Abstract
We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor’s preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for this include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.
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Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Bouchard, B.: Utility maximization on the real line under proportional transaction costs. Finance Stoch. 6, 495–516 (2002)
Bouchard, B., Pham, H.: Optimal consumption in discrete time financial models with industrial investment opportunities and non-linear returns. Ann. Appl. Probab. 15, 2393–2421 (2005)
Campi, L., Schachermayer, W.: A super-replication theorem in Kabanov’s model of transaction costs. Finance Stoch. 10, 579–596 (2006)
Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6, 133–165 (1996)
Cvitanić, J., Wang, H.: On optimal wealth under transaction costs. J. Math. Econ. 35, 223–231 (2001)
Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 259–272 (2001)
Davis, M.H.A., Norman, A.R.: Portfolio selection with transaction costs. Math. Oper. Res. 15, 676–713 (1990)
Deelstra, G., Pham, H., Touzi, N.: Dual formulation of the utility maximization problem under transaction costs. Ann. Appl. Probab. 11, 1353–1383 (2001)
Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)
Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience, New York (1964)
Guasoni, P., Rásonyi, M.: The fundamental theorem of asset pricing under transaction costs. Finance Stoch. (forthcoming)
Guasoni, P., Rásonyi, M., Schachermayer, W.: The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance (forthcoming). Available at http://www.springerlink.com/content/336w847222440318/?p=a89f53ac22794f3d9d24bc2df8cb320c&pi=13
Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995)
Kabanov, Yu.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stoch. 3, 237–248 (1999)
Kabanov, Yu.M., Last, G.: Hedging under transaction costs in currency markets: a continuous time model. Math. Finance 12, 63–70 (2002)
Kabanov, Yu.M., Stricker, Ch.: Hedging of contingent claims under transaction costs. In: Sandmann, K., Schönbucher, Ph. (eds.) Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann, pp. 125–136. Springer, Berlin (2002)
Kabanov, Yu.M., Stricker, C., Rásonyi, M.: On the closedness of sums of convex cones in L 0 and the robust no-arbitrage property. Finance Stoch. 7, 403–411 (2003)
Kamizono, K.: Hedging and optimization under transaction costs. Ph.D. Thesis, Columbia University (2001). Available at http://app.cul.columbia.edu:8080/ac/handle/10022/AC:P:4525
Kamizono, K.: Multivariate utility maximization under transaction costs. In: Akahori, J., Ogawa, S., Watanabe, S. (eds.) Stochastic Processes and Applications to Mathematical Finance: Proceedings of the Ritsumeikan International Symposium, pp. 133–149. World Scientific, Singapore (2004)
Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003)
Klein, I., Rogers, L.C.G.: Duality in optimal investment and consumption problems with market frictions. Math. Finance 17, 225–247 (2007)
Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)
Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003)
Lakner, P.: Consumption/investment and equilibrium in the presence of several commodities. Ph.D. Thesis, Columbia University (1989)
Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1969)
Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)
Owen, M.P., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009)
Rao, K.P.S.B., Rao, M.B.: Theory of Charges: A Study of Finitely Additive Measures. Academic Press, London (1983)
Richard, S.F.: Multivariate risk aversion, utility independence and separable utility functions. Manag. Sci. 22, 12–21 (1975)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1972)
Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004)
Shreve, S.E., Soner, H.M.: Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4, 609–692 (1994)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)
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Campi, L., Owen, M.P. Multivariate utility maximization with proportional transaction costs. Finance Stoch 15, 461–499 (2011). https://doi.org/10.1007/s00780-010-0125-9
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DOI: https://doi.org/10.1007/s00780-010-0125-9
Keywords
- Transaction costs
- Foreign exchange market
- Multivariate utility function
- Asymptotic satiability
- Optimal portfolio
- Duality theory
- Lagrange duality