I am trying to prove the need of a convexity adjustment to a forward rate by calculating the next expectation:
\begin{align*} P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big). \end{align*}
Where $E^{T_s}$ denotes the expectation under a T-measure with $P(t,T_s)$ as numéraire and $t_0< T_s < T_e $ and $L(T_s, T_s, T_e)$ is the libor rate observed in $T_s$ for the period between $T_s$ and $T_e$
To do it I would like to apply a change of measure so that I can calculate the expectation under a T*-measure with $P(t,T_e)$ as numéraire.
I know to do this change of measure I need to know the Radon-Nikodym derivative, so I need something like this:
\begin{align*} P(t_0, T_s)E^{T_s}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big)=P(t_0, T_s)E^{T_e}\big(\frac{dQ^{T_s}}{dQ^{T_e}}L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \end{align*} How do I know what value of $\frac{dQ^{T_s}}{dQ^{T_e}}$ changes from $Q^{T_s}$ to $Q^{T_e}$?
From what I've seen so far, the Radon-Nikodym derivative is easy to get when you have the distribution under which you are trying to calculate the expectation. For example if $X \sim N(0,1)$ with density function $f(x)$ I can calculate $E[X]$ the usual integral way, or I can introduce a measure $G$ where $g(x)$ can be the density function of say $X \sim N(0,100)$ and it would be the same if I calculate $E_g[X\frac{f(x)}{g(x)}]$ so here my Radon-Nikodym derivative is the division of two density functions. I've seen different publications in where this is used to change from one measure to another, but still I don't seem to understand how you know what value to use for each case, specially in the case I'm asking now since I'm not sure of the density functions I should be using.
The only thing that cross through my mind is that $L(T_s, T_s, T_e)$ is a martingale under $Q^{T_e}$. So perhaps I should assign it this dynamics $dL(t, T_s, T_e) = \sigma_s L(t, T_s, T_e) d W_t^s$ from there I can get a density function which would be like the $g(x)$ in my example. Then if I can find how $L(T_s, T_s, T_e)$ dynamics are under $Q^{T_s}$ maybe I could get the $f(x)$ and the division would be my Radon-Nikodym?
Much help appreciated