Common mistake 1 with Binomial option pricing

Using the real-world probability of an up move instead of the risk-neutral p.

Common mistake 2 with Binomial option pricing

Forgetting to compare continuation value with immediate exercise at every node for an American option.

Common mistake 3 with Binomial option pricing

Too few steps for a precise price. Three steps teach the method; a hundred or more are needed for accuracy.

Common mistake 4 with Binomial option pricing

Choosing u and d so that p falls outside (0, 1), which makes the tree admit arbitrage.

EconStatsCraft › Binomial option pricing

Binomial option pricing

Prices an option on a recombining tree: the up and down moves, the risk-neutral probability, every node for small trees, the early-exercise premium of an American option, and the price converging to Black–Scholes as steps are added.

The intuition

Over one short step the stock can only go up or down. With two outcomes, a mix of the stock and a bond can be built that pays exactly what the option pays in both, so the option must cost what that mix costs. Doing this step by step backwards from expiry prices the option without ever needing to know how likely an up move really is.

When to use it

Pricing American options, which Black–Scholes cannot handle, teaching risk-neutral valuation and replication, and any CFA or FRM question that asks for a one- or two-period tree.

Where the estimator comes from

Hold \Delta shares and B in bonds. Matching the option in both states gives \Delta=\frac{V_u-V_d}{S(u-d)} and B=e^{-r\Delta t}\frac{uV_d-dV_u}{u-d}. The option's value is \Delta S+B, which rearranges to V=e^{-r\Delta t}[pV_u+(1-p)V_d] with p=\frac{e^{r\Delta t}-d}{u-d}. Choosing u=e^{\sigma\sqrt{\Delta t}} and d=1/u (Cox–Ross–Rubinstein) matches the stock's volatility, and as \Delta t\to0 the European price converges to Black–Scholes.

The model

u = e^{\sigma\sqrt{\Delta t}},\quad d = \frac1u,\quad p = \frac{e^{(r-q)\Delta t}-d}{u-d},\qquad V = e^{-r\Delta t}\left[p\,V_u + (1-p)\,V_d\right]

What it assumes

The stock follows a recombining tree with constant volatility, so an up then down move lands where a down then up move does.
Markets are frictionless and the option can be replicated by the stock and a bond at every step.
p is a risk-neutral probability, not a forecast. Pricing never needs the real probability of an up move.
With enough steps the European price converges to Black–Scholes.

Statistical properties

p lies between 0 and 1 exactly when there is no arbitrage in the tree.
The price does not depend on the real probability of an up move or on investors' risk preferences.
American puts are often worth exercising early, because the strike received now earns interest.
Convergence to Black–Scholes oscillates with the number of steps, with an error of order 1/N.

How to read the output

Delta at the first node, (Vu − Vd)/(Su − Sd), is the number of shares that replicates the option over the first step.
The early-exercise premium is the American price minus the European price on the same tree. For a put deep in the money it can be large.
An American call on a stock with no dividends is never exercised early, so its premium is zero.
The convergence chart oscillates because the strike falls on or between nodes as steps are added. Averaging two neighbouring step counts smooths it.

A worked reading

A one-year American put struck at 100 on a stock at 100, with 20% volatility and a 5% rate, on a three-step tree. Each step lasts four months, so u = 1.1224, d = 0.8909 and the risk-neutral up probability is 0.5438. The tree prices the American put at 6.4996 and the European put at 6.1668, so the right to exercise early is worth 0.3328. Black–Scholes gives 5.5735 for the European put; the three-step tree is still 0.59 away, and the convergence chart shows the gap closing as steps are added.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Probability p outside 0 and 1	Steps too few for the rate and volatility	Increase the number of steps
American call worth more than European	A dividend yield is set	Correct: dividends can make early exercise optimal
Price jumps up and down as steps change	Strike falls between nodes	Use an even and an odd step count and average
Tree price far from Black–Scholes	Very few steps, or an American option	Use 200 or more steps for European pricing

Common mistakes

Using the real-world probability of an up move instead of the risk-neutral p.
Forgetting to compare continuation value with immediate exercise at every node for an American option.
Too few steps for a precise price. Three steps teach the method; a hundred or more are needed for accuracy.
Choosing u and d so that p falls outside (0, 1), which makes the tree admit arbitrage.

Where to read this properly

Cox, Ross and Rubinstein (1979), Option Pricing: A Simplified Approach.
Hull, Options, Futures and Other Derivatives (2021), ch. 13 and 21.
CFA Institute, Derivatives: Valuation of Contingent Claims (Level II).

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · Merton distance to default · EMI / home loan planner · CAPM / beta estimation · Multi-factor model · Portfolio optimization · Value at Risk & Expected Shortfall · Performance & drawdown · Time value of money & NPV/IRR · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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Binomial option pricing

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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