4 ms·

An Introduction to Stochastic Calculus (2022)

173 points by ibobev 2y ago

fancyfredbot 2y ago
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.
- vcdimension 2y ago
  Yes, you need a good tutor to help you navigate through such a complex topic.
- werdnapk 2y ago
  Most web developers don't even know what a binary tree is, nevermind rebalancing one.
- mamonster 2y ago
  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.
  - almostgotcaught 2y ago
    > 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.
- mikrl 2y ago
  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.
  - v4nn4 2y ago
    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.
    - ogogmad 2y ago
      > The comment above is probably from a bot. Wtf Is this happening?
      - bee_rider 2y ago
        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.
        deleted 2y ago
        [deleted]
      - mdp2021 2y ago
        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 ;) .
    - 2y ago
- EGreg 2y ago
  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.
  - bormaj 2y ago
    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.
  - Agingcoder 2y ago
    It depends on where you are - many large banks have their derivatives pricing libraries written in c++ or c#.
- mdp2021 2y ago
  > 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.
- whatup480 2y ago
  [dead]
deleted 2y ago
[deleted]
LostMyLogin 2y ago
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?
- kachnuv_ocasek 2y ago
  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?).
- alphazard 2y ago
  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.
  - krackers 2y ago
    If you just want some intuition, I found this previous HN submission https://jiha-kim.github.io/posts/introduction-to-stochastic-calculus/ 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.
enthdegree 2y ago
Great post, "Wiener" is misspelled a few times.
- pmdulaney 2y ago
  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.
staunton 2y ago
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.
- potbelly83 2y ago
  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.
pieisgood 2y ago
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-begin-in-approaching-stochastic-calculus https://math.stackexchange.com/questions/231712/where-to-beg...
skzv 2y ago
I took "Mathematical Methods for Quantitative Finance" from edX, which covered these topics, and found it really rewarding.
- barbarr 2y ago
  Same!
DrNosferatu 2y ago
If, you assume everything is Gaussian…
FilosofumRex 2y ago
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!

173 points by ibobev 2y ago

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.

vcdimension 2y ago
Yes, you need a good tutor to help you navigate through such a complex topic.
werdnapk 2y ago
Most web developers don't even know what a binary tree is, nevermind rebalancing one.
mamonster 2y ago
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.
- almostgotcaught 2y ago
  > 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.
mikrl 2y ago
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.
- v4nn4 2y ago
  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.
  - ogogmad 2y ago
    > The comment above is probably from a bot. Wtf Is this happening?
    - bee_rider 2y ago
      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.
      - deleted 2y ago
        [deleted]
    - mdp2021 2y ago
      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 ;) .
  - 2y ago
EGreg 2y ago
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.
- bormaj 2y ago
  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.
- Agingcoder 2y ago
  It depends on where you are - many large banks have their derivatives pricing libraries written in c++ or c#.
mdp2021 2y ago
> 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.
whatup480 2y ago
[dead]

deleted 2y ago

[deleted]

LostMyLogin 2y ago

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?

kachnuv_ocasek 2y ago
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?).
alphazard 2y ago
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.
- krackers 2y ago
  If you just want some intuition, I found this previous HN submission https://jiha-kim.github.io/posts/introduction-to-stochastic-calculus/ 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.

enthdegree 2y ago

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

pmdulaney 2y ago
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.

staunton 2y ago

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.

potbelly83 2y ago
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.

pieisgood 2y ago

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-begin-in-approaching-stochastic-calculus https://math.stackexchange.com/questions/231712/where-to-beg...

skzv 2y ago

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

barbarr 2y ago
Same!

DrNosferatu 2y ago

If, you assume everything is Gaussian…

FilosofumRex 2y ago

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!