This thesis is concerned with the calibration of affine stochastic volatility models with jumps.
This class of models encompasses most models used in practice and captures some of the common
features of market data such as jumps and heavy tail distributions of returns. Two questions arise
when one wants to calibrate such a model:
(a) How to check its theoretical consistency with the relevant market characteristics?
(b) How to calibrate it rigorously to market data, in particular to the so-called implied volatility,
which is a normalised measure of option prices?
These two questions form the backbone of this thesis, since they led to the following idea: instead
of calibrating a model using a computer-intensive global optimisation algorithm, it should be more
efficient to use a less robust—hence faster—algorithm, but with an accurate starting point. Henceforth
deriving closed-form approximation formulae for the implied-volatility should provide a way
to obtain such accurate initial points, thus ensuring a faster calibration.
In this thesis we propose such a calibration approach based on the time-asymptotics of affine
stochastic volatility models with jumps. Mathematically since this class of models is defined via
its Laplace transform, the tools we naturally use are large deviations theory as well as complex
saddle-point methods. Large deviations enable us to obtain the limiting behaviour (in small or
large time) of the implied volatility, and saddle-point methods are needed to obtain more accurate
results on the speed of convergence. We also provide numerical evidence in order to highlight the
accuracy of the closed-form approximations thus obtained, and compare them to standard pricing
methods based on real calibrated data