# Have you gone through stochastic calculus

Curious to know how many of you have gone through stochastic calculus and how many of you could understand the concepts… I have always struggled through it and still do to actually understand it, though I would very much like to. (not that I am a math geek, but quite good at math )… Do people actually use stochastic calculus or something even close to it in their jobs.?

I’ve never taken it, but I’d find it interesting.

I have never done a course in it, but I have attacked some stochastic calculus books from time to time. I find it difficult, but if you go over it again and again and again it starts to make sense. (Reminds me of a physics prof of mine who talked about learning quantum mechanics: “You just read it over and over and over again, and one day, it just makes sense.”) However, when I stop using it, it all disappears and I have to read up again to rememberify it.

Paul Wilmott Introduces Quantitative Finance is a good intro to a lot of that stuff. It’s not a course that derives everything from first principles, but it does do a good job of teaching you enough to apply it in many of its financial applications.

Stochastic calculus is mostly used in derivative pricing, and basically is about how you integrate over probability distributions that changes with time in order to come up with things like expected values and other statistics. From what I understand, you basically you end up having to integrate over two variables simultaneously (over time, and over probabilities) and in order to integrate over probabilities, you have to add some extra terms in your equations that look unintuitive to the uninitiated. From there on in, it’s mostly regular calculus, plus seeing how those extra terms change depending on what probability distribution you’ve decided to use and how to incorporate more than one random variable, etc.

I haven’t taken a class dedicated to it, but we covered Brownian motion and Ito in a derivatives class I took and have skimmed through Shreve’s books on the subject (skimmed because I don’t have the time to understand everything in it).

Unless you’re doing stuff with derivatives (or maybe graduate level work in asset pricing or something) you probably don’t need it.

As I mentioned above, I have mainly seen it used wrt options. Ito’s Lemma is like a chain rule for stochastic processes. For instance, if you have a stochastic process and create something that is a function of a stochastic process, then you can use Ito’s lemma to get the derivative of the function (well really the stochastic differential equation). If you assume the value of an option follows some function (as yet unstated), Ito will give you its SDE. The next step is to make some assumptions about holding a portfolio of the equity, cash, and some of the option (with the amount of the option held so that the value of the portfolio is always 0) and you can then derive the Black Scholes PDE.

Edit: I would second bchad’s recommendation of Paul Wilmott Introduces Quantitative Finance. That is by far the clearest book on derivatives I have read.

Hacksaw, all of you!

Yes, I took it as an undergrad. Failed it the first time and got a B+ when I retook it the next semester. I don’t remember much, but it was helpful in my derivatives course in grad school.

^What did you major in that required stochastic calculus? Or did you inflict that misery upon yourself as an elective?

It’s unusual to take stochastic calculus as an undergraduate. So, I am guessing that this was some kind of weird elective class.

Anyway, from my experience, people don’t actually use stochastic calculus much unless they are true quants or they work in a small place where people must perform multiple roles. In other places, there is usually someone else who spends a lot more time doing math and is thus better and more efficient than you.

I do think that it is worth going through the math concepts though. Even if you miss some finer points of quantitative documentation, it’s good to have a baseline level of knowledge that will allow you to understand the qualitative effects of certain models, as well as their flaws or assumptions.

A history of learning math also cumulatively adds to your quantitative ability, even if you forget specific things that you learned. You probably don’t use Pythagoras theorem in your job, but learning that, along with other math things as a kid, probably helped you think in math ways.

I was originally a dual major: math and chemistry.

Stochastic calculus is almost never used. What might be used in some quant shops is certain applications of it, typically very narrowly defined and memorized or adapted from textbooks. A lot of quant finance books and quant finance courses are also structured that way, spitting out definitions and formulas, outlining procedures and algorithms for pricing derivatives and providing no intuition behind the concepts or motivation for their development. It is a totally useless knowledge base unless you are actually working in a quant shop that requires you to commit certain peices of information to memory, so that you can use them on a daily basis kind of like a robot. That is probably due to the fact that stochastic calculus is a relatively young field compared to other branches of mathematics, and the literature and teaching methods for it are immature and underveloped.

Understanding the concepts of stochastic calculus can be very rewarding if you are fond of mathemtics and is actually useful compared to the formula memorization approach that is prevalent in most quant courses. In terms of intuition, I’d start here:

The fundamental theorem of calculus provides the following first-order approximation to a function which links derivatives and integrals: dF(X)=F’(X)dX. This works for some nice deterministic functions of bounded variation - that means that the arc length of F(X) over any interval is finite. So when you zoom in very close, it looks like a straight line.

But if X is a stochastic process, no matter how closely you zoom in, F(X) looks like a copy of itself on a larger scale (fractal) rather than a straight line. The arc length of F(X) is not finite, and first-order approximation doesn’t work any more because locally F(X) doesn’t look like a straight line. However, you can define certain “nice” stochastic processes which have bounded quadratic variation (ok - it’s random, but not too random) - then you need to change the fundamental theorem of calculus above to include a second order approximation term: dF(X)=F’(X)dX+(1/2)*F’’(X)(dX)^2. That gives rise to a new, more general definition of the ordinary integral that incorporates new classes of functions.

Done it over 10 years ago and retained less than 1%.