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Published in:
Basics of the Finite Difference Method Read chapter Read first chapter

2017 | OriginalPaper | Chapter

6. Pseudoparabolic Equations for Various Lévy Models

Author : Andrey Itkin

Published in: Pricing Derivatives Under Lévy Models

Publisher: Springer New York

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Abstract

In this chapter, following Itkin (Algorithmic Finance 3:233–250, 2014; J. Comput. Finance 19:29–70, 2016), we describe a generalization of the approach presented in Section 5.​4 For the sake of convenience, we call this method MPsDO (method of pseudodifferential operators). The idea of transforming a nonlocal jump operator into the local pseudodifferential operator here is implemented using the representation, well known in physics, of the translational operator as an operator exponential. We also establish a connection between this representation and the characteristic function of the jump process. Finally, for various popular jump models, which include models of jumps with both finite and infinite activity, and finite and infinite variation, we construct an FD scheme for solving the corresponding fractional PDE. These FD schemes provide second-order approximation in both space and time, are unconditionally stable, and preserve nonnegativity of the solution. Various numerical tests are presented to demonstrate such behavior.

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previous chapter Pseudoparabolic and Fractional Equations of Option Pricing
next chapter High-Order Splitting Methods for Forward PDEs and PIDEs
Footnotes
1
A second-order approximation could in principle be constructed as well, resulting, however, in a massive calculation of the coefficients. That probably discouraged researchers from further elaborating this approach.
 
2
So the PIDE grid is a superset of the PDE grid.
 
3
In more advanced approaches, this step could be eliminated; see [42]. Also, if the coefficients of the linear interpolation are precomputed, the overhead of performance for doing interpolation is relatively small; see [18, 19].
 
4
It is actually a double exponential operator.
 
5
The approach presented below is also applicable to single-barrier options, or to options with a nonvanilla payoff, e.g., digitals.
 
6
With a proposed improvement that reduces the total complexity of the method from O(N∕Δ s) to O(N).
 
7
For positive jumps, this could be done in a similar way. The denominator in Eq. (6.12) then changes to ϕ − 1, and the term ϕ I + aA 2 B changes to ϕ I − aA 2 F , where ϕ > 1.
 
8
Some care should be taken regarding the boundary values of A 2 C to guarantee this. Usually, introduction of ghost points at the boundaries helps to increase the accuracy of the method. Alternatively, one could use another approximation of the term \((\phi +a\triangledown )^{-1}\) in Eq. (6.11), which is (ϕ I + aA B )−1. This reduces the order of approximation from the exact second order to some order in between 1 and 2, but at the same time, significantly improves the properties of the resulting matrix J.
 
9
This method is not very accurate. But since the exact solution is not known, it provides a plausible estimate of the convergence.
 
10
All experiments were computed in Matlab with an Intel Pentium 4 CPU 3.2 GHz under x86 Windows 7 OS. Obviously, a C++ implementation provides better performance by roughly a factor of 5.
 
11
Don’t confuse this with the accuracy of the entire three steps of Strang’s algorithm, which is O(Δ τ 2). The test validates just the convergence in h, not in Δ τ.
 
12
In Itkin and Carr’s paper, jump integrals were defined on half-infinite positive and negative domains, while here they are defined on the whole infinite domain. Therefore, to prove this proposition, simply use ∫ −∞ ∞  = ∫ −∞ 0 + ∫ 0 ∞ and then apply Proposition 7 from [30].
 
13
An example of an A-stable scheme is the familiar Crank–Nicolson scheme. But we want to underline that 0 doesn’t belong to the spectrum of M, so formally, the scheme is L-stable, though with convergence properties close to those of an A-stable scheme. The formal L-stability is important, e.g., for computing the option Greeks.
 
14
As was mentioned in the introduction, in this particular case, FFT is definitely more efficient, so we provide this comparison just for illustrative purposes.
 
15
For explicit formulas to provide this translation, see [36].
 
16
They are also known as modified Bessel functions of the second kind, or Macdonald functions; see [47].
 
17
This method is not very accurate. But since the exact solution is not known, it provides a plausible estimate of the convergence.
 
18
Don’t confuse this with the accuracy of the whole three steps Strang’s algorithm, which is O(Δ τ 2). The test validates just the convergence in h, not in Δ τ.
 
19
We do this by expanding \(\cos \left (\frac{a\triangledown + b} {2}\right )\) in the denominator into a series in h.
 
20
This complexity is proportional to a certain coefficient, which in this case can be of order 10. Nevertheless, as will be shown in our tests, the methods is significantly faster than the FFT.
 
21
The standard FFT method at these values of the parameters is very sensitive to the choice of the damping factor α, and therefore, this price was computed using the cosine method of [23].
 
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Metadata
Title
Pseudoparabolic Equations for Various Lévy Models
Author
Andrey Itkin
Copyright Year
2017
Publisher
Springer New York
Book
Pricing Derivatives Under Lévy Models

Print ISBN: 978-1-4939-6790-2

Electronic ISBN: 978-1-4939-6792-6

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4939-6792-6_6

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