The Binomial Model: Option Valuation Without Probability

Imagine you are trying to figure out the fair price of an insurance policy for your car. You know that there’s a chance your car might get into an accident, and the insurance policy will pay out if it does. To price this policy, you might think you need to know the exact probability of an accident happening to you personally.

However, insurance companies often use a clever trick. They don’t necessarily need to know the precise probability of your accident. Instead, they focus on creating a portfolio of policies and hedging their overall risk. This is somewhat similar to how the Binomial Model works to value options, especially in that it doesn’t rely on knowing the actual probabilities of a stock price going up or down.

Think of an option as a special kind of contract. Let’s say you have a call option to buy shares of a company at a certain price, say $100, in three months. If, in three months, the stock price is above $100, you can exercise your option and buy the stock at $100, making a profit. If the price is below $100, you simply let the option expire worthless.

The Binomial Model simplifies the complex world of stock price movements into a series of simple steps. It imagines that over a short period, the stock price can only move in two directions: up or down. Like flipping a coin – heads it goes up, tails it goes down. Over multiple periods, this creates a branching tree of possible stock prices.

Now, here’s the surprising part. To value this option using the Binomial Model, we don’t need to guess the real probability of the stock price actually going up or down in the real world. Instead, the model works by focusing on something called ‘replication’.

Imagine we want to create a portfolio that behaves exactly like the option. We can do this using just two ingredients: the underlying stock itself and risk-free borrowing or lending, like putting money in a savings account or taking out a loan.

Let’s say we can figure out how many shares of stock we need to buy or sell and how much money we need to borrow or lend to perfectly mimic the option’s payoff in both the ‘stock price goes up’ scenario and the ‘stock price goes down’ scenario. This is the replication strategy.

If we can create such a portfolio, then the cost of creating this portfolio must be equal to the fair price of the option. Why? Because if the option is priced differently from the cost of our replicating portfolio, there would be an arbitrage opportunity. Arbitrage is like finding free money. If the option is cheaper than our portfolio, we could buy the option and sell our portfolio, locking in a risk-free profit. Conversely, if the option is more expensive, we could sell the option and buy our portfolio.

This arbitrage principle is key. It forces the option price to converge to the cost of the replicating portfolio. And the beauty is, to calculate the cost of this portfolio, we only need to know the size of the up and down movements of the stock price, the risk-free interest rate, and the time to expiration of the option. We do not need to know the actual probability of the stock going up or down.

The Binomial Model uses what are sometimes called ‘risk-neutral probabilities’. These are not the real-world probabilities of price movements. Instead, they are artificial probabilities used in the calculation to make the expected return of the stock, in this simplified two-outcome world, equal to the risk-free rate. It’s like pretending the world is ‘risk-neutral’ for calculation purposes.

Think of it like baking a cake. You might use a recipe with specific ingredient quantities, but you don’t need to know the exact probability of each grain of flour landing perfectly in the bowl. The recipe, like the Binomial Model, provides a structured way to arrive at a desired outcome, in this case, the option price, without needing to predict real-world probabilities.

In essence, the Binomial Model cleverly side-steps the difficult task of predicting actual probabilities by focusing on creating a synthetic version of the option and valuing it based on the cost of its ingredients, the stock and risk-free borrowing/lending, under the assumption of no arbitrage. This is a powerful and insightful approach to option pricing.