Equilibrium in Markets for Securities

Microeconomics for Finance

Juan F. Imbet

Master in Finance, 1st Year - Paris Dauphine University - PSL

Overview and Notation (aligned with Uncertainty)

Finite states of nature: \(s \in \{1,\dots,S\}\) with probabilities \(\pi = (\pi_1,\dots,\pi_S)\), \(\sum_s \pi_s = 1\).
\(J\) securities with time-1 state-contingent payoffs \(a_{sj}\) collected in the payoff matrix \[ A = \begin{bmatrix} a_{11} & \cdots & a_{1J} \\ \vdots & \ddots & \vdots \\ a_{S1} & \cdots & a_{SJ} \end{bmatrix} \in \mathbb{R}^{S\times J}. \]
Time-0 prices \(q = (q_1,\dots,q_J)\in\mathbb{R}^J\).
A portfolio \(x\in\mathbb{R}^J\) costs \(q\cdot x\) at \(t=0\) and delivers random payoff vector \(A x\) at \(t=1\) (a lottery, as in the uncertainty section).

Arbitrage: Definition (Linear Algebra)

An arbitrage is a portfolio \(x\in\mathbb{R}^J\) such that

\[ q\cdot x \le 0,\quad A x \ge 0 \;\text{(componentwise)}, \quad \text{and at least one strict inequality.} \]

Intuition: non-positive cost today, non-negative state payoffs with some gain for sure (or strictly negative cost with zero payoff).
Homogeneity: if \(x\) is an arbitrage, so is \(\lambda x\) for any \(\lambda>0\).
Detecting arbitrage is a feasibility LP: find \(x\) with \(A x \ge 0\) and \(q\cdot x < 0\).

Law of One Price (LoOP) and No-Arbitrage

LoOP: If two portfolios \(x,y\) have the same state-by-state payoffs, \(A x = A y\), then \(q\cdot x = q\cdot y\).
In matrix form, LoOP says pricing is a linear functional of payoffs.
Fundamental theorem (finite-state version):
- No-Arbitrage (NA) holds iff there exists a nonnegative state-price vector \(\psi \in \mathbb{R}^S_{+}\) such that \[ q = A^{\top} \psi,\quad \text{i.e. } q_j = \sum_{s=1}^S \psi_s\, a_{sj} \;\; (j=1,\dots,J). \]
- \(\psi_s\) is the Arrow–Debreu price of one unit delivered in state \(s\) (next slide).

Arrow–Debreu State Securities and State Prices

The Arrow–Debreu security for state \(s\) pays \(1\) in state \(s\) and \(0\) otherwise: payoff \(e_s \in \mathbb{R}^S\).
Its time-0 price is the state price \(\psi_s = q(e_s)\).
Stack \(\psi=(\psi_1,\dots,\psi_S)\!\in\!\mathbb{R}^S_{+}\). Pricing any asset \(j\): \[ q_j = \sum_{s=1}^S \psi_s a_{sj} = \psi^{\top} a_{\cdot j},\quad\text{or }\boxed{\; q = A^{\top} \psi\; }. \]
If markets are complete (below), \(\psi\) is unique and can be solved from observed \((A,q)\).

Complete vs Incomplete Markets

Markets are complete if payoffs of traded assets span all contingencies: \[\mathrm{rank}(A)=S.\]
- If \(J=S\) and \(\det(A)\neq 0\), then \(A\) is invertible.
- If \(J>S\) and \(\mathrm{rank}(A)=S\), many portfolios replicate the same payoff; prices are still unique by LoOP.
Consequences:
- Replication: For any target payoff \(b\in\mathbb{R}^S\), there exists \(x\) s.t. \(A x = b\).
- Pricing uniqueness: \(q(b) = \psi^{\top} b\) is uniquely defined under NA.
Incomplete markets: \(\mathrm{rank}(A)<S\).
- Not all \(b\) are replicable; state prices \(\psi\) may not be unique (set-valued) but still satisfy \(q=A^{\top}\psi\).

The Stochastic Discount Factor (SDF)

Using the probabilities \(\pi\) (as in the uncertainty section), define the SDF \(m=(m_1,\dots,m_S)\) by \[\boxed{\; m_s = \frac{\psi_s}{\pi_s} \;},\quad s=1,\dots,S.\]
Then pricing can be written as an expectation under the physical measure \(\pi\): \[ \boxed{\; q_j = \sum_{s=1}^S \pi_s\, m_s\, a_{sj} = \mathbb{E}_{\pi}[\, m\, \tilde a_j\, ]\; }. \]
In terms of gross returns \(R_j = \dfrac{\tilde a_j}{q_j}\) (random): \[\boxed{\; 1 = \mathbb{E}_{\pi}[\, m\, R_j\, ] \; }\quad \text{for all traded } j. \]
With a risk-free asset of price \(q_f\) and payoff \(1\) in all states, \(R_f = 1/q_f\) and \(1 = \mathbb{E}[m R_f] = R_f\, \mathbb{E}[m]\) so \(\mathbb{E}[m] = 1/R_f\).

Two-Period Consumption–Investment Problem and the SDF

Preferences (expected utility): \[\max_{c_0,\,x\in\mathbb{R}^J} \; u(c_0) + \beta\, \sum_{s=1}^S \pi_s\, u\big(c_1(s)\big)\]
Budget constraints: \[ c_0 + q\cdot x = w_0, \qquad c_1(s) = a_{s\cdot}\, x + y_1(s), \; s=1,\dots,S \] where \(a_{s\cdot}\) is row \(s\) of \(A\), and \(y_1(s)\) is (optional) state-\(s\) endowment.
Lagrangian with multipliers \(\lambda\) for \(t=0\), and \(\mu_s\) for \(t=1\) states: \[\mathcal{L} = u(c_0) + \beta\sum_s \pi_s u(c_1(s)) + \lambda\,(w_0 - c_0 - q\cdot x) + \sum_s \mu_s \big(a_{s\cdot}x + y_1(s) - c_1(s)\big).\]

FOCs → Euler Equation → SDF

FOCs: \[ u'(c_0) - \lambda = 0 \;\Rightarrow\; \lambda = u'(c_0), \] \[ \beta\,\pi_s\, u'(c_1(s)) - \mu_s = 0 \;\Rightarrow\; \mu_s = \beta\,\pi_s\, u'(c_1(s)), \] \[ -\lambda\, q_j + \sum_{s=1}^S \mu_s\, a_{sj} = 0 \;\Rightarrow\; q_j = \sum_s \frac{\mu_s}{\lambda} a_{sj}. \]
Identify the SDF: \[ \boxed{\; m_s = \frac{\mu_s}{\lambda\,\pi_s} = \beta\, \frac{u'(c_1(s))}{u'(c_0)} \;} \quad \Rightarrow \quad q_j = \sum_s \pi_s m_s a_{sj} = \mathbb{E}[m\, \tilde a_j]. \]
Euler equation in returns (for any traded \(j\)): \[ \boxed{\; 1 = \mathbb{E}[\, m\, R_j\, ] \;} \quad (\text{divide } q_j u'(c_0) = \beta\,\mathbb{E}[u'(c_1) a_j] \text{ by } q_j). \]
Risk-free relation: \(1 = \mathbb{E}[m\, R_f] = R_f\, \mathbb{E}[m] \Rightarrow R_f = 1/\mathbb{E}[m]\).

Interpreting Risk Premia via Covariance

Starting from the Euler equation \(\mathbb{E}[mR_j] = 1\) for any asset \(j\) and \(\mathbb{E}[mR_f] = 1\) for the risk-free asset:

Derivation:

Write the expectation of a product using covariance: \[\mathbb{E}[XY] = \mathbb{E}[X]\mathbb{E}[Y] + \mathrm{Cov}(X,Y)\]
Apply to \(\mathbb{E}[mR_j] = 1\): \[\mathbb{E}[m]\mathbb{E}[R_j] + \mathrm{Cov}(m,R_j) = 1\]
From the risk-free asset, \(\mathbb{E}[mR_f] = 1\) and since \(R_f\) is constant: \[\mathbb{E}[m] \cdot R_f = 1 \quad \Rightarrow \quad \mathbb{E}[m] = \frac{1}{R_f}\]
Substitute into step 2: \[\frac{1}{R_f}\mathbb{E}[R_j] + \mathrm{Cov}(m,R_j) = 1\]

Continuation

Multiply through by \(R_f\) and rearrange: \[\mathbb{E}[R_j] + R_f \cdot \mathrm{Cov}(m,R_j) = R_f\] \[\boxed{\mathbb{E}[R_j] - R_f = -R_f \cdot \mathrm{Cov}(m,R_j)}\]

Or equivalently, dividing by \(\mathbb{E}[m] = 1/R_f\): \[\boxed{\mathbb{E}[R_j] - R_f = -\frac{\mathrm{Cov}(m,R_j)}{\mathbb{E}[m]}}\]

Interpretation:

Assets that pay more in “bad” states (high \(m\)) have positive \(\mathrm{Cov}(m,R_j)\) and lower expected returns (insurance property).
Conversely, assets that pay in “good” states (low \(m\)) have negative covariance and require a risk premium.

Numeric Example: CRRA, Two States

Utility: \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\), with \(\gamma = 2\), \(\beta = 1\).
Probabilities: \(\pi = \left(\frac{1}{2}, \frac{1}{2}\right)\).
Consumption: \(c_0 = 1\), \(c_1 = \left(2, 1\right)\).
SDF: \(m_s = \beta \left(\frac{c_1(s)}{c_0}\right)^{-\gamma} = \left(\frac{c_1(s)}{c_0}\right)^{-2}\) \[\Rightarrow m = \left(\left(\frac{2}{1}\right)^{-2},\; \left(\frac{1}{1}\right)^{-2}\right) = \left(\frac{1}{4},\; 1\right)\]
Risk-free: \[\mathbb{E}[m] = \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 = \frac{1}{8} + \frac{1}{2} = \frac{5}{8} \quad \Rightarrow \quad R_f = \frac{1}{\mathbb{E}[m]} = \frac{8}{5}\]
Price an asset with state payoffs \(a = (1,2)\): \[ q = \mathbb{E}[m\, a] = \frac{1}{2} \cdot \frac{1}{4} \cdot 1 + \frac{1}{2} \cdot 1 \cdot 2 = \frac{1}{8} + 1 = \frac{9}{8} \] Expected payoff: \(\mathbb{E}[a] = \frac{1}{2} \cdot 1 + \frac{1}{2} \cdot 2 = \frac{3}{2}\). Expected gross return: \(\mathbb{E}[R] = \mathbb{E}\left[\frac{a}{q}\right] = \frac{1}{2} \cdot \frac{1}{9/8} + \frac{1}{2} \cdot \frac{2}{9/8} = \frac{1}{2}\left(\frac{8}{9} + \frac{16}{9}\right) = \frac{4}{3} < R_f = \frac{8}{5} = 1.6\) (pays more in state 2, the “worse” state with higher \(m\) ⇒ insurance ⇒ lower return).

Connections to Expected Utility and Risk Aversion

From decision making under uncertainty: agents evaluate lotteries \(\tilde a\) via \(\mathbb{E}[u(\tilde a)]\) (concave \(u\)).
In equilibrium, marginal utilities determine \(m\) (up to normalization), and risk aversion shapes state prices:
- “Bad” states (high marginal utility) have larger \(m_s\) and thus larger \(\psi_s\).
- Assets paying in bad states command higher prices (insurance role).

Worked Example: Solving for State Prices and the SDF

Let \(S=2\), \(J=2\) with

\[ A = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix},\quad q = \begin{bmatrix}0.9 \\ 1.4\end{bmatrix}. \]

NA implies \(q = A^{\top} \psi\). Since \(A\) has rank 2, solve for \(\psi\) uniquely: \[ A^{\top} = \begin{bmatrix}1 & 0 \\ 1 & 2\end{bmatrix},\;\; \Rightarrow\;\; \begin{cases} 0.9 = \psi_1, \\ 1.4 = \psi_1 + 2\psi_2 \end{cases} \Rightarrow (\psi_1,\psi_2) = (0.9, 0.25). \]
If \(\pi = (0.6,0.4)\), then \(m = (\psi_1/\pi_1,\psi_2/\pi_2) = (1.5, 0.625)\) and \(q_j = \mathbb{E}[m\, \tilde a_j]\).

Replication via Inversion (Complete Markets)

Given target payoff \(b\in\mathbb{R}^S\) and invertible \(A\), the unique replicating portfolio is \[\boxed{\; x = A^{-1} b \; }.\]
Price of \(b\): \(q(b) = q\cdot x = \psi^{\top} b\).
If \(J>S\) but \(\mathrm{rank}(A)=S\), replication exists and is non-unique; the price is still unique.

Exercise 1: Find an Arbitrage Strategy

States \(S=2\), assets \(J=3\) with \[ A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 0 & 2 \end{bmatrix},\quad q = \begin{bmatrix}0.9 \\ 0.4 \\ 0.9\end{bmatrix}. \]

Show there is no \(\psi\ge 0\) with \(q = A^{\top}\psi\) (hint: use the first two assets to infer \(\psi\), then check the third).
Construct an explicit arbitrage portfolio \(x\) with \(q\cdot x < 0\) and \(A x \ge 0\).

Solution sketch

From assets 1 and 2: \(q_1 = 0.9 = \psi_1 + \psi_2\) and \(q_2 = 0.4 = \psi_1 \Rightarrow \psi_1=0.4\), \(\psi_2=0.5\). Then asset 3 would imply \(q_3 = 2\psi_2 = 1.0 \neq 0.9\) ⇒ NA violated.
Replicate asset 3 using (1,2): solve \(A y = (0,2)^{\top}\) ⇒ \(y=(2,-2,0)\). Its price is \(q\cdot y = 1.0\).
Arbitrage: buy asset 3 and short its replicating portfolio \(x = e_3 - y = (-2,2,1)\).
- Cost: \(q\cdot x = -1.8 + 0.8 + 0.9 = -0.1 < 0\).
- Payoff: \(A x = 0\) (riskless free cash today) ⇒ arbitrage.

Exercise 2: Compute Arrow–Debreu Prices (State Prices)

Use the 2-state, 2-asset setup from the worked example: \[ A = \begin{bmatrix}1 & 1 \\ 0 & 2\end{bmatrix},\quad q = \begin{bmatrix}0.9 \\ 1.4\end{bmatrix}. \]

Verify \(\mathrm{rank}(A)=2\) (complete markets).
Solve \(q = A^{\top} \psi\) to obtain \(\psi\).
For \(\pi = (0.6,0.4)\), compute the SDF \(m=\psi/\pi\) and check \(q_j = \mathbb{E}[m\, \tilde a_j]\).

— Solution —

\(\psi = (0.9,0.25)\) (unique); \(m=(1.5,0.625)\); identities verify by direct substitution.

Exercise 3: Replicate a Target Payoff

With the same \(A\) as above, replicate \(b = (1,1.5)^{\top}\).

Compute \(x\) such that \(A x = b\) (use inversion or solve the linear system).
Compute its price \(q(b) = q\cdot x\) and verify \(q(b) = \psi^{\top} b\).

— Solution —

Solve \(\begin{cases} x_1 + x_2 = 1,\\ 2x_2 = 1.5\end{cases}\) ⇒ \((x_1,x_2)=(0.25,0.75)\).
Price: \(q\cdot x = 0.9\cdot 0.25 + 1.4\cdot 0.75 = 1.275\).
Also \(\psi^{\top} b = 0.9\cdot 1 + 0.25\cdot 1.5 = 1.275\).

Risk-Neutral Probabilities

Risk-neutral (or “pricing”) probabilities reweight physical probabilities to absorb investors’ risk preferences so that discounted expected payoffs under the new measure price assets.

Define the risk-neutral measure \(Q\) by \[\pi_s^{Q} = \frac{m_s\,\pi_s}{\mathbb{E}[m]},\qquad s=1,\dots,S,\] where \(m_s\) is the SDF and \(\mathbb{E}[m]=\sum_s \pi_s m_s\).
Then for any asset with payoff \(\tilde a\), \[q = \mathbb{E}[m\,\tilde a] = \mathbb{E}[m]\;\mathbb{E}_{Q}[\tilde a] = \frac{1}{R_f}\;\mathbb{E}_{Q}[\tilde a],\] since \(R_f = 1/\mathbb{E}[m]\).
Interpretation: under \(Q\) investors are effectively risk-neutral and assets are priced by discounting \(Q\)–expected payoffs at the risk-free rate. The change of measure shifts probability mass toward states that are valuable to investors (high \(m\)).

Example (from the worked example)

Use the earlier solved state prices \(\psi=(0.9,0.25)\) and physical probabilities \(\pi=(0.6,0.4)\), which give

\[m = \left(\frac{0.9}{0.6},\frac{0.25}{0.4}\right) = (1.5,0.625),\qquad \mathbb{E}[m]=0.6\cdot1.5+0.4\cdot0.625=1.15.\]

Compute the risk-neutral probabilities:

\[\pi^Q = \frac{m\circ\pi}{\mathbb{E}[m]} = \left(\frac{1.5\cdot0.6}{1.15},\frac{0.625\cdot0.4}{1.15}\right) \approx (0.783,0.217).\]

Price asset 1 with payoff \((1,0)\) under \(Q\):

\[\mathbb{E}_Q[\tilde a^{(1)}] = 0.783\cdot1 + 0.217\cdot0 = 0.783,\qquad q_1 = \frac{1}{R_f}\,\mathbb{E}_Q[\tilde a^{(1)}] = \mathbb{E}[m] \cdot 0.783 = 1.15\cdot0.783 = 0.9.\]

Similarly for asset 2 with payoff \((1,2)\) we get \(\mathbb{E}_Q[\tilde a^{(2)}] = 0.783 + 2\cdot0.217 = 1.217\), so \(q_2 = \mathbb{E}[m]\cdot 1.217 = 1.15\cdot 1.217 = 1.4\).

This shows risk-neutral probabilities shift mass toward the state(s) with high SDF and let us price by a single discounted expectation.