Binomial Option Pricing

The binomial model is the friendliest way to price an option, and the one most quant interviews start with. Instead of reaching for stochastic calculus, you pretend the stock can do only one of two things over a short slice of time: go up by some factor, or down by some factor, like a biased coin flip. Stack enough of these slices together into a tree, and you can price almost any option by a simple rule: figure out what the option is worth at the very end, then walk backward through the tree one node at a time.

One step is the whole idea

Start with the simplest case, a single step. The stock is at $S_0$ today. Over the next slice of time it either rises to $S_0 u$ (the "up" state) or falls to $S_0 d$ (the "down" state), where $u > 1$ and $d < 1$ are the up and down factors. Our option pays $f_u$ in the up state and $f_d$ in the down state, whatever its payoff rule dictates. What's it worth now?

The trick is not to guess how likely "up" is in the real world. Instead we use the risk-neutral probability:

$$p = \frac{e^{r \Delta t} - d}{u - d}.$$

Here $r$ is the risk-free rate, $\Delta t$ is the length of the time step, and $u, d$ are the up and down factors. This $p$ is the special probability under which the stock, on average, just earns the risk-free rate, exactly the risk-neutral world where prices are discounted expected payoffs. The option's value today is then that discounted expected payoff:

$$f = e^{-r \Delta t}\big[\, p\, f_u + (1 - p)\, f_d \,\big].$$

Why is $p$ allowed to ignore the real odds? Because you can replicate the option by holding a mix of the stock and a bond, and no-arbitrage forces the option's price to match that replicating portfolio. When you grind through that algebra, the real-world probability cancels and out pops $p$. The model prices off replication, not prediction.

Building the tree

For many steps, the standard Cox-Ross-Rubinstein choice ties the up and down factors to the stock's volatility $\sigma$: set $u = e^{\sigma \sqrt{\Delta t}}$ and $d = 1/u$. Then you fill in the terminal payoffs and roll backward, applying the one-step formula at every node. The beauty is that the exact same machinery handles American options, at each node you just take the larger of "value of waiting" and "value of exercising now."

Worked example

Take a one-step tree with $S_0 = 100$, up factor $u = 1.1$, down factor $d = 0.9$, and, to keep the arithmetic clean, a risk-free rate of $r = 0$ over the step. Price a call with strike $K = 100$.

First the risk-neutral probability:

$$p = \frac{e^{0} - 0.9}{1.1 - 0.9} = \frac{1 - 0.9}{0.2} = \frac{0.1}{0.2} = 0.5.$$

Now the payoffs. In the up state the stock is $100 \times 1.1 = 110$, so the call pays $\max(110 - 100, 0) = 10$. In the down state the stock is $100 \times 0.9 = 90$, so the call pays $\max(90 - 100, 0) = 0$. Discount the expected payoff (with $r = 0$, no discounting):

$$f = 0.5 \times 10 + 0.5 \times 0 = 5.$$

The call is worth $5. Notice we never asked how likely the stock really is to rise, only the replication-implied $p = 0.5$ mattered. If you'd instead (wrongly) used a real-world 70% chance of the up move, you'd get $7 and be handing out free arbitrage.

From tree to Black-Scholes

Here's the satisfying payoff. As you slice time into more and more steps, each tiny and each move governed by $\sigma$, the jagged tree smooths into a bell-shaped distribution of final prices, and the binomial price converges to the Black-Scholes formula. The two models agree in the limit; the tree is just Black-Scholes with the calculus replaced by bookkeeping. That's why practitioners reach for the binomial (or its cousin, finite differences) whenever a closed form doesn't exist, such as early-exercise or dividend-timing problems.

Common pitfalls

• Using real-world probabilities. The single most common mistake. Only the risk-neutral $p$ belongs in the pricing formula.
• Forgetting to discount each step. With a nonzero rate you must multiply by $e^{-r\Delta t}$ at every backward step, not just once at the end.
• Choosing $u$ and $d$ that don't match volatility. For a multi-step tree to converge to the right price, $u$ and $d$ have to be calibrated to $\sigma$, arbitrary factors give a tree that prices some model, just not the one you intended.
• Ignoring early exercise. For American options you must compare with the exercise value at every node, not only at expiry, or you'll underprice the option.