Skip to main content
SpringerLink
Log in

Optimal trading under constraints

  • Chapter
  • First Online:
Financial Mathematics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1656))

Abstract

These are lecture notes on the techniques and results of the theory of optimal trading for a single agent under convex constraints on his portfolio process, in a continuous-time model. A similar methodology is applied to the case of policy dependent prices, different interest rates for borrowing and lending and transaction costs problems. We study the hedging problem and the portfolio optimization problem for the investor in this market. Mathematical tools involved are those of continuous-time martingales, convex duality, forward-backward SDE’s and PDE’s.

Supported in part by the National Science Foundation under Grant NSF-DMS-95-03582. and C.I.M.E.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • BLACK, F. & SCHOLES, M. (1973) The pricing of options and corporate liabilities. J. Polit. Economy 81, 637–659.

    Article  MATH  Google Scholar 

  • BROADIE, M., CVITANIĆ, J. & SONER, H. M. (1996) On the cost of super-replication under portfolio constraints. Preprint.

    Google Scholar 

  • COX, J. & HUANG, C. F. (1989) Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory 49, 33–83.

    Article  MathSciNet  MATH  Google Scholar 

  • CVITANIĆ, J. & KARATZAS, I. (1992) Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 767–818.

    Article  MathSciNet  MATH  Google Scholar 

  • CVITANIĆ, J. & KARATZAS, I. (1995) On portfolio optimization under “draw-down constraints”. The IMA volumes in mathematics and its applications 65, 35–46.

    Article  MATH  Google Scholar 

  • CVITANIĆ, J. & KARATZAS, I. (1993) Hedging contingent claims with constrained portfolios. Ann. Appl. Probab. 3, 652–681.

    Article  MathSciNet  MATH  Google Scholar 

  • CVITANIĆ, J. & KARATZAS, I. (1996) Hedging and portfolio optimization under transaction costs: a martingale approach. Mathematical Finance 6, 133–165.

    Article  MathSciNet  MATH  Google Scholar 

  • CVITANIĆ, J. & MA, J. (1996) Hedging options for a large investor and Forward-Backward SDE’s. Annals of Applied Probability 6, 370–398.

    Article  MathSciNet  MATH  Google Scholar 

  • DAVIS, M.H.A. (1994) A general option pricing formula. Preprint, Imperial College, London.

    Google Scholar 

  • DAVIS, M. H. A. & CLARK, J. M. C. (1994) A note on super-replicating strategies. Phil. Trans. Royal Soc. London A 347, 485–494.

    Article  MATH  Google Scholar 

  • DAVIS, M. H. A. & NORMAN, A. (1990) Portfolio selection with transaction costs. Math. Operations Research 15, 676–713.

    Article  MathSciNet  MATH  Google Scholar 

  • DUFFIE, D. (1992) Dynamic Asset Pricing Theory. Princeton University Press.

    Google Scholar 

  • EKELAND, I. & TEMAN, R. (1976) Convex Analysis and Variational Problems. North-Holland, Amsterdam and Elsevier, New York.

    Google Scholar 

  • EL KAROUI, N., PENG, S. & QUENEZ, M.C. (1993) Backwards Stochastic Differential Equations in Finance and Optimization. Preprint.

    Google Scholar 

  • EL KAROUI, N. & QUENEZ, M.C. (1995) Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control & Optimization, 33, 29–66.

    Article  MathSciNet  MATH  Google Scholar 

  • FLEMING, W.H. & SONER, H.M. (1993) Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York.

    MATH  Google Scholar 

  • GROSSMAN S. J. & ZHOU, Z. (1993) Optimal investment strategies for controlling drawdowns. Math. Finance 3 (3), (1993), pp. 241–276

    Article  MATH  Google Scholar 

  • HARRISON, J. M. & KREPS, D. M. (1979) Martingales and arbitrage in multiperiod security markets. J. Econom. Theory 20, 381–408.

    Article  MathSciNet  MATH  Google Scholar 

  • HARRISON, J. M. & PLISKA, S. R. (1981) Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Appl. 11, 215–260.

    Article  MathSciNet  MATH  Google Scholar 

  • HARRISON, J. M. & PLISKA, S. R. (1983) A stochastic calculus model of continuous time trading: complete markets. Stochastic Processes and Appl. 15, 313–316.

    Article  MathSciNet  MATH  Google Scholar 

  • HE, H. & PEARSON, N. (1991) Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case. J. Econ. Theory 54, 259–304.

    Article  MathSciNet  MATH  Google Scholar 

  • JOUINI, E. & KALLAL, H. (1995) Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197.

    Article  MathSciNet  MATH  Google Scholar 

  • KARATZAS, I. (1989) Optimization problems in the theory of continuous trading. SIAM J. Control & Optimization 27, 1221–1259.

    Article  MathSciNet  MATH  Google Scholar 

  • KARATZAS, I. & KOU, S-G. (1994) On the pricing of contingent claims under constraints. To appear in The Annals of Applied Probability.

    Google Scholar 

  • KARATZAS, I., LEHOCZKY, J. P. & SHREVE, S. E. (1987) Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optimization 25, 1557–1586.

    Article  MathSciNet  MATH  Google Scholar 

  • KARATZAS, I., LEHOCZKY, J. P., SHREVE, S. E. & XU, G. L. (1991) Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control. Optimization 29, 702, 730.

    Article  MathSciNet  MATH  Google Scholar 

  • KARATZAS, I. & SHREVE, S. E. (1991) Brownian Motion and Stochastic Calculus (2nd edition), Springer-Verlag, New York.

    MATH  Google Scholar 

  • KUSUOKA, S. (1995) Limit theorem on option replication with transaction costs. Ann. Appl. Probab. 5, 198–221.

    Article  MathSciNet  MATH  Google Scholar 

  • LADYŽENSKAJA, O.A., SOLONNIKOV, V.A. & URAL’TSEVA, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, Vol. 23, American Math. Society, Providence, R.I.

    Google Scholar 

  • LEVENTAL, S. & SKOROHOD, A. V. (1995) On the possibility of hedging options in the presence of transactions costs. Preprint, Michigan State University.

    Google Scholar 

  • MA, J., PROTTER, P. & YONG, J. (1994) Solving Forward-Backward Stochastic Differential Equations Explicitly—A Four Step Scheme. Probability Theory and Related Fields, 98, 339–359

    Article  MathSciNet  MATH  Google Scholar 

  • MERTON, R.C. (1973) Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183.

    Article  MathSciNet  MATH  Google Scholar 

  • NEVEU, J. (1975) Discrete-Parameter Martingales. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • PLATEN, E. & SCHWEIZER, M. (1994) On smile and skewness. Preprint.

    Google Scholar 

  • PROTTER, P. (1990) Stochastic Integration and Differential Equations. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • ROCKAFELLAR, R.T. (1970) Convex Analysis. Princeton University Press, Princeton.

    Book  MATH  Google Scholar 

  • SHREVE, S. E. & SONER, H. M. (1994) Optimal investment and consumption with transaction costs, Ann. Appl. Probab. 4, 609–692.

    Article  MathSciNet  MATH  Google Scholar 

  • SONER, H. M., SHREVE, S. E. & CVITANIĆ, J. (1995) There is no nontrivial hedging portfolio for option pricing with transaction costs, Ann. Appl. Probab. 5, 327–355.

    Article  MathSciNet  MATH  Google Scholar 

  • XU, G. & SHREVE, S. E. (1992) A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients. II. Constant market coeficients. Ann. Appl. Probab. 2, 87–112, 314–328.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Columbia University, 10027, New York, NY

    Jakša Cvitanić

Authors
  1. Jakša Cvitanić
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Wolfgang J. Runggaldier

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag

About this chapter

Cite this chapter

Cvitanić, J. (1997). Optimal trading under constraints. In: Runggaldier, W.J. (eds) Financial Mathematics. Lecture Notes in Mathematics, vol 1656. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092000

Download citation

  • DOI: https://doi.org/10.1007/BFb0092000

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62642-8

  • Online ISBN: 978-3-540-68356-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation