14th Annual Financial Mathematics Conference: Poster Competition
The success of last years' poster competition, hosted during the annual Financial Mathematics Festival, is renewed this year !! The competition is opened to both Master and PhD students. It is way for them to present their current or past research in Mathematics and Finance. The three winners of the competition will receive a cash price, and their poster will be displayed in the hallway of the Love Building, on the first floor. Rules and additional information may be found here.
Spectral Elements Methods
Spectral element methods are numerical methods based on the expansion
of solutions into orthogonal basis functions, in the same fashion as Fourier
transform methods. Fast Fourier Transform (FFT) methods are
commonly used in finance, in particular when it is possible to relate the
characteristic function of a given process to the price of a derivative. FFT methods are based on the polynomial expansion of
solutions in the form of a truncated Fourier series to build a system of
ordinary differential equations (ODE). The accuracy of the approximation
depends on the rate of decay of the Fourier coefficients which is only of
polynomial order accuracy when the initial condition is non smooth. Spectral
element methods are also based on a truncation polynomial expansion, but use
Chebyshev or Legendre polynomials with quadrature rules to integate square
integrable functions. Its application to finance is only recent, despite
exponential convergence patterns when the solutions are sufficiently smooth.
We show below the implementation of the advection-diffusion equation in a Pipe which corresponds to the propagation of a polluant entering the pipe, at two different times.
We explain in this presentation a possible use of two dimensional spectral element methods to solve an option pricing problem related to credit derivative princing. In the framework of structural model of default, it is indeed possible to view the price of a first to default 0-coupon bond as a put option on the minimum of two underlying asset value processes.
Merton jump-diffusion and Levy copula for Option pricing
We present the application of L\'evy Copulas to Option pricing in order to model
the dependence structure between shocks occuring on related
markets. To our knowledge, this is the first time a L\'evy Copula is calibrated
using Fast Fourier Transform techniques on a finite set of European
option prices. Multidimensional L\'evy processes offer a more realistic
treatment and a better fit to market data than the geometric brownian motion
used in the Black-Scholes model. In particular, correlation matrices are not
suitable to render dependent jump discontinuities. Our model uses a
multidimensional version of Merton's jump-diffusion, with a
dependence between marginals given by a L\'evy Copula. We use the
Esscher Transform to find a common martingale measure and apply Fast Fourier Transform
techniques to reproduce the observed option prices on each asset. We expect
this model to gain popularity among credit-equity strategies and will show its
use for hedging illiquid assets.
