Stochastic differential equations with diffusion and jumps modeling currency markets |
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Authors: | Ya I Belopol’skaya S R Filimonova |
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Institution: | 1.St. Petersburg State University of Architecture and Civil Engineering,St. Petersburg,Russia |
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Abstract: | An efficient currency market with zero transaction costs is considered. The dynamics of the exchange rate in this market is
described by stochastic differential equations (SDEs) with diffusion and jumps; the latter are assumed to be described by
a Lévy process. Adjusting theoretical arbitrage-free option prices computed within these models to market option prices requires
properly choosing the coefficients in the SDEs. For this purpose, an expression for local volatility in a diffusion model
is found and a relation between local and implied volatilities is determined. For a market model with diffusion and jumps,
expressions for the local volatility and the local rate function are given. Moreover, in Merton’s model, where the jump component
is a compound Poisson process with normal jumps, a relation between the local and the implied volatilities is determined. |
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