Why does the LMM in Hull seem so different from the LMM in Brigo and Mercurio?

Question

When I look at Hull's "Options Futures and Other Derivatives" the process for $F_k(t)$ in the rolling forward risk neutral world is specified as

$\frac{dF_k(t)}{F_k(t)} = \sum^k_{i=m(t)}\frac{\delta_iF_i(t)\zeta_i(t)\zeta_k(t)}{1+\delta_iF_i(t)}dt + \zeta_k(t)dz$

where

• $F_k(t)$ is the forward rate between times $t_k$ and $t_{k+1}$
• $\zeta_k$ is the volatility of $F_k(t)$ at time t (the instantaneous volatility)
• $\delta_k$ is the compounding period between $t_k$ and $t_{k+1}$
• $m(t)$ is the index for the next reset date at time $t$

In Brigo and Mercurio's Interest Rate Models - Theory and Practice, the Lognormal Forward LIBOR Model spot-measure dynamics are specified as:

$dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)F_k(t)dZ^d_k(t)$

which becomes

$\frac{dF_k(t)}{F_k(t)} = \sigma_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)dZ^d_k(t)$

where

• $\sigma_k(t)$ is the instantaneous volatility of $F_k(t)$
• $\tau_k$ is the compounding period between $t_k$ and $t_{k+1}$
• $\beta(t)$ is the index for the next forward rate that has not expired
• $\rho_{i,j}$ is the instantaneous correlation between two forward rates $F_i(t)$ and $F_j(t)$

The only difference I can see is that Brigo and Mercurio include the correlation $\rho_{j,k}$.

How do I reconcile this difference? Does Hull make an assumption that the forward rates are not correlated?

Answer 1

Hull used a single Brownian driver. He did add, a few pages down, equation (31.15) (in my 7th edition) with $p$ independent Brownian drivers:

$$ \frac{dF_k(t)}{F_k(t)} = \sum_{i=m(t)}^k \frac{\delta_iF_i(t) \sum_{q=1}^p\zeta_{i,q}(t)\zeta_{k,q}(t)}{1+\delta_iF_i(t)} dt +\sum_{q=1}^p \zeta_{k,q}(t) dz_q $$

with $\zeta_{k,q}(t)$ the component of the volatility of $F_k(t)$ attributable to the $q$th Brownian driver.