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...99615 1.09804 0.50694 0.20597 0.08188 [81,21,1975] 1.99976 1.10794 0.51938 0.21495 0.08219 S1b [21,9,35] 1.99628 1.09831 0.50702 0.20596 0.08180 [81,21,329] 1.99977 1.10853 0.51945 0.21495 0.08217 S2 =-=[21,9,14]-=- 1.99618 1.09728 0.504978 0.20513 0.08234 [81,21,121] 1.99976 1.10768 0.51873 0.21424 0.08193 ref. [13] a [320,128,64] 2 1.10751 0.51904 0.21294 0.08181 ref. [13] b [320,128,64] 2 1.10738 0.51902 0.21...

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...e convergence properties and the treatment of the early exercise feature is nontrivial which is why we use another approach here. We formulate a linear complementarity problem (LCP) for a partial (integro-) differential equation (P(I)DE) operator for the price, discretize the P(I)DE, and then solve the resulting LCPs. Several methods have been proposed for solving the resulting LCPs. The Brennan and Schwartz algorithm [7] is a direct method for pricing American options under the Black-Scholes model; see also [8]. Numerical methods for pricing under the Heston model have been developed in [9], [10], [11], [12], [13], [14], for example. The treatment of the jumps in the Merton and Kou models have been studied in [15], [16], [17], [18], [19], [20], for example. Pricing under the Bates model has been considered in [21], [22] and under the correlated jump model in [23]. In this paper, we price American call options under the Bates model. The spatial partial derivatives in the resulting partial integro-differential operator are discretized using a seven-point finite difference stencil and the integral term is discretized using a simple quadrature rule. The Rannacher scheme [24] is employed f...

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...undary conditions on the solution is small far enough away from the boundary. Research Paper www.risk.net/journal 84 C. Chiarella and B. Kang i D 1; : : : ; N1 C 1 are Dli;1;k D 2 D l i;2;k D l i;3;k;D l i;N2C1;k D 2 Dli;N2;k D l i;N21;k ; 8k D 1; : : : ; N3 C 1; Dli;j;1 D 2 D l i;j;2 D l i;j;3;D l i;j;N3C1 D 2 Dli;j;N3 D l i;j;N31 ; 8j D 1; : : : ; N2 C 1; which is essentially an extrapolation scheme. All other derivative terms in the .v; r/-direction vanish at v D 0 or r D 0 due to the factor .v; r/ occurring in (2.4); hence, these terms do not require further treatment. We follow Ikonen and Toivanen (2007) to indicate which grid point values we use to approximate the second-order mixed derivative in order to obtain nonpositive offdiagonal weights in the finite difference stencil, which makes the matrix anM matrix as much as possible. In fact, to simplify the algorithm and take consideration of the correlations ij , in each two dimensional space, we use a seven-point stencil and the mixed derivatives when 12 > 0 are approximated as @D2 @S@v 1 2 Dl iC1;jC1;k Dl i;jC1;k .Dl iC1;j;k Dl i;j;k / S v C Dl i;j;k Dl i1;j;k .Dl i;j1;k Dl i1;j1;k / S v ; and when 12 < 0 we have @D2 @S...

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...penalty methods for pricing American two-asset options, but their differencemethods are first-order convergent. In part of the domain, the differential operator of the twoasset American option pricing model becomes a convectiondominated operator. The differential operator also contains a second-order mixed derivative term. The classical finite difference methods lead to some off-diagonal elements in the coefficient matrix of the discrete operator due to the dominating first-order derivatives and the mixed derivative. These elements can lead to nonphysical oscillations in the computed solution [17, 18]. In this paper, we present 2 Journal of Function Spaces and Applications an accurate finite difference scheme for pricing two-asset American options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, we obtain a coefficient matrix with anM-matrix property, which ensures that the solutions are oscillation-free. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. We will show that the scheme is second-order convergen...

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...king it easier to use. In [26] Ikonen and Toivanen implement their splitting technique also for American options with stochastic volatility, in this case Heston’s model [21]. In the more recent paper =-=[27]-=- Ikonen and Toivanen introduce another kind of operator splitting technique called ’componentwise operator splitting’ for the American option under stochastic volatility. The method is based on the LC...

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...pproach is to solve the problem by reformulating it as a linear complementarity problem. In order to solve each discrete complementarity problem, Ikonen and Toivanen use operator splitting methods in =-=[18]-=-, and use component-wise splitting methods in [19]. These methods are time-consuming and are typically dependent on sophisticated solver packages. In non-PDE methods, a variant of the Longstaff-Schwar...