294 11 Libor Market Model with Stochastic Volatilities
11.4 Incorporating Stochastic Volatility
AsshowninFigure(11.2), the smile/skew effect of implied volatilities of caps and
swaptions also displays a similar pattern as observed in stock option markets. It is
evident for stock options that the leptokurtic feature of the distribution of stock re-
turns mainly generates smile and skew, and as discussed in Chapter 3, stochastic
volatility models such as in Heston (1993), or in Sch
¨
obel and Zhu (1999) can ex-
plain smile/skew effect successfully, and have became the main models for smile
modeling. Accompanied by the success story for stock options, it is a natural step
to extend LMM by adopting stochastic volatilities to capture volatility smile/skew
observed in cap and swaption market.
Various extended LMMs with smile modeling are suggested and can mainly fall
into two categories: The first is the local volatility approach including constant elas-
ticity of variance (CEV) model (Andersen-Andreasen, 2000) and displaced diffu-
sion model. Both CEV model and displaced diffusion model can generate a mono-
tone down-slope skew of implied volatilities, but fail to create symmetric smile. The
second approach is to describe volatility with a stochastic process. Most models so
far with this approach use a mean-reverting square root process
´
alaHestonfor
variance to specify a part of total variance of a Libor, see Andersen and Brotherton-
Ratcliffe (ABR, 2001), Wu and Zhang (2002), Piterbarg (2003), Zhu (2007), as well
as Belomestny, Mathew and Schoenmaker (BMS, 2007). The models of ABR and
Piterbarg do not allow for a correlation between Libors and theirs stochastic volatil-
ities, and therefore lack the capability to create skew for implied cap volatilities
without adding additional model components. The zero correlation remarkably re-
stricts the ability of the proposed stochastic volatility LMMs to capture different
individual smile/skew patterns of caps and swaptions.
Hagan, Kumar, Lesniewski and Woodward (HKLW, 2002) applied a so-called
SABR model to Libor process where the volatility is governed by a geometric Brow-
nian motion which is correlated with Libor diffusion. Since SABR model admits an
explicit expansion solution for implied volatilities of caplets and swaptions, it is
very popular for brokers and traders to smooth the market implied volatilities. How-
ever, the stochastic volatility in SABR model displays no mean-reversion on the
contrary to the well-documented empirical feature of a volatility process. Driven
by a geometric Brownian motion, the stochastic volatility in SABR model is not
guaranteed to be stationary in a long way. HKLW did not provide a comprehen-
sive treatment to the application of SABR to LMM, and, for example, it is not clear
which processes should Libors and stochastic volatilities follow under an unique
forward measure. There is no link between cap pricing and swaption pricing in an
usual SABR model. In practice, SABR model as a stochastic volatility model is of-
ten used to interpolate and extrapolate market volatilities. Recently, Mercurio and
Morini (2007), Rebonato (2007), as well as Hagan and Lesniewski (2008) attempt
to fill some theoretical gaps of an isolated SABR model and build up an extended
LMM with SABR stochastic volatility allowing for non-zero correlation between
the dynamics of Libors and SARR volatilities. However, this class of models do