Swiss Finance Institute @ EPFL
The Swiss Finance Institute @ EPFL has been created to foster research in finance and to develop a strong offering of programs in finance and financial engineering at the Ecole Polytechnique Fédérale de Lausanne. The focus is on the areas within finance that have a natural interaction with mathematics, statistics, engineering, and science, namely, mathematical finance, financial econometrics, and entrepreneurial finance.
The Swiss Finance Institute @ EPFL participates in two teaching programs, The Master in Financial Engineering at EPFL, which is a highly selective 2-year master program, and The PhD in Finance, which is organized jointly with the Swiss Finance Institute and the Universities of Geneva and Lausanne.
The Swiss Finance Institute @ EPFL benefits from the institutional support of the Swiss Finance Institute, a private foundation created in 2006 by Switzerland’s banking and finance community in cooperation with leading Swiss universities, and from Swissquote, who endowed the Swissquote Chair in Quantitative Finance.
In this talk we consider two connected problems:
First, we study the classical problem of the first passage hitting density of an Ornstein-Uhlenbeck process. We give two complementary (forward and backward) formulations of this problem and provide semi-analytical solutions for both. The corresponding problems are comparable in complexity. By using the method of heat potentials, we show how to reduce these problems to linear Volterra integral equations of the second kind. For small values of t we solve these equations analytically by using Abel equation approximation; for larger t we solve them numerically. We also provide a comparison with other known methods for finding the hitting density of interest, and argue that our method has considerable advantages and provides additional valuable insights.
Second, we study the non-linear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.
By: Alexander LIPTON, SilaMoney, MIT & EPFL
By: Alberto ROSSI, University of Maryland
By: Mika KASTENHOLZ – Co-Founder