On the Numerical Approximation of Some Non-standard Volterra Integral Equations K. Nedaiasl1, A. Foroush Bastani Institute for Advanced Studies in Basic Sciences Zanjan, Iran.
American option pricing can be formulated as a free boundary problem for the Black-Scholes Equation (BSE) and as soon as the free boundary is evaluated, the option price will be known. Different approaches of investigation of the BSE, such as Fourier and Laplace transforms and Green function method give us different integral equations, different at least in their form [2, 3]. Some of them can be classified as follows ∫ t (1) x(t) = φ(x(t), t) + K(t, s, x(t), x(s))ds, t ∈ [a, b], 0
where the kernel k(t, s, x, y) is smooth or weakly singular function. The aim this paper is to introduce these classes of integral equations and other ones arising in this field. The existence issue of the Eq. (1) by the Schauder fixed point theorem is discussed. A computational method based on barycentric rational interpolatory quadrature [1, 4] is introduced for approximating Eq. (1) and for simplicity of the presentation is analyzed for the special case ∫ t (2) x(t) = y(t) + K(t, s, x(t), x(s))ds, t ∈ [a, b]. 0
Finally, the results will be compared with other methods in the financial math literature, especially an implicit Runge-Kutta discretization and methods based on fixed point iteration [5]. References [1] J.-P. Berrut, A. Hosseini, G. Klein, The Linear Barycentric Rational Quadrature Method for Volterra Integral Equations, SIAM J. Sci. Comput., 36 (2014), pp. A105– A123. [2] J.D. Evans, R. Kuske and J.B. Keller, American Options on Assets with Dividends Near Expiry, Math. Finance, 12 (2002), pp. 219–237. [3] I.J. Kim, B.G. Jang, K.T. Kim, A Simple Iterative Method for the Valuation of American Options, Quant. Financ., 13 (2013), pp. 885–895. [4] G. Klein, An Extension of the Floater–Hormann Family of Barycentric Rational Interpolants, Math. Comp., 82 (2013), pp. 2273–2292. ˇ coviˇc, Comparison of Numerical and Analytical Approximations [5] M. Lauko, D. Sevˇ of the Early Exercise Boundary of American Put Options, ANZIAM J., 51 (2010), pp. 430–448. E-mail address:
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Speaker: K. Nedaiasl 1
