Lesson 2 of 20 · 7 min

The intuition behind Black and Scholes, before any formula

The model prices an option by asking what it would cost to manufacture its payoff with stock and cash.

The insight that made option pricing tractable was not a better guess at probabilities. It was a change of question, from what is this worth to what would it cost to build.

Here is the idea. Suppose you hold some quantity of the underlying stock and some amount of borrowed cash, and you adjust that quantity continuously as the price moves. It turns out you can choose the quantity so that this package gains and loses almost exactly what the option gains and loses over the next instant. If two things have identical payoffs, they must have identical prices, or a risk-free profit exists.

So the option's price equals the cost of the replicating package. That is why the formula's two terms are what they are: one term is stock, the other is discounted cash. The model is not forecasting the stock. It is costing a manufacturing process.

This is also why the expected return on the stock does not appear anywhere in the formula, which surprises everyone at first. If you can replicate the payoff, your opinion about where the stock is heading is irrelevant to the cost of building the thing — the hedge adjusts as the price moves, and the replication works whichever way it goes. What remains is how far it wanders on the way, which is volatility, and the cost of the borrowed cash, which is the interest rate.

Hold on to the word continuously, because it is where the model's assumptions live. Continuous adjustment requires prices that move continuously, trading with no cost, and volatility that does not change. Lesson 19 examines what happens when those fail.

Key terms

Replication: Building an instrument's payoff out of simpler instruments. If two packages have the same payoff, they must have the same price.
Delta hedging: Holding the quantity of underlying that makes a package move with the option. The quantity is delta and it changes as the price moves.
No-arbitrage: The principle that a risk-free profit cannot persist. It is the argument that forces the option's price to equal the replication cost.

Worked example

One step of replication

Base case: underlying 100, strike 100, 30 days, rate 4%, volatility 25%. The model prices the call at $3.02 with a delta of 0.533.

Replicating package: 0.533 shares, part funded by borrowing
Underlying rises to $101 — package value change: +$0.533
Actual option value at $101: $3.58
Replication error over $1: $0.03
Underlying falls to $99 — package value change: −$0.533
Actual option value at $99: $2.52

The stock-and-cash package tracks the option closely over a small move, and the tracking error is what continuous re-hedging is for.

Check yourself

Why does the expected return of the stock not appear in the Black-Scholes formula?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.