Quadratic variation

What the squared steps add up to

Take a Brownian path on $[0,t]$ and cut the time into $n$ equal pieces of length $h=t/n$. In each piece the path makes a random step $\Delta W$, a normal number with mean $0$ and variance $h$. Square every step and add the squares. For a smooth curve the sum shrinks to $0$ as the pieces get smaller, because its steps are of size $h$ and their squares of size $h^{2}$. A random path is different: each squared step is about $h$ on average and there are $n$ of them, so the sum settles at $n\cdot h=t$. This limit is the quadratic variation of the path.

This one fact is what makes stochastic calculus different. In ordinary calculus a tiny step squared is so small that you drop it. Here the squared step is as large as time itself, so you must keep it. It is also something you can measure. Add up the squared daily returns of a stock and you get an estimate of its variance over that period, which is exactly how realised volatility is computed.

• Quadratic variation: the sum of squared steps. It is $0$ for a smooth curve and $t$ for Brownian motion.
• Multiplication table: $(dW)^{2}=dt$, while $dW\,dt$ and $(dt)^{2}$ are $0$.
• Realised variance is the sum of squared returns. It estimates $\sigma^{2}$ times the length of the period.

Lab · what's graded

• Add the squared steps15 pts
• A yearly volatility15 pts
• How random is the sum?15 pts
• The rule for a squared step15 pts
• Match the vocabulary15 pts
• Estimating a yearly volatility10 pts
• Why Itô has an extra term15 pts