Fractional calculus and continuous-time finance.
In financial markets, not only prices can be modelled as random variables, but also
waiting times between two consecutive transactions vary in a stochastic fashion.
We argue that the Continuous Time Random Walk (CTRW) model, formerly introduced
in Statistical Mechanics by Montroll and Weiss can provide a phenomenological description
of tick-by-tick dynamics in financial markets. We develop further theoretical arguments
based on fractional calculus and test theoretical predictions against empirical data.
Researchers involved: References:
E. Scalas, R. Gorenflo, F. Mainardi, M. Raberto.
Revisiting the derivation of the fractional diffusion equation.
Fractals 11 No. Suppl. S, February 2003, pp. 281-289.
http://xxx.lanl.gov/abs/cond-mat/0210166
M. Raberto, E. Scalas, F. Mainardi.
Waiting-times and returns in high-frequency financial data: an empirical study.
Physica A 314 No. 1-4, November 2002, pp. 749-755.
http://xxx.lanl.gov/abs/cond-mat/0203596
E. Scalas, R. Gorenflo, F. Mainardi.
Fractional calculus and continuous-time finance.
Physica A 284 No. 1-4, September 2000 , pp. 376-384.
http://xxx.lanl.gov/abs/cond-mat/0001120
F. Mainardi, M. Raberto, E. Scalas, R. Gorenflo.
Fractional calculus and continuous-time finance II: the waiting-time distribution.
Physica A 287 No. 3-4, December 2000, pp. 468-481.
http://xxx.lanl.gov/abs/cond-mat/0006454
Learning short-option valuation in the presence of rare events.
We extend the neural network approach for the valuation of financial derivatives developed
by Hutchinson et al. (1994) to the case of fat-tailed distributions of the underlying asset
returns. In order to generate a suitable input learning set, we use the method of Gorenflo
et al. based on fractional calculus. The output learning-set option price is computed by
means of a formula given by Sornette and Bouchaud and also discussed in the book by Bouchaud
and Potters.
Researchers involved: References:
M Raberto, G. Cuniberti, M. Riani, E. Scalas, F. Mainardi, G. Servizi.
Learning short-option valuation in the presence of rare events.
Int J Theor Appl Finance 3 No. 3, 2000, pp. 563-564.
http://xxx.lanl.gov/abs/cond-mat/0001253
Volatility and interest rate in financial markets.
In the Black-Scholes-Merton model for option pricing, the volatility and the risk-free
interest rate are constant. Indeed, it is well known that this is not true in general.
The purpose of this project is to empirically characterize the stochastic evolution of both
volatility and interest rate.
Researchers involved: References:
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