Binomial Option Pricing Model Explained Step by Step
The binomial option pricing model values an option by splitting the time to expiry into small steps and assuming that in each step the underlying can move only up or down. You calculate the option payoff at every end point, then work backwards to today using risk neutral probabilities.
Its main advantage over Black-Scholes is flexibility. Because you check the option value at every node, the binomial tree can price American style options where early exercise is allowed, and it can handle dividends that fall on specific dates.
The Building Blocks: Up Step, Down Step, Discounting
A one step tree needs three ingredients. The up factor u is the multiple applied if the price rises, the down factor d is applied if it falls, and the risk free rate discounts future values back to today.
A common choice sets u from volatility, with d equal to 1 divided by u, so the tree stays symmetric in percentage terms. If u is 1.05, then d is about 0.9524. Cox, Ross and Rubinstein published this construction, which is why it is often called the CRR model.
Risk Neutral Probability
The model does not use your view on direction. It uses the probability that makes the underlying grow at exactly the risk free rate, which is called the risk neutral probability p.
The formula is p equals the growth factor minus d, divided by u minus d, where the growth factor is e raised to the power rt for one step. This is not a forecast of how likely a rally is. It is the only probability under which no arbitrage exists between the option, the underlying and a risk free loan.
A One Step Nifty Example
Take Nifty at 24,000, a 24,000 strike call, one month to expiry, a risk free rate of 6.5 percent, and an illustrative up move of 5 percent. These are teaching numbers, not live prices.
| Item | Value |
|---|---|
| Up node price (24,000 x 1.05) | 25,200 |
| Down node price (24,000 x 0.9524) | 22,857 |
| Call payoff at up node | 1,200 |
| Call payoff at down node | 0 |
| Growth factor for one month | 1.0054 |
| Risk neutral probability p | 0.543 |
| Option value today | About 648 |
The last line comes from taking 0.543 times 1,200, then discounting that expected payoff back one month. One step is far too crude for real use, since it allows only two outcomes. Add a few hundred steps and the tree converges towards the Black-Scholes value.
Why the Tree Handles Early Exercise
This is the reason the model refuses to retire. Black-Scholes assumes exercise happens only at expiry, so it cannot value the right to exercise early.
In a binomial tree you already stand at every intermediate node. At each one you compare two numbers: the value of holding the option, and the payoff from exercising right away. Take whichever is higher, and the early exercise premium appears naturally.
- American puts can be worth exercising early, especially deep in the money when rates are high.
- American calls on a dividend paying stock can be worth exercising just before the ex-dividend date.
- Discrete dividends on known dates slot into specific nodes instead of being smeared as a yield.
Does This Matter for Indian Options?
Exchange traded options on NSE are European style, both index and single stock, so early exercise is not part of the picture and Black-Scholes fits well. Specifications do get revised, so check the current NSE contract specification page for lot size and settlement type rather than relying on a general article.
The tree is still worth knowing for three reasons.
- It makes risk neutral pricing intuitive in a way a closed form formula never does.
- Employee stock options in India are usually American style and are often valued with a binomial or lattice approach.
- An unusual or path dependent payoff is easier to model on a tree than in a formula.
Binomial Compared With Black-Scholes
| Feature | Binomial tree | Black-Scholes |
|---|---|---|
| Exercise style | European and American | European only |
| Output form | Numerical, step by step | Single closed form value |
| Discrete dividends | Handled directly at nodes | Approximated as a yield |
| Speed | Slower, depends on step count | Instant |
| Typical use | Teaching, American options, ESOPs | Screen pricing and implied volatility |
Frequently Asked Questions
How many steps does a binomial tree need to be accurate?
For a European option, a few hundred steps usually lands within a rupee or two of the Black-Scholes value. Accuracy improves with step count, so most implementations use several hundred and stop.
Is the risk neutral probability the real chance of a rally?
No, and confusing the two is the most common mistake with this model. It is a pricing device that forces the expected return to equal the risk free rate, so arbitrage becomes impossible.
Can I build this in a spreadsheet?
Yes, and it is a good exercise. A 10 step tree for a Nifty call fits in one sheet and shows clearly how time value shrinks towards the final column of payoffs.
Which model do brokers use for the greeks on my screen?
Almost always Black-Scholes or a close variant, since Indian options are European style and the closed form updates tick by tick across hundreds of strikes.
Key Takeaways
- The binomial model prices options by stepping a tree of up and down moves back to today.
- Risk neutral probability makes the underlying grow at the risk free rate, ruling out arbitrage.
- Checking value at every node is what lets the tree price American style early exercise.
- NSE listed options are European style, so verify contract specifications before assuming otherwise.
- With enough steps the tree converges to the Black-Scholes price for a European option.
