Classifying Stochastic Volatility Extensions of the LIBOR Market Model

Summary

The document clarifies how to classify a LIBOR market model extended with stochastic volatility. Adding a stochastic variance or volatility process to the dynamics of forward rates makes the resulting model a stochastic volatility model. The framework is flexible: the volatility process can be coupled to many underlying dynamics, including a LIBOR market model, and the extension can be described by naming both components or simply as an LMM with a stochastic volatility extension.

The response sketches general forward-rate and volatility equations and explains that the local, stochastic, or local-stochastic classification depends on how volatility scales and how the forward-rate loading depends on the state. A shifted LMM is cited as an example of a local volatility extension. The answer gives a conceptual classification guide rather than calibration details, pricing results, or a comparison of model performance. The exact category therefore depends on the chosen functional forms and parameter values, not on the LMM label alone.

Key ideas

• Coupling a stochastic volatility process to forward-rate dynamics makes the resulting LMM a stochastic volatility model.
• An LMM extension can be named by specifying both its rate model and volatility component.
• Local, stochastic, and local-stochastic labels depend on the volatility scaling and state dependence in the equations.
• A shifted LMM is presented as an example of a local volatility extension.
• The classification framework does not establish calibration quality or pricing performance.

Tags

# LIBOR market model with stochastic volatility


# 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?

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/57505

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).

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.