大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

An Alternative to Stochastic Volatility Models

Andrea Macrina (King's College London)

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大阪大学 金融・保険セミナーシリーズ 第32回 (CSFI-CRESTジョイントセミナー)

An Alternative to Stochastic Volatility Models

Andrea Macrina (King's College London)

This work is devoted to the development of conditional density models for asset prices. In particular we construct models for the conditional risk-neutral density under incomplete information. The so-called implied density models associated with option prices are proposed as an alternative to stochastic volatility models. We derive a nonlinear infinite-dimensional stochastic differential equation for the dynamics of the conditional density that is adapted to a Brownian filtration. Solutions to this differential equation are characterised by the initial density and the so-called volatility structure determining, together with the model for partial information, the corresponding class of asset price processes. We show that the conditional density process associated with the so-called “information-based asset price models” proposed by Brody, Hughston and Macrina that are constructed by use of Brownian bridges, satisfy the studied SDE over a finite time interval, for an arbitrary initial density and a particular volatility structure. We prove that the innovation process driving the information-based asset price process coincides indeed with the Brownian motion that arises in the general SDE for the conditional density. Furthermore we show that the conditional density process of the Bachelier model for asset prices in finite time is a special case of the information-based solutions to the infinite-dimensional SDE. That is, the volatility structure associated with the Bachelier model is included in the information-based class of volatility structures, and the initial density is given by the normal probability density. Another result is obtained by generalising the class of solutions to the nonlinear infinite-dimensional SDE by extending the time horizon to infinity and, at the same time, by considering volatility structures defined in terms of a deterministic function of two variables while maintaining arbitrary initial densities. Finally we demonstrate that a particular choice of the volatility structure in the infinite-time setting naturally leads to solutions of the SDE in finite time and in particular it gives rise to the information-based models for the conditional density of asset prices. (Joint work with D. Filipovic and L. P. Hughston)