An Introduction to Stochastic Calculus (2022) | Hacker News Reader

An Introduction to Stochastic Calculus (2022)

bjlkeng.io

173 pointsibobev44 comments

There's are widespread styles of teaching "mathy" subjects that completely ignore mathematical rigor (which is fine).

Often a way to do this (which I personally dislike, but it's also objectively "fine" teaching and can be done very well) relies on "manipulation of symbols" rather than "manipulation of mathematical objects". This is a bit like like learning programming in a language that has macros but no functions. Usually, this includes teaching a set of rules ("allowed manipulations") that allows proving a contradiction, the remedy being that you just don't, perhaps by relying on your "intuition" and knowledge of the problem domain (as opposed to just the math), which only comes with experience and isn't taught systematically.

The style of teaching that I find just intolerable pretends to be doing formal math, keeps telling you that rigor is important, floods you with definitions and terms, and then just does the "macro style of math" anyway, while skipping rigorous theorem statements (let alone proofs) entirely. Unfortunately, I find this article comes pretty close to this style.

Agreed. As someone with a PhD in pure math this is a pretty bad article, attempts to be informal, but then presents a bunch of theorem/defns. Not sure who his intended audience was here. Nicer approach would maybe try and approach the subject from a historical perspective, i.e. what were some of the original problems people were interested in.

Apparently this is the key to unlocking vast riches through a career as a derivatives quant. I'm told it's a requirement even though you don't really use it on the job. A bit like how you need to rebalance a binary tree to be a web developer.

Anyway now it's the key to unlocking vast riches through a career as an AI researcher too, seems like a good skill to have.

It's not extremely difficult(I mean for the most important results like Yamada-Watanabe, Girsanov, etc) if you have a good grasp on measure theory. That said, without that grasp this topic is very hellish.

The main problem for people is understanding intuitively what "quadratic variation" actually is and how that factors into the difference between a normal Riemann integral and a stochastic integral.

> not extremely difficult... if you have a good grasp on measure theory

If this were Reddit I would paste the "You got into Harvard Law? - Elle Woods" meme.

Ok it's not that hard - I did an independent study of Oksendahl in my junior year before my first measure theory class and understood most of it ok. But then again I didn't have to take exams on the material lol.

Not a quant, but I have physics training and I’m very curious about stochastic calculus and finance.

Isn’t it implicit in a lot of the work? If you’re modelling volatility you’ll need the rigorous mathematics in the back of your mind while you do so to keep you on track.

Similarly, a webdev isn’t going to use fancy tree algorithms often… but they need to understand the DOM and its structure.

Yes it's behind everything a derivatives quant would do. But I think quite a long way behind. Closed form analytic solutions using calculus only exist for relatively simple models and products. Most of the time you use it to calibrate and discretise a model and afterwards it's all Monte Carlo. What's more you can often just look that part up as models are increasingly commoditised rather than secret sauce.

Stochastic calculus is required to derive closed formulas and approximations used to calibrate SDE models. Similarly to deep learning, the secret sauce lies in the training, less in the inference. The code used by banks is closed source, and the research papers are missing said secret sauce. Calibrating models in a production environment handling correlation, multi-curves, stochastic funding, discrete dividends, etc. is not a solved problem. Interest rate derivatives modeling heavily relies on change of measure, even when using simple models.

Yes you need it, and no it’s not trivial. Not all quants need it on a daily basis though.

The comment above is probably from a bot. You do need an extensive understanding of stochastic calculus to maintain quant models code, let alone explain what it does to regulators.

The parent comment definitely violates the site guidelines.

How can you tell? They're missing the telltale sign — the em dash.

Good point. On a serious note, I probably overreacted, sorry about that. I have been working as a derivatives quant for a decade and thought the claim that stochastic calculus was not used/useful was ridiculous.

You're good bro. This is the internet; we're all here to have a good time.

I hate this em dash meme. Yes, using a totally normal bit of punctuation is a sure sign that something was written by a bot.

How so? A human is more likely to use a hyphen, an AI an em dash. Same with quotes - a human is much more likely to use ", an AI “ and ”. Typography is a differentiating signal when it's used dis-proportionally more by one group than another.

Word processors (less of an issue for Internet comments, but worth keeping in mind) but more significantly iOS (at least) and I assume Android will just swap in an em dash where needed—it is automatic.

There is probably some signal, but be a good Bayesian; we have people saying “oh, this is a bot” when there’s a huge population of mobile users with smart keyboards that are the more likely cause.

Anyway, in general I find bot-hunting annoying. Comments should be handled as comments, if someone has made a bad argument, it should be taken down as a bad argument. If it was bot-generated, it is still there to mislead people. The advantage that bots have is that they have infinite patience and nothing better in their lives to do than argue, but there have always been people like that, so hopefully readers will be able to observe that persistence!=correctness.

I'm using a stock Android keyboard - it doesn't. Perhaps that's were our differing perspectives originate. I'm updating away from AI, and toward iPhone users.

EDIT: I plugged in my prior, hit rate and false alarm rates from before updating and found that my P(AI|fancy-em) = 0.09. After updating my false alarm rate, P(AI|fancy-em) now = 0.016.

Oh wow, it is a convenient feature IMO.

> The comment above is probably from a bot.

Wtf

Is this happening?

People accusing comments they don’t agree with of being bots? Yes it has been happening for decades. Lots of folks are bad at arguing, so they make random accusations to distract from that fact.

Yes, it happens that some people create bots and have them post in these pages. They (some?) do not pass the "naïve Turing Test" though: there is one that tries to speak like an "inspiring lifecoach" and has zero juice squared. Check the shadowed posts around...

And on the other side, I have been accused a few times - writing outside expected canon (of form and content) can be sufficient.

So, bragging I will say, accusations hit both tails of the juice curve ;) .

Uh bruh. I took this class when I was 22 at NYU. Quadratic variation, brownian motion, and of course black-scholes etc. A lot of the work is based on a Japanese guy named Ito, who pioneered Ito integrals. And yes you need to know basic measure theory or probability as a prerequisite (take Math Analysis at least)

The closest I ever got to being a quant is doing an internship at a hedge fund called Concordia. They were just using Excel and VBA for credit default swaps back in the day. I then ended up at Bloomberg building their front end in C++ which st that time was a huge compiled binary.

I quickly exited that world and realized I enjoy building web applications. Had been doing that ever since. Guess turning $220 billion into $223 billion wasnt my idea of fun.

What you need as the key is Python, ML, SciKit, etc.

Adding to this, stochastic calculus matters more for modeling volatility/interest rates/derivatives. As you mention, Python/ML are more than suitable for many other areas within quant finance like optimization, algo development, signal research, etc.

It depends on where you are - many large banks have their derivatives pricing libraries written in c++ or c#.

Most web developers don't even know what a binary tree is, nevermind rebalancing one.

Yes, you need a good tutor to help you navigate through such a complex topic.

> now it's the key ... as an AI researcher

...For the moment. We will have to return to controlled processes at some stage - pure stochastic (using stochastic processes alone) is not adequate for precise questions requiring correct answers.

Only very little ago an LLM stated General Zhukov as German (probably because he had been the scourge of the German army - enough of a relation to make of something its substantive opposite in a weak mind). Imagine if we had that "method" applied to serous things.

Quant finance (let alone finance) is not math, and should not be studied as such.

Browse this excellent & concise book, which starts with a few practical problems to test your math background; if you pass, it'll take from Forwards, to Bermudan Swaptions in only about 150 pages!

Blyth, S.J. (2013), “An Introduction to Quantitative Finance”

Fun factoid - Blyth was the former head of Harvard's Endowment and Stats prof. He taught Stat-123 which was a jr level class at Harvard. He'd put on IR options trades via Bloomberg chat in the middle of his lectures in real time!

I took "Mathematical Methods for Quantitative Finance" from edX, which covered these topics, and found it really rewarding.

Same!

Does anyone have a solid road map of what to learn to get to the point where learning stochastic calculus is possible? I have a CS degree that was obtained 8-10 years ago. What are the prerequisites?

A few weeks ago I decided I wanted to get into this so I started self-studying probability theory (with measure theory) [0] as a bridge to start in on stochastic calculus [1]

I think the hardest part of self-studying anything that has some formal math foundations is knowing _what_ to pay attention to. There's so much in just the first chapter of the probability book. Is having a general understanding of set theory enough or should I actually know how to prove a function is a singular function?

That's why I often like to find a university course with lectures posted online so I can use that as a rough guideline for what's important, but I haven't quite found that yet for stochastic calculus. Would love if someone coul point me to one.

[0]: https://www.amazon.com/dp/3030976815 [1]: https://www.amazon.com/dp/9811247560

Same background here. I finally got into stochastic calculus last year thanks to a local college course (after several unsuccessful attempts on my own).

You need at least

1. a basic grasp of classical calculus, measure theory and topology

2. solid understanding of probability theory

3. basics of stochastic processes

I believe you should be able to dive in from there. It's good to have an idea where you're heading as well (mathematical finance and modelling and pricing derivatives? Bayesian inference and MCMC? statistical physics?).

If you want to understand the language of stochastic calculus as mathematicians have formalized it, then you need all of their jargon. Probability, Diff Eqs, Integrals, and Derivatives. If you are trying to tick a box on a resume, then that's what you have to do. If you have a CS degree then you have a little slice of Probability from combinatorics and information theory. You'll have to build up from there.

Stochastic Calculus was invented to understand stochastic processes analytically rather than experimentally. If you just want to build an intuition for stochastic processes, you should skip all that and start playing with Monte Carlo simulations, which you can do easily in Excel, Mathematica, or Python. Other programming languages will work too, but those technologies are the easiest to go from 0 to MC simulation in a short amount of time.

If you just want some intuition, I found this previous HN submission https://jiha-kim.github.io/posts/introduction-to-stochastic-... pretty approachable at giving you some key ideas without being too rigorous. It's not useful for doing calculating anything practical of course but it can either be a starting point or just a way to satisfy that curiosity.

You should learn calculus and differential equations, and then some probability. At that point you should learn a bit of measure theory and then stochastic calculus builds on all that. Stochastic calculus is basically just weird calculus. It has an additional differential dW and the chain rule is more complex (for the Ito formulation. Stratonovich is different but not by much)

From there you study the behavior of various forms of stochastic differential equations that are intended to model certain situations. Then, you make this cool connection between stochastic differential equations and ordinary differential equations that describe the evolution of the corresponding probability distributions. There’s lots of other stuff but those are the hits.

From a CS background, several people I know have raved about the following book[1], of which will be friendly and useful for future needs anyway in the field. The first part of the book is what appears to be a pretty good refresher path.

IMHO working through that book will make you practice with enough basic calc to make moving on to stochastic calculus fairly easy.

[1] Performance Modeling and Design of Computer Systems: Queueing Theory in Action - Mor Harchol-Balter

https://www.cs.cmu.edu/~harchol/PerformanceModeling/book.htm...

There's a book on financial calculus by Rennie and Baxter [0] that gave me very good intuition on the ideas behind option pricing. It starts with the binomial model and moves on to using stochastic calculus. If you get into the topic you'll want to read more in depth books, but this may be a good place to start.

[0] https://www.goodreads.com/book/show/307698.Financial_Calculu...

I'm not a practitioner, so read with some skepticism, but here's my list:

* Calculus

* Real Analysis

* Statistical Mechanics

* Probability

I'm not sure I have any good recommendations for Calculus, but for real analysis, I would recommend "The Way of Analysis" by Strichartz [0].

I don't have good recommendations for books on statistical mechanics, as I haven't found a book that isn't entrenched in coming from a physics perspective and teaches the underlying methods and algorithms. The best I can recommend is "Complexity and Criticality" by Christensen and Moloney [1], but it's pretty far afield of statistical mechanics and the like. Simulating percolation, the Ising model and ricepiles uses a lot of the same methods as financial simulation (MCMC, etc.).

For probability, I would recommend "Probability and Computing" by Mitzenmacher and Upfal [2], "Probability ..." by Durrett [3] and Feller Vol. 1 and 2 [4] [5] for reference.

I also would recommend "Frequently asked questions in Quantitative Finance" by Wilmott [6].

Also know that there's a quantitative finance SO [7] that might be helpful.

[0] https://www.amazon.com/Analysis-Revised-Jones-Bartlett-Mathe...

[1] https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-Colle...

[2] https://www.amazon.com/Probability-Computing-Randomization-P...

[3] https://www.amazon.com/Probability-Theory-Examples-Durrett-H...

[4] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[5] https://www.amazon.com/Introduction-Probability-Theory-Appli...

[6] https://www.amazon.com/Frequently-Asked-Questions-Quantitati...

[7] https://quant.stackexchange.com/

In the far past I was also curious about learning Stochastic Calculus. So I asked math stack exchange and got some thoughtful answers that may still be relevant. https://math.stackexchange.com/questions/231712/where-to-beg...

Great post, "Wiener" is misspelled a few times.

I remember that it's "ie" not "ei" by imagining (and this is probably true) that Wiener means "someone from Vienna", and I have no problem remembering how to spell Vienna.

If, you assume everything is Gaussian…

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