# Are no arbitrage models and equilibrium models equivalent?

This YouTube video from WHU (starting from 3:50) claims that no-arbitrage models (such as Black-Scholes and HJM) are equivalent to equilibrium models (such as CAPM or C-CAPM).

He uses the Euler equation and the stochastic discount factor (SDF) as arguments which apparently link the two types of models.

I understand that an equilibrium requires the absence of arbitrage (to clear the market) but I don't see why any market without arbitrage is automatically in equilibrium.

Put differently, I thought that, for example, the Black-Scholes model can be derived by merely assuming that the market is free of arbitrage. I understand that the BS model can also be derived from the CAPM but that these additional assumptions ($$\mu$$-$$\sigma$$ based agents $$\Leftrightarrow$$ quadratic utility function $$\Leftrightarrow$$ linear SDF) are not necessary and the poor empirical performance of the CAPM does not directly impact the validity of the Black-Scholes model (which is flawed for other reasons).

Of course, Black and Scholes assume that (log-)returns are normally distributed and this may be in line with a quadratic utility function but we can derive similar option pricing formulae using completely different distributions with more parameters (CEV and such).

...no-arbitrage models (such as Black-Scholes and HJM) are equivalent to equilibrium models (such as CAPM or C-CAPM).

Short Answer Yes, for models where asset prices are assumed to be Ito semimartingales (where the martingale part is a Brownian integral), although a more general argument is needed than that suggested by the special cases typically encountered in finance.

Clearly, no-arbitrage is a necessary condition for general equilibrium. To claim equivalence is then to say it is also sufficient, i.e. given any price process $$P_t$$ and a density process $$D_t$$ such that the discounted price $$e^{-rt} P_t$$ is a martingale after change of measure by $$D_t$$, one needs to find a (say, representative) investor $$u$$ and a equilibrium consumption process $$c_t$$ such that

$$e^{-\beta t} u'(c_t) = \lambda e^{-rt} D_t, \quad\quad (*)$$ for some $$\lambda > 0$$. In other words, one needs a "SDF/marginal utility representation" for equivalent martingale measure densities.

The equation $$(*)$$ is the usual relationship $$\mbox{marginal utility} \; \propto \mbox{price}.$$ If you write down a heuristic Lagrangian, $$\lambda$$ is the Lagrange multiplier in the FOC. In general, FOC's of this type is only necessary for optimality of $$c_t$$. If $$(*)$$ is sufficient for optimality of $$c_t$$, you can take the dividend to be $$c_t$$ in $$(*)$$ and $$P_t$$ becomes the equilibrium price faced by the representative investor $$u$$.

With certain assumptions on $$u$$---such as concavity and the Inada condition, Karatzas, Lehoczky, Shreve (1987) showed that this can be done when $$P_t$$ is an Ito semimartingale and market is complete. (See also Cox and Huang (1989).) The rigorous argument makes use of convex duality and is referred to as the martingale duality method in mathematical finance.

The Ito semimartingale case certainly covers many---perhaps most--- asset pricing models in finance. In fact, asset prices are usually assumed to follow a very special Ito semimartingale---geometric Brownian motion, where the risk neutral density $$D_t$$ is itself an exponential martingale. Then $$(*)$$ takes the special form $$\frac{dM}{M} = - r dt + \frac{dD}{D},$$ where $$M$$ is the SDF. One can then take the representative investor to be CRRA and, since powers of exponential martingales are still exponential martingales, back out an SDF $$M$$ without referring to a more general argument.

For example, as you already point out, the Black-Scholes formula for pricing European call option in complete markets can be recovered from the Lucas asset pricing model, where the equilibrium cum-dividend return process follows $$\frac{d P + D dt}{P} = (\mu + \delta) dt + \sigma dW$$ with $$\delta$$ being the endogenous dividend-price ratio $$\frac{D}{P}$$. The underlying for the call being priced is the Lucas tree.

I don't know whether the martingale duality method has been extended to general semimartingales. When market is incomplete, after some browsing it appears that only the case of terminal utility has been investigated and it is shown that certain constraints must be placed on $$u$$.

• Thanks for the great answer. Do you know an arbitrage free model in which no utility function can give rise to an appropriate (linear) pricing function? So basically a counterexample for the  utility representation theorem ''. It may have to be a pathological model? In the simplest case, we'd need a functional form for the SDF which cannot be expressed as $\beta\frac{u'(c_{t+1})}{u'(c_t)}$, regardless which utility function we choose? – Alex Jul 16 at 20:52
• Can't you take the representative agent simply to maximize the expectation under the equivalent martingale measure? – Michael Greinecker Jul 16 at 21:06
• Doesn't the equivalent martingale measure depend on the utility function to start with? $P_t=E^P[SDF\cdot X]=\frac{1}{R_f}E^Q[X]$, but $Q$ is now a merger of the real world probabilities ($P$) and the SDF (which depends on preferences/utility function)? – Alex Jul 16 at 22:49
• @Alex After some browsing, it would appear the claim holds for a pretty large class of no-arbitrage models---specifically Ito semimartingales, although the approach is more general then the special cases one typically encounters in finance. Therefore the previous answer has to be revised (my apologies to folks). As for your second question, the answer is no, preference/utility do not enter fundamental theorems of asset pricing in any way. – Michael Jul 19 at 0:18
• @MichaelGreinecker It seems that you would only be backing out an indirect utility function by doing that. – Michael Jul 19 at 0:19