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

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


  • $F_k(t)$ 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

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


  • $\sigma_k(t)$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.