在這篇論文中，我們將介紹布萊克-休斯方程（美式買權）的高階緊緻方法。我們先計算自由邊界上的函數跳躍條件。藉由跳躍條件，我們可以更精準地找出自由邊界的位置。我們首先介紹二階收歛的方法，然後推廣至高階緊緻的方法。

In this thesis, we introduce high order compact methods for the Black-Scholes equation of American call options. We compute some jump conditions depend on the free boundary. By applying the jump conditions, we can locate the free boundary more precisely. At first we would introduce a second order method, and then attempt to a higher order method with compact scheme.

1 Introduction . . . . . . . . . . . . . . . . . . . . . 1
2 The Black-Scholes equation of the American call option 2
2.1 Problem state . . . . . . . . . . . . . . . . . . . 2
2.2 Some properties of the solution . . . . . . . . . . 3
3 Introduction of the method of Han & Wu . . . . . . . . 4
4 Second order modified method . . . . . . . . . . . . . 7
4.1 Second order backward differentiation formula (BDF2) 7
4.2 Modified value of right boundary condition . . . . . 7
4.3 The case while Sb() crossing the space lattice . . . 10
4.4 Some problems near the expiry . . . . . . . . . . . 11
4.5 Algorithm . . . . . . . . . . . . . . . . . . . . . 12
4.6 Numerical results . . . . . . . . . . . . . . . . . 14
5 Remove singularity by asymptotic expansion . . . . . . 17
6 High order compact method (3rd order) . . . . . . . . 21
6.1 Fundamental scheme . . . . . . . . . . . . . . . . . 21
6.2 Jump values of the interface . . . . . . . . . . . . 22
6.3 Idea of first 3 time steps . . . . . . . . . . . . . 24
6.4 Numerical results for 3rd order . . . . . . . . . . 24
7 4th order scheme . . . . . . . . . . . . . . . . . . . 25
7.1 Fundamental scheme . . . . . . . . . . . . . . . . . 25
7.2 Jump value of the interface . . . . . . . . . . . . 25
7.3 Equation for variable d . . . . . . . . . . . . . . 26
7.4 Idea of first 4 time steps . . . . . . . . . . . . . 27
7.5 Numerical result for 4th order . . . . . . . . . . . 28
8 Graphs of solutions . . . . . . . . . . . . . . . . . 29
9 Conclusion . . . . . . . . . . . . . . . . . . . . . . 32

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