Multinomial method for option pricing under Variance Gamma

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.

KEYWORDS:

• American option
• Lévy processes
• moment matching
• multinomial tree
• Variance Gamma

2010 MSC SUBJECT CLASSIFICATIONS:

• 91G20
• 60G51
• 60G57
• 60J75
• 60J22
• 60J10
• 45K05

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 $\tilde{\sigma }$ because σ will be used for the VG process.

2 Usually, the Gamma distribution is parametrized by a shape and scale positive parameters $T\sim \mathrm{\Gamma }(\rho ,\zeta )$. The Gamma process $T_t\sim \mathrm{\Gamma }(\rho t,\zeta )$ has pdf $f_{T_t}\left(x\right)=\left({\zeta }^{-\rho t}/\mathrm{\Gamma }\left(\rho t\right)\right)x^{\rho t-1}{\mathrm{e}}^{-x/\zeta }$ and has moments $\mathbb{E}\left[T_t\right]=\rho \zeta t$ and $\mathrm{Var}\left[T_t\right]=\rho {\zeta }^{2}t$. Here we use a parametrization as in [18] such that $\mathbb{E}\left[T_t\right]=\mu t$ and $\mathrm{Var}\left[T_t\right]=\kappa t$, so $\zeta =\kappa /\mu$, $\rho ={\mu }^2/\kappa$.

3 In [18], 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 [22].

5 To obtain the correction term ω we have to find the exponential moment of $X_t$ using its characteristic function: $\mathbb{E}\left[{\mathrm{e}}^{X_t}\right]={\phi }_{X_t}(-i)={\mathrm{e}}^{-\omega t}$ .

6 We use the bar over κ, to distinguish the kurtosis from the variance of the gamma process κ.

7 Remind that $\mathrm{Skew}\left[X\right]={\mu }_3/{\mu }_2^{3/2}$ and $\mathrm{Kurt}\left[X\right]={\mu }_4/{\mu }_2^2$, with ${\mu }_i$ the central ith moment. Remind also that ${\mu }_3={\mu }_3^{\prime }-3{\mu }^{\prime }{\mu }_2^{\prime }+2{\mu }^{\prime 3}$ and ${\mu }_4={\mu }_4^{\prime }-4{\mu }^{\prime }{\mu }_3^{\prime }+6{\mu }^{\prime 2}{\mu }_2^{\prime }-3{\mu }^{\prime 4}$.

Additional information

Funding

This research was supported by the European Union in the FP7-PEOPLE-2012-ITN project STRIKE – Novel Methods in Computational Finance (304617), and by CEMAPRE MULTI/00491, financed by FCT/MEC through Portuguese national funds.