Skip to main content
SpringerLink
Log in

Risk measure pricing and hedging in incomplete markets

  • Research Article
  • Published:
Annals of Finance Aims and scope Submit manuscript

Abstract

This article attempts to extend the complete market option pricing theory to incomplete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. I will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average VaR will be shown.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acerbi C., Tasche D. (2002). On the coherence of expected shortfall. J Banking Finance 26, 1487–1503

    Article  Google Scholar 

  • Artzner P., Delbaen F., Eber J.-M., Heath D. (1999). Coherent measures of risk. Math Finance 9, 203–228

    Article  Google Scholar 

  • Barrieu, P., El Karoui, N.: Optimal derivatives design under dynamic risk measures. Mathematics of Finance, Contemporary Mathematics (A.M.S. Proceedings 351) (2004)

  • Barrieu P., El Karoui N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance Stochastics 9, 269–298

    Article  Google Scholar 

  • Bellamy N., Jeanblanc M. (2000). Incompleteness of markets driven by a mixed diffusion. Finance Stochastics 4, 209–222

    Article  Google Scholar 

  • Belini F., Fritelli M. (2002). On the existence of minimax martingale measures. Math Finance 12, 1–21

    Article  Google Scholar 

  • Carr P., Geman H., Madan D. (2001). Pricing and hedging in incomplete markets. J Financ Econ 62, 131–167

    Article  Google Scholar 

  • Cvitanić J., Karatzas I. (1999). On dynamic measures of risk. Finance Stochastics 3, 451–482

    Article  Google Scholar 

  • Davis M.H.A. (1997). Option pricing in incomplete markets. In: Dempster M.A.H., Pliska S.R (ed). Mathematics of derivative securities. Cambridge University Press, Cambridge, pp. 216–226

    Google Scholar 

  • Delbaen F. (2002). Coherent risk measures on general probability spaces. Advances in finance and stochastics: Essays in honour of Dieter Sondermann. Springer, Berlin Heidelberg New York, pp. 1–37

    Google Scholar 

  • Delbaen F., Schachermayer W. (1994). A general version of the fundamental theorem of asset pricing. Math Annalen 300, 463–520

    Article  Google Scholar 

  • Delbaen F., Schachermayer W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math Annalen 312, 215–250

    Article  Google Scholar 

  • Duffie D., Pan J. (1997). An overview of value at risk. J Derivatives 4, 7–49

    Google Scholar 

  • Eberlein E., Jacod J. (1997). On the range of options prices. Finance Stochastics 1, 131–140

    Article  Google Scholar 

  • El Karoui N., Quenez M-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J Control Optim 33, 29–66

    Article  Google Scholar 

  • Foldes L. (2000). Valuation and martingale properties of shadow prices: an exposition. J Econ Dynam and Control 24, 1641–1701

    Article  Google Scholar 

  • Föllmer H., Leukert P. (2000). Efficient hedging: cost versus shortfall risk. Finance Stochastics 4, 117–146

    Article  Google Scholar 

  • Föllmer H., Kabanov Y.M. (2002). Optional decomposition and Lagrange multipliers. Finance Stochastics 2, 69–81

    Article  Google Scholar 

  • Föllmer H., Schied A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics 6, 429–447

    Article  Google Scholar 

  • Föllmer H., Schied A. (2002). Robust preferences and convex measures of risk. Advances in Finance and Stochastics. Springer, Berlin Heidelberg New York, pp. 39–56

    Google Scholar 

  • Föllmer H., Schied A. (2002). Stochastic finance - an introduction in discrete time. Studies in Mathematics, Vol. 27. de Gruyter, Amsterdam

    Google Scholar 

  • Föllmer H., Schweizer M. (1991). Hedging of contingent claims under incomplete information. In: Davis M.H.A., Elliott R.J (ed). Applied Stochastic Analysis. Gordon and Breach, New York, pp. 389–414

    Google Scholar 

  • Frittelli M. (2002). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math Finance 10, 39–52

    Article  Google Scholar 

  • Frittelli M. (2000). Introduction to a theory of value coherent with the no-arbitrage principle. Finance Stochastics 4, 275–297

    Article  Google Scholar 

  • Gerber H.U., Shiu E.S.W. (1994). Option pricing by Esscher transforms (with discussion). Trans Soc Actuaries 46, 99–191

    Google Scholar 

  • Goll T., Rüschendorf L. (2001). Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stochastics 5, 557–581

    Article  Google Scholar 

  • Harrison J.M., Kreps D. (1979). Martingales and arbitrage in multiperiod security markets. J Econ Theory 2, 381–408

    Article  Google Scholar 

  • Harrison J.M., Pliska S.R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process Appl 11, 215–260

    Article  Google Scholar 

  • Harrison J.M., Pliska S.R. (1983). A stochastic calculus model of continuous trading: complete markets. Stochastic Process Appl 15, 313–316

    Article  Google Scholar 

  • Henderson V. (2002). Valuation of claims on nontraded assets using utility maximization. Math Finance 12, 351–373

    Article  Google Scholar 

  • Henderson, V., Hobson, D.: Utility indifference pricing - an overview. Indifference Pricing. PUP (to appear) (2005)

  • Hodges S.D., Neuberger A. (1989). Optimal replication of contingent claims under transaction costs. Rev Futures Markets 8, 222–239

    Google Scholar 

  • Hugonnier J., Kramkov D., Schachermayer W. (2004). On the utility based pricing of contingent claims in incomplete markets. Math Finance 15, 203-212

    Article  MathSciNet  Google Scholar 

  • Kallsen J. (2002). Derivative pricing based on local utlity maximization. Finance Stochastics 6, 115–140

    Article  Google Scholar 

  • Karatzas I., Kou S.G. (1996). On the pricing of contingent claims under constraints. Ann Appl Probab 6, 321–369

    Article  Google Scholar 

  • Kirch M.: Efficient hedging in incomplete markets under model uncertainty. Ph.D. thesis, Humboldt University (2002)

  • Klöppel, S., Schweizer, M.: Dynamic utility indifference valuation via convex risk measures. preprint, ETH (2005)

  • Kramkov D.O. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab Theory Relat Fields 105: 459–479

    Article  Google Scholar 

  • Larsen K., Pirvu T., Shreve S.E., Tütüncü R. (2005). Satisfying convex risk limits by trading. Finance Stochastics 9, 177–195

    Article  Google Scholar 

  • Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Annals of Applied Probability (to appear) (2005)

  • Miyahara Y. (2001). [Geometric Lévy Process & MEMM] Pricing model and related estimation problems. Asia-Pacific Financial Markets 8, 45–60

    Article  Google Scholar 

  • Merton R. (1973). Theory of rational option pricing. Bell J Econ 4, 141–183

    Article  Google Scholar 

  • Musiela M., Zariphopoulou T. (2004a). An example of indifference prices under exponential preferences. Finance Stochastics 8, 229–239

    Article  Google Scholar 

  • Musiela M., Zariphopoulou T. (2004b). A valuation algorithm for indifference prices in incomplete markets. Finance Stochastics 8, 399–414

    Article  Google Scholar 

  • Musiela, M., Zariphopoulou, T.: Derivative pricing, investment management and the term structure of exponential utilities: the case of binomial model. Indifference Pricing. PUP (to appear) (2005)

  • Rockafellar R.T., Uryasev S. (2002). Conditional value-at-risk for general loss distributions. J Banking Finance 26, 1443–1471

    Article  Google Scholar 

  • Rouge R., El Karoui N. (2000). Pricing via utility maximisation and entropy. Math Finance 10, 259–276

    Article  Google Scholar 

  • Sekine J. (2004). Dynamic minimization of worst conditional expectation of shortfall. Math Finance 14, 605–618

    Article  Google Scholar 

  • Xu, M.: Minimizing shortfall risk using duality approach - an application to partial hedging in incomplete markets. Ph.D. thesis, Carnegie Mellon University (2004)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina, 9201 University City Boulevard, Charlotte, NC, 28223, USA

    Mingxin Xu

Authors
  1. Mingxin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingxin Xu.

Additional information

The work of Mingxin Xu is supported by the National Science Foundation under grant SES-0518869. I would like to thank Steven Shreve for insightful comments, especially his suggestions to extend the pricing idea from using shortfall risk measure to coherent ones, and to study its relationship to utility based derivative pricing. The comments from the associate editor and the anonymous referee have reshaped the paper into its current version. The paper has benefited from discussions with Freddy Delbaen, Jan Večeř, David Heath, Dmitry Kramkov, Peter Carr, and Joel Avrin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, M. Risk measure pricing and hedging in incomplete markets. Annals of Finance 2, 51–71 (2006). https://doi.org/10.1007/s10436-005-0023-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10436-005-0023-x

Keywords and Phrases

JEL Classification Numbers

Search

Navigation