Library

Stochastic calculus

8 items in the Thuztra Learn library carry the topic stochastic calculus: 5 reference articles, 1 interview question and 2 resources. They are gathered here so one idea can be met in every form it takes, which is usually the difference between recognising a term and being able to use it.

6 of them have pages of their own, linked below. Everything else is listed with the summary its author wrote. The one problem on this topic is set out under its own heading and quoted in full.

Problems in full

Quoted exactly as the library sets them, so the topic is something you can try rather than something you have read about.

Squares that settle, absolute values that do not

Let $W$ be a standard Brownian motion. Fix a horizon $T$ and cut $[0, T]$ into $n$ equal intervals of length $\Delta = T/n$ , writing $\Delta_i W = W_{(i+1)\Delta} - W_{i\Delta}$ .

Consider three sums:

Q_n = \sum_{i=0}^{n-1} \left( \Delta_i W \right)^2, \qquad V_n = \sum_{i=0}^{n-1} \left| \Delta_i W \right|, \qquad C_n = \sum_{i=0}^{n-1} \left| \Delta_i W \right|^3

Say what each one does as $n$ grows. For $Q_n$ , also give its variance at finite $n$ ...

The rest of the question is for members

Reference articles

The Bachelier model

Arithmetic Brownian motion for prices. The closed form, why negative rates and negative oil brought it back, and the assumption that fails on equities.

The Black-Scholes PDE

Where the pricing equation comes from, what the gamma-theta trade really pays, and the four assumptions that fail the moment a hedge goes on in a real book.

Ito's lemma

The chain rule for processes with non-zero quadratic variation, why the second derivative appears at first order, and the four places the theorem stops applying.

Quadratic variation

Earn the squared-increment limit properly, then watch Ito's lemma fall out of an exact algebraic identity. The second-order term is not a correction to anything.