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.
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 [Citation5], 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
.