# LIBOR market model with stochastic volatility

I have read that there are 3 types of pricing models: local volatility, stochastic volatility and stochastic-local volatility models (LSV).

I am now looking at interest rates exotics pricing models and I see that LIBOR market model (LMM) is the market standard for simple exotics. But given this model cannot fit the smile since you are just simulating all the forward rates under the same measure, via a series of drift corrections, the solution is adding stochastic volatility to LMM to price more complex structures.

But how would you classify this model given that we can either have Local or Stochastic vol models (or mix of the two, as in LSV)? Does LMM with stochastic volatility fall under the LSV category?

Yes, a stochastic volatility SDE can be coupled with any underlying SDE (GBM, diffusion, mean reverting, LMM, etc.).

Once stochastic volatility is present, the model earns the right to be labeled 'SV model'.

In its name, one may want to specify the names of both SDE's, like in the SABR LMM example found here, or just call it LMM with SV extension.

Similarly, LMM with LV extension (shifted LMM is one of those), LMM with LSV extension etc.

Note: A generic coupled SDE extending LMM would be:

$$dL^n_t = v_t^\gamma \phi(t, L^n_t) \lambda_n(t)^\intercal dW^{T_{n+1}}_t$$ $$dv_t = \kappa (\theta -v_t) dt + \eta(t) \psi(v_t) dB_t$$

So the LV, SV and LSV classification would dependent on the values of $$\gamma$$ (usually $$0$$, $$0.5$$, or $$1$$) and the shapes of $$\phi$$ (state dependent and maybe also time dependent, possibly in a non-separable way).

• Thank you for your answer. So for example SABR-LMM can be considered a SV model but not a LSV? Given it is a mix of two models (LMM and SV) I thought it would fall in the LSV category. But I guess the LMM per se is not a LV model, right? – Diuoo Aug 20 '20 at 19:52
• I added a note that might help. – ir7 Aug 20 '20 at 21:31
• Ignore my other comment. I guess the only 'basic' model would be the normal LMM ($\phi$ set to $1$ and $\gamma =0$). – ir7 Aug 20 '20 at 21:50
• In theory it should be possible to make the LMM have a LV feature? Instead of having constant volatility (as in the standard LMM) or stochastic (as in SABR-LMM) one could make the volatility a deterministic function of underlying and time. Any reason why this has not become a market standard? – Diuoo Aug 21 '20 at 7:26
• Volatility of the form $\phi(L^n_t)\lambda_n(t)$ ($\gamma =0$) is definitely used. At least for the usual choices $\phi(x)= x$ (lognormal), or $ax+b$ (shifted/displaced lognormal), or $x^\beta$ (CEV, $0<\beta <1$). More complex $\phi$ (in time and state) would lead to lack of tractability. Also LV's don't have good control/intuition of forward vol skew (which leads to keeping $\phi$ simpler but adding SV). – ir7 Aug 21 '20 at 8:03