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Robustness of the Black-Scholes approach in the case of options on several assets. (English) Zbl 0957.35063

In this paper the robustness of the Black-Scholes formula with respect to stochastic volatility in the case of European multiasset derivatives is analysed. The problem is the following: we have a riskless asset whose value is supposed constant through time, and \(n\) risky assets whose prices \(S= (S^{1},\dots, S^{n})\) follow the dynamic \[ dS_{t}=\overline S_{t}\sigma_{t}dW_{t}, \] where \(W\) is an \(n\)-dimensional Brownian motion under the so called forward-neutral measure, the matrix process \(\sigma\) takes values in a some closed bounded set, and it is used the notation \(\overline s= \text{diag}(s)\) for \(s\in \mathbb{R}^{n}.\) An agent wants to price and hedge a European contingent claim whose payoff is a continuous deterministic function \(h\) with polynomial growth, calculated in \(S_{T}.\) Since the market could be incomplete because of the stochastic volatility \(\sigma\) and the agent is not able to hedge the volatility, he chooses to hedge the option by using the superhedging approach. The Markov superstrategies are characterized, and it is shown that they are linked to a nonlinear PDE, called the Black-Scholes-Barenblatt (BSB) equation. This equation is the Hamilton-Jacobi-Bellman equation of an optimal control problem, which has a nice financial interpretation. Then the optimization problem included in the BSB equation is analysed and some sufficient conditions for reduction of the BSB equation to a linear Black-Scholes equation is given. Some examples are presented.

MSC:

35K55 Nonlinear parabolic equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
60G35 Signal detection and filtering (aspects of stochastic processes)
91B99 Mathematical economics
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