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Super-replication in stochastic volatility models under portfolio constraints

Part of: Stochastic analysis Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Jakša Cvitanić ,
Huyên Pham  and
Nizar Touzi
Jakša Cvitanić*
Affiliation:
Columbia University
Huyên Pham*
Affiliation:
Université Marne-la-Vallée and CREST
Nizar Touzi*
Affiliation:
CEREMADE and CREST
*
∗Postal address: Department of Statistics, Columbia University, 2990 Broadway, New York, NY 10027. Email address: cj@stat.columbia.edu
∗∗Postal address: Equipe d'Analyse et de Mathématiques Appliquées, Université Marne-la-Vallée, Cité Descartes, 5 Boulevard Descartes, Marne-la-Vallée Cedex, 77454, France.
∗∗∗Postal address: CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
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Abstract

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

Keywords

Stochastic volatilityportfolio constraintshedging optionsviscosity solution

MSC classification

Primary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 523 - 545
Copyright
Copyright © Applied Probability Trust 1999 

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References

Avellaneda, M., Levy, A., and Paras, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2, 73–88.Google Scholar
Bergman, Y. Z., Grundy, B. D., and Wiener, Z. (1996). General properties of option prices. Preprint, Wharton School, University of Pennsylvania,Google Scholar
Broadie, M., Cvitanić, J., and Soner, M. (1996). Optimal replication of contingent claims under portfolio constraints. To appear in Rev. Financial Studies.Google Scholar
Buckdahn, R., and Hu, Y. (1997). Pricing of American contingent claims with jump stock price and constrained portfolios. Preprint.Google Scholar
Crandall, M. G., Ishii, H., and Lions, P. L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67.Google Scholar
Cvitanić, J., and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl. Prob., 3, 652–681.Google Scholar
El Karoui, N., Jeanblanc-Picqué, M., and Shreve, S. E. (1996). Robustness of the Black-Scholes formula. Preprint, Carnegie Mellon University,Google Scholar
El Karoui, N., and Quenez, M. C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim., 33, 29–66.Google Scholar
Fleming, W. H., and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.Google Scholar
Frey, R., and Sin, C. A. (1997). Bounds on European option prices under stochastic volatility. Preprint, ETH, Zurich.Google Scholar
Hull, J., and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance, 42, 281–300.Google Scholar
Jouini, E., and Kallal, H. (1995). Arbitrage in securities markets with short-sales constraints. Math. Finance, 3, 197–232.CrossRefGoogle Scholar
Karatzas, I., and Kou, S. (1996). On the pricing of contingent claims under constraints. Ann. Appl. Prob. 6, 321–369.Google Scholar
Karatzas, I., and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, Berlin.Google Scholar
Lions, P. -L. (1983). Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, parts I and II. Commun. P. D. E. 8, 1101–1174 and 1229–1276.CrossRefGoogle Scholar
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press, Princeton, NJ.Google Scholar
Wiggins, J. B. (1987). Option values under stochastic volatility: Theory and empirical estimates. J. Financial Economics 19, 351–372.CrossRefGoogle Scholar