arXiv subjects

Christoph Reisinger

Publications and source records attributed to Christoph Reisinger.

At least 91 records · Page 5Linked to original sources

arXiv2016

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.

math.NA↗

arXiv2016

The non-locality of Markov chain approximations to two-dimensional diffusions

In this short paper, we consider discrete-time Markov chains on lattices as approximations to continuous-time diffusion processes. The approximations can be interpreted as finite difference schemes for the generator of the process. We derive conditions on the diffusion coefficients which permit transition probabilities to match locally first and second moments. We derive a novel formula which expresses how the matching becomes more difficult for larger (absolute) correlations and strongly anisotropic processes, such that instantaneous moves to more distant neighbours on the lattice have to be allowed. Roughly speaking, for non-zero correlations, the distance covered in one timestep is proportional to the ratio of volatilities in the two directions. We discuss the implications to Markov decision processes and the convergence analysis of approximations to Hamilton-Jacobi-Bellman equations in the Barles-Souganidis framework.

math.PR↗

arXiv2016

Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets

We study the Heston-Cox-Ingersoll-Ross++ stochastic-local volatility model in the context of foreign exchange markets and propose a Monte Carlo simulation scheme which combines the full truncation Euler scheme for the stochastic volatility component and the stochastic domestic and foreign short interest rates with the log-Euler scheme for the exchange rate. We establish the exponential integrability of full truncation Euler approximations for the Cox-Ingersoll-Ross process and find a lower bound on the explosion time of these exponential moments. Under a full correlation structure and a realistic set of assumptions on the so-called leverage function, we prove the strong convergence of the exchange rate approximations and deduce the convergence of Monte Carlo estimators for a number of vanilla and path-dependent options. Then, we perform a series of numerical experiments for an autocallable barrier dual currency note.

q-fin.CP↗

arXiv2016

A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection

We derive a forward equation for arbitrage-free barrier option prices, in terms of Markovian projections of the stochastic volatility process, in continuous semi-martingale models. This provides a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted as the canonical extension of local volatility for barrier options. Alternatively, a forward partial-integro differential equation (PIDE) is introduced which provides up-and-out call prices, under a Brunick-Shreve model, for the complete set of strikes, barriers and maturities in one solution step. Similar to the vanilla forward PDE, the above-named forward PIDE can serve as a building block for an efficient calibration routine including barrier option quotes. We provide a discretisation scheme for the PIDE as well as a numerical validation.

q-fin.MF↗

arXiv2016

A partial Fourier transform method for a class of hypoelliptic Kolmogorov equations

We consider hypoelliptic Kolmogorov equations in $n+1$ spatial dimensions, with $n\geq 1$, where the differential operator in the first $n$ spatial variables featuring in the equation is second-order elliptic, and with respect to the $(n+1)$st spatial variable the equation contains a pure transport term only and is therefore first-order hyperbolic. If the two differential operators, in the first $n$ and in the $(n+1)$st co-ordinate directions, do not commute, we benefit from hypoelliptic regularization in time, and the solution for $t>0$ is smooth even for a Dirac initial datum prescribed at $t=0$. We study specifically the case where the coefficients depend only on the first $n$ variables. In that case, a Fourier transform in the last variable and standard central finite difference approximation in the other variables can be applied for the numerical solution. We prove second-order convergence in the spatial mesh size for the model hypoelliptic equation $\frac{\partial u}{\partial t} + x \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2}$ subject to the initial condition $u(x,y,0) = δ(x) δ(y)$, with $(x,y) \in \mathbb{R} \times\mathbb{R}$ and $t>0$, proposed by Kolmogorov, and for an extension with $n=2$. We also demonstrate exponential convergence of an approximation of the inverse Fourier transform based on the trapezium rule. Lastly, we apply the method to a PDE arising in mathematical finance, which models the distribution of the hedging error under a mis-specified derivative pricing model.

math.NA↗

arXiv2016

A mixed Monte Carlo and PDE variance reduction method for foreign exchange options under the Heston-CIR model

In this paper, the valuation of European and path-dependent options in foreign exchange (FX) markets is considered when the currency exchange rate evolves according to the Heston model combined with the Cox-Ingersoll-Ross dynamics for the stochastic domestic and foreign short interest rates. The mixed Monte Carlo/PDE method requires that we simulate only the paths of the squared volatility and the two interest rates, while an "inner" Black-Scholes-type expectation is evaluated by means of a PDE. This can lead to a substantial variance reduction and complexity improvements under certain circumstances depending on the contract and the model parameters. In this work, we establish the uniform boundedness of moments of the exchange rate process and its approximation, and prove strong convergence in $L^p$ ($p\geq1$) of the latter. Then, we carry out a variance reduction analysis and obtain accurate approximations for quantities of interest. All theoretical contributions can be extended to multi-factor short rates in a straightforward manner. Finally, we illustrate the efficiency of the method for the four-factor Heston-CIR model through a detailed quantitative assessment.

q-fin.CP↗

arXiv2016

Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases.

math.NA↗

arXiv2015

Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process

We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler-Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.

q-fin.CP↗

arXiv2014

Modelling of dependence in high-dimensional financial time series by cluster-derived canonical vines

We extend existing models in the financial literature by introducing a cluster-derived canonical vine (CDCV) copula model for capturing high dimensional dependence between financial time series. This model utilises a simplified market-sector vine copula framework similar to those introduced by Heinen and Valdesogo (2008) and Brechmann and Czado (2013), which can be applied by conditioning asset time series on a market-sector hierarchy of indexes. While this has been shown by the aforementioned authors to control the excessive parameterisation of vine copulas in high dimensions, their models have relied on the provision of externally sourced market and sector indexes, limiting their wider applicability due to the imposition of restrictions on the number and composition of such sectors. By implementing the CDCV model, we demonstrate that such reliance on external indexes is redundant as we can achieve equivalent or improved performance by deriving a hierarchy of indexes directly from a clustering of the asset time series, thus abstracting the modelling process from the underlying data.

q-fin.ST↗

arXiv2013

Numerical Valuation of Derivatives in High-Dimensional Settings via PDE Expansions

In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and uses principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. The approximation is related to anchored analysis of variance (ANOVA) decompositions and is expected to be accurate whenever the covariance matrix has one or few dominating eigenvalues. A main purpose of the present article is to give a careful analysis of the numerical accuracy and computational complexity compared to state-of-the-art Monte Carlo methods on the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems. We are able to demonstrate that for problems with medium to high dimensionality and moderate time horizons the presented PDE method delivers results comparable in accuracy to the MC methods considered here in similar or (often significantly) faster runtime.

q-fin.CP↗

arXiv2013

The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.

q-fin.CP↗

arXiv2012

Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative

In this article, we extend a Milstein finite difference scheme introduced in [Giles & Reisinger(2011)] for a certain linear stochastic partial differential equation (SPDE), to semi- and fully implicit timestepping as introduced by [Szpruch(2010)] for SDEs. We combine standard finite difference Fourier analysis for PDEs with the linear stability analysis in [Buckwar & Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The results show that Crank-Nicolson timestepping for the principal part of the drift with a partially implicit but negatively weighted double Itô integral gives unconditional stability over all parameter values, and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries.

math.NA↗

arXiv2012

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.

math.NA↗

arXiv2011

Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems

In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.

q-fin.CP↗

arXiv2011

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 [Forsyth & Labahn, 2007], is an extremely simple generic algorithm for solving linear complementarity problems resulting from the finite difference and finite element approximation of American options. We show that, in general, O(N) is an upper and lower bound on the number of iterations needed to solve a discrete LCP of size N. If embedded in a class of standard discretisations with M time steps, the overall complexity of American option pricing is indeed only O(N(M+N)), and, therefore, for M N, identical to the pricing of European options, which is O(MN). We also discuss the numerical properties and robustness with respect to model parameters in relation to penalty and projected relaxation methods.

q-fin.CP↗

arXiv2011

Analysis of Linear Difference Schemes in the Sparse Grid Combination Technique

Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretisation error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretisations. We obtain overall error formulae of the type $ε= \mathcal{O} (h^p |\log h|^{d-1})$ and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions.

math.NA↗

arXiv2011

Stochastic evolution equations in portfolio credit modelling with applications to exotic credit products

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 pre and post credit crunch. Finally, we give further examples illustrating the valuation of exotic credit products, specifically forward starting CDOs.

q-fin.PR↗

arXiv2010

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

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.

q-fin.CP↗