Superreplication in stochastic volatility models and optimal stopping
The pricing and hedging of derivative securities is nowadays well-understood in the context of the classical Black-Scholes model of geometric Brownian motion. However, recent empirical research has produced a lot of statistical evidence that is difficult to reconcile with the assumption of independent and normally distributed asset returns. Researches have therefore attempted to build models for asset price fluctuations that are flexible enough to cope with these empirical deficiencies of the Black-Scholes model. In particular, a lot of work has been devoted to relaxing the assumption of constant volatility in the Black- Scholes model and there is a growing literature on Stochastic Volatility models (SV-models). In this class of models the Stochastic Differential Equation (SDE) that governs the asset price process is driven by a Brownian motion, but the diffusion coefficient of this SDE is modelled as a stochastic process which is only imperfectly correlated to the Brownian motion driving the asset price process. It is well-known that SV-models are incomplete, i.e. one cannot replicate the payoff of a typical derivative by dynamic trading in the underlying risky asset (``the stock'') and in some riskless money market account. The author restricts himselves to SV-models where the range of the volatility is bounded. Under this additional assumption ``nontrivial'' superhedging strategies for a large class of derivatives whose payoff may even be path-dependent are obtained. These strategies are universal in the sense that they depend only on the bounds imposed on the volatility and not on a particular parametric model for the volatility dynamics. The value process of the superhedging strategy is characterized by an optimal-stopping problem in the context of the Black-Scholes model. Roughly speaking the result of the paper can be phrased as follows: the value of a superhedging strategy for a European type derivative under stochastic volatility equals the value of a corresponding American type derivative under constant volatility. The proof is based on probabilistic arguments. The main tools are the optional decomposition theorem by \textit{N. El Karoui} and \textit{M.-C. Quenez} [SIAM J. Control Optim. 33, 27-66 (1995; Zbl 0831.90010)]\ or \textit{D. Kramkov} [Probab. Theory Relat. Fields 105, 459-479 (1996; Zbl 0853.60041)]\ and the results on time-change for continuous martingales. This approach is illustrated by examples and simulations.
scientific article; zbMATH DE number 1524265