Absolutely Continuous Measures and the Radon-Nikodym Derivative

Black-Scholes option pricing computes

e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+]

— but where does the measure

\mathbb{Q}

come from, and what does it mean to change probability measures on path space? The answer is the Radon-Nikodym theorem: whenever two probability measures

\mathbb{Q}

and

\mathbb{P}

agree on which events are impossible (i.e.

\mathbb{Q} \ll \mathbb{P}

— "

\mathbb{Q}

is absolutely continuous with respect to

\mathbb{P}

"), there exists a non-negative random variable

Z = d\mathbb{Q}/d\mathbb{P}

such that

\mathbb{Q}(A) = \mathbb{E}^{\mathbb{P}}[Z \mathbf{1}_A]

for every event

A

This density $Z$ is the conversion factor between the two measures. Every $\mathbb{Q}$ -expectation is a $\mathbb{P}$ -expectation weighted by $Z$ :

\mathbb{E}^{\mathbb{Q}}[Y] = \mathbb{E}^{\mathbb{P}}[Y \cdot Z].

In derivatives pricing,

Z

is the stochastic discount factor or pricing kernel — the random variable that converts physical-measure expectations into risk-neutral ones. The specific form it takes (the exponential martingale

Z_t = e^{\theta W_t - \theta^2 t/2}

) is derived in the next note.

Learning objectives. By the end of this note you should be able to:

State the definitions of absolute continuity ( $\mathbb{Q} \ll \mathbb{P}$ ), equivalence ( $\mathbb{Q} \sim \mathbb{P}$ ), and singularity ( $\mathbb{Q} \perp \mathbb{P}$ ).
State the Radon-Nikodym theorem and identify $d\mathbb{Q}/d\mathbb{P}$ .
Compute $\mathbb{Q}$ -expectations from $\mathbb{P}$ -expectations using the density.
Explain why two Wiener measures with different variances are singular.
Apply the conditional expectation formula under measure change.

In this lesson

Absolute Continuity and Singularity
The Radon-Nikodym Theorem
Changing Expectations
The Density Process

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