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Fixed Income and Interest-Rate Models / No-Arbitrage Pricing and the Risk-Neutral Framework

Martingale Measures and the Market Price of Risk

The fundamental pricing equation prices bonds as conditional expectations under a risk-neutral measure Q\mathbb{Q}. But where does Q\mathbb{Q} come from? The answer is Girsanov's theorem: Q\mathbb{Q} is obtained from the real-world measure P\mathbb{P} by shifting the Brownian motion by a process γ(t)\gamma(t), called the market price of risk. The shift γ(t)\gamma(t) is the extra expected return (per unit of volatility) that investors demand for bearing interest rate risk — the formal, continuous-time version of the risk premium studied in Expectations, Convexity, and Risk Premium.

This note covers three things: (1) the definition of an equivalent martingale measure (EMM) and its link to no-arbitrage; (2) Girsanov's theorem, which constructs Q\mathbb{Q} from γ\gamma; (3) the market-price-of-risk equation, which pins down which γ\gamma is consistent with observed excess returns. The result is a tight constraint: in a model with nn assets and dd Brownian drivers, the market price of risk is the same vector γ\gamma for every asset.

Learning objectives. By the end of this note you should be able to:
  • define an equivalent (local) martingale measure (EMM, ELMM) and state its connection to no-arbitrage;
  • state Girsanov's theorem (existence of the new measure, transformation of Brownian motion);
  • identify the market price of risk vector −γ-\gamma and derive it from the drift-excess-return condition;
  • state the first and second fundamental theorems of asset pricing and their relationship to uniqueness of the EMM;
  • explain why the market price of risk is the same for every bond in the market.

In this lesson

    1. The financial market model
    1. Equivalent martingale measures
    1. Girsanov's theorem — constructing Q\mathbb{Q}
    1. The market price of risk
    1. The second fundamental theorem
    1. Bond dynamics under Q\mathbb{Q}

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