ADI schemes for pricing American-style options under the Heston model
Karel In 't Hout (University of Antwerp, Belgium)
Joint work with Tinne Haentjens

Tuesday June 3, 14:00-14:30 | session 2.1 | Computational Finance | room AB

We consider the valuation of American-style options in the Heston stochastic volatility model by numerically solving partial differential complementarity problems (PDCPs). First, the relevant time-dependent advection-diffusion-reaction operator is numerically discretized on a two-dimensional spatial domain by applying suitable finite difference schemes. Subsequently, for the time discretization of the obtained semidiscrete PDCPs, we propose the adaptation of Alternating Direction Implicit (ADI) schemes. The adaptation of such schemes is novel in the literature. It is achieved by employing a recent splitting technique of Ikonen and Toivanen [2]. Through numerical experiments for challenging applications, the novel schemes are shown to be highly effective. Also a relevant convergence theorem is proved.

[1] T. Haentjens and K.J. in 't Hout: ADI schemes for pricing American options under the Heston model. Submitted for publication (2013).
[2] S. Ikonen and J. Toivanen, Operator splitting methods for pricing American options under stochastic volatility, Numer. Math. 113 (2009) 299-324.

Efficient pricing in jump models and calibration to American options
Kathrin Glau (Technische Universität München, Germany)
Joint work with Ernst Eberlein and Maximilian Gaß

Tuesday June 3, 14:30-15:00 | session 2.1 | Computational Finance | room AB

Modeling and pricing of financial instruments as well as model calibration are some of the major tasks of mathematical finance. In particular, calibration to market prices requires efficient pricing methods. Commonly, calibration procedures are designed for European options and a number of efficient algorithms have been developed in academia and practice. However, most of the frequently traded stock options are of American type, which has to be incorporated in realistic calibration procedures. In contrast to European options, those of American type are path dependent. Therefore, efficient pricing methods for path dependent options are indispensable. Moreover, model uncertainty leads to the simultaneous use of various models ranging from stochastic volatility to models driven by pure jump processes. Essentially three approaches to compute option prices in these models are being used: Monte Carlo simulation, Fourier based valuation methods and solving the related partial differential equation (PDE). Monte Carlo simulation is typically too slow for calibration purposes. Fourier techniques have been proven to be efficient in Lévy models, especially for pricing European options. PDE methods, however, can be used for pricing not only European but also various path dependent options. Moreover, this can be achieved with the same computational and implementational effort. Applying PDE methods in models driven by jump processes, instead of partial differential equations, partial integro-differential equations (PIDEs) have to be solved. Due to the integral part, standard techniques generally lead to inefficient algorithms. Combining Fourier and PIDE-techniques with model reduction, we achieve flexibility of the pricing method towards different option and model types, and at the same time efficiency and computational feasibility. This also allows us to calibrate jump models to American options, thus avoiding de-Americanization procedures, which are typically used by practitioners. We present the theoretical framework for models driven by time-inhomogeneous Lévy processes. The potential of the method is illustrated by numerical results concerning the calibration of pure jump models to American option prices.

Fast option pricing based on a principal component analysis using adaptive finite differences in space and discontinuous Galerkin in time
Lina Von Sydow (Uppsala University, Sweden)
Joint work with Erik Lehto, Paria Ghafari and Mats Wångersjö

Tuesday June 3, 15:00-15:30 | session 2.1 | Computational Finance | room AB

We consider the numerical pricing of European multi-asset options. The standard way to price such options is to use a Monte-Carlo method to solve the stochastic differential equations for the underlying assets. Due to the slow convergence of these methods we solve the high-dimensional Black-Scholes equation. A straight-forward discretization of this equation leads to the so-called curse of dimensionality. One way to circumvent this curse is to use a dimension reduction technique based on a principal component analysis and asymptotic expansions, [1,2,3]. This way, we only have to solve 1 one-dimensional problem and d-1 two-dimensional problems where d is the number of dimensions (= number of underlying assets) to the original problem. This is in general a large reduction in the complexity of the problem.
The remaining problems are discretized in space using second-order finite differences. To further reduce the arithmetic complexity of the algorithm we introduce adaptivity in space by estimating the discretization error using Richardson-extrapolation, [4]. Thereby, we can place gridpoints where they are most needed for accuracy reasons.
In time we use a discontinuous Galerkin discretization. This type of discretizations have shown to be highly efficient for this type of problems [5,6].
We will present numerical results demonstrating the efficiency of the suggested method for options issued on correlated assets.

[1] C. Reisinger and G. Wittum, Efficient Hierarchical Approximation of high-dimensional Option Pricing Problems. SIAM J. on Sci. Comput., 29(2007), pp. 440-458.
[2] E. Ekedahl, E. Hansander and E. Lehto, Dimension Reduction for the Black-Scholes Equation - Alleviating the Curse of Dimensionality, Dept. of Information Technology, Uppsala University, 2007.
[3] P. Ghafari, Dimension Reduction and Adaptivity to Price Basket Options, U.U.D.M. project report; 2013:3, Dept. of Mathematics, Uppsala University, 2013.
[4] J. Persson and L. von Sydow, Pricing European multi-asset options using a space-time adaptive FD-method, Comput. Vis. Sci., 10(2007), pp. 173-183.
[5] A. Matache, C. Schwab, and T. Wihler, Fast Numerical Solution of Parabolic Integro-Differential Equations with Applications in Finance, SIAM J. Sci. Comput., 27(2005), pp. 369–393.
[6] L. von Sydow, On discontinuous Galerkin for time integration in option pricing problems with adaptive finite differences in space, In Numerical Analysis and Applied Mathematics: ICNAAM 2013, volume 1558 of AIP Conference Proceedings, pp 2373-2376, 2013.

PIDE methods for pricing and calibration of Lévy models
Maximilian Gass (Technical University of Munich, Germany)
Joint work with Kathrin Glau

Tuesday June 3, 15:30-16:00 | session 2.1 | Computational Finance | room AB

It is well known, that Lévy processes are well suited for modeling stocks, volatilities and other typical underlyings due to their ability to incorporate jumps and heavy tails. We are concerned with pricing and calibration in Lévy models using PIDE methods. For European option pricing, Fourier methods offer a feasible solution. For pricing American and other exotic options, however, this solution is not applicable. Instead, we focus on solving the PIDE numerically. Efficient methods in this context are developed by Schwab et. al. based on a Wavelet-Galerkin scheme. Their implementation relies on the specific form of the Lévy measure.
We propose a method avoiding the direct usage of the Lévy measure. The approach is based on a paper by Eberlein and Glau (2013) and combines Fourier and PDE techniques. It exploits the connection between the operator of the PIDE and the symbol of the underlying Lévy process which is known in closed form in many cases. We demonstrate the applicability of the approach for pricing and calibration purposes. In particular, the method allows calibrating Lévy models to American option prices. We discuss a Galerkin implementation and present numerical results.