Complete–market models of stochastic volatility

By Mark H. A. Davis. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2004

52 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/cc2ccebb0663ad82d83ede2d45ef95404e1d68cc

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1. Marco Di Francesco
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Citations in the corpus

• Path dependent volatility (2008, background): “Then the essential and hard problem is to determine the " correct " hedging strategy: indeed it is well-known that the hedge parameters are strongly model-dependent even for call and put options (cf. Davis 2004; Cont 2006).”
• On the complete model with stochastic volatility by Hobson and Rogers (2004, background): “…present or past values of the underlying price, and multifactor models that allow for exogenous stochastic influence (see, for instance, the monograph by Epps [11] and the recent papers by Barucci, Malliavin, Mancino, Renò and Thalmaier [2], Davis [7] for comprehensive bibliographic references).”