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Arbitrage in continuous complete markets

Part of: Applications Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Eckhard Platen
Eckhard Platen*
Affiliation:
University of Technology, Sydney
*
∗ Postal address: School of Finance and Economics and School of Mathematical Sciences, University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia. Email address: eckhard.platen@uts.edu.au
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Abstract

This paper introduces a benchmark approach for the modelling of continuous, complete financial markets, when an equivalent risk-neutral measure does not exist. This approach is based on the unique characterization of a benchmark portfolio, the growth optimal portfolio, which is obtained via a generalization of the mutual fund theorem. The discounted growth optimal portfolio with minimum variance drift is shown to follow a Bessel process of dimension four. Some form of arbitrage can be explicitly modelled by arbitrage amounts. Fair contingent claim prices are derived as conditional expectations under the real world probability measure. The Heath-Jarrow-Morton forward rate equation remains valid despite the absence of an equivalent risk neutral measure.

Keywords

Continuous financial marketarbitrage amountmutual fund theoremgrowth optimal portfolionuméraire portfoliocontingent claim pricingforward rate equation

MSC classification

Secondary: 60G30: Continuity and singularity of induced measures 62P20: Applications to economics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 3 , September 2002 , pp. 540 - 558
Copyright
Copyright © Applied Probability Trust 2002 

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References

Artzner, P. (1997). On the numeraire portfolio. In Mathematics of Derivative Securities (Pubs Newton Inst. 15), eds Dempster, M. A. H. and Pliska, S. R., Cambridge University Press, pp. 53–58.Google Scholar
Bajeux-Besnainou, I. and Portait, R. (1997). The numeraire portfolio: a new perspective on financial theory. Europ. J. Finance 3, 291–309.Google Scholar
Becherer, D. (2001). The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5, 327–341.CrossRefGoogle Scholar
Bühlmann, H., (1992). Stochastic discounting. Insurance Math. Econom. 11, 113–127.Google Scholar
Bühlmann, H., (1995). Life insurance with stochastic interest rates. In Financial Risk and Insurance, ed. Ottaviani, G., Springer, Berlin, pp. 1–24.Google Scholar
Bühlmann, H. and Platen, E. (2002). A discrete time benchmark approach for finance and insurance. QFRG Research Paper 74, University of Technology, Sydney.Google Scholar
Bühlmann, H., Delbaen, F., Embrechts, P. and Shiryaev, A. (1998). On Esscher transforms in discrete finance models. ASTIN Bull. 28, 171–186.Google Scholar
Constatinides, G. M. (1992). A theory of the nominal structure of interest rates. Rev. Financial Studies 5, 531–552.Google Scholar
Davis, M. H. A. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities (Pubs Newton Inst. 15), eds Dempster, M. A. H. and Pliska, S. R., Cambridge University Press, pp. 227–254.Google Scholar
Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.Google Scholar
Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250.Google Scholar
Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press.Google Scholar
Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (Stoch. Monogr. 5), eds Davis, M. and Elliott, R., Gordon and Breach, New York, pp. 389–414.Google Scholar
Föllmer, H. and Sondermann, D. (1986). Hedging of non-redundant contingent claims. In Contributions to Mathematical Economics, eds Hildebrandt, W. and Mas-Colell, A., North Holland, Amsterdam, pp. 205–223.Google Scholar
Geman, H., El Karoui, N. and Rochet, J. C. (1995). Changes of numéraire, changes of probability measures and pricing of options. J. Appl. Prob. 32, 443–458.CrossRefGoogle Scholar
Harrison, J. M. and Kreps, D. M. (1979). Martingale and arbitrage in multiperiod securities markets. J. Econom. Theory 20, 381–408.Google Scholar
Harrison, J. M. and Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260.Google Scholar
Heath, D. and Platen, E. (2002). Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In Recent Developments in Mathematical Finance, ed. Yong, J., World Scientific, River Edge, NJ, pp. 117–126.Google Scholar
Heath, D., Jarrow, R. and Morton, A. (1992). Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105.Google Scholar
Hurst, S. R. and Platen, E. (1997). The marginal distributions of returns and volatility. In L_1-Statistical Procedures and Related Topics (Lecture Notes IMS Monogr. 31), ed. Dodge, Y.. Institute of Mathematical Statistics, Hayward, CA, pp. 301–314.Google Scholar
Jacod, J., Méléard, S. and Protter, P. (2000). Explicit form and robustness of martingale representations. Ann. Prob. 28, 1747–1780.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance (Appl. Math. 39). Springer, Berlin.Google Scholar
Kelly, J. R. (1956). A new interpretation of information rate. Bell Systems Tech. J. 35, 917–926.Google Scholar
Khanna, A. and Kulldorff, M. (1999). A generalization of the mutual fund theorem. Finance Stoch. 3, 167–185.Google Scholar
Korn, R. (2001). Value preserving strategies and a general framework for local approaches to optimal portfolios. Math. Finance 10, 227–241.Google Scholar
Korn, R. and Schäl, M. (1999). On value preserving and growth-optimal portfolios. Math. Meth. Operat. Res. 50, 189–218.Google Scholar
Kramkov, D. O. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Prob. 9, 904–950.Google Scholar
Long, J. B. (1990). The numeraire portfolio. J. Financial Econom. 26, 29–69.Google Scholar
Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investment. John Wiley, New York.Google Scholar
Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413.Google Scholar
Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica 41, 867–888.Google Scholar
Platen, E. (2001). A minimal financial market model. In Mathematical Finance, eds Kohlmann, M. and Tang, S., Birkhäuser, Basel, pp. 293–301.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.Google Scholar
Rogers, L. C. G. (1997). The potential approach to the term structure of interest rates and their exchange rates. Math. Finance 7, 157–176.Google Scholar
Ross, S. A. (1976). The arbitrage theory of capital asset pricing. J. Econom. Theory 13, 341–360.Google Scholar