Talks

Christian Bayer: Smoothing the payoff for efficient computation of basket options

We consider the problem of pricing basket options in a multivariate Black-Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 25. (Joint work with Markus Siebenmorgen und Raul Tempone).

Emmanuil Georgoulis: Reduced complexity space-time discontinuous Galerkin methods for degenerate parabolic problems

We present a new hp-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic and, in general, degenerate evolution equations on general spatial meshes which consist of prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree, say p, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete hp-dG scheme using fewer degrees of freedom for each time step, compared to standard dG time-stepping schemes employing tensorized space-time basis, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with arbitrary number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments.

Kathrin Glau: Parametric Option Pricing by Interpolation

Model calibration requires fast and accurate numerical methods. In the current paradigm, semi-closed pricing formulas for liquid options are seen as a prerequisite for modelling financial asset evolution. Thus attention is restricted to stochastic processes that are simple enough to allow for straightforward expressions of the pricing formulas. This obviously imposes a severe modelling restriction. However, rising demands to include more realistic features, for instance stylized facts on asymptotics of the implied volatility surface, compel us to consider a wider class of processes and hence more complex models. We therefore propose numerical techniques to reduce the computational complexity of the resulting pricing tasks. In this talk we focus on interpolation of option prices in the parameter space. Both the theoretical and experimental results show highly promising gains in efficiency. The efficiency gain is particularly high in cases where Monte Carlo simulation is required and prices need to be computed for a large number of parameter constellations.

[1] Chebyshev Interpolation for Parametric Option Pricing, M. Gaß, K. Glau, M. Mahlstedt and M. Mair (2016), http://arxiv.org/abs/1505.04648.

Emmanuel Gobet: Price expansion in local-stochastic volatility model: vanilla and barrier options.

Analytical approximations of option prices under various models have become very popular in the last decade, because they give simple closed formulas allowing for fast calibration. In this work we consider the case of local-stochastic volatility models where the stochastic volatility is a square root process and the local volatility part is of general form (possibly including term-structure). We derive expansions of call-put option price, implied volatility and we discuss the extension to barrier options. Our formulas are fast to compute compared to the Monte Carlo approach, with a very competitive accuracy for a large range of strikes. Joint work with Romain Bompis (CACIB).

Ralf Korn: Pricing of cliquet options and their use in life insurance products

Annika Lang: Simulating weak convergence rates for SPDE approximations

The finding of weak convergence rates for approximations of solutions is one of the current and still partly open problems in the numerical analysis of stochastic partial differential equations. The confirmation of the existing theory as well as of conjectured rates with simulations has hardly been done and has not been successful for equations driven by multiplicative noise so far. In this talk it is discussed why the standard methods fail even for toy examples to recover theoretical rates and new error estimators are introduced that allow for simulations of weak convergence rates for SPDEs driven by multiplicative noise. This is joint work with Andreas Petersson.

Mikko Pakkanen: Hybrid scheme for moving average processes and random fields

The hybrid scheme can be seen as a generic approach to simulation of white noise-driven moving average processes and random fields that have been specified using a kernel function with power-law behavior near the origin. Such a kernel function produces a process/field whose realizations are rougher or smoother, in terms of Hölder regularity, than those of a Brownian motion/sheet, which is a useful feature for example in stochastic modelling of financial market volatility or turbulent velocity fields in physics. In my talk, I will first review the original one-parameter hybrid scheme, developed in collaboration with M. Bennedsen and A. Lunde [arXiv:1507.03004]. I will then present some recent joint work with C. Heinrich and A. Veraart that adapts the hybrid scheme for two-parameter, volatility-modulated moving average random fields.

Lina von Sydow: Efficient and accurate pricing of options

In this talk we will discuss numerical pricing of options issued on a basket of underlying assets or assets with stochastic volatility. We will present the challenges with such problems and present a method based on radial basis functions generated finite differences that is well suited for this purpose. We will also present a benchmarking project where a large number of methods to numerically price options are compared http://www.it.uu.se/research/scientific_computing/project/compfin/benchop. A first paper has been published and there is ongoing work on the more challenging problems described in this talk.

Michaela Szölgyenyi: Numerical methods for SDEs with discontinuous drift appearing in mathematical finance

When solving certain stochastic control problems in insurance mathematics or mathematical finance, the optimal control policy sometimes turns out to be of threshold type, meaning that the control depends on the controlled process in a discontinuous way. The stochastic differential equations (SDEs) modeling the underlying process then typically have a discontinuous drift coefficient. This motivates the study of a more general class of such SDEs. We prove an existence and uniqueness result, based on a certain transformation of the state space by which the drift is “made continuous”. As a consequence the transform becomes useful for the construction of a numerical method. The resulting scheme is proven to converge with strong order 1/2. This is the first scheme for which strong convergence is proven for such a general class of SDEs with discontinuous drift. As a next step, the transformation method is applied to prove strong convergence with positive rate of the Euler-Maruyama scheme for this class of SDEs. Joint work with G. Leobacher (KFU Graz).

Nizar Touzi: Branching diffusion representation of nonlinear PDEs

We provide a representation of the solution of a semilinear partial differential equation by means of a branching diffusion. Unlike the backward SDE approach, such a representation induces a purely forward Monte Carlo method. We also provide applications to the unbiased simulation of stochastic differential equations, and extensions to general initial value problems.