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

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.

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.