Abstract
The paper develops a general discrete-time framework for asset pricing and hedging in financial markets with proportional transaction costs and trading constraints. The framework is suggested by analogies between dynamic models of financial markets and (stochastic versions of) the von Neumann–Gale model of economic growth. The main results are hedging criteria stated in terms of “dual variables” – consistent prices and consistent discount factors. It is shown how these results can be applied to specialized models involving transaction costs and portfolio restrictions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aliprantis C., Brown D.J., Burkinshaw O. (1989) Existence and optimality of competitive equilibria. Springer, Berlin Heidelberg New York
Arkin V.I., Evstigneev I.V. (1987) Stochastic models of control and economic dynamics. Academic Press, London
Bensaid B., Lesne J.-P., Pagès H., Scheinkman J. (1992) Derivative asset pricing with transaction costs. Math Finance 2: 63–86
Birchenhall C., Grout P. (1984) Mathematics for modern economics. Philip Allan Publishers, Oxford
Björk T. (1998) Arbitrage theory in continuous time. Oxford University Press, Oxford
Carassus L., Pham H., Touzi N. (2001) No arbitrage in discrete time under portfolio constraints. Math Finance 11: 315–329
Cherny, A.S.: General arbitrage pricing model: transaction costs. Working paper, Faculty of mechanics and mathematics, Moscow State University (2005)
Cvitanić J., Karatzas I. (1996) Hedging and portfolio optimization under transaction costs: A martingale approach. Math Finance 6: 133–165
Dalang R.C., Morton A., Willinger W. (1990) Equivalent martingale measures and no-arbitrage in stochastic securities market model. Stochastics Stochastics Rep 29: 185–201
Dynkin E.B. (1971) Some probability models for a developing economy. Sov Math (Doklady) 12: 1422–1425
Dynkin E.B., Yushkevich A.A. (1979) Controlled Markov processes and their applications. Springer, New York
El Karoui N., Quenez M.-C. (1995) Dynamic programming and the pricing of contingent claims in an incomplete market. SIAM J Control 33: 29–66
Evstigneev I.V., Schenk-Hoppé K.R. (2006) The von Neumann–Gale growth model and its stochastic generalization. In: Le Van C., Dana R.-A., Mitra T., Nishimura K.(eds). Handbook of optimal growth. Springer, Berlin Heidelberg New York
Evstigneev I.V., Schürger K., Taksar M.I. (2004) On the fundamental theorem of asset pricing: random constraints and bang-bang no-arbitrage criteria. Math Finance 14: 201–221
Evstigneev, I.V., Taksar, M.I.: A general framework for arbitrage pricing and hedging theorems in models of financial markets. Working Paper SUNYSB-AMS-00-09, Department of applied mathematics and statistics, State University of New York at Stony Brook (2000)
Föllmer H., Schied A. (2002) Stochastic finance: an introduction in discrete time. Walter de Gruyter, Berlin
Föllmer H., Kramkov D. (1997) Optional decomposition under constraints. Probability Theory and Related Fields 109: 1–25
Gale D. (1956) A closed linear model of production. In: Kuhn H.W., et al. (eds) Linear inequalities and related systems. annals of mathematical studies, vol. 38., pp. 285–303. Princeton: Princeton University Press
Harrison J.M., Kreps D.M. (1979) Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20: 381–408
Harrison J.M., Pliska S. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process Appl 11: 215–260
Jouini E. (2000) Price functionals with bid-ask spreads: an axiomatic approach. J Math Econ 34: 547–558
Jouini E. (2001) Arbitrage and control problems in finance: a presentation. J Math Econ 35: 167–183
Jouini E., Kallal H. (1995a) Martingales and arbitrage in securities markets with transaction costs. J Econ Theory 66: 178–197
Jouini E., Kallal H. (1995b) Arbitrage in securities markets with short-sales constraints. Math Finance 3: 197–232
Jouini E., Kallal H. (1999) Viability and equilibrium in securities markets with frictions. Math Finance 9: 275–292
Jouini E., Napp C. (2001) Arbitrage and investment opportunities. Finance Stochastics 5: 305–325
Kabanov Y.M. (1999) Hedging and liquidation under transaction costs in currency markets. Finance Stochastics 3: 237–248
Kabanov Y.M. (2001) The arbitrage theory. In: Jouini E., vitanić J., Musiela M. (eds), Handbooks in mathematical finance. Option pricing, interest rates and risk management. Cambridge University Press, Cambridge, pp. 3–42
Kabanov Y.M., Stricker C. (2001) The Harrison–Pliska arbitrage pricing theorem under transaction costs. J Math Econ 35: 185–196
Kreps D.M. (1981) Arbitrage and equilibrium in economies with infinitely many commodities. J Math Econ 8: 15–35
Luenberger D.G. (1969) Optimization by vector space methods. Wiley, New York
Napp C. (2001) Pricing issues with investment flows: applications to market models with frictions. J Math Econ 35: 383–408
Pham H., Touzi N. (1999) The fundamental theorem of asset pricing under cone constraints. J Math Econ 31: 265–279
Pliska S.R. (1997) Introduction to mathematical finance: discrete time models. Blackwell, Oxford
Radner R.: Balanced stochastic growth at the maximum rate. In: Contributions to the von Neumann growth model, Zeitschrift für Nationalökonomie, Suppl. 1, 39–53 (1971)
Rockafellar R.T. (1970) Convex analysis. Princeton University Press, Princeton
Schachermayer W. (2004) The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Mathematical Finance 14: 19–48
Soner H., Shreve S., Cvitanić J. (1995) There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann Appl Prob 5: 327–355
Stettner L. (2002) Discrete time markets with transaction costs. In: Yong J. (ed) Recent developments in mathematical finance pp. 168–180. Singapore World Scientific
von Neumann, J.: Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In: Ergebnisse eines Mathematischen Kolloquiums, Vol. 8, pp. 73–83. Leipzig Franz–Deuticke (1937) [An English translation: A model of general economic equilibrium, Rev Econ Stud 13: 1–9 (1945–1946)]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dempster, M.A.H., Evstigneev, I.V. & Taksar, M.I. Asset Pricing and Hedging in Financial Markets with Transaction Costs: An Approach Based on the Von Neumann–Gale Model. Annals of Finance 2, 327–355 (2006). https://doi.org/10.1007/s10436-006-0042-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10436-006-0042-2
Keywords
- Asset pricing
- Hedging
- Transaction costs
- Trading constraints
- Von Neumann–Gale model
- Consistent valuation systems