Asset Pricing Theory
Paris Dauphine University-PSL
The view about efficiency in financial markets accepted in the 60s and 70s states that assets quickly react to new information.
A portfolio is a linear combination of assets, using our notation, given \(N\) assets with payoffs \(X^{(1)}, X^{(2)}, \ldots, X^{(N)}\) where each \(X^{(i)}\) is a vector, a portfolio is defined by weights \(\theta = (\theta_1, \theta_2, \ldots, \theta_N)\).
The expected payoff of the portfolio in each state of nature is given by the weighted sum of the payoffs of the individual assets: \[ X^P(s) = \sum_{i=1}^N \theta_i X^{(i)}(s) \]
These weights do not need to sum to one when we think about payoffs but they will sum to one when we talk about returns.
An arbitrage opportunity is a portfolio with zero or negative price, but non-negative expected payoffs in all states of nature (and positive in at least one state). In other words, it is a risk-free profit opportunity that arises when the market misprices an asset or a portfolio of assets.
A formal definition of an arbitrage states that there exist weights \(\theta = (\theta_1, \theta_2, \ldots, \theta_N)\) such that:
The price of the portfolio is zero or negative: \[ P = \sum_{i=1}^N \theta_i P^{(i)} \leq 0 \]
The expected payoff of the portfolio is non-negative in all states of nature (and positive in at least one state): \[ X^P(s) = \sum_{i=1}^N \theta_i X^{(i)}(s) \geq 0 \quad \forall s \in S \]
Strictly positive payoff with positive probability for at least one state: \[ \mathbb{P}(X^P(s) > 0) > 0 \]
An arbitrage strategy is the solution to the following system of equations
\[ \begin{align} A\vec{\theta} &\geq 0 \\ \vec{p}^T \vec{\theta} &\leq 0 \\ 1^T A \vec{\theta} &\geq \epsilon \end{align} \]
where \(A\) is the matrix of payoffs, \(\vec{p}\) is the vector of prices, and \(\epsilon\) is a small positive number. This deals with the strict inequality imposed by condition 3.
Arbitrage portfolio: \(\theta=(1, -1)\).
Market Dynamics and Equilibrium:
We work in a one–period economy. Let \(\mathcal{X}\) be the linear space of payoffs at \(1\) (e.g. bounded random variables \(L^\infty(\Omega,\mathcal{F},\mathbb{P})\)).
Assets \(i=1,\dots,n\) have prices \(p_i\) and payoffs \(X_i \in \mathcal{X}\).
The pricing map on portfolios is linear: \[ f\!\left(\sum_i \theta_i X_i\right) = \sum_i \theta_i p_i. \]
Define the cone of zero-cost or negative payoffs: \[ \mathcal{C} = \Bigl\{\, Y=\sum_i \theta_i X_i : \sum_i \theta_i p_i \leq 0 \,\Bigr\}. \] This is the set of payoffs that can be obtained at zero or negative cost.
Economic idea:
Under NA, the set of non positive costs \(\mathcal{C}\) cannot produce strictly positive payoffs.
That is, \(\mathcal{C}\) intersects the positive cone of payoffs \(\mathcal{X}_+\) only at the origin: \[
\mathcal{C} \cap \mathcal{X}_+ = \{0\}.
\]
Mathematical idea:
Both \(\mathcal{C}\) and \(\mathcal{X}_+\) are convex.
If two convex sets intersect only at \(0\), the Separating Hyperplane Theorem guarantees the existence of a linear functional \(\varphi\) that “separates” them.
Result:
There exists a nonzero linear functional \(\varphi:\mathcal{X}\to\mathbb{R}\) such that \[
\varphi(Y) \leq 0 \quad \forall Y \in \mathcal{C},
\qquad
X \ge 0 \implies \varphi(X) \ge 0.
\]
Interpretation:
\(\varphi\) is a positive pricing rule: it prices every portfolio consistently with NA and vanishes on zero-cost trades.
Geometrically, \(\varphi\) is the “supporting hyperplane” separating feasible trades from arbitrage opportunities.
Step 2: Positive linear functionals on payoff spaces can be represented as expectations.
Theorem (Riesz / Yosida–Hewitt):
If \(\varphi:\mathcal{X}\to\mathbb{R}\) is positive and continuous on \(L^p\) (e.g. \(L^2\)), then there exists a nonnegative random variable \(M\) such that \[
\varphi(X) = \mathbb{E}[M X], \quad \forall X \in \mathcal{X}.
\]
Let \(S = \{s_1,\dots,s_n\}\) with probabilities \(\pi(s_i) > 0\).
Thus \(\psi\) is an expectation with respect to the random variable \(M\).
Step 1. Separation:
NA \(\implies\) \(\mathcal{C} \cap \mathcal{X}_+ = \{0\}\).
By the separating hyperplane theorem, there exists \(\varphi\) with \[
\varphi(Y) \leq 0 \;\;\forall Y \in \mathcal{C}, \qquad
X \ge 0 \implies \varphi(X) \ge 0.
\]
Step 2. Representation:
By the representation theorem,
\[
\varphi(X) = \mathbb{E}[M X]
\]
For any portfolio \(\theta\), \[ f\!\left(\sum_i \theta_i X^{(i)}\right) = \sum_i \theta_i p^{(i)}. \]
But \(f=\varphi\), so \[ \sum_i \theta_i p^{(i)} = \mathbb{E}\!\left[M \sum_i \theta_i X^{(i)}\right]. \]
Therefore, asset by asset: \[ p^{(i)} = \mathbb{E}[M X^{(i)}], \quad i=1,\ldots,N. \]
A market is complete if any payoff vector \(X \in \mathbb{R}^{|S|}\) can be replicated by a portfolio of traded assets.
Formally:
The span of traded payoffs \(\{X^{(1)}, \ldots, X^{(N)}\}\) is the whole space \(\mathbb{R}^{|S|}\).
Implication:
For any payoff \(Y=(Y(s_1), \ldots, Y(s_{|S|}))\), there exist weights \(\theta=(\theta_1,\ldots,\theta_N)\) such that \[
Y(s) = \sum_{i=1}^N \theta_i X^{(i)}(s), \quad \forall s \in S.
\]
Arrow–Debreu Securities:
An Arrow–Debreu security \(\delta^{(s)}\) pays \(1\) in state \(s\) and \(0\) in all other states: \[ \delta^{(s)}(s') = \begin{cases} 1 & \text{if } s'=s, \\ 0 & \text{otherwise.} \end{cases} \]
If markets are complete, each \(\delta^{(s)}\) can be replicated with some portfolio \(\theta^{(s)}\).
The price of \(\delta^{(s)}\) is the state price \(q(s)\).
Historical Note:
Stack payoffs in the matrix \(A \in \mathbb{R}^{|S|\times N}\) with entries \[ A_{s i} = X^{(i)}(s). \]
Completeness \(\iff \operatorname{rank}(A) = |S|\) (full row rank).
Then, for each state \(s\), there exists a portfolio \(\theta^{(s)}\) such that \[ A\,\theta^{(s)} = \delta^{(s)}. \]
Its price is the state price: \[ q(s) = \sum_{i=1}^N \theta^{(s)}_i \, p^{(i)}. \]
Any payoff \(X \in \mathbb{R}^{|S|}\) can be decomposed as \[ X = \sum_{s\in S} X(s)\, \delta^{(s)}. \]
By linearity of pricing: \[ P(X) = \sum_{s\in S} X(s)\, q(s). \]
Equivalently, in probability form: \[ P(X) = \sum_{s\in S} \pi(s)\, M(s)\, X(s), \quad \text{where } M(s) = \tfrac{q(s)}{\pi(s)}. \]
In Arrow-Debreu framework, the price of any asset is: \[ P(X) = \sum_{s \in S} X(s) \, q(s) \]
The risk-free asset pays \(1\) in all states, so its price is: \[ \sum_{s \in S} q(s) = \frac{1}{1 + r} \] where \(r\) is the risk-free rate.
Define risk-neutral probabilities: \[ \pi^{\mathbb{Q}}(s) = \frac{q(s)}{\sum_{s' \in S} q(s)} = q(s) (1 + r) \]
Then, the price becomes: \[ P(X) = \sum_{s \in S} X(s) \, q(s) = \sum_{s \in S} X(s) \, \frac{\pi^{\mathbb{Q}}(s)}{1 + r} = \frac{1}{1 + r} \sum_{s \in S} X(s) \, \pi^{\mathbb{Q}}(s) = \frac{1}{1 + r} \mathbb{E}^{\mathbb{Q}}[X] \]
This is the change of measure from physical probabilities \(\pi\) to risk-neutral probabilities \(\pi^{\mathbb{Q}}\).
Uniqueness in Complete Markets:
Suppose \(M\) and \(M'\) both price all traded assets: \[ p^{(i)} = \mathbb{E}[M X^{(i)}] = \mathbb{E}[M' X^{(i)}], \quad i=1,\ldots,N. \]
In state–price form: \(A^\top q = p\) and \(A^\top q' = p\),
where for each state \(s\), \[
q(s) = \pi(s)\,M(s),
\qquad
q'(s) = \pi(s)\,M'(s).
\]
Subtracting: \(A^\top (q-q') = 0\).
If markets are complete, \(\ker(A^\top)=\{0\}\) \(\;\Rightarrow\;\) \(q=q'\).
With \(\pi(s)>0\) for all \(s\), it follows that \(M(s) = M'(s)\).
Extension to Continuous States:
What do we need to extend these concepts to infinite dimensional spaces?
Define inner product:
\[
\langle X, Y \rangle = \sum_{s\in S} \pi(s)\,X(s)Y(s).
\]
Span of traded payoffs:
\[
\mathcal{S} = \{ A\theta : \theta \in \mathbb{R}^N \}.
\]
Any SDF \(M\) decomposes uniquely as
\[
M = M_{\parallel} + M_{\perp},
\quad M_{\parallel} \in \mathcal{S},\; M_{\perp} \in \mathcal{S}^\perp.
\]
For any traded payoff \(Y \in \mathcal{S}\),
\[
\mathbb{E}[M Y] = \mathbb{E}[M_{\parallel} Y].
\]
Key Insight: Only the projection of \(M\) onto \(\mathcal{S}\) matters for pricing traded assets.
Mathematical Decomposition: \[ M = M_{\parallel} + M_{\perp} \] where:
\(M_{\parallel} = \text{proj}_{\mathcal{S}}(M)\) is the orthogonal projection of \(M\) onto \(\mathcal{S}\)
\(M_{\perp} \in \mathcal{S}^{\perp}\) is the orthogonal component such that \(\langle M_{\perp}, Y \rangle = 0\) for any \(Y \in \mathcal{S}\)
Pricing Implications:
For any traded payoff \(Y \in \mathcal{S}\): \[ \langle M, Y \rangle = \langle M_{\parallel} + M_{\perp}, Y \rangle = \langle M_{\parallel}, Y \rangle + \underbrace{\langle M_{\perp}, Y \rangle}_{=0} = \langle M_{\parallel}, Y \rangle \]
Consequence: All SDFs with the same projection \(M_{\parallel}\) yield identical prices: \[ M_1 = M_{\parallel} + M_{\perp}^{(1)}, \quad M_2 = M_{\parallel} + M_{\perp}^{(2)} \quad \Rightarrow \quad \langle M_1, Y \rangle = \langle M_2, Y \rangle \text{ for all } Y \in \mathcal{S} \]
Asset Pricing Theory - Day 1