Skip to content
Financial Instruments, Markets and Conventions / A First Look at Options: Payoffs, Bounds, and the Binomial Model

The One-Period Binomial Model: Replication, Delta Hedging, and Risk-Neutral Pricing

The no-arbitrage bounds of the previous note are constraints on option prices but not formulas. To price an option exactly, we need a model — a description of how stock prices evolve. The one-period binomial model is the simplest possible such model: the stock can move to one of exactly two values over the option's life.
That simplicity is not a weakness. It is precisely the two-outcome structure that allows us to replicate the option exactly with a portfolio of stock and cash — and therefore to price the option without making any assumption about investor risk preferences. The resulting formula introduces three ideas that persist throughout all of derivatives pricing:
  1. Delta (Δ\Delta): the number of shares needed to hedge one short option — the sensitivity of the option's value to the stock price.
  2. Risk-neutral probabilities: the unique probabilities under which all assets earn the risk-free rate; discounting at rr under these probabilities gives the correct price in any world, not just a risk-neutral one.
  3. The no-arbitrage price = cost of the replicating portfolio: an option's price is the cost of the hedge, not a reflection of subjective views on direction.
Learning objectives. After this note you should be able to:
  • set up the one-period binomial tree and label the four terminal values;
  • derive the delta Δ\Delta from the condition that the replicating portfolio is riskless;
  • price a European option by discounting the risk-free portfolio;
  • derive the risk-neutral probability pp and verify that it makes the stock grow at rr;
  • price a European call or put using the risk-neutral pricing formula;
  • explain why the stock's real-world expected return is irrelevant to option pricing.

In this lesson

  1. The one-period binomial lattice
  2. The replicating portfolio and delta
  3. Risk-neutral valuation
  4. The delta connection
  5. Pricing a put in the one-period model

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Financial Instruments, Markets and Conventions syllabus.