Exercise: Radon-Nikodym in a Binomial Model — Pricing via Change of Measure

Prerequisites: Radon-Nikodym Theorem

Problem

Consider a two-period binomial stock model. At each period, the stock price multiplies by $u = 1.1$ (up) or $d = 0.9$ (down). The real-world probability of up is $p = 0.6$. The per-period risk-free rate is $r = 0$. The initial stock price is $S_0 = 100$.

1. Compute the risk-neutral up-probability $q = (1 + r - d)/(u - d)$.

2. List the eight elementary outcomes in $\Omega = \{u, d\}^2$ (there are only $4$ distinct outcomes in this 2-period model; list those). Compute $\mathbb{P}(\omega)$ and $\mathbb{Q}(\omega)$ for each.

3. Compute $Z(\omega) = d\mathbb{Q}/d\mathbb{P}(\omega)$ for each outcome. Verify $\mathbb{E}^{\mathbb{P}}[Z] = 1$.

4. Consider a European call with strike $K = 100$. Compute its price $C_0 = \mathbb{E}^{\mathbb{Q}}[(S_2 - K)_+]$ using (a) the direct $\mathbb{Q}$-expectation, and (b) the change-of-measure identity $\mathbb{E}^{\mathbb{Q}}[(S_2 - K)_+] = \mathbb{E}^{\mathbb{P}}[Z\cdot (S_2 - K)_+]$. Verify they match.

Hint

The $4$ distinct terminal prices are $S_2 \in \{121, 99, 99, 81\}$ — but note that two different paths ($ud$ and $du$) lead to $S_2 = 99$.