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Robustness of Delta Hedging for Path-Dependent Options in Local Volatility Models

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Alexander Schied  and
Mitja Stadje
Alexander Schied*
Affiliation:
TU Berlin
Mitja Stadje*
Affiliation:
Princeton University
*
∗Postal address: Institut für Mathematik, TU Berlin, MA 7-4, Strasse des 17. Juni 136, 10623 Berlin, Germany. Email address: [email protected]
∗∗Postal address: Department of Operations Research and Financial Engineering, Princeton University, Engineering Quadrangle, Princeton, NJ 08544, USA. Email address: [email protected]
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Abstract

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We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.

Keywords

Delta hedginglocal volatilityrobustnessdirectional convexity

MSC classification

Secondary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 865 - 879
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Bergenthum, J. and Rüschendorf, L. (2006). Comparison of option prices in semimartingale models. Finance Stoch. 10, 222–249.CrossRefGoogle Scholar
[2] Bühler, H. (2005). Volatility markets. Consistent modeling, hedging and practical implementation. , TU Berlin.Google Scholar
[3] Bühler, H. (2006). Consistent variance swap models. Finance Stoch. 10, 178–203.Google Scholar
[4] Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7, 913–934.Google Scholar
[5] Davis, M. (2004). Complete-market models of stochastic volatility. Proc. R. Soc. London A 460, 11–26.Google Scholar
[6] Dupire, B. (1997). Pricing and hedging with smiles. In Mathematics of Derivative Securities (Cambridge, 1995; Publ. Newton Inst. 15), Cambridge University Press, pp. 103–111.Google Scholar
[7] Ekström, E., Janson, S. and Tysk, J. (2005). Superreplication of options on several underlying assets. J. Appl. Prob. 42, 27–38.Google Scholar
[8] El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. (1998). Robustness of the Black and Scholes formula. Math. Finance 8, 93–126.Google Scholar
[9] Föllmer, H. (1981). Calcul d'Itô sans probabilités. In Séminaire de probabilités XV (Lecture Notes Math. 850), Springer, Berlin, pp. 143–150.Google Scholar
[10] Föllmer, H. (1991). Probabilistic aspects of options. Rolf Nevanlinna Institute Reports B6, Helsinki.Google Scholar
[11] Föllmer, H. (2001). Probabilistic aspects of financial risk. In European Congress of Mathematics (Barcelona 2000; Progr. Math. 201), Vol. I, Birkhäuser, Basel, pp. 21–36.Google Scholar
[12] Föllmer, H. and Schied, A. (2004). Stochastic Finance. An Introduction in Discrete Time (de Gruyter Studies Math. 27), 2nd edn. De Gruyter, Berlin.Google Scholar
[13] Gushchin, A. and Mordecki, È. (2002). Bounds on option prices for semimartingale market models. Proc. Steklov Inst. Math. 2002, 73–113.Google Scholar
[14] Gyöngy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Prob. Theory Relat. Fields 71, 501–516.Google Scholar
[15] Hajek, B. (1985). Mean stochastic comparison of diffusions. Z. Wahrscheinlichkeitsth. 68, 315–329.Google Scholar
[16] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd edn. Cambridge University Press.Google Scholar
[17] Hobson, D. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Prob. 8, 193–205.CrossRefGoogle Scholar
[18] Janson, S. and Tysk, J. (2003). Volatility time and properties of option prices. Ann. Appl. Prob. 13, 890–913.Google Scholar
[19] Janson, S. and Tysk, J. (2004). Preservation of convexity to solutions of parabolic equations. J. Differential Equat. 206, 182–226.Google Scholar
[20] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, Berlin.Google Scholar
[21] Lyons, T. (1995). Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133.CrossRefGoogle Scholar
[22] Schied, A. (2005). Optimal investments for robust utility functionals in complete market models. Math. Operat. Res. 30, 750–764.Google Scholar
[23] Stadje, M. (2005). Convexity of option prices in a local volatility model. , TU Berlin.Google Scholar
[24] Wright, E. M. (1954). An inequality for convex functions. Amer. Math. Monthly 61, 620–622.Google Scholar