DH 3rd floor SR

1. It can generate flexible credit spread curves.
2. It leads to flexible implied volatility curves, thus providing a link between credit spread and implied volatility.
3. It implies that high tech firms tend to have very little debts.
4. It yields analytical solutions for debt and equity values.

In this talk, we present several numerical issues, that we currently pursue,

related to accurate approximation of option prices. Next to the numerical

solution of the Black-Scholes equation by means of accurate finite differences

and an analytic coordinate transformation, we present results for options under

the Variance Gamma Process with a grid transformation. The techniques are

evaluated for European and American options.

This talk attempts to survey key aspects of the mathematics that has been developed in recent years towards an explicit understanding of the structure of exponential functionals of Brownian motion, starting with work of Yor's in the 1990s

It is known that a continuous path of bounded variation

can be reconstructed from a sequence of its iterated integrals (called the signature) in a similar way to a function on the circle being reconstructed from its Fourier coefficients. We study the radius of convergence of the corresponding logarithmic signature for paths in an arbitrary Banach space. This convergence has important consequences for control theory (in particular, it can be used for computing the logarithm of a flow)and the efficiency of numerical approximations to solutions of SDEs. We also discuss the nonlinear structure of the space of logarithmic signatures and the problem of reconstructing a path by its signature.