Overview — No-Arbitrage Pricing and the Risk-Neutral Framework

The bootstrapped yield curve describes where rates are today. What it cannot tell you is how the curve will move tomorrow, or how to price an option that depends on that movement. For that, you need a no-arbitrage dynamic model — a probability space, a short-rate process, a risk-neutral measure, and the fundamental pricing equation that ties them together.

This topic builds the theoretical engine that powers every interest rate model in the unit. The three lessons move from the simplest object (the money-market account and the short rate) to the most powerful tool (the change of numeraire), via the central concept (the risk-neutral measure and the market price of risk). The derivation sub-note works through the Radon-Nikodym density and its application to bond option pricing — the engine under the hood of the

T

-forward measure.

By the end of this topic you should be able to:

define the money-market account $B(t)$ and the short rate $r(t)$ and state the fundamental pricing equation for a $T$ -bond;
define an equivalent martingale measure (EMM) $\mathbb{Q}$ , state Girsanov's theorem, and identify the market price of risk $\gamma$ as the Girsanov kernel;
derive the market price of risk equation from the condition that discounted prices be $\mathbb{Q}$ -martingales;
define the $T$ -forward measure $\mathbb{Q}^T$ using the $T$ -bond as numeraire and state the change-of-numeraire formula;
prove that $P(t,S)/P(t,T)$ is a $\mathbb{Q}^T$ -martingale;
state the T-forward measure pricing formula $\pi(t) = P(t,T)\,\mathbb{E}^{\mathbb{Q}^T}[X \mid \mathcal{F}_t]$ and use it to price a European bond option;
compute the Radon-Nikodym density of $\mathbb{Q}^T$ with respect to $\mathbb{Q}$ and use it to evaluate the two probabilities in the Margrabe-style bond option formula.

In this lesson

Contents
The big picture

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