Exercise: Building the Probability Space for a Two-Step Binomial Tree

Prerequisites: Probability Space, Binomial Tree Model

Problem

A stock is worth $S_0 = 100$ at time 0. At each step it moves up by a factor $u = 1.1$ or down by a factor $d = 0.9$. There are two time steps, so the tree has paths $uu$, $ud$, $du$, $dd$.

The risk-free rate is $r = 0.05$ per step (continuously compounded).

1. Write down the sample space $\Omega$ for this two-step model. What does each element $\omega \in \Omega$ represent?

2. Write down the full power-set sigma-algebra $\mathcal{F}$. How many elements does it have?

3. Under the real-world measure $\mathbb{P}$, each up-move has probability $p = 0.6$ and each down-move has probability $1 - p = 0.4$, independently. Assign $\mathbb{P}$ to each element of $\Omega$.

4. Find the risk-neutral measure $\mathbb{Q}$ by solving for the risk-neutral probability $q$ of an up-move at each step. (Assume the stock pays no dividends and that $d < e^r < u$.)

5. Compare $\mathbb{P}$ and $\mathbb{Q}$ on the event $A = \{uu\}$. Are they equal? Are they equivalent (i.e., do they agree on which events have probability zero)? What does equivalence of measures mean in this context?

Hint

For part 4, the risk-neutral condition is that the discounted stock price $e^{-r}S_t$ is a martingale under $\mathbb{Q}$. At each node, this gives: