Option Pricing with Differential Interest Rates

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Abstract
The classic option pricing model is generalized to a more realistic, imperfect, dynamically incomplete capital market with different interest rates for borrowing and for lending and a return differential between long and short positions in stock. It is found that, in the absence of arbitrage opportunities, the equilibrium price of any contingent claim must lie within an arbitrage-band. The boundaries of an arbitrage-band are computed as solutions to a quasi-linear partial differential equation, and, in general, each end-point of such a band depends on both interest rates for borrowing and for lending. This, in turn, implies that the vector of concurrent equilibrium prices of different contingent claims - even claims that are written on different underlying assets - must lie within a computable arbitrage-oval in the price space.

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