Markus Holtz

Dimension-wise integration of high-dimensional functions with applications to finance

We present a new general class of methods for the computation of high-dimensional integrals. The quadrature schemes result by truncation and discretization of the anchored-ANOVA decomposition. They are designed to exploit low effective dimensions and include sparse grid methods as special case. To derive bounds for the resulting modelling and discretization errors, we introduce effective dimensions for the anchored-ANOVA decomposition. We show that the new methods can be applied in locally adaptive and dimension-adaptive ways and demonstrate their efficiency by numerical experiments with high-dimensional integrals from finance.

Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.

Sparse Grid Quadrature

This chapter is concerned with sparse grid (SG) quadrature methods. These methods are constructed using certain combinations of tensor products of one-dimensional quadrature rules. They can exploit the smoothness of f, overcome the curse of dimension to a certain extent and profit from low effective dimensions, see, e.g., [16, 44, 45, 57, 116, 146].

B-Spline-Based Monotone Multigrid Methods

For the efficient numerical solution of elliptic variational inequalities on closed convex sets, multigrid methods based on piecewise linear finite elements have been investigated over the past decades. Essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements. On the other hand, there are a number of problems which profit from higher order approximations. Among these are the problem of pricing American options, formulated as a parabolic boundary value problem involving Black-Scholes’ equation with a free boundary. In addition to computing the free boundary (the optimal exercise price of the option) of particular importance are accurate pointwise derivatives of the value of the stock option up to order two, the so-called Greek letters. In this paper, we propose a monotone multigrid method for discretizations in terms of B-splines of arbitrary order to solve elliptic variational inequalities on a closed convex set. In order to maintain monotonicity (upper bound) and quasi optimality (lower bound) of the coarse grid corrections, we propose an optimized coarse grid correction (OCGC) algorithm which is based on B-spline expansion coefficients. We prove that the OCGC algorithm is of optimal complexity of the degrees of freedom of the coarse grid and, therefore, the resulting monotone multigrid method is of asymptotically optimal multigrid complexity. Finally, the method is applied to a standard model for the valuation of American options. In particular, it is shown that a discretization based on B-splines of order four enables us to compute the second derivative of the value of the stock option to high precision.

Valuation of Performance-Dependent Options

Performance‐dependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. This paper presents a novel approach to the valuation of general performance‐dependent options. To this end, a multidimensional Black–Scholes model is used to describe the temporal development of the asset prices. The martingale approach then yields the fair price of such options as a multidimensional integral whose dimension is the number of stochastic processes used in the model. The integrand is typically discontinuous, which makes accurate solutions difficult to achieve by numerical approaches, though. Using tools from computational geometry, a pricing formula is derived which only involves the evaluation of several smooth multivariate normal distributions. This way, performance‐dependent options can efficiently be priced even for high‐dimensional problems as is shown by numerical results.

Geometric Tools For The Valuation OfPerformance-dependent Options

In this paper, we describe several methods for the valuation of performance-dependent options. Thereby, we use a multidimensional Black-Scholes model for the temporal development of the asset prices. The martingale approach then yields the fair price as a multidimensional integral whose dimension is the number of stochastic processes in the model. The integrand is typically discontinuous, though, which makes accurate solutions difficult to achieve by numerical approaches. However, using tools from computational geometry we are able to derive a pricing formula which only involves the evaluation of smooth multivariate normal distributions. This way, performance-dependent options can efficiently be priced even for high-dimensional problems as it is shown by numerical results.

Numerical Simulation for Asset-Liability Management in Life Insurance

New regulations and stronger competitions have increased the demand for stochastic asset-liability management (ALM) models for insurance companies in recent years. In this article, we propose a discrete time ALM model for the simulation of simplified balance sheets of life insurance products. The model incorporates the most important life insurance product characteristics, the surrender of contracts, a reserve-dependent bonus declaration, a dynamic asset allocation and a two-factor stochastic capital market. All terms arising in the model can be calculated recursively which allows an easy implementation and efficient evaluation of the model equations. The modular design of the model permits straightforward modifications and extensions to handle specific requirements. In practise, the simulation of stochastic ALM models is usually performed by Monte Carlo methods which suffer from relatively low convergence rates and often very long run times, though. As alternatives to Monte Carlo simulation, we here propose deterministic integration schemes, such as quasi-Monte Carlo and sparse grid methods for the numerical simulation of such models. Their efficiency is demonstrated by numerical examples which show that the deterministic methods often perform much better than Monte Carlo simulation as well as by theoretical considerations which show that ALM problems are often of low effective dimension.

Dimension Reduction and Smoothing

Numerical quadrature methods can profit from smoothness and low effective dimension as we showed in Chapter 3 and Chapter 4. Integrals that arise in applications from finance often have kinks or even jumps, however. Moreover, they often have a high truncation dimension. We here focus on several different approaches which aim to reduce the effective dimension or to smooth the resulting integrands. In this way, the efficiency of the numerical quadrature methods can be improved in many cases as we finally show in Chapter 6 by numerical experiments.

Validation and Applications

In this chapter, we present various numerical experiments with different applications from finance. We study the performance of the different numerical quadrature methods from Chapter 3 and Chapter 4 and investigate the impact of the different approaches for dimension reduction and smoothing from Chapter 5. Parts of this chapter are taken from [58].

Dimension-wise Decompositions

In this chapter, we introduce the classical ANOVA and the anchored-ANOVA decomposition of a multivariate function f. Based on these decompositions, we then define different notions of effective dimensions of f and derive error bounds for approximation and integration.