ABSTRACT
This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma (VG) process. The continuous time VG process is approximated by a continuous time process with the same first four cumulants and then discretized in time and space. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised before the expiration date. Numerical computations of European and American options are presented and compared with results obtained with finite differences method and with the Black–Scholes prices.
Acknowledgements
Our sincere thanks are for the Department of Mathematics of ISEG and CEMAPRE, University of Lisbon, http://cemapre.iseg.ulisboa.pt/. We wish also to acknowledge all the members of the STRIKE network, http://www.itn-strike.eu/.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Nicola Cantarutti http://orcid.org/0000-0002-0441-7159
Notes
1 The diffusion coefficient is usually called σ. Here we use because σ will be used for the VG process.
2 Usually, the Gamma distribution is parametrized by a shape and scale positive parameters . The Gamma process
has pdf
and has moments
and
. Here we use a parametrization as in [Citation18] such that
and
, so
,
.
3 In [Citation18], the authors derive the expression for the Lévy measure by representing the VG process as the difference between two Gamma processes.
4 See Example 8.10 in [Citation22].
5 To obtain the correction term ω we have to find the exponential moment of using its characteristic function:
.
6 We use the bar over κ, to distinguish the kurtosis from the variance of the gamma process κ.
7 Remind that and
, with
the central ith moment. Remind also that
and
.