Abstract
One of the best known and widely used numerical methods to solve partial differential equations in finance and elsewhere is the finite difference method.
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- 1.
In the following we will often use the abbreviation PDE for “partial differential equation”, as is common practice in the related literature.
- 2.
The sign \(\widehat {=}\) means “corresponds to”.
- 3.
For the sake of simplicity, the price of the forward contract is given for the case of a flat interest rate term structure, a flat dividend yield curve and no discrete dividend payments. However, this relation also holds for interest rates and dividend yields which are time-dependent.
- 4.
As already pointed out in the section on Dirichlet boundary conditions, the solution is at best as exact as the given boundary conditions. This is important in all cases where there are only approximations to the boundary conditions RU(t) and RL(t) available.
- 5.
One of the rare exceptions to this rule is a power option without cap, with a pay off profile proportional to S2. Such options are of minor relevance in praxis.
- 6.
Independent of the number of time steps.
- 7.
Experience shows that increasing/decreasing the step size in a regular manner works often well.
References
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L.B.G. Andersen, R. Brotherton-Ratcliffe, The equity option volatility smile: an implicit finite-difference approach. J. Comput. Finance 1(2), 5–37 (1998)
L.B.G. Andersen, V.V. Piterbarg, Interest Rate Modeling (Atlantic Financial Press, New York, London, 2010)
N. Anderson, F. Breedon, M. Deacon, et al., Estimating and Interpreting the Yield Curve (Wiley, Chichester, 1996)
D.F. Babbel, C.B. Merrill, Valuation of Interest-Sensitive Instruments (Society of Actuaries, Schaumburg, IL, 1996)
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Deutsch, HP., Beinker, M.W. (2019). Numerical Solutions Using Finite Differences. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_10
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DOI: https://doi.org/10.1007/978-3-030-22899-6_10
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