How do we derive the Radon-Nikodym derivative for T-forward measures?

Question

Let $Q^{T_e}$ denote the $T_e$-forward measure and let $Q^{T_p}$ denote the $T_p$-forward measure.

I have seen the following Radon-Nikodym derivative being used in derivations. For $0 \le t \le T_p$, \begin{align*} \eta_t \equiv \frac{dQ^{T_p}}{dQ^{T_e}}\mid_{t} = \frac{P(t, T_p)P(0, T_e)}{P(0, T_p)P(t, T_e)}. \end{align*}

How do we derive this formula? This seems very different from the Radon-Nikodym derivative specified in Girsanov's theorem for changing measures.

Answer 1

To give some background, I give you some pieces from Geman, El Karoui and Rochet (1995) which use a change of numéraire. A fundamental observation is the change of measure formula $\mathbb{E}^\mu[X]=\mathbb{E}^\nu\left[\frac{\mathrm{d}\mu}{\mathrm{d}\nu} X\right]$.

Defintion 2 defines what a numéraire is.

A numéraire is a price process $X(t)$ almost surely strictly positive for each $t\in[0, T]$.

Basically, numéraires are assets which are used to measure the prices of all other assets. Often, one just uses a risk-free bank account. But there are other possibilities, particularly in the interest rate world.

Assumption 1 assumes that there exists an asset a numéraire with associatted martingale measure.

There exists a non-dividend-paying asset $n(t)$ and a probability $\pi$ equivalent to the initial probability $\mathbb{P}$ such that for any basic security $S_k$ without intermediate payments, the price of $S_k$ relative to $n$, i.e. $S_k(t)/n(t)$ is a local martingale with respect to $\pi$. By convention, we will take $n(0) = 1$.

The key part of the paper, and the answer to your question, is given in theorem 1.

Let $X(t)$ be a non-dividend paying numeraire such that $X(t)$ $n$-martingale. Then there exists a probability measure $Q_X$ defined by its Radon-Nikodym derivative with respect to $\pi$ $$\frac{\mathrm{d}Q_X}{\mathrm{d}\pi}\Bigg|\mathcal{F}_T = \frac{X(T)n(0)}{X(0)n(T)}$$ such that

1. the basic securities prices are $Q_X$-local martingales,
2. if a contingent claim $H$ has a fair price under $(n, \pi)$, then it has a fair price $(X, Q_X)$ and the hedging portfolio is the same.

So, replacing $X$ and $n$ by your bond prices (and $T$ by $t$) yield the sought Radon-Nikodym derivative in your question.

As an application, Section 4.1 derives the time zero price of a European-style call option expiring at time $T_0$ with strike price $K$ written on a default-free zero-coupon bond maturing at time $T_1>T_0$ as $$C(0)=P(0,T_1)\mathbb{Q}^{T_1}[A]-KP(0,T_0)\mathbb{Q}^{T_0}[A],$$ where $A$ is the exercise set and $\mathbb{Q}^T$ is the $T$-forward measure using a bond maturing at time $T$ as numéraire.