earlier quoted context omitted.

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

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.

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