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Diffusion approximation in past dependent models and applications to option pricing. (English) Zbl 0734.60035

This work is concerned with the diffusion approximation for a certain class of processes satisfying equations with past dependent coefficients and its applications to a model of option pricing. It is assumed that the dependence on the past is through the quadratic variation process. In the first part, the authors prove that the pairs given by the processes and their quadratic variations converge to a limit process, and the second component is its quadratic variation. It is shown that the limiting quadratic variation process satisfies a deterministic delay equation, and hence if the initial condition is deterministic, the entire limiting quadratic variation process is deterministic, which in turn implies the limit process is a Gauss-Markov diffusion. In the second part of the paper, applying the results of the first part to a model of option pricing, the authors obtain a generalized Black and Scholes formula.

MSC:

60F17 Functional limit theorems; invariance principles
91G20 Derivative securities (option pricing, hedging, etc.)
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