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

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?

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.