Skip to content
Derivatives Pricing and Computational Finance / Foundations of Computational Option Pricing

Market Model and Risk-Neutral Valuation

This note answers the central conceptual question of the course: why is the price of an option a discounted expectation, and expectation under which probability? The answer comes from replication — the option's price is the cost of the portfolio that hedges it — and the risk-neutral measure Q\mathbb{Q} emerges not as an assumption but as a theorem. We derive everything in the cleanest possible setting (one period, two states), then identify the continuous-time analogue: discounted prices must be martingales under Q\mathbb{Q}.

If you understand this note properly, you will never again be tempted to plug real-world probabilities into a pricing formula — and you will know exactly why the stock's drift μ\mu never appears in an option price.

Learning objectives. By the end of this note you should be able to:
  • state the standard frictionless market model and its assumptions;
  • derive the one-period risk-neutral price by constructing the replicating hedge;
  • explain what the risk-neutral measure is, where it comes from, and why mispricing against it is an arbitrage;
  • define martingales, verify that Brownian motion is one, and derive the risk-neutral drift by requiring St/BtS_t/B_t to be a martingale;
  • state the risk-neutral valuation formula and list the four computational attacks on it.

In this lesson

  1. 2.1 Model of the financial market
  2. 2.2 Replication in a one-period model
  3. 2.3 What is the risk-neutral measure?
  4. 2.4 Martingales and the fundamental pricing theorem
  5. 2.5 Risk-neutral valuation formula
  6. 2.6 The fundamental theorems

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 Derivatives Pricing and Computational Finance syllabus.