Exercise: Computing $q$ in a Two-Period Binomial Tree

Prerequisites: Risk-Neutral Measure, Binomial Tree Model

Problem

Consider a two-period binomial model with the following parameters:

• Initial stock price $S_0 = 100$
• Up factor $u = 1.1$, down factor $d = 0.9$ (so each period multiplies the price by $u$ or $d$)
• Each period has length $\Delta t = 0.5$ years
• Continuously compounded risk-free rate $r = 0.04$ per year

1. Compute the single-period risk-neutral probability $q$ such that $e^{-r\Delta t}\mathbb{E}^{\mathbb{Q}}[S_{t+\Delta t} \mid S_t] = S_t$.
2. Draw (or write out) the tree of stock prices at times $0$, $\Delta t$, and $2\Delta t$. Compute the risk-neutral probability of each terminal state.
3. Price a European call with strike $K = 100$ and maturity $T = 2\Delta t = 1$ year.
4. Verify directly that under the real-world measure $\mathbb{P}$ with $p = 0.6$, the discounted stock price is not a martingale — compute $\mathbb{E}^{\mathbb{P}}[e^{-r\Delta t}S_{t+\Delta t} \mid S_t]$ and compare it to $S_t$.

Hint

For part 1, solve $q u + (1-q)d = e^{r\Delta t}$ for $q$. For part 2, note that the up-up, up-down, down-up, and down-down states have risk-neutral probabilities $q^2$, $q(1-q)$, $(1-q)q$, $(1-q)^2$, and the middle two states both land at the same stock price $S_0 u d$.