354 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVESDeterministic orlocal volatility(LV) models assume that the asset follows
an Itô diffusion with a volatility function that is a deterministic (non-linear)
function of the underlying state variables. Derman and Kani (1994) and, in
particular, Dupire (1994) succeeded in showing that, if noise is driven by
a single Brownian motion, it is possible to uniquely determine the form of
volatility function out of the market prices of traded options. This approach
is often referred to as “implied volatility theory”.
An alternative, more sophisticated framework consists of assuming that
the volatility itself is driven by an independent noise source, usually a
Wiener process. This approach, known asstochastic volatility(SV) modeling,
was introduced and developed by Hull and White (1987), Stein and Stein
(1991) and Heston (1993) among others. SV models have recently become
very popular in the industry since they provide a simple (yet satisfactory)
approach to the modeling of the smile dynamics while LV models are less
appealing from this point of view (Andersen and Andreasen, 2000a).
Finally, a generalization of BS can be achieved by relaxing the assump-
tion of diffusive-type dynamics and by assuming the presence of jumps
in the asset, in the volatility or in both. From a pure statistical perspec-
tive,jump-diffusion(JD) models (possibly with the inclusion of stochastic
volatility features, or SVJD models) are in excellent agreement with empir-
ical observations, as shown for instance in Eraker, Johannes and Polson
(2003). Unfortunately, pricing and hedging in presence of jumps is in gen-
eral a much harder task to achieve than in pure diffusion models (Duffie,
Pan and Singleton, 2000; Föllmer and Schweizer, 1991).
Despite the differences among these three approaches, as some authors
have observed (see for instance Schoutens, Simons and Tistaert, 2003), they
are all capable of reproducing the observed shape of the implied volatility
surface.^1 In fact, one can always parametrically adjust a given model to
enforce its unconditional probability distributions to be very “close” to a
pre-assigned marginal distribution to the point that the resulting differences
would be indistinguishable from a practical point of view. As is well-known,
however, two processes with different conditional distributions give rise
to totally different sample paths even if their unconditional distributions
are the same. Therefore, one might argue that the price of certain path-
dependent options and, more importantly, the dynamic risk-management
are both heavily affected by model selection. This is indeed the case as the
results of this chapter suggest.
This remark is particularly relevant in the context of model implied cal-
ibration, a common and well-known market practice. To avoid potential
arbitrage opportunities and identify the market price of risk practitioners
“calibrate” their models to the market prices of vanilla European options
(i.e. the smile) and use the resulting dynamics to evaluate, by arbitrage,
more complex derivatives whose prices are not directly available in the mar-
ket. In doing so, one implicitly assumes that the market contains enough