Radon-Nikodym on a Coin

Take a stock at 100 that moves to 120 or 90 each step, with $r = 0$. Say the real coin is fair, $p = \tfrac12$. For pricing we use a second coin, the one that makes the stock a fair game: $120\tilde p + 90(1 - \tilde p) = 100$ gives $\tilde p = \tfrac13$.

Toss twice. Both coins see the same four paths and weight them differently:

Neither measure calls any path impossible. Two measures that agree on which events are impossible are equivalent, and that is what lets you divide one by the other. The path-by-path ratio is the Radon-Nikodym derivative of $\tilde{\mathbb P}$ with respect to $\mathbb P$:

On a finite tree it is just a quotient. The word "derivative" comes from the general case, such as Brownian paths, where every single path has probability zero. There $Z$ is pinned down by events instead of single paths, through $\tilde{\mathbb P}(A) = E[Z\,\mathbf 1_A]$.

Two facts follow at once. $Z > 0$ on every path, and

Here: $\tfrac14(\tfrac49 + \tfrac89 + \tfrac89 + \tfrac{16}9) = \tfrac14 \cdot \tfrac{36}{9} = 1$. Throughout, $E$ with no tilde means the real measure $\mathbb P$, and $\tilde E$ means $\tilde{\mathbb P}$.