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 would not be an arbitrage-free model. 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, an arbitrage free model. 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:

- They 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 Treasury securities different from those quoted in the market.
- They were not arbitrage-free models.

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 an arbitrage-free model. It 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—including the Libor Market Model and the Swap Market Model—are arbitrage-free models.

## References

- Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities,
*Journal of Political Economy*, 81, 637-654. - Cox, John, Jonathan Ingersoll, and Stephen Ross (1985), A theory of the term structure of interest rates,
*Econometrica*, 53 (2), 385–407. - Heath, David, Robert Jarrow, and Andrew Morton (1992), Bond pricing and the term structure of interest rates,
*Econometrica*, 60 (1), 77–105. - Ho, Thomas and Sang-Bin Lee (1986), Term structure movements and pricing interest rate contingent claims,
*Journal of Finance*, 41 (5), 1011–1029. - Rendleman, Richard and Brit J Bartter (1980). The pricing of options on debt securities,
*Journal of Financial and Quantitative Analysis*, 15 (1), 11-24. - Vasicek, Oldrich (1977), An equilibrium characterization of the term structure,
*Journal of Financial Economics*, 5 (8), 177–188.