Choosing the Numeraire for a Forward Rate Coupon Payment
Summary
This document clarifies how the payment date determines the numeraire term when pricing a floating-rate coupon under a forward measure. The example has a rate fixed at an earlier date and paid later, with the zero-coupon bond maturing on the payment date used as numeraire. At that payment date the bond value is one, so the expectation contains the rate itself; under the stated setup, the coupon value is the payment-date bond price multiplied by the forward rate observed at the initial time.
The key distinction is between when the rate is fixed and when the cash flow is paid. If the rate is both fixed and paid on the earlier date, while the numeraire matures later, the bond price at the fixing date remains inside the expectation. That setup can produce a convexity adjustment. The explanation addresses this coupon-timing example; it does not derive a general pricing framework or cover additional features such as spreads, day-count conventions, or stochastic collateral terms.
Key ideas
- A forward measure is associated with the maturity of its numeraire bond.
- For a coupon fixed at one date and paid later, the numeraire bond is worth one on the payment date.
- The fixing date and payment date must be distinguished when writing the pricing expectation.
- A rate fixed and paid earlier can retain a bond-price term that leads to a convexity adjustment.
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# Martingale measure result application for interest rates under T-forward measure?
# Martingale measure result application for interest rates under T-forward measure?
I've got a question about the way the equivalent martingale measure result is used for pricing derivatives. Hull states the result as the next equality:
\begin{align*} f_o = g_0 E^{g}\big(\frac{f_T}{g_T}\mid \mathcal{F}_{t_0}\big) \end{align*}
Given that $f_T$ has some dynamics which are depedent on $g_T$ dynamics's volatility.
So what I understand is that as long as $f_T$ has the correct dynamics I can divide by $g_T$ and get the price of any derivative.
As an example, for a call with payoff $max(S_T-K,0)$ I can choose $g_0$ as the money market account with $g_0 = 1$ and $g_T = e^{rT}$ (assuming constant r). Then to price the option I would use the result like this:
\begin{align*} f_o = E^{r}\big(\frac{max(S_T-K,0)}{e^{rT}}\mid \mathcal{F}_{t_0}\big) \end{align*}
Solving this with the correct dynamics ($\mu=r$ for $S_T$) would lead us to Black and Scholes formula.
Now, in the case of interest rates I know that under a $T^*$-measure with numeraire as $P(t,T^*)$ and $T<T^*$ the forward interest rate $R(T,T,T^*)$ as seen in time $T$ is a martingale, that is:
\begin{align*} R(t_0,T,T^*) = E^{T^*}\big(R(T,T,T^*)\mid \mathcal{F}_{t_0}\big) \end{align*}
However, if I wanted to apply the same logic I used in the example before to value a derivative that pays the T-forward interest rate in time $T^*$ I would go on and do this:
\begin{align*} f_o = P(0,T^*)E^{T^*}\big(\frac{R(T,T,T^*)}{P(T,T^*)}\mid \mathcal{F}_{t_0}\big) \end{align*}
But I get the term $P(T,T^*)$ which doesn't seem right because I believe the correct valuation is:
\begin{align*} f_o = P(0,T^*)E^{T^*}\big(R(T,T,T^*)\mid \mathcal{F}_{t_0}\big)=P(0,T^*)R(t_0,T,T^*) \end{align*}
Should I be using $g_T=P(T^*,T^*)=1$ ? That doesn't seems correct to me since it's $g_T$ not $g_T^*$
I saw in here (What is the correct convexity adjustment for an Interest Rate Swap with unnatural reset lag?) and it looks like $P(T_p, T_p)$ is being used even thought the rate is observed in $T_s$
What am I missing?
Much help appreciated
## Answer by Antoine Conze (score 2, accepted)
https://quant.stackexchange.com/a/37415
Do not confuse the fixing date $T$ and the payment date $T^*$. In your example you are valuing a floating coupon that fixes on $T$ and pays $R(T, T, T^*)$ on $T^*$, and you are using the $T^*$ zero coupon bond as numeraire, so the PV is computed as $$ p_0 = P(0, T^*)E^{T^*}\left[\frac{R(T, T, T^*)}{P(T^*,T^*)} \right] = P(0, T^*)E^{T^*}\left[R(T, T, T^*)\right]=P(0, T^*) R(0, T, T^*) $$ The floating rate $R(T, T, T^*)$ is fixed on $T$ but it is paid on $T^*$, so the denominator in the expectation is $P(T^*,T^*)$. If the floating rate was fixed AND paid on $T$ (as would be the case with an in arrears fixing) then the denominator would be $P(T,T^*)$ and that would lead to a convexity adjustment.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.