An arbitrage-free model is a financial engineering model that assigns prices to derivatives or other instruments in such a way that it is impossible to construct arbitrages between two or more of those prices. For example, if an option pricing formula assigned prices to put and call options that violated put-call parity, that model would not be arbitrage-free. While technically not required by the definition, arbitrage-free models used for trading are generally calibrated to one or more market prices to preclude arbitrages between prices assigned by the model and those quoted prices. The Black-Scholes (1973) option pricing formula is, of course, arbitrage free. Problems arose more with term structure models developed for pricing fixed income derivatives during the 1980s. Early term structure models—including Vasicek (1977), Rendleman and Bartter (1980), and Cox, Ingersoll and Ross (1985)—were equilibrium models. They had two shortcomings:
The Ho and Lee (1986) model was the first term structure model to solve these problems. It could be calibrated to the current term structure, so it ascribed prices to Treasury securities that were the same as those observed in the market. Also, it was arbitrage-free. The model is one example of a larger class of arbitrage-free models specified by Heath, Jarrow and Morton (1992). Today, the theory of arbitrage-free term structure modeling is well developed. All standard models used in trading are arbitrage-free.
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constructed a current equilibrium term structure that was generally
different from the actual term structure observed in the market. For this
reason, they ascribed prices to
