Controlled random fields, von Neumann–Gale dynamics and multimarket hedging with risk

Abstract

We develop a model of asset pricing and hedging for interconnected financial markets with frictions – transaction costs and portfolio constraints. The model is based on a control theory for random fields on a directed graph. Market dynamics are described by using von Neumann–Gale dynamical systems first considered in connection with the modelling of economic growth [13,24]. The main results are hedging criteria stated in terms of risk-acceptable portfolios and consistent price systems, extending the classical superreplication criteria formulated in terms of equivalent martingale measures.

Keywords:

• hedging
• multimarket models
• von Neumann–Gale dynamical systems
• controlled random fields
• consistent price systems

Keywords:

• 91B28
• 91B26
• 91B25
• 60G60

Acknowledgements

The authors are grateful to H. Föllmer, D.B. Rokhlin and W. Schachermayer for valuable discussions.

Notes

^1. The notion of a consistent price system extends the notion of an equivalent martingale measure (see [5], section 9).

^2. We assume that dividends on short stock positions should be returned. Dividend rates for long and short positions may be different, e.g. when some assets pay dividends in a currency different from asset 1 and there is a bid–ask spread in the exchange rates.

^3. A random -measurable closed cone in is a mapping such that is a closed cone in for each and for any open set .