在這篇論文，我們利用有限體積的方法去解一個歐式選擇權價值用隨機的股價波動度的二維 Black-Scholes 方程。整個過程，首先我們將 Black-Scholes 方程用張量(或是矩陣)擴散係數公式化為傳統的形式。接著我們根據在一維的 Black-Scholes 方程提出的合適方法，使用有限體積法在公式化後的方程上。我們證明整個離散方程系統是一個 M-矩陣 來表示這個方法是單調的。最後數值實驗的結果將會表示出完成這個方法是有用的論證。

In this paper we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conversative form. 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 presented.

1. Introduction..................1
2. The Continuous Problem........2
3. The finite volume method......4
4. Numerical experiments........11
5. Conclusion...................13

1. D.N. de G. Allen and R.V. Southwell,
"Relaxation methods applied to determine the motion, in two dimensions,
of a viscous fluid past a fixed cylinder",
Quart. J. Mech. Appl. Math., 8, 129 (1955).
2. F.~Black and M.~Scholes.
The pricing of options and corporate liabilities.
J. Political Economy, 81:637--659, 1973.
3. G.~Courtadon,
A more accurate finite difference approximation for the valuation of
options.
J. Financial Economics Quant. Anal, 17:697--703, 1882.
4. J. Hull and A. White, The Pricing of Options on Assets with Stochastic Volatilities, The Journal of Finance, 42, Issue 2 (1987) 281-300.
5. J.J.H. Miller and S. Wang, "A new non-conforming
Petrov-Galerkin method with triangular elements for a singularly
perturbed advection-diffusion problem",
IMA J. Numer. Anal., 14, 257--276 (1994).
6. J.J.H. Miller and S. Wang,
"An exponentially fitted finite element volume method
for the numerical solution of 2D unsteady incompressible flow
problems", J. Comput. Phys., 115, No.1 (1994) 56--64.
7. L.C.G. Rogers and D. Tallay,
Numerical Methods in Finance, Cambridge University Press (1997).
8. E. Schwartz, "The valuation of warrants: implementing a new approach", J. Financial Economics, 13, 79-93 (1977).
9. R.S. Varga, Matrix Iterative Analysis,
Prentice-Hall, Englewood Cliffs, NJ (1962).
10. S. Wang,
"A novel exponentially fitted triangular finite element
method for an advection-diffusion problem
with boundary layers", J. Comp. Phys., 134 (1997) 253--260.
11. S. Wang, A Novel Fitted Finite Volume Method for the
Black-Scholes Equation Governing Option Pricing, IMA J. Numer. Anal., to appear.
12. P.~Wilmott, J.~Dewynne, and S.~Howison.
Option pricing: mathematical models and computation.
Oxford Financial Press, Oxford, 1993.
13. R.~Zvan, P.A.~Forsyth and K.R.~Vetzal.
Penalty methods for {A}merican options with stochastic volatility. J. Comput. Appl. Math., 91(2):199--218, 1998.