Binomial Tree Option Pricing › The Cox-Ross-Rubinstein Model

The choice of up and down factors $u$ and $d$ determines how well the discrete model approximates continuous-time dynamics. These parameters are not arbitrary; they must be chosen carefully to ensure that the discrete binomial process converges to the continuous geometric Brownian motion as the number of steps increases. The Cox-Ross-Rubinstein (CRR) parameterization, introduced in their seminal 1979 paper, sets:

$$\begin{aligned} u &= e^{\sigma\sqrt{\Delta t}} \\ d &= e^{-\sigma\sqrt{\Delta t}} = \frac{1}{u} \end{aligned}$$

where:

• $u, d$: up and down factors
• $\sigma$: annualized volatility
• $\Delta t$: time step size ($T/N$)

The relationship $d = 1/u$ ensures that the tree recombines: an up move followed by a down move returns the stock to its original price ($ud = 1$). The exponential form reflects the multiplicative nature of stock returns, and the dependence on $\sigma\sqrt{\Delta t}$ ensures that the volatility of the discrete model matches the volatility of the continuous model.

The CRR parameters are chosen to match the first two moments of the log-return distribution under the continuous-time model. This moment-matching approach ensures that the discrete model faithfully reproduces the statistical properties of the continuous model it approximates.

In continuous time, over a small interval $\Delta t$, the log-return has:

• Mean: $(r - \frac{\sigma^2}{2})\Delta t$
• Variance: $\sigma^2 \Delta t$

The mean reflects the drift of the log-price process, which under the risk-neutral measure equals the risk-free rate minus a convexity adjustment. The variance reflects the accumulated randomness over the time interval and grows proportionally with time.

In the binomial model, the log-return is either $\ln(u) = \sigma\sqrt{\Delta t}$ (with probability $p$) or $\ln(d) = -\sigma\sqrt{\Delta t}$ (with probability $1-p$). These are the only two possible outcomes in the discrete model. The variance of this discrete log-return is:

$$\begin{aligned} \text{Var}[\ln(S_{t+\Delta t}/S_t)] &= p(1-p)[\ln(u) - \ln(d)]^2 \\ &= p(1-p)[\sigma\sqrt{\Delta t} - (-\sigma\sqrt{\Delta t})]^2 && \text{(substitute CRR parameters)} \\ &= p(1-p)(2\sigma\sqrt{\Delta t})^2 && \text{(simplify)} \end{aligned}$$

where:

• $S_t, S_{t+\Delta t}$: stock prices at time $t$ and $t+\Delta t$
• $p(1-p)$: variance of a Bernoulli variable
• $\ln(u) - \ln(d)$: difference in log prices between up and down states
• $2\sigma\sqrt{\Delta t}$: magnitude of the log price jump
• $p$: risk-neutral probability
• $\sigma$: volatility
• $\Delta t$: time step
• $u, d$: up and down factors

With the CRR choice and as $\Delta t \to 0$, we can approximate the risk-neutral probability using Taylor series expansions. This analysis reveals how the probability approaches a specific limiting value:

$$\begin{aligned} p &= \frac{e^{r\Delta t} - d}{u - d} \\ &\approx \frac{(1 + r\Delta t) - (1 - \sigma\sqrt{\Delta t} + \frac{1}{2}\sigma^2\Delta t)}{2\sigma\sqrt{\Delta t}} && \text{(Taylor expansion)} \\ &= \frac{\sigma\sqrt{\Delta t} + (r - \frac{1}{2}\sigma^2)\Delta t}{2\sigma\sqrt{\Delta t}} && \text{(simplify numerator)} \\ &= \frac{1}{2} + \frac{r - \sigma^2/2}{2\sigma}\sqrt{\Delta t} && \text{(divide by denominator)} \end{aligned}$$

where:

• $p$: risk-neutral probability
• $u, d$: up and down factors
• $r - \sigma^2/2$: drift adjustment term
• $\Delta t$: time step size
• $\sqrt{\Delta t}$: scaling factor for the drift
• $r$: risk-free interest rate
• $\sigma$: volatility

This derivation reveals that the risk-neutral probability approaches 1/2 as the time step shrinks, with a small correction proportional to $\sqrt{\Delta t}$. The correction term involves the drift rate $r - \sigma^2/2$, which is the expected growth rate of the log-price under the risk-neutral measure. This means $p(1-p) \to 1/4$ as $\Delta t \to 0$, giving variance $4 \times 1/4 \times \sigma^2 \Delta t = \sigma^2 \Delta t$, which matches the continuous-time variance.

Other parameterizations exist with different properties:

• Jarrow-Rudd (JR): Sets $p = 0.5$ exactly and adjusts $u$ and $d$ to match moments. This gives equal probabilities but asymmetric up/down moves.

• Tian: Matches the first three moments of the price distribution, potentially providing faster convergence.

For most practical purposes, the CRR parameterization works well and has become the standard choice. Its mathematical elegance, intuitive interpretation, and proven convergence properties make it the preferred approach in both academic and industry applications.