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Superreplication of Options on Several Underlying Assets

Part of: Parabolic equations and systems Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Erik Ekström ,
Svante Janson  and
Johan Tysk
Erik Ekström*
Affiliation:
Uppsala University
Svante Janson*
Affiliation:
Uppsala University
Johan Tysk*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden.
∗Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden.
∗Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden.
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Abstract

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We investigate the conditions on a hedger, who overestimates the (time- and level-dependent) volatility, to superreplicate a convex claim on several underlying assets. It is shown that the classic Black-Scholes model is the only model, within a large class, for which overestimation of the volatility yields the desired superreplication property. This is in contrast to the one-dimensional case, in which it is known that overestimation of the volatility with any time- and level-dependent model guarantees superreplication of convex claims.

Keywords

Parabolic equationsuperreplicationconvexityoption

MSC classification

Secondary: 60H05: Stochastic integrals 35K10: Second-order parabolic equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 27 - 38
Copyright
© Applied Probability Trust 2005 

Footnotes

∗∗∗

Partially supported by the Swedish Research Council.

References

Avellaneda, M., Levy, A. and Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2, 73–88.Google Scholar
Bergenthum, J. and Rüschendorf, L. (2004). Comparison of option prices in semimartingale models. Preprint, Freiburg University.Google Scholar
Bergman, Y. Z., Grundy, D. B. and Wiener, Z. (1996). General properties of option prices. J. Finance 51, 1573–1610.Google Scholar
Ekström, E. (2004). Properties of American option prices. Stoch. Process. Appl. 114, 265–278.CrossRefGoogle Scholar
El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. (1998). Robustness of the Black and Scholes formula. Math. Finance 8, 93–126.Google Scholar
Gozzi, F. and Vargiolu, T. (2002a). Superreplication of European multiasset derivatives with bounded stochastic volatility. Math. Meth. Operat. Res. 55, 69–91.CrossRefGoogle Scholar
Gozzi, F. and Vargiolu, T. (2002b). On the superreplication approach for European interest rate derivatives. In Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999; Progress Prob. 52), eds Dalang, R. C. and Russo, F., Birkhöuser, Basel, pp. 173–188.Google Scholar
Hobson, D. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Prob.ö 8, 193–205.CrossRefGoogle Scholar
Janson, S. and Tysk, J. (2003). Volatility time and properties of option prices. Ann. Appl. Prob. 13, 890–913.Google Scholar
Janson, S. and Tysk, J. (2004). Preservation of convexity of solutions to parabolic equations. J. Differential Equat. 206, 182–226.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance (Appl. Math. 39). Springer, New York.Google Scholar
Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133.CrossRefGoogle Scholar
Martini, M. (1995). Propagation de la convexitö par des semi-groupes martingaliens sur la demi-droite. , Evry University.Google Scholar
Martini, M. (1999). Propagation of convexity by Markovian and martingalian semigroups. Potential Anal. 10, 133–175.Google Scholar
Romagnoli, S. and Vargiolu, T. (2000). Robustness of the Black–Scholes approach in the case of options on several assets. Finance Stoch. 4, 325–341.Google Scholar
Vargiolu, T. (2001). Existence, uniqueness and smoothness for the Black–Scholes–Barenblatt equation. Tech. Rep., Department of Tech. Rep. Google Scholar