<p>Market is complete in the Luca economy, with some caveats.</p>

<h2><strong>Market is complete (only) with respect to dividend-contingent claims</strong></h2>

<p>Consider the Lucas economy with one tree.
The representative investor's portfolio can be viewed as a portfolio of
dividend-contingent Arrow-Debreu securities. In particular, markets are complete with respect to dividend-contingent claims.</p>

<p>If <span class="math-container">$D_t$</span> is the exogenous dividend process and the investor has utility function <span class="math-container">$u$</span>, the equilibrium price is characterized by the asset pricing (Euler) equation
<span class="math-container">$$
P_t = \sum_{\omega} e^{- \beta }\frac{ u'(D_{t+1})}{u'(D_t)} (\omega)) P_{t+1}(\omega) \cdot p(\omega)
$$</span>
where <span class="math-container">$\omega$</span> is possible state of the world that occurs at date-<span class="math-container">$t+1$</span> with probability <span class="math-container">$p(\omega)$</span>.
This decomposes the representative investor's portfolio---the Lucas tree---as a portfolio of Arrow-Debreu securities.</p>

<p>The date-<span class="math-container">$t$</span> price <span class="math-container">$Q(\omega')$</span> of the AD security whose date-<span class="math-container">$t+1$</span> payoff is <span class="math-container">$1_{\{ \omega = \omega' \}}$</span> is
<span class="math-container">$$
e^{- \beta }\frac{ u'(D_{t+1})}{u'(D_t)} (\omega')) \cdot p(\omega').
$$</span>
The Lucas tree is a portfolio whose date-<span class="math-container">$t+1$</span> pay-off at state <span class="math-container">$\omega$</span> is <span class="math-container">$P_{t+1}(\omega)$</span>. Therefore, like <em>any</em> portfolio in <em>any</em> AD economy, its date-<span class="math-container">$t$</span> price is sum of 
<span class="math-container">$$
Q(\omega) \times P_{t+1}(\omega),
$$</span>
over all possible states <span class="math-container">$\omega$</span>.</p>

<p>It is therefore clear that market is complete with respect to <span class="math-container">$\omega$</span>-contingent claims.
In principle, any investor in this economy has access to a full menu of AD securities, and any portfolio can be replicated by a portfolio of AD securities.
The equilibrium AD prices/stochastic discount factor is such that the optimal portfolio for the representative investor is the Lucas tree itself. </p>

<p>Suppose now there is a risk-free asset/saving technology.
If interest rate <span class="math-container">$r$</span> is given by 
<span class="math-container">$$
e^{-r} = \sum_{\omega} Q(\omega),
$$</span>
then the representative investor would not choose to lend or borrow. 
This is the no-arbitrage condition for the zero coupon bond.
(If, say, <span class="math-container">$e^{-r} &lt; \sum_{\omega} Q(\omega)$</span>, the investor will attempt to save some of his wealth <span class="math-container">$P_t$</span> and only hold a fraction of the tree. There would be an excess demand of bonds.)
Therefore <span class="math-container">$r$</span> is the equilibrium interest rate that clears the bond market.</p>

<p>The asset pricing equation
<span class="math-container">\begin{align}
P_t &amp; = \sum_{\omega} e^{- \beta  + r}\frac{ u'(D_{t+1})}{u'(D_t)} (\omega)) p(\omega) \cdot e^{-r} P_{t+1}(\omega) \\
    &amp; = \sum_{\omega} \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')}  \cdot e^{-r} P_{t+1}(\omega) \quad (1)
\end{align}</span>
now says that, in a market with risk-free rate <span class="math-container">$r$</span> as well as AD securities, 
the Lucas tree is the representative investor's optimal portfolio.</p>

<h2><strong>No-trade equilibrium</strong></h2>

<p>As is true for any representative investor model, the equilibrium is a no-trade equilibrium.
At each date-<span class="math-container">$t$</span>, there are AD markets for <span class="math-container">$t+1$</span> claims.
In equilibrium, no trade occurs---the equilibrium prices in these markets are precisely the prices the ensures this market clearing condition holds on the equilibrium path.
But one can price derivatives using the stochastic discount factor/pricing kernel just  the same. In the standard continuous-time formulation of the Lucas model, one can recover the Black-Scholes formula.</p>

<h2><strong>Non-completeness w.r.t. non-consumption risk</strong></h2>

<p>In the Lucas economy, only assets that are needed are those that enable the representative investor to hedge his consumption risk.
Market is <em>not</em> complete with respect to different states for which the dividend yield of the Lucas tree is the same. </p>

<p>Indeed, it has no reason to be.
In order to be induced to hold the asset (Lucas tree), the price of the asset must compensate the risk-averse representative investor for his consumption risk.
In equilibrium, he consumes the dividend. This means price, therefore the stochastic discount factor/pricing kernel, must be a function of dividend.
On the other hand, the representative investor does not care about risk unrelated to his consumption.</p>

<p>For example, if possible states of the world are given by <span class="math-container">$(\omega, \omega')$</span>, and dividend <span class="math-container">$D(\omega)$</span> only depends on <span class="math-container">$\omega$</span>, then market is only complete with respect to <span class="math-container">$\omega$</span>-contingent claims.
This is referred to as "man-made uncertainty" in  Ljungqvist and  Sargent.</p>

<h2>Black-Scholes formula in Lucas economy</h2>

<p>In the standard continuous-time formulation of the Lucas model, one can recover 
the Black-Scholes formula for pricing European call option in complete markets.
This is an example of market completeness with respect to dividend-contingent claims.</p>

<p>(In the language of mathematical finance, derivatives in complete markets are priced via the <em>risk-neutral measure</em>. Discounted prices are martingales under the risk-neutral measure. 
This is already reflected in the discrete-time asset pricing equation <span class="math-container">$(1)$</span> from above:
<span class="math-container">\begin{align}
P_t &amp; = \sum_{\omega} \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')}  \cdot e^{-r} P_{t+1}(\omega).
\end{align}</span>
The risk-neutral measure is <span class="math-container">$q(\omega) = \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')}$</span>; it differs from the SDF by the discount factor <span class="math-container">$e^{-r}$</span>.
The Black-Scholes formula is a continuous-time version of this.
)</p>

<p>Suppose the exogenous dividend process is given by
<span class="math-container">$$
\frac{d D}{D} = \mu dt + \sigma dW
$$</span>
where <span class="math-container">$W$</span> is standard Brownian motion, and the investor maximizes expected utility
<span class="math-container">$$
E[\int_0^{\infty} e^{-\beta t} u(c_t) dt],
$$</span>
over consumption stream <span class="math-container">$c_t$</span> adapted with respect to the filtration generated by <span class="math-container">$W$</span>.</p>

<p>The asset pricing equation is 
<span class="math-container">$$
P_0 = E[\int_0^{\infty} \frac{ e^{-\beta t} u'(D_t)}{u'(D_0)} D_t dt].
$$</span>
Assume <span class="math-container">$u$</span> is the CRRA utility <span class="math-container">$u(c) = \frac{1}{1-\gamma} c^{1-\gamma}$</span>.
Then <span class="math-container">$P_0$</span>, which is just an expectation, can be computed directly:
<span class="math-container">$$
\frac{P_0}{D_0} = \frac{1}{-\beta + (1-\gamma) (\mu + \frac{1}{2} \sigma^2) + \frac{1}{2} (1-\gamma)^2 \sigma^2} \equiv \frac{1}{\delta}.
$$</span>
So in equilibrium, price-dividend ratio <span class="math-container">$\frac{P}{D}$</span> is constant <span class="math-container">$\frac{1}{\delta}$</span> and price follows
<span class="math-container">$$
\frac{d P}{P} = \mu dt + \sigma dW. 
$$</span>
The <em>cum-dividend</em> return process is
<span class="math-container">$$
\frac{d P + D dt}{P} =  (\mu + \delta) dt + \sigma dW. \quad (2)
$$</span></p>

<p>Suppose the equilibrium interest rate is <span class="math-container">$r$</span> in this economy, and time-0 price of Lucas tree is <span class="math-container">$P$</span>. Let <span class="math-container">$E$</span> be the equilibrium price of an European call on the Lucas tree entered at time-<span class="math-container">$0$</span> maturing at time-<span class="math-container">$t$</span> with strike <span class="math-container">$K$</span>. <strong><span class="math-container">$E$</span> is given precisely by the standard Black-Scholes formula <span class="math-container">$C(r,P, K, t)$</span> for European call.</strong></p>

<p>This follows from direct calculations of the pricing kernel <span class="math-container">$M = u'(D_t)$</span>: Simply plugging in gives
<span class="math-container">$$
\frac{dM}{M} = (-\beta - \gamma \mu + \frac{1}{2}\gamma (1+\gamma) \sigma^2) dt - \gamma \sigma dW.
$$</span>
Therefore equilibrium interest rate is
<span class="math-container">$$
r = \beta + \gamma \mu - \frac{1}{2}\gamma (1+\gamma) \sigma^2
$$</span>
with standard observations regarding the three terms on the right-hand side---they reflect time preference, intertemporal substitution, and precautionary saving.
Therefore <span class="math-container">$\gamma \sigma = \frac{(\mu + \delta) - r}{\sigma}$</span>
and
<span class="math-container">$$
\frac{dM}{M} = -r dt - \frac{(\mu + \delta) - r}{\sigma} dW.
$$</span>
So the cum-dividend price process <span class="math-container">$(2)$</span>, after discounting by <span class="math-container">$e^{-rt}$</span>, is a martingale under risk-neutral density
<span class="math-container">$$
\frac{dL}{L} = - \frac{(\mu + \delta) - r}{\sigma} dW.
$$</span>
This is exactly the Black-Scholes setting for pricing derivatives and the price of European call follows accordingly.</p>

<h2>Market completeness in continuous-time</h2>

<p>The mathematical statement that gives market completeness in continuous-time models is the <strong>Martingale Representation Theorem</strong>, which says every martingale with respect to a <em>Brownian</em> filtration can be represented as an Ito integral with respect to the Brownian generating that filtration.</p>

<p>The result is <em>not</em> true for general filtrations, i.e. if <span class="math-container">$W_t$</span> is a <span class="math-container">$(\mathcal{F_t})$</span>-Brownian motion, it is not true in general that every <span class="math-container">$(\mathcal{F_t})$</span>-martingale is a <span class="math-container">$dW$</span>-integral.</p>

<p>This is consistent with the economic statement that market is complete only with respect to dividend-contingent claims.
In the Lucas/Black-Scholes example, if <span class="math-container">$W_t$</span> is a <span class="math-container">$(\mathcal{F_t})$</span>-Brownian motion, then equilibrium price and SDF are measurable not only respect to <span class="math-container">$(\mathcal{F_t})$</span> but the minimal filtration generated <span class="math-container">$W_t$</span>. In general, the minimal filtration is smaller than <span class="math-container">$(\mathcal{F_t})$</span>.
Portfolio payoffs that can be hedged/replicated are only those that are measurable with respect to dividend (in equilibrium, price/SDF), i.e. the minimal filtration.</p>

CC BY-SA 4.0
2020-05-30T11:24:14.160Z
2020-06-02T01:19:18.857Z
2020-06-02T01:19:18.857Z
Michael
<p>It seems to me that we might as well say that markets are complete. It seems to me to be somewhat inconsequential since this is a representative agent model and that market clearing requires that the only assets that the agent may hold in positive net supply are the <span class="math-container">$n$</span> trees. The prices of all Arrow-Debreu securities may be inferred from shadow prices. Suppose the optimization problem is rewritten such that the agent may invest in the <span class="math-container">$n$</span> trees plus the set of all Arrow-Debreu securities. (There will be redundancies---I'm hand-waving a little.) The market clearing conditions say that, in equilibrium, the agent must have a unit holding in each of the <span class="math-container">$n$</span> trees and zero holdings in the rest of the assets and the price of each of these securities in equilibrium must be set to support these holdings. Since market completeness is a matter of whether or not the agent may trade in all of these securities, we may interpret the situation a two different ways: either the agent <em>can</em> invest in the Arrow-Debreu securities (as well as the trees) but the AD securities must be in zero net supply or the agent <em>cannot</em> invest in the AD securities (only in the trees). In either case, the result is the same.</p>

<p>On the other hand, if you had a model with more than one agent and they had different endowment processes, then the question of market completeness would matter for equilibrium outcomes.</p>

<p><strong>Reference</strong></p>

<p>Take a look at the <a href="https://www.ssc.wisc.edu/~nwilliam/Econ714_files/lect1-18.pdf" rel="nofollow noreferrer">notes here</a>.</p>

<blockquote>
  <p>We have used the equilibrium of the model to determine a single price,
  the price of a tree. With trading in trees, markets are complete.
  However we can introduce additional (redundant) assets and use the
  same approach to determine their prices.</p>
</blockquote>

<p><a href="https://i.sstatic.net/zAGLF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/zAGLF.png" alt="enter image description here"></a>
<a href="https://i.sstatic.net/e4L0Q.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/e4L0Q.png" alt="enter image description here"></a></p>

<p><strong>Additional Note:</strong></p>

<p>Your question is somewhat related to the question asked here: <a href="https://economics.stackexchange.com/q/1616/59">"Complete Markets in Continuous Time"</a>. See the solutions given there. </p>

<p>In addition, you can also find a discussion in these <a href="https://www.princeton.edu/~markus/teaching/Eco525/07%20MultiperiodModel_a.pdf" rel="nofollow noreferrer">notes</a> by Markus Brunnermeier:</p>

<blockquote>
  <p>If the horizon is infinite, the number of events is also infinite.
  Does that imply that we need an infinite number of assets to make the
  market complete? </p>
  
  <ul>
  <li>List item
  
  <ul>
  <li>Do we need assets with all possible times to maturity and events to have a complete market?</li>
  </ul></li>
  <li>No. Dynamic completion.
  
  <ul>
  <li>Arrow (1953) and Guesnerie and Jaffray (1974)</li>
  </ul></li>
  </ul>
</blockquote>

<p>The conclusion, which is in the slide titled "How many assets to complete market?", is that, in discrete time, it depends on the "branching number" (see the slides) and that, in continuous time, you can get complete markets simply with a risk-free bond and <span class="math-container">$n$</span> independent risky assets where <span class="math-container">$n$</span> is the number of independent Brownian motion underlying the economy.</p>

CC BY-SA 4.0
2019-03-19T19:38:54.810Z
Thank you for the helpful answer. One last question -- the original paper is in _discrete_ time, _continuous_ state (not vice versa). Does the argument still go through in a similar way in that particular setting as well?
mathsquestions1
CC BY-SA 4.0
2019-03-20T06:45:17.903Z
You're right. My apologies. I've rewritten the answer the best I could. It may still have errors, though. Let me know what you think.
jmbejara
CC BY-SA 4.0
2019-03-19T04:20:38.943Z
2019-03-21T20:54:08.193Z
2019-03-21T20:54:08.193Z
jmbejara
<p>The Lucas (1978) asset pricing model seems to be one of the workhorse models in finance / asset pricing models. It also seems to be the case that the environment, with claims to <span class="math-container">$n$</span> (exogenous) productive units traded, features complete markets. I have also seen another version of this model with the agent trading a single stock and a single bond that the instructor told me had complete markets.</p>

<p>To me, this is not obvious at all -- the agent does not have access to Arrow securities for each state, or something of that sort. In fact, in the original paper, the states are continuous, and this makes it even harder for me think about market completeness.</p>

<p>Can anyone tell me why (or perhaps why not) this model features market completeness?</p>

<p>Any help would be greatly appreciated (or pointers to relevant links would work). Thanks!</p>

<p><strong>Edit #1:</strong> ----------------------------------------</p>

<p>Just to clarify, while I understand that in equilibrium the net holdings (of all the identical agents) must be zero, my concern was that having access to AD securities vs. not having access may have asset pricing implications. What I mean by this is that although the outcome in terms of asset holdings must (almost by construction) be equal in any case, but perhaps the presence of AD securities may distort the prices of other assets (e.g. equity claims "trees").</p>

<p>Intuitively, perhaps the price of trees must be different in the presence of AD securities to offset the agent's desire to hold the AD securities in equilibrium. My next logical step in determining whether or not this is the case, was to think about whether the markets were complete or not in the first place -- if markets were already complete, then the additional AD securities are "redundant" is some sense, and prices should not change. If markets are incomplete, my initial intuition was that this may possibly have consequences (not in asset positions, but in prices).</p>

<p>To me, this is not obvious at all, and something I would really like to see a proof of. </p>

CC BY-SA 4.0
2019-03-21T20:51:00.163Z
Thanks for the edit. My thoughts are that the proof is trivial. Consider one model where the agent can only invest in the trees. Trying to write notation compactly (loosely speaking), let this first model be $\max_{c, x} U(\{c\})$ st $\{c,x\} \in \Gamma$ . Then consider another model where the agent invests in trees and AD securities that are in zero net supply. This is $\max_{c,x,a} U(\{c\})$ st $\{c,x\} \in \Gamma$ and st $a_{i,t} = 0$ $\forall i,t$, where $a_{it}$ are the AD security holdings for state $i$ at time $t$. The models should result in the exact same outcomes, right?
jmbejara
CC BY-SA 4.0
2019-03-25T22:52:40.947Z
Thanks for the continuous update! This has been very helpful, and I have accepted your answer.
mathsquestions1
CC BY-SA 4.0
2019-03-26T01:57:48.383Z
No problem! Good question (and feedback)!
jmbejara
<asset-pricing>
Does the Lucas (1978) asset pricing model feature complete markets?
mathsquestions1
# Does the Lucas (1978) asset pricing model feature complete markets?

(Asked by: username_0 on 2019-03-18)

The Lucas (1978) asset pricing model seems to be one of the workhorse models in finance / asset pricing models. It also seems to be the case that the environment, with claims to $n$ (exogenous) productive units traded, features complete markets. I have also seen another version of this model with the agent trading a single stock and a single bond that the instructor told me had complete markets.

To me, this is not obvious at all -- the agent does not have access to Arrow securities for each state, or something of that sort. In fact, in the original paper, the states are continuous, and this makes it even harder for me think about market completeness.

Can anyone tell me why (or perhaps why not) this model features market completeness?

Any help would be greatly appreciated (or pointers to relevant links would work). Thanks!

**Edit #1:** ----------------------------------------

Just to clarify, while I understand that in equilibrium the net holdings (of all the identical agents) must be zero, my concern was that having access to AD securities vs. not having access may have asset pricing implications. What I mean by this is that although the outcome in terms of asset holdings must (almost by construction) be equal in any case, but perhaps the presence of AD securities may distort the prices of other assets (e.g. equity claims "trees").

Intuitively, perhaps the price of trees must be different in the presence of AD securities to offset the agent's desire to hold the AD securities in equilibrium. My next logical step in determining whether or not this is the case, was to think about whether the markets were complete or not in the first place -- if markets were already complete, then the additional AD securities are "redundant" is some sense, and prices should not change. If markets are incomplete, my initial intuition was that this may possibly have consequences (not in asset positions, but in prices).

To me, this is not obvious at all, and something I would really like to see a proof of.

**Tags:** asset-pricing

> **Comments:**
> **username_1**: Thanks for the edit. My thoughts are that the proof is trivial. Consider one model where the agent can only invest in the trees. Trying to write notation compactly (loosely speaking), let this first model be $\max_{c, x} U(\{c\})$ st $\{c,x\} \in \Gamma$ . Then consider another model where the agent invests in trees and AD securities that are in zero net supply. This is $\max_{c,x,a} U(\{c\})$ st $\{c,x\} \in \Gamma$ and st $a_{i,t} = 0$ $\forall i,t$, where $a_{it}$ are the AD security holdings for state $i$ at time $t$. The models should result in the exact same outcomes, right?  
> **username_0**: Thanks for the continuous update! This has been very helpful, and I have accepted your answer.  
> **username_1**: No problem! Good question (and feedback)!  

---

## 2 answers

### Answer by username_2 on 2020-05-30. Score: 1 | Accepted answer

Market is complete in the Luca economy, with some caveats.

## **Market is complete (only) with respect to dividend-contingent claims**

Consider the Lucas economy with one tree.
The representative investor's portfolio can be viewed as a portfolio of
dividend-contingent Arrow-Debreu securities. In particular, markets are complete with respect to dividend-contingent claims.

If $D\_t$ is the exogenous dividend process and the investor has utility function $u$, the equilibrium price is characterized by the asset pricing (Euler) equation
$$
P\_t = \sum\_{\omega} e^{- \beta }\frac{ u'(D\_{t+1})}{u'(D\_t)} (\omega)) P\_{t+1}(\omega) \cdot p(\omega)
$$
where $\omega$ is possible state of the world that occurs at date-$t+1$ with probability $p(\omega)$.
This decomposes the representative investor's portfolio---the Lucas tree---as a portfolio of Arrow-Debreu securities.

The date-$t$ price $Q(\omega')$ of the AD security whose date-$t+1$ payoff is $1\_{\{ \omega = \omega' \}}$ is
$$
e^{- \beta }\frac{ u'(D\_{t+1})}{u'(D\_t)} (\omega')) \cdot p(\omega').
$$
The Lucas tree is a portfolio whose date-$t+1$ pay-off at state $\omega$ is $P\_{t+1}(\omega)$. Therefore, like *any* portfolio in *any* AD economy, its date-$t$ price is sum of
$$
Q(\omega) \times P\_{t+1}(\omega),
$$
over all possible states $\omega$.

It is therefore clear that market is complete with respect to $\omega$-contingent claims.
In principle, any investor in this economy has access to a full menu of AD securities, and any portfolio can be replicated by a portfolio of AD securities.
The equilibrium AD prices/stochastic discount factor is such that the optimal portfolio for the representative investor is the Lucas tree itself.

Suppose now there is a risk-free asset/saving technology.
If interest rate $r$ is given by
$$
e^{-r} = \sum\_{\omega} Q(\omega),
$$
then the representative investor would not choose to lend or borrow.
This is the no-arbitrage condition for the zero coupon bond.
(If, say, $e^{-r} < \sum\_{\omega} Q(\omega)$, the investor will attempt to save some of his wealth $P\_t$ and only hold a fraction of the tree. There would be an excess demand of bonds.)
Therefore $r$ is the equilibrium interest rate that clears the bond market.

The asset pricing equation
\begin{align}
P\_t & = \sum\_{\omega} e^{- \beta + r}\frac{ u'(D\_{t+1})}{u'(D\_t)} (\omega)) p(\omega) \cdot e^{-r} P\_{t+1}(\omega) \\
& = \sum\_{\omega} \frac{ Q(\omega)}{\sum\_{\omega'} Q(\omega')} \cdot e^{-r} P\_{t+1}(\omega) \quad (1)
\end{align}
now says that, in a market with risk-free rate $r$ as well as AD securities,
the Lucas tree is the representative investor's optimal portfolio.

## **No-trade equilibrium**

As is true for any representative investor model, the equilibrium is a no-trade equilibrium.
At each date-$t$, there are AD markets for $t+1$ claims.
In equilibrium, no trade occurs---the equilibrium prices in these markets are precisely the prices the ensures this market clearing condition holds on the equilibrium path.
But one can price derivatives using the stochastic discount factor/pricing kernel just the same. In the standard continuous-time formulation of the Lucas model, one can recover the Black-Scholes formula.

## **Non-completeness w.r.t. non-consumption risk**

In the Lucas economy, only assets that are needed are those that enable the representative investor to hedge his consumption risk.
Market is *not* complete with respect to different states for which the dividend yield of the Lucas tree is the same.

Indeed, it has no reason to be.
In order to be induced to hold the asset (Lucas tree), the price of the asset must compensate the risk-averse representative investor for his consumption risk.
In equilibrium, he consumes the dividend. This means price, therefore the stochastic discount factor/pricing kernel, must be a function of dividend.
On the other hand, the representative investor does not care about risk unrelated to his consumption.

For example, if possible states of the world are given by $(\omega, \omega')$, and dividend $D(\omega)$ only depends on $\omega$, then market is only complete with respect to $\omega$-contingent claims.
This is referred to as "man-made uncertainty" in Ljungqvist and Sargent.

## Black-Scholes formula in Lucas economy

In the standard continuous-time formulation of the Lucas model, one can recover
the Black-Scholes formula for pricing European call option in complete markets.
This is an example of market completeness with respect to dividend-contingent claims.

(In the language of mathematical finance, derivatives in complete markets are priced via the *risk-neutral measure*. Discounted prices are martingales under the risk-neutral measure.
This is already reflected in the discrete-time asset pricing equation $(1)$ from above:
\begin{align}
P\_t & = \sum\_{\omega} \frac{ Q(\omega)}{\sum\_{\omega'} Q(\omega')} \cdot e^{-r} P\_{t+1}(\omega).
\end{align}
The risk-neutral measure is $q(\omega) = \frac{ Q(\omega)}{\sum\_{\omega'} Q(\omega')}$; it differs from the SDF by the discount factor $e^{-r}$.
The Black-Scholes formula is a continuous-time version of this.
)

Suppose the exogenous dividend process is given by
$$
\frac{d D}{D} = \mu dt + \sigma dW
$$
where $W$ is standard Brownian motion, and the investor maximizes expected utility
$$
E[\int\_0^{\infty} e^{-\beta t} u(c\_t) dt],
$$
over consumption stream $c\_t$ adapted with respect to the filtration generated by $W$.

The asset pricing equation is
$$
P\_0 = E[\int\_0^{\infty} \frac{ e^{-\beta t} u'(D\_t)}{u'(D\_0)} D\_t dt].
$$
Assume $u$ is the CRRA utility $u(c) = \frac{1}{1-\gamma} c^{1-\gamma}$.
Then $P\_0$, which is just an expectation, can be computed directly:
$$
\frac{P\_0}{D\_0} = \frac{1}{-\beta + (1-\gamma) (\mu + \frac{1}{2} \sigma^2) + \frac{1}{2} (1-\gamma)^2 \sigma^2} \equiv \frac{1}{\delta}.
$$
So in equilibrium, price-dividend ratio $\frac{P}{D}$ is constant $\frac{1}{\delta}$ and price follows
$$
\frac{d P}{P} = \mu dt + \sigma dW.
$$
The *cum-dividend* return process is
$$
\frac{d P + D dt}{P} = (\mu + \delta) dt + \sigma dW. \quad (2)
$$

Suppose the equilibrium interest rate is $r$ in this economy, and time-0 price of Lucas tree is $P$. Let $E$ be the equilibrium price of an European call on the Lucas tree entered at time-$0$ maturing at time-$t$ with strike $K$. **$E$ is given precisely by the standard Black-Scholes formula $C(r,P, K, t)$ for European call.**

This follows from direct calculations of the pricing kernel $M = u'(D\_t)$: Simply plugging in gives
$$
\frac{dM}{M} = (-\beta - \gamma \mu + \frac{1}{2}\gamma (1+\gamma) \sigma^2) dt - \gamma \sigma dW.
$$
Therefore equilibrium interest rate is
$$
r = \beta + \gamma \mu - \frac{1}{2}\gamma (1+\gamma) \sigma^2
$$
with standard observations regarding the three terms on the right-hand side---they reflect time preference, intertemporal substitution, and precautionary saving.
Therefore $\gamma \sigma = \frac{(\mu + \delta) - r}{\sigma}$
and
$$
\frac{dM}{M} = -r dt - \frac{(\mu + \delta) - r}{\sigma} dW.
$$
So the cum-dividend price process $(2)$, after discounting by $e^{-rt}$, is a martingale under risk-neutral density
$$
\frac{dL}{L} = - \frac{(\mu + \delta) - r}{\sigma} dW.
$$
This is exactly the Black-Scholes setting for pricing derivatives and the price of European call follows accordingly.

## Market completeness in continuous-time

The mathematical statement that gives market completeness in continuous-time models is the **Martingale Representation Theorem**, which says every martingale with respect to a *Brownian* filtration can be represented as an Ito integral with respect to the Brownian generating that filtration.

The result is *not* true for general filtrations, i.e. if $W\_t$ is a $(\mathcal{F\_t})$-Brownian motion, it is not true in general that every $(\mathcal{F\_t})$-martingale is a $dW$-integral.

This is consistent with the economic statement that market is complete only with respect to dividend-contingent claims.
In the Lucas/Black-Scholes example, if $W\_t$ is a $(\mathcal{F\_t})$-Brownian motion, then equilibrium price and SDF are measurable not only respect to $(\mathcal{F\_t})$ but the minimal filtration generated $W\_t$. In general, the minimal filtration is smaller than $(\mathcal{F\_t})$.
Portfolio payoffs that can be hedged/replicated are only those that are measurable with respect to dividend (in equilibrium, price/SDF), i.e. the minimal filtration.


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### Answer by username_1 on 2019-03-19. Score: 4

It seems to me that we might as well say that markets are complete. It seems to me to be somewhat inconsequential since this is a representative agent model and that market clearing requires that the only assets that the agent may hold in positive net supply are the $n$ trees. The prices of all Arrow-Debreu securities may be inferred from shadow prices. Suppose the optimization problem is rewritten such that the agent may invest in the $n$ trees plus the set of all Arrow-Debreu securities. (There will be redundancies---I'm hand-waving a little.) The market clearing conditions say that, in equilibrium, the agent must have a unit holding in each of the $n$ trees and zero holdings in the rest of the assets and the price of each of these securities in equilibrium must be set to support these holdings. Since market completeness is a matter of whether or not the agent may trade in all of these securities, we may interpret the situation a two different ways: either the agent *can* invest in the Arrow-Debreu securities (as well as the trees) but the AD securities must be in zero net supply or the agent *cannot* invest in the AD securities (only in the trees). In either case, the result is the same.

On the other hand, if you had a model with more than one agent and they had different endowment processes, then the question of market completeness would matter for equilibrium outcomes.

**Reference**

Take a look at the [notes here](https://www.ssc.wisc.edu/~nwilliam/Econ714_files/lect1-18.pdf).

> We have used the equilibrium of the model to determine a single price,
> the price of a tree. With trading in trees, markets are complete.
> However we can introduce additional (redundant) assets and use the
> same approach to determine their prices.

[![enter image description here](https://i.sstatic.net/zAGLF.png)](https://i.sstatic.net/zAGLF.png)
[![enter image description here](https://i.sstatic.net/e4L0Q.png)](https://i.sstatic.net/e4L0Q.png)

**Additional Note:**

Your question is somewhat related to the question asked here: ["Complete Markets in Continuous Time"](https://economics.stackexchange.com/q/1616/59). See the solutions given there.

In addition, you can also find a discussion in these [notes](https://www.princeton.edu/~markus/teaching/Eco525/07%20MultiperiodModel_a.pdf) by Markus Brunnermeier:

> If the horizon is infinite, the number of events is also infinite.
> Does that imply that we need an infinite number of assets to make the
> market complete?
>
> * List item
>   + Do we need assets with all possible times to maturity and events to have a complete market?
> * No. Dynamic completion.
>   + Arrow (1953) and Guesnerie and Jaffray (1974)

The conclusion, which is in the slide titled "How many assets to complete market?", is that, in discrete time, it depends on the "branching number" (see the slides) and that, in continuous time, you can get complete markets simply with a risk-free bond and $n$ independent risky assets where $n$ is the number of independent Brownian motion underlying the economy.

> **Comments:**
> **username_0**: Thank you for the helpful answer. One last question -- the original paper is in _discrete_ time, _continuous_ state (not vice versa). Does the argument still go through in a similar way in that particular setting as well?  
> **username_1**: You're right. My apologies. I've rewritten the answer the best I could. It may still have errors, though. Let me know what you think.  

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