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A Continuous Time Financial Market with a Poisson Process as Transaction Timer


We study the influence of transactions on investors' portfolio values under the assumption that the investors' transaction times are determined by Poisson point processes, whose intensity measures can naturally be interpreted as transaction frequencies. We give lower and upper bounds on the expectations of portfolio values in terms of transaction intensities, and prove that there exist a sequence of portfolio fractions and a transaction frequency which maximize the expectation of the portfolio value for a finite horizon. We also give bounds on transaction frequencies for preventing the investors from losing money. Then the optimal transaction strategies for finite and infinite time horizons and the asymptotic effects of making transactions are discussed based on the concept of a benefit function of transactions. Finally, we investigate the influence of transactions on financial markets, with the market mean rates of return and volatilities being connected with the transaction frequency. We show that the market becomes unprofitable in a finite time if an overwhelming amount of transactions is involved and the market is suitable for some limited transactions when its trade capacity does not increase beyond any limit at a relatively high speed. Our models and simulations illustrate how the investors' collective action affects the financial market.

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  1. [1]

    S. Albeverio, L.J. Lao and X.L. Zhao, On-line portfolio selection strategy with predictions in the presence of transaction costs, Math. Methods Oper. Res. (2001) to appear.

  2. [2]

    S. Albeverio and V. Steblovskaya, A model of financial market with several interacting assets (Extending the multidimensional Black-Scholes model), Preprint No. 609, Institute of Applied Mathematics, University of Bonn (1999), to appear in Finance and Stochastics (2001).

  3. [3]

    A. Cadenillas and S.R. Pliska, Optimal trading of a security when there are taxes and transaction costs, Finance Stochastics 2 (1999) 137-165.

    Google Scholar 

  4. [4]

    T. Cover, Universal portfolio, Math. Finance 1 (1991) 1-29.

    Google Scholar 

  5. [5]

    M.H.A. Davis and A.R. Norman, Portfolio selection with transaction costs, Math. Oper. Res. 15 (1990) 676-713.

    Google Scholar 

  6. [6]

    D. Duffie, Dynamic Asset Pricing Theory (Princeton Univ. Press, Princeton, NJ, 1992).

    Google Scholar 

  7. [7]

    D. Duffie and T. Sun, Transaction costs and portfolio choice in a discrete-continuous time setting, J. Econom. Dyn. Control 14 (1990) 35-51.

    Google Scholar 

  8. [8]

    J. Hull, Options, Futures, and Other Derivative Securities (Prentice-Hall, Englewood Cliffs, NJ, 1993).

    Google Scholar 

  9. [9]

    K. Itô and H. McKean, Diffusion Processes and Their Sample Paths (Springer, Berlin, 1965).

    Google Scholar 

  10. [10]

    I. Karatzas, Lectures on the Mathematics of Finance (Amer. Math. Soc., Providence, RI, 1997).

    Google Scholar 

  11. [11]

    I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus (Springer, New York, 1988).

    Google Scholar 

  12. [12]

    R. Korn, Optimal Portfolios (World Scientific, Singapore, 1997).

    Google Scholar 

  13. [13]

    S. Kusuoka, Limit theorem on option replication cost with transaction costs, Ann. Appl. Probab. 5(1) (1995) 198-221.

    Google Scholar 

  14. [14]

    G. Last and A. Brandt, Marked Point Processes on the Real Line: The Dynamic Approach (Springer, New York, 1995).

    Google Scholar 

  15. [15]

    R.C. Merton, Optimum consumption and portfolio rules in a continuous time model, J. Econom. Theory 3 (1971) 373-413.

    Google Scholar 

  16. [16]

    A.J. Morton and S.R. Pliska, Optimal portfolio management with fixed transaction costs, Math. Finance 5(4) (1995) 337-356.

    Google Scholar 

  17. [17]

    L.C.G. Rogers and D. Talay, Numerical Methods in Finance (Cambridge Univ. Press, Cambridge, 1997).

    Google Scholar 

  18. [18]

    W. Shaw, Modelling Financial Derivatives with Mathematica (Cambridge Univ. Press, Cambridge, 1998).

    Google Scholar 

  19. [19]

    S.E. Shreve and H.M. Soner, Optimal investment and consumption with transaction costs, Ann. Appl. Probab. 4(3) (1994) 609-692.

    Google Scholar 

  20. [20]

    D. Sondermann, Reinsurance in arbitrage-free markets, Insurance Math. Econom. 10(3) (1998) 191-202.

    Google Scholar 

  21. [21]

    A. Tourin and T. Zariphopoulou, Viscosity solutions and numerical schemes for investment/consumption models with transaction costs, in: Numerical Methods in Finance, eds. L.C.G. Rogers and D. Talay (Cambridge Univ. Press, Cambridge, 1997) pp. 245-269.

    Google Scholar 

  22. [22]

    K. Yosida, Functional Analysis (Springer, Berlin, 1968).

    Google Scholar 

  23. [23]

    X.L. Zhang, Valuation of American option in a jump-diffusion models, in: Numerical Methods in Finance, eds. L.C.G. Rogers and D. Talay (Cambridge Univ. Press, Cambridge, 1997) pp. 93-114.

    Google Scholar 

  24. [24]

    X.L. Zhao, Introduction toMeasure-Valued Branching Processes (in Chinese) (Science Press, Beijing, 2000).

    Google Scholar 

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Albeverio, S., Lao, LJ. & Zhao, XL. A Continuous Time Financial Market with a Poisson Process as Transaction Timer. Advances in Computational Mathematics 16, 305–330 (2002).

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  • portfolio management
  • transaction costs
  • utility functions
  • Poisson point processes
  • Monte-Carlo simulations