Exercise: Two-state binomial pricing via risk-neutral valuation

A one-period binomial model: $S_0 = 100$. At time $T = 1$, $S_T \in \{120, 80\}$ (up or down). Risk-free rate $r = 0$ (so $B_t = 1$). Real-world probability of "up" is $p = 0.6$.

Tasks

1. Find $\mathbb{Q}$. Determine the risk-neutral probability $q$ of "up" such that $S_t$ is a $\mathbb{Q}$-martingale, i.e., $\mathbb{E}^{\mathbb{Q}}[S_T] = S_0$.
2. Verify $\mathbb{Q} \sim \mathbb{P}$. Are $\mathbb{P}$ and $\mathbb{Q}$ equivalent? Compute the Radon-Nikodym derivative $d\mathbb{Q}/d\mathbb{P}$ on each state.
3. Price a call with strike $K = 100$. Use risk-neutral valuation directly.
4. Replicate. Find the replicating portfolio $(\phi^S, \phi^B)$ — units of stock and bond — and verify the cost equals the price from (3).
5. Sanity check. What happens if you instead used $p = 0.6$ (the real-world probability) to price? Why does the answer differ from the risk-neutral price, and which is "correct"?