Mathematical and Statistical Methods for Actuarial Sciences and Finance

Tempered stable distributions and processes in finance: numerical analysis 41

general there is no constructive method to find the subordinator process that changes
the time of the Brownian motion; that is we do not know the processTtsuch that the
TSαprocessXtcan be rewritten asWT(t)[7]. The shot noise representation allows
one to generate any TSαprocess.

5 Conclusions

In this work, we have focused our attention on the practical implementation of nu-
merical methods involving the use of TSαdistributions and processes in the field of
finance. Basic definitions are given and a possible algorithm to approximate the den-
sity function is proposed. Furthermore, a general Monte Carlo method is developed
with a look at option pricing.

References

1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, with Formulas,
   Graphs, and Mathematical Tables. Dover Publications, Mineola, NY (1974)


2. Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Springer,
   New York (2007)


3. Becken, W., Schmelcher, P.: The analytic continuation of the Gaussian hypergeometric
   function 2F1 (a, b; c; z) for arbitrary parameters. J. Comp. Appl. Math. 126(1–2), 449–478
   (2000)


4. Boyarchenko, S.I., Levendorski ̆ı, S.: Non-Gaussian Merton-Black-Scholes Theory. World
   Scientific, River Edge, NJ (2002)


5. Carr, P., Geman, H., Madan, D.B., Yor, M.: The Fine Structure of Asset Returns: An
   Empirical Investigation. J. Bus. 75(2), 305–332 (2002)


6. Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finan.
   2(4), 61–73 (1999)


7. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall / CRC
   Press, London (2004)


8. Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. 2. Wiley,
   Princeton, 3rd edition (1971)


9. Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. Siam,
   Philadelphia (2007)


10. Houdre, C., Kawai, R.: On fractional tempered stable motion. Stoch. Proc. Appl. 116(8), ́
    1161–1184 (2006)


11. Janicki, A., Weron, A.: Simulation and Chaotic Behavior ofα-stable Stochastic Processes.
    Marcel Dekker, New York (1994)


12. Kawai, R.: An importance sampling method based on the density transformation of L ́evy
    processes. Monte Carlo Meth. Appl. 12(2), 171–186 (2006)


13. Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: Financial market models with l ́evy
    processes and time-varying volatility. J. Banking Finan. 32(7),1363–1378 (2008)


14. Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J.: A new tempered stable distribution
    and its application to finance. In: Bol, G., Rachev, S.T., Wuerth, R. (eds.) Risk Assessment:
    Decisions in Banking and Finance, pp. 77–119. Physika Verlag, Springer, Heidelberg
    (2009)