Deriving the Change of Measure Between Bond Forward Measures
Summary
The document sets up a derivation for converting between the forward probability measures associated with zero-coupon bonds maturing at two different dates. It uses the defining martingale property of asset prices divided by the relevant bond numeraire, then compares conditional expectations at the earlier maturity to propose a Radon–Nikodym derivative at time t.
The proposed density is expressed as a ratio of bond prices evaluated at t and at the earlier maturity. The text ends by asking whether this result is correct; it includes no response or independent verification. The argument is therefore an attempted derivation rather than a confirmed result. Its setup also relies on maturity and filtration conditions and on the numeraire-change relationship being applied consistently, details that are not examined in the document. Readers should treat the formula as a question to check against the standard change-of-numeraire theorem, rather than as established guidance.
Key ideas
- A forward measure is defined using a zero-coupon bond as its numeraire.
- The document compares asset-price martingales under measures tied to two bond maturities.
- It proposes a bond-price ratio as the density for changing between the measures at time t.
- The proposed derivation is posed as a question and is not verified in the text.
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Full text
# Change of numeraire between t1-forward mesure and t2-forward mesure
# Change of numeraire between t1-forward mesure and t2-forward mesure
Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$.
Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$.
I am trying to deduce $\frac{d\mathbb{Q}_{t_1}}{d\mathbb{Q}_{t_2}}|_t$. Here is my reasoning :
Let denote $V(t)$ the price of an asset at time t.
By definition $\frac{V(t)}{B(t,t_1)}$ is a martingal process under $\mathbb{Q}_{t_1}$ for $t_0\leq t \leq t_1$.
Similarly, $\frac{V(t)}{B(t,t_2)}$ is a martingal process $\mathbb{Q}_{t_2}$ for $t_0\leq t \leq t_2$.
Hence we could write :
$$\frac{V(t)}{B(t,t_2)}=\mathbb{E}^{\mathbb{Q}_{t_2}}\left ( \frac{V(t_1)}{B(t_1,t_2)} |\mathbb{F}_{t} \right )$$
$$\frac{V(t)}{B(t,t_1)}=\mathbb{E}^{\mathbb{Q}_{t_1}}\left ( \frac{V(t_1)}{B(t_1,t_1)} |\mathbb{F}_{t} \right )$$
Since $B(t,t_1)$ and $B(t,t_2)$ are $\mathbb{F}_{t}$ mesurable we have:
$$\mathbb{E}^{\mathbb{Q}_{t_2}}\left ( \frac{B(t,t_2)}{B(t_1,t_2)}V(t_1) |\mathbb{F}_{t} \right )=\mathbb{E}^{\mathbb{Q}_{t_1}}\left ( \frac{B(t,t_1)}{B(t_1,t_1)}V(t_1) |\mathbb{F}_{t} \right )=\mathbb{E}^{\mathbb{Q}_{t_2}}\left (\frac{d\mathbb{Q}_{t_1}}{d\mathbb{Q}_{t_2}}|_t \frac{B(t,t_1)}{B(t_1,t_1)}V(t_1) |\mathbb{F}_{t} \right ) $$
Hence we deduce that :
$$\frac{d\mathbb{Q}_{t_1}}{d\mathbb{Q}_{t_2}}|_t=\frac{B(t,t_2)}{B(t_1,t_2)}\frac{B(t_1,t_1)}{B(t,t_1)}$$
Question : Is my result correct ?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.