Black Scholes maths

gerry87

Would anybody be able to roughly explain the ito calculus/brownian motion/weiner process.

dXt =α(Xt,t)dt+σ(Xt,t)dWt

How I understand it so far:

Xt is say, a stock price process and is continuous in time but kind of discrete/kinked (what would be a better way to describe it?) so it's not differentiable/integrable. So to integrate it you break it down to two separate functions, α(Xt,t) which is a continuous trend term and σ(Xt,t) is a variance term (??). Wt is the wiener process within the variance term which you integrate with respect to.

Do I have this at all right? Would anybody be able to explain it to me better? What exactly is the wiener process?

Fremen

The wiener process represents "noise" in the calculation. If the coefficient of dW were set to zero, you would have a deterministic differential equation.

The dW notation you're using is really a kind of shorthand, it's not really rigorous. As you said, the wiener process has the property that is is continuous, yet nowhere differentiable. This is because if you pick a small portion of it and zoom in, it looks like another wiener process(it's a fractal), so it's jagged everywhere .

You can't differentiate a wiener process, but a guy named Kyoshi ito found a nice way to define integration with respect to the Wiener process. Hopefully you know what a normally distributed random variable is. If you have a function f(x,t) which you want to integrate with respect to W_t, (from 0 to T) you pick numbers t_1 to t_n (where t_1 = 0 and t_n = T) which for the sake of simplicity we'll assume are spaced out equally with spacing Delta_t.

The ito integral is then approximated by

Sum(from 1 to n) f(x,t_i)N_i(0, Delta t)

where N_i(0,Delta t) are independent, normally distributed random variables with mean 0 and variance delta t, indexed by i. Then as the number of terms in the sum increase and the spacings Delta_t go to 0, the value of this sum converges to the ito integral.

In other words, it's the limit of a linear combination of normal random variables.

Informally, you can think of your SDE as meaning "the change in X is given by some function a(x,t) by the change in time, plus the function sigma(x,t) times some normal random variable with a really small variance."

If sigma is deterministic, then the integral will be normally distributed. However, in your case, sigma depends on X_t, so it turns out that X is a lognormal process. You can show this using Ito's lemma on Y_t = log(X_t)

The wiener process is also known as Brownian motion, it might be worth reading the wikipedia page on that. Feel free to post here again if you have more questions.

Edit: If you're genuinely interested, I can't recommend "Stochastic calculus for finance Vol. II" by Steven Shreve highly enough. It's the best mathematics textbook I've ever read. It's self-contained with plenty of examples, and the maths isn't too intense. Should be accessible to a bright commerce student or maths undergrad.

gerry87

Great! Thanks for that, it's starting to make more sense, have to read through it a few more times before i can attempt some questions!

Is the wiener process a similar kind of thing to a riemann sum?

I'll definitely have a look around for that book.

Fremen

Think of a wiener process as a random function. With some constraints, every point on the function is a random variable, which is why the function is so jagged.

Another roughly accurate way to think of the wiener process is as a random walk. Suppose we're tossing a coin, and I pay you a euro every time there's a head, and you pay me one whenever there's a tail. After we've been playing this game for a looooong time, the graph of your wealth over time will look like a wiener process. More accurately, your wealth will look like one sample path of the wiener process. There are a huge number of other ways our game could have turned out, each corresponding to different wiener sample paths. The wiener process itself is the set of all possible wealth histories, and as such, the limit of a random step function.

Integration with respect to a wiener process works a bit like riemann sums though, yeah.