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Chapter 4: Measure Change

This chapter develops the framework for changing probability measures in continuous time, which underlies modern derivative pricing and connects stochastic calculus to financial economics.

The central idea is:

Pricing is achieved by changing the probability measure so that discounted asset prices become martingales.

Rather than introducing new pricing principles, this chapter shows how the measure-change mechanism implements the no-arbitrage framework developed earlier.

Conceptual Roadmap

The chapter follows a strict progression:

flowchart LR
A[Martingale Machinery]
--> B[Girsanov Theorem]
--> C[Risk-Neutral Measure]
--> D[Financial Insights]

Structure of the Chapter

4.1 Martingale Machinery

We build the mathematical foundation required for measure change.

  • Local martingales and why they matter
  • Integrability conditions (Novikov, Kazamaki)
  • Stochastic exponential as the Radon–Nikodym density
  • Martingale representation theorem
  • A unifying principle: controlling when local martingales are true martingales

This section answers: when does a candidate density actually define a valid probability measure?

4.2 Girsanov's Theorem

We introduce the mechanism of measure change.

  • Intuition: drift lives in the measure, not the paths
  • Financial meaning of drift adjustment
  • Formal statement of Girsanov's theorem
  • Proof via stochastic exponentials

The key result: changing the measure removes drift while preserving volatility and sample paths.

4.3 Risk-Neutral Measure

We apply Girsanov's theorem to construct pricing measures.

I. Financial Meaning

  • Martingales and no-arbitrage
  • Risk-neutral valuation principle

II. Construction

  • Building the risk-neutral measure
  • Market price of risk
  • Concrete examples

III. Extensions

  • Numéraire and measure change
  • Forward measures

This section answers: how does measure change produce the pricing measure used in finance?

4.4 Financial Insights

We interpret the framework from an economic and practical perspective.

I. Economic Foundation (WHY)

  • Stochastic discount factor (SDF)
  • Connection to CAPM and factor models

II. Measure Change Interpretation (HOW)

  • Physical vs risk-neutral probabilities
  • Risk premium decomposition

III. Pricing vs Hedging (WHAT)

  • Relationship between expectations and replication

IV. Practice (REALITY)

  • How practitioners actually use the framework

V. Limits (BOUNDARIES)

  • When measure change fails (incompleteness, bubbles, strict local martingales)

What This Chapter Achieves

By the end of this chapter, the reader understands that:

  • measure change is the mechanism behind risk-neutral pricing
  • Girsanov's theorem explains how drift is removed
  • pricing depends on volatility and payoff, not physical drift
  • martingale representation underpins hedging
  • different measures reflect different economic viewpoints, not different prices

Role in the Book

This chapter provides the bridge between stochastic calculus (Chapters 2–3) and pricing and PDE methods (Chapters 5–6). It supplies the machinery used for:

  • Black–Scholes derivation
  • Feynman–Kac representation
  • Interest-rate modeling via forward measures