Continuous-time statistics and generalized relaxation equations

Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.Comment: 12 pages, submitted to EPJ

A class of CTRWs: Compound fractional Poisson processes

This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known L\'evy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson process, where the Poisson counting process is replaced by the Mittag-Leffler counting process also known as fractional Poisson process. This process is no longer Markovian and L\'evy. However, several analytical results are available and some of them are discussed here. The functional limit of the compound Poisson process is an $\alpha$-stable L\'evy process, whereas in the case of the compound fractional Poisson process, one gets an $\alpha$-stable L\'evy process subordinated to the fractional Poisson process.Comment: 23 pages. To be published in a World Scientific book edited by Ralf Metzle

Five Years of Continuous-time Random Walks in Econophysics

This paper is a short review on the application of continuos-time random walks to Econophysics in the last five years.Comment: 14 pages. Paper presented at WEHIA 2004, Kyoto, Japa

Five Years of Continuous-time Random Walks in Econophysics

This paper is a short review on the application of continuos-time random walks to Econophysics in the last five years.Duration; Continuous-time random walk; Fractional calculus; Statistical finance

A note on intraday option pricing

Compound renewal processes can be used as an approximate phenomenological model of tick-by-tick price fluctuations. An exact and explicit general formula is derived for the martingale price of a European call option written on a compound renewal process. The option price is obtained using the direct method of indicator functions. The applicability of this result is discussed

A parsimonious model for intraday European option pricing

A stochastic model for pure-jump diffusion (the compound renewal process) can be used as a zero-order approximation and as a phenomenological description of tick-by-tick price fluctuations. This leads to an exact and explicit general formula for the martingale price of a European call option. A complete derivation of this result is presented by means of elementary probabilistic tools.Comment: Submitted to Economics E-Journal: http://www.economics-ejournal.org/economics/discussionpapers/2012-1

A note on intraday option pricing

Compound renewal processes can be used as an approximate phenomenological model of tick-by-tick price fluctuations. An exact and explicit general formula is derived for the martingale price of a European call option written on a compound renewal process. The option price is obtained using the direct method of indicator functions. The applicability of this result is discussed

The art of fitting financial time series with Levy stable distributions

This paper illustrates a procedure for fitting financial data with $\alpha$-stable distributions. After using all the available methods to evaluate the distribution parameters, one can qualitatively select the best estimate and run some goodness-of-fit tests on this estimate, in order to quantitatively assess its quality. It turns out that, for the two investigated data sets (MIB30 and DJIA from 2000 to present), an $\alpha$-stable fit of log-returns is reasonably good.Comment: 17 pages, 10 figures, 2 tables. Paper presented at the DDAP4 conference, Pohang, Korea, July 2006. Submitted to Journal of Korean Physical Societ

The art of fitting financial time series with Levy stable distributions

This paper illustrates a procedure for fitting financial data with alpha-stable distributions. After using all the available methods to evaluate the distribution parameters, one can qualitatively select the best estimate and run some goodness-of-fit tests on this estimate, in order to quantitatively assess its quality. It turns out that, for the two investigated data sets (MIB30 and DJIA from 2000 to present), an alpha-stable fit of log-returns is reasonably good.finance; statistical methods; stable distributions