It is well known that Gaussian distributions can be a reasonable zero order approximation in modelling financial data. It has been observed that a much better approximation can be achieved by Levy distributions. The superiority of Levy distributions over Gaussian can be twofold. First, Levy distributions allowed for big jumps which are frequently observed on real markets. Second, they have fat tails which again agrees with observations. Levy distribution has however one unpleasant property: infinite moments.

To remove the bad property while retaining other good ones, Mantegna and Stanley (1994) introduced truncated Levy distributions. Then Koponen (1995) defined a different truncation which is better suited for analytical treatment. Financial data analyzed in recent years show however quite complicated behaviour. The empirical distributions of returns on real markets are symmetric near the origin but with much fatter left tail than the right one (cf. Cont at al (1997), Matacz (2000)). This behaviour cannot be obtained by Koponen's family. A satisfactory solution of this difficulty has been proposed by Boyarchenko and Levendrovskii (2000,2002).

Since the distribution function of the Levy process cannot be represented analytically the discussion is restricted to the characteristic function of the process. The starting point is the Levy-Khintchine formula, which for a purely non-Gaussian driftless process X_t has the form

φ_t(k)=E[exp(izk)-1-z/(1+z²)], (1)

where expectation is taken with respect to the Levy measure Π(dz) corresponding to the underlying Levy process X_t.

The idea of Koponen was to take a Levy measure for which the defined process has bounded variation. The measure with such a property can be defined by the following density function

f(z)=c_+/- exp(-λ|z|) |z|^-1-α, (2)

where c₊ and c_- correspond to positive and negative values of z, respectively. These constants are responsible for the asymmetry of the Levy process and λ is the cut-off parameter, which gives finite variance. In particular letting λ to zero we obtain a standard Levy process.

Formula (2) gives a truncated Levy process which only partially suits the analysis of financial data. To obtain a symmetry near the origin we have to assume c₊=c_-. But then also tails became symmetric. To get nonsymmetric tails we should assume different cut-off parameters for positive and negative values of z. This was already suggested by Matacz, who however was unable to figure out how to insert these two parameters to the Koponen formula. The same idea was risen also by Boyarchenko and Levendrovskii, who introduced abstract nonsymmetric Levy measures.

We have applied that approach to the density function of Koponen proposing its nonsymmetric version

f(z)=c_+/- exp(-λ_+/-|z|) |z|^-1-α. (3)

The obvious advantage of this formula is its simplicity, which allows for analytical calculation of the integral in formula (1). Long calculations lead to the following simple expression

φ_t(k)= tΓ(-α)(c_-(λ_-+ik)^α-c_-λ_-^α+c₊(λ₊-ik)^α-c₊λ₊^α) (4)

Taking in this expression c₊=c_- but different values of λ₊ and λ_- we can obtain a truncated Levy process which is symmetric near the origin but possesses nonsymmetric tails.

The obtained formula has been calibrated to the index data on Warsaw Stock Exchange (strictly speaking to WIG20 data). We have found quite satisfactory agreement and good fit of truncated Levy distribution to empirical data.

Essentially there is no obstacle in using that calibration to option pricing. It is however not a very reasonable project for near future as options traded on WSE are not liquid enough to reflect noarbitrage prices.


References

1. Boyarchenko S.I. and Levendorskii S.Z. - Option pricing for truncated Levy processes, Int.J.Theor.Appl.Finance 3 (2000), 549-552.
2. Boyarchenko S.I. and Levendorskii S.Z. - Non-Gaussian Merton-Black-Scholes Theory, World Scientific 2002.
3. Cont R., Potters M. and Bouchaud J.P. - Scaling in stock market data: stable laws and beyond, in Scale Invariance and beyond, B.Dubrulle, F.Graner, D.Sornette Eds. Springer 1997, pp.75-85.
4. Koponen I. - Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Phys.Rev. E 52 (1995), 1197-1199.
5. Matacz A. - Financial modeling and option theory with the truncated Levy process, Int.J.Theor.Appl.Finance 3 (2000), 143-160.
6. Montegna R.N. and Stanley H.E. - Stochastic process with ultraslow convergence to a Gaussian: the truncated Levy flight, Phys.Rev.Lett. 73 (1994), 2946-2949.

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