Abstract
Evaluation of European-style derivatives can be reduced to solving initial value or initial-boundary value problems of parabolic partial differential equations. This chapter discusses numerical methods for such problems. If an American option problem is formulated as a linear complementarity problem, then the only difference between solving a European option and an American option is that if the solution obtained by the partial differential equation does not satisfy the constraint at some point, then the solution of the PDE at the point should be replaced by the value determined from the constraint condition. Such methods are usually referred to as projected methods for American-style derivatives. Therefore, the two methods are very close, and we also study the projected methods in this chapter.
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Notes
- 1.
We also know that because the Black–Scholes equation holds, \(\mbox{ E}_{D}\left [S_{n+1}\right ] =\mathrm{ {e}}^{(r-D_{0})\Delta t}S_{n}\) and \(\mbox{ E}_{D}\left [S_{n+1}^{2}\right ] =\mathrm{ {e}}^{[2(r-D_{0})+{\sigma }^{2}]\Delta t}S_{n}^{2}\) should be true (see Problem 39 of Chap. 2).
- 2.
- 3.
In his paper, he assumes D 0 = 0. However, it is not difficult to generalize that result to the case with D 0≠0.
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Zhu, Yl., Wu, X., Chern, IL., Sun, Zz. (2013). Initial-Boundary Value and LC Problems. In: Derivative Securities and Difference Methods. Springer Finance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7306-0_8
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DOI: https://doi.org/10.1007/978-1-4614-7306-0_8
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