Ito Calculus by peatar

Great idea for a thread, but I’m noting a serious lack of mathematics here so I’ll give it a go. Sorry for the sensational title, but it’s always hard to get people interested in mathematics.

(伊藤Ito calculus or Why Wall Street likes physicists

At first sight some sort of calculus for randomness, financial mathematics and physics may not have too much in common, so let me (very briefly) motivate theuse of randomness in financial mathematics (how it connects to physics will be dealt with later on) before it gets more technical:
Assume you want to model the price of some stock (let’s take some company that sells plants), i.e. you want to find a formula that takes everything you know about the market, your competitors, etc. and outputs what the prize of that stock will be tomorrow. Of course there is no way to do that with certainty (an unexpected drought may harm business, another competitor may get drunk and suddenly randomly start selling all of their stocks and in fact there are thousands of competitors so there are always some unforeseeable fluctuations, etc.), but not all is lost! We still can, at least to some degree, say something about how likely what outcome is, but I’ll leave the question on how exactly to model things to the financial mathematicians/economists.

What do we want to calculate? For now let us assume the size of interest is a function of something we cannot really control/predict (i.e. something we model as random). So we are interested in some F(B_t), where B_t is a Wiener process (think of it as a function that, for each time t, randomly moves in any possible direction with the same probability) and F is some smooth function (i.e. all derivatives exist and are bounded), then we are interested in the expectation EF(B_t) (=if we ran this random process an infinite number of times, what would its average be?).
How is it connected to physics? We find d/dt EF(B_t) = 1/2 E[d^2/dx^2 F(B_t)]. This basically follows from writing the Taylor expansion of F(B_{t+dt}) = F(B_t)+d/dx F(B_t)dB_t+1/2 d^2/dx^2 F(B_t)|dB_t|^2+O(|dB_t|^3), taking expectations on both sides and letting dt go to zero, while noting that EdB_t=0. Now this already looks almost like the heat equation and indeed, the probability density function ρ, implicitly defined by EF(B_t)=\int F(x)ρ(t,x)dx, is a solution to the heat equation d/dt ρ = 1/2 d^2/dx^2 (which can be seen by using the above result, the definition and integration by parts).
In particular the evolution of such a probability density function is given by the Fokker-Planck equation.
What does this have to do with Ito calculus? The “Ito-way” of putting this would be to write dF(B_t)=d/dx F(B_t)dB_t + 1/2 d^2/dx^2 F(B_t)dt. In fact the Ito calculus is something far more general and elaborate—maybe I will add a section on stochastic integration in the future.
How is this different from ordinary calculus? Usually this formula is compared to the usual chain rule dF(X_t)=d/dx F(X_t)dX_t which holds if the mapping t->X_t is smooth, but fails for Wiener processes since they are “only” (almost surely) continuous, but also (almost surely, almost everywhere) not differentiable, so the usual linearization argument X_{t+dt}≈X_t+dt*X’_t, used to derive the classical formulas, does not hold.

So how does this answer the question why Wall Street likes physicists?

  • The Ito calculus is used to derive the Black-Scholes equation (which entailed a Nobel (memorial) prize in economics in 1997) and people working with/on this model should know a quite a bit of the mathematical tools employed there (some of which happen to come from physics).
  • Obviously every physicist is (or at least should be) familiar with the heat equation and also have heard something about the Fokker-Planck equation. There’s also a fair share of them who still work on problems involving those formulas (the Fokker-Planck equation also appears in quantum mechanics) and hence are able to quickly grasp the underlying models used in financial mathematics. Apart from that, some of us mathematicians (who struggle to find fancy names like physicists do) simply call quantum mechanics “noncommutative probability theory” (there’s a reason for it!) and I think it’s somewhat reasonable to assume that probability theory has many applications in finance as well.

Warning: I don’t really know much about financial mathematics so if there are any economists around: Let me know about eventual mistakes.
Sources: Links in the text and Terry Tao’s blog.

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