Collectively fluctuating assets in the presence of arbitrage opportunities, and option pricing
A.N. Adamchuk a
S.E. Esipov b
a The University of Chicago, 5801 South Ellis Ave., Chicago, Illinois, 60637, USA
b Department of Physics and James Frank Institute, University of Chicago, Chicago, Illinois, USA
Methods of functional analysis are applied to describe collectively fluctuating default-free pure discount bonds subject to trading-related noise which generates arbitrage opportunities. Two key elements of the model are: (i) the naturally incorporated fixed bond price at maturity which is achieved by making use of only those fluctuating paths of price motion which terminate at a specified final condition, and (ii) the most attractive arbitrage opportunities between bonds with close maturities, with modeled a local linear approximation. The model can be written in different closed forms as a stochastic partial differential equation. The functional Black-Scholes equation for contingent claims is derived, and a connection with the conventional methods of option valuation is indicated.