<blockquote>
<p>...no-arbitrage models (such as Black-Scholes and HJM) are equivalent to
equilibrium models (such as CAPM or C-CAPM).</p>
</blockquote>
<p><em>Short Answer</em> 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.</p>
<p>Clearly, no-arbitrage is a necessary condition for general equilibrium.
To claim equivalence is then to say it is also sufficient, i.e. given <em>any</em> price process <span class="math-container">$P_t$</span> and a density process <span class="math-container">$D_t$</span> such that the discounted price <span class="math-container">$e^{-rt} P_t$</span>
is a martingale after change of measure by <span class="math-container">$D_t$</span>, one needs to find
a (say, representative) investor <span class="math-container">$u$</span> and a equilibrium consumption process <span class="math-container">$c_t$</span> such that</p>
<p><span class="math-container">$$
e^{-\beta t} u'(c_t) = \lambda e^{-rt} D_t, \quad\quad (*)
$$</span>
for some <span class="math-container">$\lambda &gt; 0$</span>.
In other words, one needs a &quot;SDF/marginal utility representation&quot; for equivalent martingale measure densities.</p>
<p>The equation <span class="math-container">$(*)$</span> is the usual relationship
<span class="math-container">$$
\mbox{marginal utility} \; \propto \mbox{price}.
$$</span>
If you write down a heuristic Lagrangian, <span class="math-container">$\lambda$</span> is the Lagrange multiplier in the FOC.
In general, FOC's of this type is only necessary for optimality of <span class="math-container">$c_t$</span>.
If <span class="math-container">$(*)$</span> is sufficient for optimality of <span class="math-container">$c_t$</span>, you can take the dividend to be <span class="math-container">$c_t$</span> in <span class="math-container">$(*)$</span> and <span class="math-container">$P_t$</span> becomes the equilibrium price faced by the representative investor <span class="math-container">$u$</span>.</p>
<p>With certain assumptions on <span class="math-container">$u$</span>---such as concavity and the Inada condition,
<a href="https://epubs.siam.org/doi/abs/10.1137/0325086" rel="nofollow noreferrer">Karatzas, Lehoczky, Shreve (1987)</a> showed that this can be done when <span class="math-container">$P_t$</span> is an Ito semimartingale and market is complete. (See also <a href="https://www.sciencedirect.com/science/article/pii/0022053189900677" rel="nofollow noreferrer">Cox and Huang (1989)</a>.) The rigorous argument makes use of convex duality and is referred to as the <em>martingale duality method</em> in mathematical finance.</p>
<p>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 <span class="math-container">$D_t$</span> is itself an exponential martingale.
Then <span class="math-container">$(*)$</span> takes the special form
<span class="math-container">$$
\frac{dM}{M} = - r dt + \frac{dD}{D},
$$</span>
where <span class="math-container">$M$</span> 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 <span class="math-container">$M$</span> without referring to a more general argument.</p>
<p>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
<span class="math-container">$$
\frac{d P + D dt}{P} =  (\mu + \delta) dt + \sigma dW
$$</span>
with <span class="math-container">$\delta$</span> being the endogenous dividend-price ratio <span class="math-container">$\frac{D}{P}$</span>.
The underlying for the call being priced is the Lucas tree.</p>
<p>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 <span class="math-container">$u$</span>.</p>

CC BY-SA 4.0
2020-07-16T20:52:43.577Z
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
CC BY-SA 4.0
2020-07-16T21:06:24.807Z
Can't you take the representative agent simply to maximize the expectation under the equivalent martingale measure?
Michael Greinecker
CC BY-SA 4.0
2020-07-16T22:49:58.440Z
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
CC BY-SA 4.0
2020-07-19T00:18:29.200Z
@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
CC BY-SA 4.0
2020-07-19T00:19:48.377Z
@MichaelGreinecker It seems that you would only be backing out an indirect utility function by doing that.
Michael
CC BY-SA 4.0
2020-07-19T10:06:00.637Z
@MichaeThe FTAP states that existence (no-arbitrage) and uniqueness (completeness) are independent of utility functions, but I can write $P_t=\sum_\omega SDF(\omega)\cdot X(\omega) = \frac{1}{R_f}\sum_\omega q(\omega)\cdot X(\omega)$, where the risk-neutral probabilities $q(\omega)$ depend on the SDF and thus on risk-aversion/utility? In the Black-Scholes mode, $Q$ is unqiue and ``risk-neutral'', but defined via the SDF and hence in terms of the real-world drift $\mu$.
Alex
CC BY-SA 4.0
2020-07-19T23:39:09.937Z
@Alex "...but I can write $P_t=\sum_\omega SDF(\omega)\cdot X(\omega)$..."---that's precisely the question at hand, whether the risk neutral density admits a representation as SDF. The claim that you "can write ..." does not follow trivially, in reasonable generality. (Assume market completeness so that the EMM is unique. The incomplete case is more involved, as stated in the answer.)
Michael
CC BY-SA 4.0
2020-07-20T12:41:24.913Z
@Michael So the no free lunch without vanishing risk conditions for Itô semimartingale price processes is equivalent to the existence of an equivalent martingale measure (EMM) by the 1st FTAP. The SDF's existence follows from the Riesz representation theorem and linearity of prices (law of one price). So every EMM corresponds to an SDF. So, irrespective of completeness, no arbitrage $\approx$ (at least one) EMM $\approx$ (at least one) SDF. Your answer then says that for most cases, the SDFs can be written as marginal utilities and thus no arbitrage model $\approx$ equilibrium model?
Alex
CC BY-SA 4.0
2020-07-20T19:06:59.250Z
If one defines the SDF to be a marginal utility process, then its existence does *not* follows from Riesz representation theorem. (Actually it's not Riesz representation theorem that is used for general formulations of FFTAP. Riesz representation holds for bounded functionals on Hilbert spaces. In general, one needs a Hahn-Banach type result---or its geometric version, separating hyperplanes). And, when market is incomplete, utility maximization requires super-replication and even more functional-analytic considerations are required. Thus far only the terminal payoff case has been considered.
Michael
CC BY-SA 4.0
2020-07-16T20:09:07.423Z
2020-07-19T23:45:10.517Z
2020-07-19T23:45:10.517Z
Michael
<p>This <a href="https://www.youtube.com/watch?v=kHSNu6sfrfM" rel="nofollow noreferrer">YouTube video from WHU</a> (starting from 3:50) claims that <strong>no-arbitrage models</strong> (such as Black-Scholes and HJM) <strong>are equivalent to equilibrium models</strong> (such as CAPM or C-CAPM).</p>
<p>He uses the Euler equation and the stochastic discount factor (SDF) as arguments which apparently link the two types of models.</p>
<p>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.</p>
<p>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 (<span class="math-container">$\mu$</span>-<span class="math-container">$\sigma$</span> based agents <span class="math-container">$\Leftrightarrow$</span> quadratic utility function <span class="math-container">$\Leftrightarrow$</span> linear SDF) are <em>not necessary</em> 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).</p>
<p>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).</p>

<finance><financial-economics><general-equilibrium><asset-pricing><arbitrage>
Are no arbitrage models and equilibrium models equivalent?
Alex
# Are no arbitrage models and equilibrium models equivalent?

(Asked by: username_0 on 2020-07-14)

This [YouTube video from WHU](https://www.youtube.com/watch?v=kHSNu6sfrfM) (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).

**Tags:** finance, financial-economics, general-equilibrium, asset-pricing, arbitrage


---

## 1 answers

### Answer by username_1 on 2020-07-16. Score: 6 | Accepted answer

> ...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)](https://epubs.siam.org/doi/abs/10.1137/0325086) showed that this can be done when $P\_t$ is an Ito semimartingale and market is complete. (See also [Cox and Huang (1989)](https://www.sciencedirect.com/science/article/pii/0022053189900677).) 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 $u$.

> **Comments:**
> **username_0**: 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?  
> **username_2**: Can't you take the representative agent simply to maximize the expectation under the equivalent martingale measure?  
> **username_0**: 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)?  
> **username_1**: @username_0 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.  
> **username_1**: @username_2 It seems that you would only be backing out an indirect utility function by doing that.  
> **username_0**: @MichaeThe FTAP states that existence (no-arbitrage) and uniqueness (completeness) are independent of utility functions, but I can write $P_t=\sum_\omega SDF(\omega)\cdot X(\omega) = \frac{1}{R_f}\sum_\omega q(\omega)\cdot X(\omega)$, where the risk-neutral probabilities $q(\omega)$ depend on the SDF and thus on risk-aversion/utility? In the Black-Scholes mode, $Q$ is unqiue and ``risk-neutral'', but defined via the SDF and hence in terms of the real-world drift $\mu$.  
> **username_1**: @username_0 "...but I can write $P_t=\sum_\omega SDF(\omega)\cdot X(\omega)$..."---that's precisely the question at hand, whether the risk neutral density admits a representation as SDF. The claim that you "can write ..." does not follow trivially, in reasonable generality. (Assume market completeness so that the EMM is unique. The incomplete case is more involved, as stated in the answer.)  
> **username_0**: @username_1 So the no free lunch without vanishing risk conditions for Itô semimartingale price processes is equivalent to the existence of an equivalent martingale measure (EMM) by the 1st FTAP. The SDF's existence follows from the Riesz representation theorem and linearity of prices (law of one price). So every EMM corresponds to an SDF. So, irrespective of completeness, no arbitrage $\approx$ (at least one) EMM $\approx$ (at least one) SDF. Your answer then says that for most cases, the SDFs can be written as marginal utilities and thus no arbitrage model $\approx$ equilibrium model?  
> **username_1**: If one defines the SDF to be a marginal utility process, then its existence does *not* follows from Riesz representation theorem. (Actually it's not Riesz representation theorem that is used for general formulations of FFTAP. Riesz representation holds for bounded functionals on Hilbert spaces. In general, one needs a Hahn-Banach type result---or its geometric version, separating hyperplanes). And, when market is incomplete, utility maximization requires super-replication and even more functional-analytic considerations are required. Thus far only the terminal payoff case has been considered.  

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