Superreplication in stochastic volatility models and optimal stopping

Abstract. In this paper we discuss the superreplication of derivatives in a stochastic volatility model under the additional assumption that the volatility follows a bounded process. We characterize the value process of our superhedging strategy by an optimal-stopping problem in the context of the Black-Scholes model which is similar to the optimal stopping problem that arises in the pricing of American-type derivatives. Our proof is based on probabilistic arguments. We study the minimality of these superhedging strategies and discuss PDE-characterizations of the value function of our superhedging strategy. We illustrate our approach by examples and simulations.

[1]  R. Frey,et al.  Bounds on European Option Prices under Stochastic Volatility , 1999 .

[2]  Maurizio Pratelli,et al.  Functional convergence of Snell envelopes: Applications to American options approximations , 1998, Finance Stochastics.

[3]  N. Krylov.  Controlled Diffusion Processes , 1980 .

[4]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

[5]  D. Lamberton,et al.  Variational inequalities and the pricing of American options , 1990 .

[6]  Hans Föllmer,et al.  Quantile hedging , 1999, Finance Stochastics.

[7]  R. Frey.  Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility , 1997 .

[8]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[9]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[10]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[11]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[12]  D. Lamberton,et al.  Sur l'approximation des réduites , 1990 .

[13]  K. Sandmann,et al.  A Discrete Time Approach for European and American Barrier Options , 1995 .

[14]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[15]  Antonio Roma,et al.  Stochastic Volatility Option Pricing , 1994, Journal of Financial and Quantitative Analysis.

[16]  R. Myneni.  The Pricing of the American Option , 1992 .

[17]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[18]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[19]  Jakša Cvitanić,et al.  Super-replication in stochastic volatility models under portfolio constraints , 1999, Journal of Applied Probability.

[20]  Marco Avellaneda,et al.  Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model , 1996 .

[21]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[22]  D. Kramkov.  Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets , 1996 .

[23]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[24]  Freddy Delbaen,et al.  REPRESENTING MARTINGALE MEASURES WHEN ASSET PRICES ARE CONTINUOUS AND BOUNDED , 1992 .