Complete–market models of stochastic volatility

In the Black–Scholes option–pricing theory, asset prices are modelled as geometric Brownian motion with a fixed volatility parameter σ, and option prices are determined as functions of the underlying asset price. Options are in principle redundant in that their exercise values can be replicated by trading in the underlying. However, it is an empirical fact that the prices of exchange–traded options do not correspond to a fixed value of σ as the theory requires. This paper proposes a modelling framework in which certain options are non–redundant: these options and the underlying are modelled as autonomous financial assets, linked only by the boundary condition at exercise. A geometric condition is given, under which a complete market is obtained in this way, giving a consistent theory under which traded options as well as the underlying asset are used as hedging instruments.

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