Exercise: The Sigma-Algebra Generated by an Indicator

Prerequisites: Sigma-Algebras, Random Variables

Problem

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $A \in \mathcal{F}$ be a fixed event. Define the indicator random variable:

1. Describe $\sigma(X)$ explicitly — list every set it contains.
2. How many elements does $\sigma(X)$ have? How does your answer change if $A = \emptyset$ or $A = \Omega$?
3. Now consider a digital call on a stock with payoff $D = \mathbf{1}_{\{S_T > K\}}$ where $S_T$ is the terminal stock price and $K$ is the strike. Describe $\sigma(D)$ in terms of events involving $S_T$, and explain what information about $S_T$ is encoded in $\sigma(D)$ versus $\sigma(S_T)$.

Hint

For part 1, think about which Borel sets of $\mathbb{R}$ pull back to non-trivial subsets of $\Omega$ under $X$. Only a handful of cases matter — $\{0\}, \{1\}, \{0, 1\}, \emptyset$ — because $X$ only takes two values.