Pricing Americans with finite-difference • Hacker News Board

Re: Pricing Americans with finite-difference

#41

Post by Agingcoder » Sat, Dec 23, 2023, 7:04 PM UTC

quanto wrote: post #24

fakesson wrote: post #21

Earlier quoted context omitted.

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)

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 )

Re: Pricing Americans with finite-difference

#42

Post by mikrl » Sat, Dec 23, 2023, 8:48 PM UTC

gituliar wrote: post #34

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.

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.

Re: Pricing Americans with finite-difference

#43

Post by smabie » Sat, Dec 23, 2023, 10:04 PM UTC

luca3v wrote: post #37

smabie wrote: post #23

Earlier quoted context omitted.

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…

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

Re: Pricing Americans with finite-difference

#44

Post by logicchains » Sun, Dec 24, 2023, 8:43 AM UTC

gituliar wrote: post #35

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.

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

https://www.aimspress.com/aimspress-data/qfe/2023/2/PDF/QFE-... is an example