Option Pricing with Differential Interest Rates: Arbitrage-Bands Beget Arbitrage-Ovals

Yaacov Z. Bergman

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, or of a portfolio of such claims, 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 oval in the price space.

September, 1993
Published in: 
Review of Financial Studies 8 (1995), 475-500