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LIBOR market model

It is common to model the LIBOR rates as geometric Brownian motions and to use the ZCB as a numeraire. The martingale-measure corresponding to the numeraire $p(t,S)$ is called the forward measure, here denoted $\mathbb{Q}^S$. Since \[ L_t[T,S]=\frac{1}{S-T}\frac{p(t,T)-p(t,S)}{p(t,S)}\] it is the ratio of a linear combination of the traded assets and the numeraire. It is thus a martingale under the $\mathbb{Q}^S$ measure. We now assume that \[ L_t[T,S]=L_t[T,S]\sigma(t,T)\text{d} W^{\mathbb{Q}^S}(t),~0\leq t \leq T\] where $\sigma(t,T)$ is deterministic function and $W^{\mathbb{Q}^S}(t)$ is a $\mathbb{Q}^S$ Brownian motion. This gives that \begin{eqnarray*} \text{d} L_T[T,S]&=&L_t[T,S]e^{-\frac{1}{2}\int_t^T \sigma(u,T)^2\text{d} u+\int_t^T\sigma(u,T)\text{d} W^{\mathbb{Q}^S}(u)}\\ &\stackrel{\scriptsize{d}}{=}&L_t[T,S]e^{-\frac{1}{2}\Sigma^2_{t,T}+\Sigma_{t,T}G}, \end{eqnarray*} where $G\in \text{N}(0,1)$ and $\Sigma^2_{t,T}=\int_t^T \sigma(u,T)^2\text{d} u$. The LIBOR market model is consistent in the following sense:
Given a tenor structure $\bar{S}=[S_0,S_1,S_2,\ldots,S_n]$ it is possible to define all the corresponding LIBOR rates of the time intervals $[S_0,S_1],[S_1,S_2],\ldots,[S_{n-1},S_n]$ in one common $n-dim$ model in a consistent way using the $\mathbb{Q}^{S_n}$ dynamics, that is the martingale measure corresponding to th e numeraire $p(t,S_n)$.
We can then change measure for each of the LIBOR rates $L_t[S_{i-1},S_i],~i=1,2,\ldots,n$ so that they get $\mathbb{Q}^{S_i}$ dynamics of the same type as above (see Björk 27.4 for the details).
So we get for $i=1,2,\ldots,n$ the following $\mathbb{Q}^{S_i}$ dynamics \[ \text{d} L_t[S_{i-1},S_i]=L_t[S_{i-1},S_i]\sigma(t,S_{i-1})\text{d} W^{\mathbb{Q}^{S_i}}(t),~0\leq t \leq S_{i-1}\] where $\sigma(t,S_{i-1})$ is deterministic function and $W^{\mathbb{Q}^{S_i}}(t)$ is a $\mathbb{Q}^{S_i}$ Brownian motion.

 

Questions: Magnus Wiktorsson
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