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DISCRETIZATION SCHEMES FOR SUBORDINATED PROCESSES CLAUDIO ALBANESE AND ALEXEY KUZNETSOV
 

Summary: DISCRETIZATION SCHEMES FOR SUBORDINATED PROCESSES
CLAUDIO ALBANESE AND ALEXEY KUZNETSOV
ABSTRACT. We introduce a new class of continuous time lattices which are suitable for local Levy and
stochastic volatility processes and use them to construct numerical discretizations for the corresponding
partial integro-differential equations. Transition probabilities are computed either analytically or by means
of numerical linear algebra.
1. INTRODUCTION
The upsurge of interest in option pricing models based on jump processes, notably by Merton [1],
Eberlein [2] and Madan et al. [3], motivated the development of suitable lattice approximation schemes.
In the continuous limit, price functions solve partial integro-differential equations (PIDE's). In the con-
text of the variance-gamma model of [4] and [3], Madan and Hirsa introduce in [5] a lattice discretization
scheme for the integral kernel of the relevant PIDE. In [6], Petersdorff and Schwab propose a wavelet
compression method for the resulting matrices and stability conditions appear in [7].
Lattice models for jump processes are more subtle to implement than the analogue models for dif-
fusions. In the latter case, simple trinomial models are suitable (see Fig. 1) and transition probabilities
can be set by matching the first two moments, drift and volatility. However, if the underlying process
has jumps, moments higher than the second have to be fine tuned, the hopping range is not limited to
nearest neighbors and scaling laws are more complex. The problem is all the more delicate for models
combining jumps with state dependent local volatility which lack of translation invariance.
In this article we introduce continuous time lattice models which may be regarded as time changed

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics