Radon–Nikodym Densities for Bond Numeraire Changes
Summary
The document examines whether a time-indexed expression involving a zero-coupon bond price and the money-market account is the Radon–Nikodym derivative for switching to the bond numeraire. Its key correction is that the density between probability measures is set at the initial time: express the terminal discount factor divided by its initial conditional expectation, then use that initial value as the derivative. A valid density must be nonnegative and have expectation one under the original measure.
The question’s proposed checks use the wrong measure and time-indexed quantity. A process that is identically one would be a trivial martingale, but that does not establish the claimed change of measure; nor is a single expectation under the new measure the relevant density condition. The response gives the forward-measure framing but no derivation of the full density process or discussion of technical assumptions such as integrability and filtration conditions, so those details would need separate treatment.
Key ideas
- A change of measure is specified by a nonnegative density with expectation one under the original measure.
- For the bond forward measure, the density is formed from the terminal discount factor and its initial expectation.
- A density process at time t is obtained by conditioning the terminal density on the information available at t.
- A martingale claim under the new measure does not by itself establish the proposed Radon–Nikodym derivative.
Tags
Full text
# Verifying if a Function is a Radon-Nikodym Derivative for changing the numeraire
# Verifying if a Function is a Radon-Nikodym Derivative for changing the numeraire
I am analyzing the following function within a financial mathematics framework:
$$ f(t) = \dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)} $$
where: $$ B(t; S) := \mathbb{E}_{t}^{\mathbb{P}} \left[\exp\left(-\int_{t}^{S}r_{f}(u)\, du\right)\right] $$ and $$ m(t) := \exp\left(\int_{0}^{t} r_{f}(u)\, du\right) $$ Definitions:
- $B(t; S)$ represents the price at time $t$ of a zero-coupon bond maturing at time $S>t$, given the information available at time $t$.
- $m(t)$ is a discount factor related to the risk-free interest rate $r_f(t)$.
I want to determine whether this function can be considered as the Radon-Nikodym derivative $$\dfrac{d\mathbb{P}^{S}}{d\mathbb{P}}_{|\mathfrak{F}_{t}}$$ where $\mathbb{P}^{S}$ is the new probability measure associated to the new numeraire $B(t;S)$ and $\mathfrak{F}_{t}$ is a filtration.
#### My Approach:
Notice that in order to define $\dfrac{B(S; S) \cdot m(t)}{B(t; S) \cdot m(S)}$ as the Radon-Nikodym derivative $\dfrac{d\mathbb{P}^{S}}{d\mathbb{P}}_{|\mathfrak{F}_{t}}$, it is sufficient to verify that:
- The expression is a $\mathbb{P}^{S}$-martingale.
- It has a $\mathbb{P}^{S}$-expectation equal to one.
Here’s how these conditions are met:
- $B(S; S)$ is equal to 1.
- The expectation of $\frac{m(t)}{m(S)}$ under $\mathbb{P}^{S}$ is equal to $B(t; S)$.
These properties ensure that the expression has a $\mathbb{P}^{S}$-expectation equal to one. Moreover, since the expression is equal to one for every $t$, this implies that it is also a $\mathbb{P}^{S}$-martingale.
#### Questions:
- Additional Considerations: Are there other properties or conditions that I should consider in this context?
Any feedback or insights would be greatly appreciated.
Thank you!
## Answer by Wei (score 0)
https://quant.stackexchange.com/a/80402
I note that $$f(t) = \frac{D(S)}{\mathbb E^\mathbb P _t[D(S)]}$$ where $$D(S) = \exp\left(-\int_0^Sr_f(u)du\right).$$
What you actually want is to use $f(0)$ as your Radon-Nikodym derivative. All that is needed for this to be a Radon-Nikodym derivative between probability measures is that it is non-negative and $E^\mathbb P _0 [f(0)] = 1$, which is plainly true. For more info, see: https://en.wikipedia.org/wiki/Forward_measure.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.