Abstract
We study a Gamma-modulated diffusion process as a long-memory generalization of the standard Black-Scholes model. This model introduces a time dependent volatility. The option pricing problem associated with this type of processes is computed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bansal R., Yaron A.: Risks for the long run: a potential resolution of asset pricing puzzles. J. Financ. 59(4), 1481–1509 (2004)
Barkoulas J., Labys W., Onochie J.: Long memory in future prices. Financ. Rev. 34, 91–100 (1999)
Black F., Scholes M.: The pricing of options and corporate liabilities. J. Political Econ. 81, 635–654 (1973)
Campbell J., Cochrane J.: By force of habit: a consumption based explanation of aggregate stock market behavior. J. Political Econ. 107(2), 205–251 (1999)
Chan T.: Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab. 9(2), 504–528 (1999)
Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Chapman & Hall, mathematical series 2 (2004)
Duffie D.: Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton, NJ (1996)
Eraker J.: Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J. Financ. 59(3), 1367–1404 (2004)
Evans I.G.: Bayesian estimation of parameters of a multivariate normal distribution. J. Roy. Stat. Soc. Ser. B 27, 279–283 (1965)
Finlay R., Seneta E.: Stationary-increment student and variance-gamma processes. J. Appl. Probab. 43(2), 441–453 (2006)
Finlay R., Seneta E.: Stationary-increment variance-Gamma and t models: simulation and parameter estimation. Int. stat. Rev. 76(2), 167–186 (2008)
Floros C., Jaffry S., Valle G.: Long memory in the Portuguese stock market. Stud. Econ. Financ. 24(3), 220–232 (2007)
Follmer, H., Sondermann, D.: Hedging of nonredundant contingent claims. In: Hiloen-Brand, W., Mas-Colell, A. (eds.) Contributions to Mathematical Economics, pp. 205–223. North-Holland, Amsterdam
Follmer, H., Schweizer, M.: Hedging of contingent claims under incomplete information. Applied stochastic analysis (London, 1989), 389–414, Stochastics Monogr., 5, Gordon and Breach, New York (1991)
Frey R.: Risk minimization with incomplete information in a model for high-frequency data. Math. Financ. 10(2), 215–225 (2000)
Galea M., Ma J., Torres S.: Price calculation for power exponential jump-diffusion models—a Hermite series approach. Contemp. Math. Vol. 336, 137–160 (2003)
Granger C.W.J.: Long memory relationships and the aggregation of dynamic models. J. Econom. 14, 227–238 (1980)
Granger C.W.J., Joyeux R.: An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1, 15–29 (1980)
Hansen L., Heaton J., Li N.: Consumption strikes back? Measuring long run risk. J. Political Econ. 116(2), 260–302 (2008)
Harrison J.M., Pliska S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)
Henderson, V., Hobson, D.: Indifference pricing—an overview. In: Carmona, R. (ed.) Indifference Pricing Theory and Applications, chap. 2, pp. 44–72. Princeton University Press, Princeton (2008)
Heyde C.C., Leonenko N.N.: Student process. Adv. Appl. Prob. 37, 342–365 (2005)
Hurst H.E.: Long-term storage capacity of reservoirs. Trans. Am. Soc. Civil Eng. 116, 770–808 (1951)
Hurst, H.E.: Methods of using long-term storage in reservoirs. In: Proceedings of the Institute of Civil Engineers, vol. 1, pp. 519–543 (1956)
Jacquier E., Polson N., Rossi P.E.: Bayesian analysis of stochastic volatility model. J. Bus. Econ. Stat. 20(1), 69–87 (2002)
Kou S.G., Wang H.: Double exponential jump diffusion model. Manag. Sci. 50(9), 1178–1192 (2002)
Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall/CRC, Financial Mathematics Series (2000)
Mandelbrot B.B., Ness J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
Merton R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1976)
Protter P.: Stochastic Integration and Differential Equations A New Approach Applications of Mathematics (New York), vol. 21. Springer, Berlin (1990)
Protter P., Talay D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25(1), 393–423 (1997)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, vol. 293. Springer, Berlin (2005)
Shreve S.: Stochastic Calculus for Finance II Continuous-Time Models. Springer finance Series, New York (2004)
Sun W., Rachev S., Fabozzi F., Kalev P.: Fractals in trade duration: capturing long-range dependence and heavy tailedness in modeling trade duration. Ann. Financ. 4, 217–241 (2008)
Tehranchi M.: Explicit solution of some utility maximization problems in incomplete markets. Stoch. Process. Appl. 114(1), 1568–1590 (2004)
Torres S., Tudor C.A.: Donsker type theorem for the Rosenblatt process and a binary market model. Stochas. Anal. Appl. 27(3), 555–573 (2009)
Tudor C.A.: Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12, 230–257 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iglesias, P., San Martín, J., Torres, S. et al. Option pricing under a Gamma-modulated diffusion process. Ann Finance 7, 199–219 (2011). https://doi.org/10.1007/s10436-011-0176-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10436-011-0176-8