Processus de Lévy en Finance - Laboratoire de Probabilités et ...

130 CHAPTER 3. NUMERICAL IMPLEMENTATION
The implied volatility Σ(T ) of the hypothetical option with maturity T was then interpolated
using the following formula:
Σ 2 (T ) = Σ 2 (T 1 ) T 2 − T
T 1 − T + Σ2 (T 2 ) T − T 1
T 2 − T 1
.
As we have seen, in an exponential Lévy model the implied volatility of an option which is at
the money and has fixed maturity must not depend on time or stock price. Figure 3.13 shows
that in reality this is not so: both graphs are rapidly varying random functions.
This simple test shows that real markets do not have the “sticky moneyness” property:
arrival of new information can alter the form of the smile. The exponential Lévy models are
therefore “not random enough” to account for the time evolution of the smile.
Moreover,
models based on additive processes, that is, time-inhomogeneous processes with independent
increments, although they perform well in calibrating the term structure of implied volatilities
for a given date [27], are not likely to describe the time evolution of the smile correctly since in
these models the future form of the smile is still a deterministic function of its present shape
[27]. To describe the time evolution of the smile in a consistent manner, one may need to
introduce additional stochastic factors (e.g. stochastic volatility) [9, 10, 22].
One of the important conclusions of this chapter is that at least in the domain of stock index
options, to which we have applied our tests, exponential Lévy models, although they provide a
considerable improvement compared to Black-Scholes model, do not allow to calibrate the term
structure of implied volatilities with sufficient precision. Our calibration algorithm therefore
should not be seen as a method to find an exponential Lévy model consistent with all available
market data but rather as a way to construct a Lévy measure, implied by option prices of a
particular maturity. It can therefore be extended to other models where jumps are represented,
in a nonparametric way, by a Lévy measure. Its many possible applications include, among
others,
• Interpolation of option prices for a single maturity;
• Calibration, using short-maturity data, of the jump part of hybrid models, including both
jumps and stochastic volatility;
• Separate calibration of Lévy densities for different maturities in order to determine the
correct pattern of time dependence of Lévy measure for additive processes.