The Binomial Model Part 1: Building Intuition for Option Pricing
Chapter 15 of Wilmott’s book introduces the binomial model, and honestly it might be the single most important chapter for building intuition about how option pricing actually works. Forget stochastic calculus for a moment. This model uses nothing more than basic arithmetic, and yet it arrives at exactly the same answers as Black-Scholes.
The Simplest Possible Stock Model
Imagine a stock worth $100 today. Tomorrow it can do one of two things: go up to $101 or go down to $99. That is it. Two possible outcomes. The probability of going up is 0.6, and down is 0.4.
Now add a call option with a strike of $100, expiring tomorrow. If the stock goes to $101, the option pays $1 (101 minus 100). If the stock falls to $99, the option pays nothing.
Quick question: what is this option worth today?
If you said $0.60 (just the expected payoff: 0.6 times $1 plus 0.4 times $0), you fell into the trap. The correct answer is $0.50.
Why Expected Value Is Wrong
Here is where things get interesting. Build a portfolio: buy one option and sell half a share of stock.
If the stock goes up to $101: the portfolio is worth $1 (from the option) minus $50.50 (half of 101) = minus $49.50.
If the stock goes down to $99: the portfolio is worth $0 (option expires worthless) minus $49.50 (half of 99) = minus $49.50.
Both outcomes give exactly the same portfolio value. We built a risk-free portfolio. Since interest rates are zero in this example, it must be worth minus $49.50 today as well. That means: option value minus half of $100 = minus $49.50. So the option value = $0.50.
The probability of going up or down never entered the calculation. This is the fundamental insight that makes the entire derivatives industry possible.
Delta Hedging in the Binomial World
That number “half a share” we used for hedging? It has a name: delta (written as the Greek letter, capital delta). In general, delta equals the difference in option payoffs divided by the difference in stock prices:
Delta = (V+ minus V-) / (uS minus vS)
where V+ is the option value if the stock goes up, V- is the option value if it goes down, u is the up multiplier, and v is the down multiplier.
Delta tells you exactly how many shares to hold against your option to eliminate all risk. In our example: (1 minus 0) / (101 minus 99) = 1/2.
This is the same delta hedging concept from Black-Scholes, just expressed in the simplest possible setting.
Non-Zero Interest Rates
When interest rates are not zero, the logic is identical. You still delta hedge to create a risk-free portfolio. The only difference is that you now discount the future portfolio value back to today. If the risk-free rate is r, you multiply by the discount factor exp(-r times dt).
The option price changes slightly because of this discounting, but the probability of the stock going up or down still does not appear in the formula.
The Stock Is “Mispriced” on Purpose
Here is something that confuses people. If the stock has a 0.6 probability of going to $101 and 0.4 of going to $99, the expected stock price tomorrow is $100.20. But the stock trades at $100 today. Seems wrong?
It is not wrong. We pay less than the expected future value because the stock is risky. We demand a positive expected return as compensation for taking that risk. This is the whole foundation of the risk-return tradeoff you will see in portfolio theory.
The stock and the bank deposit (risk-free) sit on a risk-return diagram. If you split your money between them, you can reach any point on the line connecting the two. The option also sits on this same line, which means its payoff can be replicated using stock and cash.
Complete Markets
When the option lies on the same risk-return line as the stock and cash, we say markets are complete. This is a technical way of saying: options are redundant. Anything an option can do, a combination of stock and cash can do too.
The practical consequence is huge. If options can be replicated, they can be priced without knowing probabilities. You just need to know the range of possible stock prices (the volatility), not how likely each price is.
The Risk-Neutral World
Wilmott introduces a clever thought experiment here. Imagine a parallel universe where nobody cares about risk. In this “risk-neutral world,” people use simple expectations to price everything.
If the stock is at $100 and can go to $101 or $99, and people price by expectation alone, then the probability must be 50/50 for the stock to be “correctly priced” at $100.
Using these risk-neutral probabilities (50% up, 50% down), the option value is: 0.5 times $1 + 0.5 times $0 = $0.50.
The same answer we got by hedging! Two wrong assumptions (no risk aversion, and pricing by expectation) cancel each other out. This trick always works, and it is the basis of risk-neutral pricing.
When interest rates are non-zero, the risk-neutral probability is calculated from:
p’ = (exp(r dt) minus v) / (u minus v)
The risk-neutral probability p’ looks like the real probability p, except the drift rate mu is replaced by the interest rate r. The interest rate plays two roles: it is used for discounting, and it replaces the growth rate in the random walk.
The General Formula
Using symbols, the stock starts at S and either goes to uS (up) or vS (down). We pick u, v, and probability p to match the real stock’s drift and volatility. One common choice:
- u = 1 + sigma times sqrt(dt) + mu times dt
- v = 1 minus sigma times sqrt(dt) + mu times dt
- p = 1/2
The option pricing equation becomes:
V = exp(-r dt) times [p’ times V+ + (1 minus p’) times V-]
This says: the option value today is the discounted risk-neutral expected value of the option tomorrow. Simple as that. No partial differential equations, no stochastic calculus. Just weighted averages and discounting.
What Wilmott Loves and Hates About This Model
Wilmott is honest about the binomial model. He loves it for the intuition it gives. The ideas of delta hedging, risk elimination, and risk-neutral pricing come through clearly in a setting where you can work everything out with a calculator.
But he also calls the model “poor” as a description of real stock behavior. A stock that can only go to one of two prices tomorrow? Clearly unrealistic. And as a numerical method, he says it is “prehistoric” compared to modern finite-difference methods.
His advice: learn the binomial model for the ideas. Then move on to better tools for actual calculations.
Key Takeaways
- Option prices do not depend on the probability of the stock going up or down. Only the range of movement (volatility) matters.
- Delta hedging creates a risk-free portfolio by holding the right ratio of options and stock.
- Risk-neutral pricing gives the same answer as hedging. The option value is the discounted expected payoff under “fake” probabilities where the stock grows at the risk-free rate.
- Markets are complete in the binomial model, meaning options can be perfectly replicated with stock and cash.
- The binomial model is great for intuition but limited for real-world calculations.
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