Valuing an Earlier Zero-Coupon Bond Under a Later Bond Measure
Summary
The document poses a fixed-income valuation question involving two zero-coupon bonds with different maturities and the forward rate between them. It assumes that this forward rate is lognormally distributed under the measure associated with the later-maturity bond. Using the relationship between the two bond prices and the forward rate, the author proposes valuing the earlier bond by taking an expectation under that measure.
The author’s uncertainty focuses on whether the bond-price ratio is a martingale only up to the earlier maturity and whether that supports the proposed expectation. The text contains no answer or verification, so the calculation should be treated as a question rather than an established result. It does not discuss the precise numeraire argument, integrability conditions, or the scope of the measure change. Those details matter when justifying the valuation and the expectation used.
Key ideas
- The setup relates two zero-coupon bond prices through their forward rate.
- The forward rate is assumed lognormal under the measure associated with the later bond.
- The proposed valuation uses the earlier bond price relative to the later bond as a martingale candidate.
- The document asks whether the martingale argument supports the expectation but provides no resolution.
Tags
Full text
# How to find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$
# How to find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$
How do I find the risk neutral valuation of $P(T_{1})$ und the measure $\mathbb Q^{P(T_{2})}$, where $P(T_{1})$ and $P(T_{2})$ refer to the $T_{1}$ and $T_{2}$ zero coupon bond with $0 < T_{1} < T_{2}$, and $L(T_{1},T_{2},t)$ be the associated forward rate. The forward rate is assumed to be follow lognormal law under $\mathbb Q ^{P(T_{2})}$. I first note that $P(T_{1},t)=P(T_{2},t)(1+L(T_{1},T_{2};t)(T_{2}-T_{1}))$ such that it can be seen as a function of two traded instruments in the $T_{2}$ market, i.e. $P(T_{1},t)=f(P(T_{2},t),L(T_{1},T_{2};t))$. Therefore, I think(?) that $(\frac{P(T_{1},t)}{P(T_{2},t)})_{0\leq t \leq T_{1}}$ is a $\mathbb Q ^{P(T_{2})}$ martingale. Such that the risk-neutral valuation looks like
$$V(0)\stackrel{?}{=}P(T_{2},0)\mathbb E ^{\mathbb Q^{P(T_{2})}}\left[\frac{P(T_{1},T_{1})}{P(T_{2},T_{1})}\right]= P(T_{2},0)\mathbb E ^{\mathbb Q^{P(T_{2})}}\left[\frac{1}{P(T_{2},T_{1})}\right]=P(T_{2},0)\mathbb E ^{\mathbb Q^{P(T_{2})}}\left[1+L(T_{1},T_{2};T_{1})(T_{2}-T_{1})\right]=P(T_{2},0)(1+L(T_{1},T_{2};0)(T_{2}-T_{1}))$$
Am I not sure on the first equality (where I put the question mark) as I have only used the fact that $\frac{P(T_{1},t)}{P(T_{2},t)}$ is a martingale up to time $T_{1}$ but not necessarily up to $T_{2}$.
I think I may be making this harder than it is, but I just want to have sound reasoning as to why I can/cannot do the above. Thanks!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.