Reconciling Hull and Brigo–Mercurio LMM Correlation Terms
Summary
The document compares forward-rate dynamics in Hull’s LIBOR market model (LMM) and Brigo and Mercurio’s formulation. Hull’s displayed equation uses a single Brownian driver, so the drift term has no separate correlation coefficient: dependence is implicit in the shared driver. Brigo and Mercurio write the dynamics using rate-specific Brownian motions and include their instantaneous correlations explicitly.
The answer points to Hull’s extension with multiple independent Brownian drivers. In that version, the drift contains a sum of products of volatility components across the drivers, which accounts for covariance and reconciles the formulations. The excerpt therefore does not imply that Hull assumes uncorrelated rates; the initial equation is a one-driver specification. It provides no broader discussion of calibration, model assumptions, or empirical evidence, and the multiple-driver equation is reported from a particular edition of Hull’s book.
Key ideas
- A single shared Brownian driver can encode dependence without an explicit correlation parameter.
- Brigo and Mercurio express correlations between forward rates directly in the drift.
- Hull also presents a multiple-driver version whose drift uses cross-products of volatility components.
- The apparent discrepancy reflects different Brownian-driver specifications, not necessarily an assumption of uncorrelated rates.
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Full text
# Why does the LMM in Hull seem so different from the LMM in Brigo and Mercurio?
# 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?
## Answer by ir7 (score 3, accepted)
https://quant.stackexchange.com/a/66072
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.