Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

BR: Ito Calculus – I

Time to get down to business. Having build up the necessary intuition that GBM seems to be the way to go, the next step is build up the toolkit to work with BM. We need to make sure that we can take derivatives, do integration and the like with in our Brownian world.

Well, the fact that it needs mention itself suggests that probably the rules of ordinary calculus won’t work. That’s correct. When dealing with BMs, we need to be extra careful when doing calculus. BR next develop the intuition of why ordinary calculus is not useful when dealing with BM.

BR take the examples of some simplest Newtonian differentials and remind “…that though ODEs are powerful construction tools, they are also dangerous ones. There are plenty of bad ODE which we haven’t a clue how to explore”.

In fact, I very much liked their example where they zoom in on a general curve, and ‘show’ that differentiable functions “…are at heart built from straight line segments”.

Their further discussion is fine, and helps ease one into thinking about ODEs – but from that point on, I think it’s sufficient to argue that because BMs are nowhere differentiable, this makes it even more difficult to rely on Newtonian calculus – zooming in on a BM doesn’t produce a straight line! It’s self similar.

But self similarity, BR note, in itself is not a curse. In fact:

…this self similarity is ideal for a building block – we could build global Brownian motion out of lots of Brownian motion segments. And we could build general random processes from small segments of Brownian motion (suitably scaled). If we built using straight line segments  (suitably scaled) too, we could include Newtonian functions itself.

From there they take the jump to introducing the first Stochastic Differential Equation (SDE) [till this point they are talking the language of ODEs, and it’s not too abrupt]:

$dX_t = \sigma_t dW_t + \mu_t dt$

The “noisiness” $\sigma_t$ is called the volatility, and $\mu_t$ the drift of the process $X_t$. For processes such as above to be well-defined, it’s required that the variables of $X_t, \sigma_t, \mu_t$ depend only on the history/filtration $F_t$, but not the future. In that case, the variables are called $F_t - adapted$.