# For which instruments performs SABR/LMM better than LMM?

For which class of instruments the SABR/LIBOR Market Model does perform better than the classical LIBOR Market Model?

## The LIBOR Market Model

The LIBOR Market Model — also known as Brace, Gatarek, Musiela model — is an interest rate model capable of reproducing the correlation structure of forward rates. One-factor models are unable to reproduce this structure and therefore cannot price accurately derivatives whose prices reflect these correlations. A typical example of such derivatives are swaps paying a non-linear function of the difference two swap rates for two different maturities.

The model is constructed by using a family of LIBOR rates: $L_0(t), \ldots, L_n(t)$, where $L_i(t)$ is LIBOR forward rate starting at $t_i$ and ending at $t_{i+1}$, following

$$dL_i(t)=\sigma_i(t)L_i(t)dW^{i+1}(t).$$

## The SABR LIBOR-Market Model

An important flaw of the LMM is known as sticky volatilities: if the model is calibrated in a highly volatile market it assumes that this high volatility lasts forever, which leads to inaccurate results.

The SABR LMM attempts to address this issue. In this model, each LIBOR rate is assumed to follow a log-normal dynamic having stochastic volatility:

$$dL_i(t)=\sigma_i(t)L_i^{\beta_i}(t)dW^{i+1}(t) \\d\sigma_i(t) = \alpha_i \sigma_i(t)dZ(t)\\ <dW,dZ> = \rho dt$$

• Could you provide brief description for the mentioned models? SDEs are welcomed. – Bruno Sep 12 '14 at 12:05
• To my knowledge, most swaptions traders peruse the SABR or extended SABR model. – Matthias Wolf Sep 15 '14 at 7:07
• I'd say that nobody uses the Classical LMM, as it doesn't allow to calibrate a smile. But there are a lot of extensions of the volatility functions which could be used with the LMM. We could divide them into: Local Volatility Functions (LVFs) and Stochastic VFs. As LVF one could use Constant Elasticity Variance, Displaced Lognormal or quadratic VF. SVF group includes various setups where VF is stochastic. IMHO, SABR is computationally heavy, and I'd use Piterbarg's SVF see 15.2.5 – Bruno Sep 15 '14 at 13:26

The choice of a model depends on what inputs you have, the complexity allowed (e.g. calculation time restrictions) and what you want to infer from it.

The development of the LMM adressed the mathematical difficulty of finding a joint model for all Libor forwards and was a great achievement in the late 90'. But at that time the distribution of the Libors was no longer assumed to be log-normal (the so-called skew and smile appeared).

Since then various extensions were discussed: stoch-vol (SABR, Heston,...), jumps, local-vol,...

Swaption traders use SABR a lot but $F_i$ is the swap rate of the specific option (no joint model is used).

LMM models were used for the ultra-complex products (path-dependent and callable) which are not so important any more.

So, the advantage of the SABR-LMM over the standard LMM is its capability to produce a smile, but things get terribly complicated in terms of calibration and dynamics (forward smile).

Quick answer: all non-vanilla instruments with a rate component. It is particularly relevant in the context of portfolio risk aggregation (containing multiple instruments with different maturities, strikes, etc). Individual instruments can be often priced using "locally calibrated" models, but their risk metrics may be inconsistent with each other.