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Capital Asset Pricing for Markets with Intensity Based Jumps

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Stochastic Finance

Summary

This paper proposes a unified framework for portfolio optimization, derivative pricing, modeling and risk measurement in financial markets with security price processes that exhibit intensity based jumps. It is based on the natural assumption that investors prefer more for less, in the sense that for two given portfolios with the same variance of its increments, the one with the higher expected increment is preferred. If one additionally assumes that the market together with its monetary authority acts to maximize the long term growth of the market portfolio, then this portfolio exhibits a very particular dynamics. In a market without jumps the resulting dynamics equals that of the growth optimal portfolio (GOP). Conditions are formulated under which the well-known capital asset pricing model is generalized for markets with intensity based jumps. Furthermore, the Markowitz efficient frontier and the Sharpe ratio are recovered in this continuous time setting. In this paper the numeraire for derivative pricing is chosen to be the GOP. Primary security account prices, when expressed in units of the GOP, turn out to be supermartingales. In the proposed framework an equivalent risk neutral martingale measure need not exist. Fair derivative prices are obtained as conditional expectations of future payoff structures under the real world probability measure. The concept of fair pricing is shown to generalize the classical risk neutral and the actuarial net present value pricing methodologies.

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References

  1. Bajeux-Besnainou, I. & R. Portait (1997). The numeraire portfolio: A new perspective on financial theory. The European Journal of Finance 3, 291–309.

    Article  Google Scholar 

  2. Bühlmann, H. (1992). Stochastic discounting. Insurance: Mathematics and Economics 11, 113–127.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bühlmann, H. (1995). Life insurance with stochastic interest rates. In G. Ottaviani (Ed.), Financial Risk and Insurance, pp. 1–24. Springer.

    Google Scholar 

  4. Christensen, M. M. & E. Platen (2004). A general benchmark model for stochastic jump sizes. Technical report, University of Technology, Sydney. QFRC Research Paper 139, to appear in Stochastic Analysis and Applications.

    Google Scholar 

  5. Cochrane, J. H. (2001). Asset Pricing. Princeton University Press.

    Google Scholar 

  6. Constatinides, G. M. (1992). A theory of the nominal structure of interest rates. Rev. Financial Studies 5, 531–552.

    Article  Google Scholar 

  7. Cont, R. & P. Tankov (2004). Financial Modelling with Jump Processes. Financial Mathematics Series. Chapman & Hall/CRC.

    Google Scholar 

  8. Delbaen, F. & W. Schachermayer (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.

    Article  MathSciNet  Google Scholar 

  9. Dimson, E., P. Marsh, & M. Staunton (2002). Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press.

    Google Scholar 

  10. Duffie, D. (2001). Dynamic Asset Pricing Theory (3rd ed.). Princeton, University Press.

    Google Scholar 

  11. Duffie, D. & K. Singleton (2003). Credit Risk: Pricing, Measurement, and Management. Princeton Series in Finance. Princeton University Press.

    Google Scholar 

  12. Föllmer, H. & M. Schweizer (1991). Hedging of contingent claims under in-complete information. In M. Davis and R. Elliott (Eds.), Applied Stochastic Analysis, Volume 5 of Stochastics Monogr., pp. 389–414. Gordon and Breach, London/New York.

    Google Scholar 

  13. Gerber, H. U. (1990). Life Insurance Mathematics. Springer, Berlin.

    Google Scholar 

  14. Harrison, J. M. & D. M. Kreps (1979). Martingale and arbitrage in multiperiod securities markets. J. Economic Theory 20, 381–408.

    Article  MathSciNet  Google Scholar 

  15. Harrison, J. M. & S. R. Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11(3), 215–260.

    Article  MathSciNet  Google Scholar 

  16. Heath, D. & E. Platen (2002a). Consistent pricing and hedging for a modified constant elasticity of variance model. Quant. Finance. 2(6), 459–467.

    Article  MathSciNet  Google Scholar 

  17. Heath, D. & E. Platen (2002b). Perfect hedging of index derivatives under a minimal market model. Int. J. Theor. Appl. Finance 5(7), 757–774.

    Article  MathSciNet  Google Scholar 

  18. Heath, D. & E. Platen (2002c). Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In J. Yong (Ed.), Recent Developments in Mathematical Finance, pp. 117–126. World Scientific.

    Google Scholar 

  19. Karatzas, I. & S. E. Shreve (1998). Methods of Mathematical Finance, Volume 39 of Appl. Math. Springer.

    Google Scholar 

  20. Kelly, J. R. (1956). A new interpretation of information rate. Bell Syst. Techn. J. 35, 917–926.

    Google Scholar 

  21. Korn, R. & M. Schäl (1999). On value preserving and growth-optimal portfolios. Math. Methods Oper. Res. 50(2), 189–218.

    Article  MathSciNet  Google Scholar 

  22. Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econom. Statist. 47, 13–37.

    Article  Google Scholar 

  23. Long, J. B. (1990). The numeraire portfolio. J. Financial Economics 26, 29–69.

    Article  Google Scholar 

  24. Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investment. Wiley, New York.

    Google Scholar 

  25. Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, 867–888.

    MATH  MathSciNet  Google Scholar 

  26. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. J. Financial Economics 2, 125–144.

    Article  Google Scholar 

  27. Miller, S. & E. Platen (2004). A two-factor model for low interest rate regimes. Technical report, University of Technology, Sydney. QFRC Research Paper 130, to appear in Asia-Pacific Financial Markets 11(1).

    Google Scholar 

  28. Platen, E. (2001). A minimal financial market model. In Trends in Mathematics, pp. 293–301. Birkhäuser.

    Google Scholar 

  29. Platen, E. (2002). Arbitrage in continuous complete markets. Adv. in Appl. Probab. 34(3), 540–558.

    Article  MATH  MathSciNet  Google Scholar 

  30. Platen, E. (2004a). A class of complete benchmark models with intensity based jumps. J. Appl. Probab. 41, 19–34.

    Article  MATH  MathSciNet  Google Scholar 

  31. Platen, E. (2004b). Diversified portfolios with jumps in a benchmark framework. Technical report, University of Technology, Sydney. QFRC Research Paper 129, to appear in Asia-Pacific Financial Markets 11(1).

    Google Scholar 

  32. Platen, E. (2004c). Modeling the volatility and expected value of a diversified world index. Int. J. Theor. Appl. Finance 7(4), 511–529.

    Article  MATH  MathSciNet  Google Scholar 

  33. Platen, E. (2004d). Pricing and hedging for incomplete jump diffusion benchmark models. In Mathematics of Finance, Volume 351 of Contemporary Mathematics, pp. 287–301. American Mathematical Society.

    MATH  MathSciNet  Google Scholar 

  34. Protter, P. (2004). Stochastic Integration and Differential Equations (2nd ed.). Springer.

    Google Scholar 

  35. Revuz, D. & M. Yor (1999). Continuous Martingales and Brownian Motion (3rd ed.). Springer.

    Google Scholar 

  36. Rogers, L. C. G. & D. Williams (2000). Diffusions, Markov Processes and Martingales: ltô Calculus (2nd ed.), Volume 2 of Cambridge Mathematical Library. Cambridge University Press.

    Google Scholar 

  37. Ross, S. A. (1976). The arbitrage theory of capital asset pricing. J. Economic Theory 13, 341–360.

    Article  Google Scholar 

  38. Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. J. Finance 19, 425–442.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Finance & Economics and Department of Mathematical Sciences, University of Technology Sydney, PO Box 123, Broadway, NSW, 2007, Australia

    Eckhard Platen

Authors
  1. Eckhard Platen
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Editor information

Editors and Affiliations

  1. Steklov Mathematical Institute, Moscow, Russia

    A. N. Shiryaev

  2. ISEG, Universidade Técnica de Lisboa, Portugal

    M. R. Grossinho

  3. Universidade de Coimbra, Portugal

    P. E. Oliveira

  4. FCT, Universidade Nova de Lisboa, Monte de Caparica, Portugal

    M. L. Esquível

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© 2006 Springer Science+Business Media, Inc.

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Platen, E. (2006). Capital Asset Pricing for Markets with Intensity Based Jumps. In: Shiryaev, A.N., Grossinho, M.R., Oliveira, P.E., Esquível, M.L. (eds) Stochastic Finance. Springer, Boston, MA. https://doi.org/10.1007/0-387-28359-5_5

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