LIBOR Market Model - Model Dynamic

Model Dynamic

The LIBOR market model models a set of forward rates, as lognormal processes. Under the respective -Forward measure

dL_j(t) = \sigma_j(t) L_j(t) dW^{Q_{T_j}}(t)\text{.}

Here, denotes the forward rate for the period . For each single forward rate the model corresponds to the Black model. The novelty is that, in contrast to the Black model, the LIBOR market model describes the dynamic of a whole family of forward rates under a common measure. The question now is how to switch between the different -Forward measures. By means of the multivariate Girsanov's theorem one can show that

dW^{Q_{T_j}}(t) = \begin{cases} dW^{Q_{T_p}}(t) - \sum\limits_{k=j+1}^{p} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \qquad j < p \\ dW^{Q_{T_p}}(t) \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad j = p \\ dW^{Q_{T_p}}(t) + \sum\limits_{k=p}^{j-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_k(t) dt \qquad \quad j > p \\ \end{cases}

and

dL_j(t) = \begin{cases} L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) - L_j(t)\sum\limits_{k=j+1}^{p} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \qquad j <p\\ L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad j = p \\ L_j(t){\sigma}_j(t)dW^{Q_{T_{p}}}(t) + L_j(t)\sum\limits_{k=p}^{j-1} \frac{\delta L_k(t)}{1 + \delta L_k(t)} {\sigma}_j(t){\sigma}_k(t){\rho}_{jk}dt \quad \qquad j > p\\ \end{cases}

Read more about this topic:  LIBOR Market Model

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