Abstract
In this chapter, we construct second-order (in both space and time) FD schemes for forward PDEs and PIDEs consistent with the corresponding FD schemes for the backward PDEs and PIDEs considered in previous chapters. In this context, consistency means that the option prices obtained by solving both the forward and backward equations coincide up to some tolerance. This approach is partly inspired by Andreasen and Huge (RISK, 66–71, 2011), whose authors reported a pair of consistent finite difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility model. We extend their approach by constructing schemes that are of second order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.
Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.
Hermann Weyl.
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Notes
- 1.
In other words, option prices for multiple strikes given the spot price can be computed by solving just one forward equation.
- 2.
If \(\mathcal{A} = \mathcal{A}(t)\), we solve in multiple steps in time Δ t using, for instance, a piecewise linear approximation of \(\mathcal{A}\) and the solution in Eq. (7.2).
- 3.
It is worth mentioning that the consistent forward scheme for the Craig–Sneyd scheme was implemented in 2009 by Michael Konikov as part of the Numerix library.
- 4.
We recall that a standard Brownian motion already has paths of infinite variation. Therefore, the Lévy process in Eq. (5.2) has infinite variation, since it contains a continuous martingale component. However, here we refer to the infinite variation that comes from the jumps.
- 5.
Since instantaneous variance is not a martingale, the upper boundary of β can be extended to infinity.
- 6.
Here for simplicity we consider only Dirichlet boundary conditions.
- 7.
Assuming that we know all necessary strikes in advance.
- 8.
If the ex-dividend date falls between two nodes of the temporal grid, a common approach is to move it to the grid node either forward (and adjusting the dividend amount to the forward value of the dividend at that date) or backward (and discounting the dividend amount back to that date).
- 9.
We allow the coefficients of the LSV model to be time-dependent. However, they are assumed to be piecewise constant at every time step.
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Itkin, A. (2017). High-Order Splitting Methods for Forward PDEs and PIDEs. In: Pricing Derivatives Under Lévy Models . Pseudo-Differential Operators, vol 12. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-6792-6_7
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