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Coupled continuous time random walks in finance

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Author Info

  • Meerschaert, Mark M.
  • Scalas, Enrico

Abstract

Continuous time random walks (CTRWs) are used in physics to model anomalous diffusion, by incorporating a random waiting time between particle jumps. In finance, the particle jumps are log-returns and the waiting times measure delay between transactions. These two random variables (log-return and waiting time) are typically not independent. For these coupled CTRW models, we can now compute the limiting stochastic process (just like Brownian motion is the limit of a simple random walk), even in the case of heavy-tailed (power-law) price jumps and/or waiting times. The probability density functions for this limit process solve fractional partial differential equations. In some cases, these equations can be explicitly solved to yield descriptions of long-term price changes, based on a high-resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log-returns. In the heavy-tailed case, this involves operator stable space–time random vectors that generalize the familiar stable models. In this paper, we will review the fundamental theory and present two applications with tick-by-tick stock and futures data.

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Bibliographic Info

Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

Volume (Year): 370 (2006)
Issue (Month): 1 ()
Pages: 114-118

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Handle: RePEc:eee:phsmap:v:370:y:2006:i:1:p:114-118

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Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

Related research

Keywords: Anomalous diffusion; Continuous time random walks; Heavy tails; Fractional calculus;

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References

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  1. Marco Raberto & Enrico Scalas & Francesco Mainardi, 2004. "Waiting-times and returns in high-frequency financial data: an empirical study," Finance 0411014, EconWPA.
  2. Francesco Mainardi & Marco Raberto & Rudolf Gorenflo & Enrico Scalas, 2004. "Fractional calculus and continuous-time finance II: the waiting- time distribution," Finance 0411008, EconWPA.
  3. Enrico Scalas & Rudolf Gorenflo & Francesco Mainardi, 2000. "Fractional calculus and continuous-time finance," Papers cond-mat/0001120, arXiv.org.
  4. Bertram, William K, 2004. "An empirical investigation of Australian Stock Exchange data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 533-546.
  5. Scheffler, Hans-Peter, 1999. "On estimation of the spectral measure of certain nonnormal operator stable laws," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 385-392, July.
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Citations

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Cited by:
  1. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2013. "Fractal dimension results for continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1083-1093.
  2. Cohen, Serge & Meerschaert, Mark M. & Rosinski, Jan, 2010. "Modeling and simulation with operator scaling," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2390-2411, December.
  3. Greenwood, Priscilla E. & Schick, Anton & Wefelmeyer, Wolfgang, 2011. "Estimating the inter-arrival time density of Markov renewal processes under structural assumptions on the transition distribution," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 277-282, February.
  4. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
  5. Schumer, Rina & Baeumer, Boris & Meerschaert, Mark M., 2011. "Extremal behavior of a coupled continuous time random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(3), pages 505-511.
  6. Scalas, Enrico & Politi, Mauro, 2012. "A parsimonious model for intraday European option pricing," Economics Discussion Papers 2012-14, Kiel Institute for the World Economy.
  7. Straka, P. & Henry, B.I., 2011. "Lagging and leading coupled continuous time random walks, renewal times and their joint limits," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 324-336, February.

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