CRR And Jarrow-Rudd Tree Parameters

A binomial tree needs three numbers per step: how far up, how far down, and with what probability. There are infinitely many ways to pick those three numbers so that, as the number of steps grows, the tree converges to the same lognormal distribution. Two choices became standard because each simplifies a different piece of arithmetic — Cox-Ross-Rubinstein (CRR) makes the up and down moves mirror images of each other, and Jarrow-Rudd (JR) makes the probability of each move exactly one half.

Two ways to draw the same grid

Imagine drawing graph paper for a hiking map. One convention makes every grid square exactly the same size and lets the hiker's odds of going left or right vary by terrain (CRR: fixed step sizes, solved-for probability). A second convention insists the hiker always has 50/50 odds at every junction, and instead makes some grid squares wider than others to reflect that (JR: fixed 50/50 probability, solved-for step sizes). Both maps describe the same mountain in the end; they just put the complexity in different places.

$$\text{CRR:}\quad u = e^{\sigma\sqrt{\Delta t}}, \; d = \frac{1}{u}, \; p = \frac{e^{r\Delta t} - d}{u - d}$$ $$\text{Jarrow-Rudd:}\quad u = e^{(r - \frac12\sigma^2)\Delta t + \sigma\sqrt{\Delta t}}, \; d = e^{(r - \frac12\sigma^2)\Delta t - \sigma\sqrt{\Delta t}}, \; p = \tfrac12$$

In plain English: CRR fixes $u$ and $d$ as symmetric reciprocals ($u \times d = 1$) and solves for whatever probability $p$ makes the tree's average growth rate match the risk-free rate. Jarrow-Rudd instead fixes the coin flip at exactly fair odds and shifts the up and down move sizes themselves — both by the same drift term, offset by $\pm\sigma\sqrt{\Delta t}$ — so that the average of the two outcomes matches the right growth rate instead.

Worked example 1: comparing the two parameterisations. Take $\sigma=25\%$, $r=5\%$, $\Delta t=1$ month $=1/12$. For CRR: $u=e^{0.25\sqrt{1/12}}=e^{0.0722}=1.0749$, $d=1/1.0749=0.9303$, and $p=(e^{0.05/12}-0.9303)/(1.0749-0.9303)=(1.00417-0.9303)/0.1446=0.0739/0.1446=0.5111$. For Jarrow-Rudd: the drift term is $(0.05-0.03125)/12=0.001563$, so $u=e^{0.001563+0.0722}=e^{0.0737}=1.0765$ and $d=e^{0.001563-0.0722}=e^{-0.0706}=0.9318$, with $p$ fixed at $0.5$ exactly. Notice CRR's up-move (1.0749) and JR's up-move (1.0765) are close but not identical — CRR pushed the asymmetry into probability (0.5111 vs 0.5), JR pushed it into the move sizes instead.

Worked example 2: they still converge to the same price. Price a 3-month at-the-money call, $S=K=100$ (a $100 stock), using both trees with, say, 50 steps each. The Black-Scholes benchmark is about $5.71. Both CRR and JR trees typically land within a few cents of that value at 50 steps, and both converge to it exactly as the step count grows — the difference between them shows up in how they converge (CRR tends to oscillate around the true value from one step-count to the next; JR tends to approach more smoothly but can be more sensitive to a very coarse number of steps), not in what they converge to.

Both trees are discrete approximations of the same continuous path shown above — as steps increase, the CRR ladder and the JR ladder both flatten into this same random walk.

What this means in practice

CRR remains the default in most textbooks and production pricers because its symmetric $u=1/d$ structure is simple to code and recombines cleanly (an up-then-down move lands on the same node as a down-then-up move). Jarrow-Rudd is preferred when a fixed, intuitive 50/50 branching probability matters more than symmetric step sizes — for instance, when building the tree by hand or explaining it to someone new to the method.