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Abstract:
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 parts 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 Black-Scholes equation for contingent claims is derived, and a connection with the conventional methods of option valuation is indicated.

See also a related paper by A.N.Adamchuk, S.Adamchuk and S.E.Esipov
"Arbitrage Relaxation of Instruments with Temporal Constraints"

Abstract:
The classical "no-arbitrage" argument assumption is based on perfect replication of instruments. In fixed-income markets there are many instruments for which replication is costly and/or time-consuming. This work is based on a modeling approach where finite times are needed to perform arbitrage. Termed "arbitrage times" they provide measures of market liquidity. From a mathematical stance, the models in complete markets can be recovered in the limit of vanishing arbitrage times.

There are indications that pricing in fixed-income markets will soon be based on a VaR or actuarial approach to profit and loss distributions modeled with static or dynamic positions in fixed-income instruments. Such models require a "seed" process similar to risky log-normal motion for the underlying security which is widely used in conventional derivatives.

A mechanism of almost certain local arbitrage between bonds with close maturities is suggested which is effective in forming the term structure of interest rates. A "seed" interest rate model is proposed and calibrated using 1994 U.S. interest rates. Current term structure (i.e., market information about future) is incorporated as one of the boundary conditions in this model, while the "arbitrage times" come from historical data. Unlike in risk-neutral term structure models the fluctuating forward rates explicitly reflect market fundamentals.

Finally, we emphasize that there is no real difference between future and past in the model since the stochastic process is updated in additional time dimension, called "artificial time." We think that this is the minimal theoretical structure which accommodates initial, final and running boundary conditions.

See related paper by A.N. Adamchuk, and S.E. Esipov "Collectively Fluctuating Assets in the Presence of Arbitrage Opportunities and Option Pricing"