Analysis of a finite volume element method for a degenerate parabolic equation in the zero-coupon bond pricing
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Analysis of a finite volume element method for a degenerate parabolic equation in the zero-coupon bond pricing T. Chernogorova · R. Valkov
Received: 29 October 2013 / Revised: 4 March 2014 / Accepted: 11 March 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract We construct and analyze a stable exponentially fitted numerical scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Introducing weighted Sobolev spaces, we present the Gärding coercivity and the weak maximum principle for the differential solution. The differential problem is discretized by a fitted finite volume element method resolving the degeneration. We derive coercivity of the discrete bilinear form as we also show that the fully discrete system matrix is essentially of positive type which implies the maximum principle for the implicit time stepping. Numerical experiments validate the theoretical results. Keywords Degenerate parabolic equation · Gärding coercivity · Finite volume element method · M-matrix · Stability · Convergence Mathematics Subject Classification
35K65 · 65M08 · 65M12
1 Introduction Derivative securities are nowadays indispensable tools for traders on financial markets as they are used to hedge risk exposures. The majority of the financial derivatives are today modeled by a variations of the celebrated Black-Scholes model (Mikula et al. 2011; Wilmott et al. 1995) while closed-form solutions are available only for a minority of these models. Numerical analysis of one- and multi-factor derivative products such as bonds, bond options, interest rate caps, swap options in the presence of unpredictable interest rate is of crucial practical importance. A large number of publications on numerical pricing of financial derivatives have been issued in the recent years Communicated by Josselin Garnier. T. Chernogorova · R. Valkov (B) Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria e-mail: [email protected] T. Chernogorova e-mail: [email protected]
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T. Chernogorova, R. Valkov
(Angermann and Wang 2007; Cen and Le 2011; Chernogorova and Stehlíková 2012; Chernogorova and Valkov 2011; Ramírez-Espinoza and Ehrhardt 2013; Zhou et al. 2011). A bond is a contract, paid for up front, that yields a known amount on a known date in the future, the maturity date t = T . Bonds are issued both by government and companies. The main purpose of a bond issue is the raising of capital, and the up-front premium can be thought of as a loan to the government or the company. The bond may also pay a known cash dividend (the coupon) at fixed times during the life of the contract. If there is no coupon the bond is known as a zero-coupon bond (ZCB). A detailed discussion on the bond pricing is given in Mikula et al. (2011), Wilmott et al. (1995), Zhu et al. (2004). Analogously to the derivation of the Black-Scholes equation, the problem of ZCB pricing can be reduced to the following partial differential equation (Deng et al. 2010; Stampfli and Goodman 200
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