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I've been working with a convexity adjustment for an interest rate payoff and the next question came to me:

The usual problem that gives rise to the convexity adjustment I'm referring to is as follows:

Consider a series of dates,

\begin{align*} 0 \leq t_0 \leq T_s < T_p < T_e, \end{align*}

Where $T_p$ is the time of payment, $T_s$ is the time when the interest rate is observed and $T_e$ is the time where the interest of the forward rate ends.

If a payment was to ve priced in time $T_p$, one would calculate the expectation over the measure $Q_{T_p}$, as follows:

\begin{align*} P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{align*}

Now changing from the measure $Q_{T_p}$ to the natural measure $Q_{T_e}$ would induce an extra term in the expectation that would lead us to the convexity adjustment:

\begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{dQ_{T_p}}{dQ_{t_e}}L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{P(t_0, T_e)}{P(t_0, T_p)P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\frac{1}{P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_p, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_s, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big) \end{align*}

Now, what if the date sequence was like this:

\begin{align*} 0 \leq t_0 \leq T_s < T_e < T_p, \end{align*}

With the payment made in $T_p$ after the interest rate term ends. Would a change of measure and an adjustment be needed to price the payoff? I'd think that if that's the case it would be something like this:

\begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ =&\ P(t_0, T_e)E^{T_e}\Big(P(T_e, T_p) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\frac{1}{1+ \Delta_e^p L(T_e, T_e, T_p)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ \end{align*}

But I don't see how a change of measure with the same logic as the former case would induce this term. On the other hand, it might be possible that since the payment is done after $T_e$ it doesn't make much sense to make the adjustment since the interest period is over (in theory) and one would price the payoff up to $T_e$ and maybe discount it from $T_p$.

I hope I made myself clear,

Much help appreciated

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A convexity adjustment arises regardless of $T_p$ being before or after $T_e$: $$ E^{T_p}\left[L(T_s, T_s, T_e) \right] = E^{T_e}\left[\frac{dQ^{T_p}}{dQ^{T_e}}L(T_s, T_s, T_e) \right] = E^{T_e}\left[\frac{P(t_0, T_e) P(T_s,T_p)}{P(t_0, T_p)P(T_s,T_e)}L(T_s, T_s, T_e) \right] $$ If $T_p < T_e$ then $\frac{P(T_s,T_p)}{P(T_s,T_e)} = 1 + \Delta_p^e L(T_s, T_p, T_e)$ and you get your original formula.

If $T_p > T_e$ then $\frac{P(T_s,T_p)}{P(T_s,T_e)} = \frac{1}{1 + \Delta_e^p L(T_s, T_e, T_p)}$ and you get the formula you were looking for.

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