一開始我們先公式化Black-Scholes方程並且表示出它的收斂性. 然後我們找到一個基底函數使得我們得以公式化成Petrov-Galerkin有限元素法. 我們證明這樣的雙線性模式是coercive與連續的. 我們也說這樣的離散解有order h的誤差. 最後我們也用數值模擬了整個二維模型並且與一維模型做比較. 我們也得到這樣的二維模型是滿足M矩陣.

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The Continuous Problem and its Solvability . . .. . 4
2.1 The Continuous Problem . . . . . . . . . . . . . . . . .. . 4
2.2 The Existence and Uniqueness . . . . . . . . . . . . . . . 6
2.2.1 Coercivity of the Bilinear Form B(·, ·; t) . . . . .. . . . 7
2.2.2 Continuity of the Bilinear Form B(·, ·; t) . . . . . . . . 8
3 The Finite Element Method . . . .11
3.1 The Finite Element Formulation of the Discretization Scheme . . . . . 11
3.2 Stability and Error Analysis of the Method . . . . . . 15
3.2.1 Lower Bound for the Bilinear Form B(·, ·) . . . . . 16
3.2.2 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 The Finite Volume Method. . . . . . . . . 25
4.1 The Formulation for the Flux κ(f) on Different Intervals . . . . . . . . 26
4.2 The Finite Volume System . . . . . . . . . .. . . . . . . . 29
4.3 Boundary Conditions and Payoff Conditions . . . 32
5 Numerical Experiments . . . . . . . . 34
5.1 Ramp Payoff Final Condition . . . . . . . . . . . . . . .. . . 34
5.2 Cash-or-Nothing Final Condition . . . . . . . . . . . . . 37
5.3 Portfolio of Options . . . . . . . . . . . . . . .. . . . . . . 39
5.4 2-D and 1-D Simulations . . . . . . . . . . .. . . . . . . . . 40
6 Conclusions . . . . 45
Bibliography. . . . 46

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