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An Operator Splitting Method for Pricing American Options Samuli Ikonen 1 and Jari Toivanen 2 1 Nordea Markets, FI-00020 Nordea, Finland Samuli.Ikonen@nordea.com 2 Department of Mathematical Information Technology, P.O. Box 35 (Agora), FI-40014 University of Jyväskylä, Finland Jari.Toivanen@mit.jyu.fi Summary. Pricing American options using partial (integro-)differential equation based methods leads to linear complementarity problems (LCPs). The numerical solution of these problems resulting from the Black–Scholes model, Kou’s jumpdiffusion model, and Heston’s stochastic volatility model are considered. The finite difference discretization is described. The solutions of the discrete LCPs are approximated using an operator splitting method which separates the linear problem and the early exercise constraint to two fractional steps. The numerical experiments demonstrate that the prices of options can be computed in a few milliseconds on aPC. 1 Introduction Since 1973 Black, Scholes, and Merton developed models for pricing options in [BS73, Mer73] and, on the other hand, the Chicago Board Options Exchange started to operate, the trading of options has grown to tremendous scale. Basic options give either the right to sell (put) or buy (call) the underlying asset with the strike price. European options can be exercised only at the expiry time while American options can be exercised anytime before the expiry. The Black–Scholes partial differential equation (PDE) describes the evolution of the option price in time for European options. In order to avoid arbitrage opportunities with an American option, the so-called early exercise constraint has to be posed on its value. Combining this constraint with the PDE leads to a linear complementarity problem (LCP). For European options it is generally possible to derive formulas for their price, but American options usually need to be priced numerically. This paper considers the solution of these pricing problems. For the general discussion on these topics, we refer to the books [AP05, CT04, TR00, Wil98]. The Black–Scholes model [BS73] assumes a constant volatility for all options with different strike prices and expiry times on the same underlying asset. In practice, this does not hold in the markets. One possibility to make
280 S. Ikonen and J. Toivanen the prices consistent with the markets is to define the volatility as a function of time and the value of the underlying asset, and then calibrate this function; see [Dup94], for example. In 1976, Merton suggested to add jumps to the model of the underlying asset in [Mer73]. This jump-diffusion model helps to explain a good part of difference between the market prices and the ones given by the Black–Scholes model with a constant volatility. Since then there has been growing activity to incorporate jumps to the model; see [CT04] and references therein. One of the models used in this paper is Kou’s jumpdiffusion model. Another generalization is to make the volatility a stochastic process. Several such multifactor models have been proposed; see [FPS00], for example. Here Heston’s stochastic volatility model [Hes93] is used. One can also combine stochastic volatility and jump models like in [Bat96, DPS00], for example. Several ways to solve the discretized LCPs resulting from pricing American options have been described in the literature. Maybe the most common method is the project SOR iteration proposed in [Cry71]. This method is fairly generic and easy to implement, but its convergence rate degrades as grids are refined. For one-dimensional PDE models the resulting LCPs can be solved very efficiently using the direct algorithm in [BS77] if the matrix is a tridiagonal M-matrix and the solution has suitable form. The full matrices resulting from jump-diffusion models require special techniques in order to obtain efficient algorithms. The papers [AO05, AA00, CV05, MSW05] study the numerical pricing of European options, and in [dFL04, dFV05, Toi06] the pricing of American options is considered. For higher-dimensional problems like the ones resulting from Heston’s model multigrid methods have been considered in [BC83, CP99, Oos03, RW04], for example. An alternative way is to approximate the LCPs using a penalty method [FV02, ZFV98]. This leads to a sequence system of linear equations with varying matrices. With this approach the constraints are always slightly violated. With a fairly similar Lagrange multiplier method [AP05, HIK03, IK06, IT06b] it can be guaranteed that the constraints are satisfied. This paper considers an operator splitting method proposed for the Black– Scholes model in [IT04a]. The method was applied to Heston’s model and analyzed in [IT04b], and for Kou’s model it was applied in [Toi06]. The basic idea of this method is to split a time step with a LCP to two fractional time steps. The first fractional step requires a system of linear equations to be solved and the second one enforces the early exercise constraint. The update to satisfy the constraint is simple and, thus, the main computational burden will be the solution linear systems. A similar approach is commonly used to treat the incompressibility condition in the computational fluid dynamics; see [Glo03], for example. The operator splitting method has two obvious benefits. There are several efficient methods available for solving resulting systems of linear equations while only a few methods are available for the original LCPs and they usually cannot compete in the efficiency. Secondly the operator splitting method is easier to implement than an efficient LCP solver. This
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Let a h and b h denote the bilinear
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Table 1. Primal DG for transport Di
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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