Ito Derivation for Products of Correlated Forward Rates

Summary

The document asks how to derive the product of two forward rates with different reset times when each follows a lognormal diffusion. The rates are driven by correlated Brownian motions, and the displayed expression includes both stochastic terms and variance adjustments. The questioner correctly identifies the cross-variation term, which depends on the correlation between the Brownian drivers, but is unsure why the expression also contains separate negative half-variance terms.

Those terms arise when solving each rate’s stochastic differential equation in exponential form: each rate accumulates its own lognormal drift correction over the interval, while correlation affects the product’s joint stochastic behavior. The document presents the question and model setup but no answer or derivation, so it does not resolve the calculation or establish broader assumptions about the rates’ dynamics. Readers would need to derive the individual solutions and multiply them to complete the explanation.

Key ideas

• Each forward rate is modeled as a lognormal diffusion driven by a Brownian motion.
• The two Brownian drivers have correlation that contributes to the product’s cross-variation.
• The displayed expression includes a separate variance correction for each rate.
• The document poses the derivation question but provides no answer.

Tags

# How to calculate the product of forward rates with different reset times using Ito's lemma?


# How to calculate the product of forward rates with different reset times using Ito's lemma?












I am curious about a calculation I saw in this question.

Specifically in this equation:

\begin{align*} &\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\ =&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\frac{\sigma_s^2}{2}(T_s-t_0) -\frac{\sigma_p^2}{2}(T_s-t_0) + \sigma_s\big(W_{T_s}^s -W_{t_0}^s\big) + \sigma_p\Big(\rho \big(W_{T_s}^s - W_{t_0}^s\big) + \sqrt{1-\rho^2}\big(W_{T_s}^p - W_{t_0}^p\big)\Big)}. \end{align*}

I'm trying to prove it using Ito's lemma and the dynamics:

\begin{align*} dL(t, T_s, T_e) &= \sigma_s L(t, T_s, T_e) d W_t^s,\\ dL(t, T_p, T_e) &= \sigma_p L(t, T_p, T_e)d\Big(\rho W_t^s + \sqrt{1-\rho^2}W_t^p\Big), \end{align*}

But when I apply Ito's lemma I end up computing

\begin{align*} dL(t, T_s, T_e)dL(t, T_p, T_e) &= \sigma_s \sigma_p L(t, T_s, T_e)L(t, T_p, T_e) \rho dt \end{align*}

And I don't know where the next term is coming from:

\begin{align*} -\frac{\sigma_s^2}{2}(T_s-t_0)-\frac{\sigma_p^2}{2}(T_s-t_0) \end{align*}

Can someone help me with the calculation? Thanks

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