Abstract
This paper considers diversified portfolios in a sequence of jump diffusion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversified portfolios are provided. Under realistic assumptions, it is shown that diversified portfolios approximate the GOP without requiring any major model specifications. This provides a basis for systematic use of diversified stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ansel, J. P. and Stricker, C. (1994) Couverture des actifs contingents, Ann. Inst. H. Poincaré Probab. Stat. 30, 303–315.
Artzner, P. (1997) On the numeraire portfolio. Mathematics of Derivative Securities, Cambridge University Press, Cambridge, pp. 53–58.
Bajeux-Besnainou, I. and Portait, R. (1997) The numeraire portfolio: A new perspective on financial theory, Eur. J. Finance 3, 291–309.
Basle (1995) Planned Supplement to the Capital Accord to Incorporate Market Risk. Basle Committee on Banking and Supervision, Basle, Switzerland.
Becherer, D. (2001) The numeraire portfolio for unbounded semimartingales, Finance Stoch. 5, 327–341.
Björk, T. and Näslund, B. (1998) Diversified portfolios in continuous time, Eur. Finance Rev. 1, 361–387.
Breymann, W., Kelly, L. and Platen, E. (2004) Intraday Empirical Analysis and Modeling of Diversified World Stock Indices, Technical Report, University of Technology, Sydney, QFRC Research, p. 125.
Constatinides, G. M. (1992) A theory of the nominal structure of interest rates, Rev. Financ. Studies 5, 531–552.
Delbaen, F. and Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing, Math. Ann. 300, 463–520.
Delbaen, F. and Schachermayer, W. (1998) The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann. 312, 215–250.
Duffie, D. (2001) Dynamic Asset Pricing Theory (3rd ed.), Princeton University Press, Princeton.
Fernholz, E. R. (2002) Stochastic Portfolio Theory, Vol. 48 of Applications of Mathematics, Springer, New York.
Föllmer, H. and Schweizer, M. (1991) Hedging of contingent claims under incomplete information. In M. Davis and R. Elliott (eds.), Applied Stochastic Analysis, Vol. 5 of Stochastics Monographs, Gordon and Breach, London/New York, pp. 389–414.
Föllmer, H. and Sondermann, D. (1986) Hedging of non-redundant contingent claims. In W. Hildebrandt and A. Mas-Colell (eds.), Contributions to Mathematical Economics, North Holland, pp. 205–223.
Goll, T. and Kallsen, J. (2003) A complete explicit solution to the log-optimal portfolio problem, Adv. Appl. Probab. 13(2), 774–799.
Guan, L. K., Liu, X. and Chong, T. K. (2004) Asymptotic dynamics and VaR of large diversified portfolios in a jump-diffusion market, Quant. Finance 4(2), 129–139.
Harrison, J. M. and Kreps, D. M. (1979) Martingale and arbitrage in multiperiod securities markets, J. Econ. Theory 20, 381–408.
Harrison, J. M. and Pliska, S. R. (1981) Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11(3), 215–260.
Heath, D. and Platen, E. (2002) Consistent pricing and hedging for a modified constant elasticity of variance model, Quant. Finance 2(6), 459–467.
Heath, D. and Platen, E. (2003) Pricing of index options under a minimal market model with lognormal scaling, Quant. Finance 3(6), 442–450.
Hofmann, N. and Platen, E. (2000) Approximating large diversified portfolios, Math. Finance 10(1), 77–88.
Karatzas, I. and Shreve, S. E. (1998) Methods of Mathematical Finance, Vol. 39 of Applications of Mathematics, Springer, New York.
Kelly, J. R. (1956) A new interpretation of information rate, Bell Syst. Tech. J. 35, 917–926.
Kramkov, D. O. and Schachermayer, W. (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab. 9, 904–950.
Long, J. B. (1990) The numeraire portfolio, J. Financ. Econ. 26, 29–69.
Markowitz, H. (1959) Portfolio Selection: Efficient Diversification of Investment, Wiley, New York.
Merton, R. C. (1973) An intertemporal capital asset pricing model, Econometrica 41, 867–888.
Platen, E. (2001) A minimal financial market model. In Trends in Mathematics, Birkhäuser, pp. 293–301.
Platen, E. (2002) Arbitrage in continuous complete markets, Adv. Appl. Probab. 34(3), 540–558.
Platen, E. (2004a) A benchmark framework for risk management. In Stochastic Processes and Applications to Mathematical Finance, Proceedings of the Ritsumeikan International Symposium, World Scientific, Singapore, pp. 305–335.
Platen, E. (2004b) A class of complete benchmark models with intensity based jumps, J. Appl. Probab. 41, 19–34.
Platen, E. (2004c) Modeling the volatility and expected value of a diversified world index, Int. J. Theor. Appl. Finance 7(4), 511–529.
Platen, E. and Stahl, G. (2003) A structure for general and specific market risk, Comput. Statist. 18(3), 355–373.
Protter, P. (1990). Stochastic Integration and Differential Equations, Springer, Berlin.
Rogers, L. C. G. (1997) The potential approach to the term structure of interest rates and their exchange rates, Math. Finance 7, 157–176.
Ross, S. A. (1976) The arbitrage theory of capital asset pricing, J. Econ. Theory 13, 341–360.
Sharpe, W. F. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk, J.~Finance 19, 425–442.
Author information
Authors and Affiliations
Corresponding author
Additional information
1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20
JEL Classification: G10, G13
Rights and permissions
About this article
Cite this article
Platen, E. Diversified Portfolios with Jumps in a Benchmark Framework. Asia-Pacific Finan Markets 11, 1–22 (2004). https://doi.org/10.1007/s10690-005-4253-8
Issue Date:
DOI: https://doi.org/10.1007/s10690-005-4253-8