Christoph Reisinger

Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems

A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretization can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted.

Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.

Stochastic Evolution Equations in Portfolio Credit Modelling

We consider a structural credit model for a large portfolio of credit risky assets where the correlation is due to a market factor. By considering the large portfolio limit of this system we show the existence of a density process for the asset values. This density evolves according to a stochastic partial differential equation, and we establish existence and uniqueness for the solution taking values in a suitable function space. The loss function of the portfolio is then a function of the evolution of this density at the default boundary. We develop numerical methods for pricing and calibration of the model to credit indices and consider its performance before and after the credit crunch.

Modelling bonds and credit default swaps using a structural model with contagion

This paper develops a two-dimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default swap spreads and corporate bond yields. The credit dependence structure is influenced by both a longer-term correlation structure as well as by the possibility of default contagion. In this way, the model is able to generate a diverse range of shapes for the term structure of credit spreads using realistic values for input parameters.

Modeling basket credit default swaps with default contagion

The specification of a realistic dependence structure is key to the pricing of multi-name credit derivatives. We value small kth-to-default CDS baskets in the presence of asset correlation and default contagion. Using a first-passage framework, firm values are modeled as correlated geometric Brownian motions with exponential default thresholds. Idiosyncratic links between companies are incorporated through a contagion mechanism whereby a default event leads to jumps in volatility at related entities. Our framework allows for default causality and is extremely flexible, enabling us to evaluate the spread impact of firm value correlations and credit contagion for symmetric and asymmetric baskets.

On the Use of Policy Iteration as an Easy Way of Pricing American Options

In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [P. A. Forsyth and G. Labahn, J. Comput. Finance, 11 (2007), pp. 1--44], is an extremely simple generic algorithm for solving linear complementarity problems (LCPs) resulting from the finite difference and finite element approximation of American options. We show that, in general,

Multilevel Simulation of Functionals of Bernoulli Random Variables with Application to Basket Credit Derivatives

O(N)

Robust calibration of financial models using Bayesian estimators

is an upper and a lower bound on the number of iterations needed to solve a discrete LCP of size

A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

N

A forward equation for barrier options under the Brunick & Shreve Markovian projection

. If embedded in a class of standard discretizations with