# What is the Difference Between the Equilibrium Approach and the No-Arbitrage Approach to Modelling?

**Short answer**

Equilibrium models balance supply and demand, they require knowledge of investor preferences and probabilities. No-arbitrage models price one instrument by relating it to the prices of other instruments.

**Example**

The Vasicek interest rate model can be calibrated to historical data. It can therefore be thought of as a representation of an equilibrium model. But it will rarely match traded prices. Perhaps it would therefore be a good trading model. The BGM model matches market prices each day and therefore suggests that there are never any profitable trading opportunities.

**Long answer**

Equilibrium models represent a balance of supply and demand. As with models of equilibria in other, non-financial, contexts there may be a single equilibrium point, or multiple, or perhaps no equilibrium possible at all. And equilibrium points may be stable such that any small perturbation away from equilibrium will be corrected (a ball in a valley), or unstable such that a small perturbation will grow (a ball on the top of a hill). The price output by an equilibrium model is supposedly correct in an absolute sense.

Genuine equilibrium models in economics usually require probabilities for future outcomes, and a representation of the preferences of investors. The latter perhaps quantified by utility functions. In practice neither of these is usually available, and so the equilibrium models tend to be of more academic than practical interest.

No-arbitrage, or arbitrage-free, models represent the point at which there aren't any arbitrage profits to be made. If the same future payoffs and probabilities can be made with two different portfolios then the two portfolios must both have the same value today, otherwise there would be an arbitrage. In quantitative finance the obvious example of the two portfolios is that of an option on the one hand and a cash and dynamically rebalanced stock position on the other. The end result being the pricing of the option relative to the price of the underlying asset. The probabilities associated with future stock prices falls out of the calculation and preferences are never needed. When no-arbitrage pricing is possible it tends to be used in practice. The price output by a no-arbitrage model is supposedly correct in a relative sense.

For no-arbitrage pricing to work we need to have markets that are complete, so that we can price one contract in terms of others. If markets are not complete and we have sources of risk that are unhedgeable then we need to be able to quantify the relevant market price of risk. This is a way of consistently relating prices of derivatives with the same source of unhedgeable risk, a stochastic volatility for example.

Both the equilibrium and no-arbitrage models suffer from problems concerning parameter stability.

In the fixed-income world, examples of equilibrium models are Vasicek, CIR, Fong & Vasicek. These have parameters which are constant, and which can be estimated from time series data. The problem with these is that they permit very simple arbitrage because the prices that they output for bonds will rarely match traded prices. Now the prices may be correct based on the statistics of the past but are they correct going forward? The models of Ho & Lee and Hull & White are a cross between the equilibrium models and no-arbitrage models. Superficially they look very similar to the former but by making one or more of the parameters time dependent they can be calibrated to market prices and so supposedly remove arbitrage opportunities. But still, if the parameters, be they constant or functions, are not stable then we will have arbitrage. But the question is whether that arbitrage is foreseeable. The interest rate models of HJM and BGM match market prices each day and are therefore even more in the no-arbitrage camp.

**References and Further Reading**

Brace, A, Gatarek, D & Musiela, M 1997 The market model of interest rate dynamics. *Mathematical Finance* 7 127-154

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term structure of interest rates. *Econometrica* 53 385-467

Fong, G & Vasicek, O 1991, Interest rate volatility as a stochastic factor. Working Paper

Heath, D, Jarrow, R & Morton, A 1992 bond pricing and the term structure of interest rates: a new methodology. *Econometrica* 60 77-105

Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. *Journal of Finance* 42 1129-1142

Hull, JC & White, A 1990 pricing interest rate derivative securities. *Review of Financial Studies* 3 573-592

Vasicek, OA 1977 An equilibrium characterization of the term structure. *Journal of Financial Economics* 5 177-188