# Hedging American contingent claims with constrained portfolios

## Author Info

• Ioannis Karatzas

()
(Departments of Mathematics and Statistics, Columbia University, New York, NY 10027, USA)

• (*), S. G. Kou

()
(Department of Statistics, University of Michigan, Mason Hall, Ann Arbor, MI 48109-1027, USA Manuscript)

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## Abstract

The valuation theory for American Contingent Claims, due to Bensoussan (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice, including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price $u_0$; this is expressed as the supremum, over all stopping times, of the claim's expected discounted value under the equivalent martingale measure. In the presence of constraints, $\{u_0\}$ is replaced by an entire interval $[h_{\rm low}, h_{\rm up}]$ of arbitrage-free prices, with endpoints characterized as $h_{\rm low} = \inf_{\nu\in{\cal D}} u_\nu, h_{\rm up} = \sup_{\nu\in{\cal D}} u_\nu$. Here $u_\nu$ is the analogue of $u_0$, the arbitrage-free price with unconstrained portfolios, in an auxiliary market model ${\cal M}_\nu$; and the family $\{{\cal M}_\nu\}_{\nu\in{\cal D}}$ is suitably chosen, to contain the original model and to reflect the constraints on portfolios. For several such constraints, explicit computations of the endpoints are carried out in the case of the American call-option. The analysis involves novel results in martingale theory (including simultaneous Doob-Meyer decompositions), optimal stopping and stochastic control problems, stochastic games, and uses tools from convex analysis.

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## Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 2 (1998)
Issue (Month): 3 ()
Pages: 215-258

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Handle: RePEc:spr:finsto:v:2:y:1998:i:3:p:215-258

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## Related research

Keywords: Contingent claims; hedging; pricing; arbitrage; constrained markets; incomplete markets; different interest rates; Black-Scholes formula; optimal stopping; free boundary; stochastic control; stochastic games; equivalent martingale measures; simultaneous Doob-Meyer decompositions.;

Find related papers by JEL classification:

• G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
• D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
• C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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## Citations

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Cited by:
1. Karatzas, Ioannis & Ocone, Daniel, 2002. "A leavable bounded-velocity stochastic control problem," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 31-51, May.
2. Erhan Bayraktar & Song Yao, 2013. "On the Robust Optimal Stopping Problem," Papers 1301.0091, arXiv.org, revised Apr 2014.
3. Zhou, Qing & Wu, Weixing & Wang, Zengwu, 2008. "Cooperative hedging with a higher interest rate for borrowing," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 609-616, April.
4. Erhan Bayraktar & Zhou Zhou, 2012. "On controller-stopper problems with jumps and their applications to indifference pricing of American options," Papers 1212.4894, arXiv.org, revised Nov 2013.
5. Kuhn, Christoph, 2002. "Pricing contingent claims in incomplete markets when the holder can choose among different payoffs," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 215-233, October.
6. Frank Riedel & Frederik Herzberg, 2013. "Existence of Financial Equilibria in Continuous Time with Potentially Complete Markets," Working Papers 443, Bielefeld University, Center for Mathematical Economics.
7. Alet Roux, 2007. "The fundamental theorem of asset pricing under proportional transaction costs," Papers 0710.2758, arXiv.org.
8. Haluk Yener, 2012. "Maximising Survival, Growth, and Goal Reaching Under Borrowing Constraints," Papers 1209.6385, arXiv.org.
9. M. Pınar & A. Camcı, 2012. "An Integer Programming Model for Pricing American Contingent Claims under Transaction Costs," Computational Economics, Society for Computational Economics, vol. 39(1), pages 1-12, January.
10. Tatjana Chudjakow & Jörg Vorbrink, 2009. "Exercise Strategies for American Exotic Options under Ambiguity," Working Papers 421, Bielefeld University, Center for Mathematical Economics.
11. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742 Elsevier.
12. Erhan Bayraktar & Arash Fahim, 2011. "A Stochastic Approximation for Fully Nonlinear Free Boundary Parabolic Problems," Papers 1109.5752, arXiv.org, revised Nov 2013.
13. Henderson, Vicky & Hobson, David, 2007. "Horizon-unbiased utility functions," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1621-1641, November.
14. Matos, Joao Amaro de & Lacerda, Ana, 2004. "Dry Markets and Superreplication Bounds of American Derivatives," FEUNL Working Paper Series wp461, Universidade Nova de Lisboa, Faculdade de Economia.
15. Erhan Bayraktar & Yu-Jui Huang, 2010. "On the Multi-Dimensional Controller and Stopper Games," Papers 1009.0932, arXiv.org, revised Jan 2013.
16. Aliprantis, Charalambos D. & Polyrakis, Yiannis A. & Tourky, Rabee, 2002. "The cheapest hedge," Journal of Mathematical Economics, Elsevier, vol. 37(4), pages 269-295, July.

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