The Geometry of the Payoff Space

This notebook develops the geometric machinery underlying the theory of the stochastic discount factor. We work throughout in a finite-dimensional probability space, where every random variable is simply a vector of state-contingent payoffs and expectation defines an inner product. This inner-product structure turns the space of payoffs into a Euclidean space, making geometric tools available for economic problems.

Three results organize the notebook. First, we characterize projections: the projection of a payoff y onto a subspace M is the element of M closest to y, and its residual is orthogonal to every element of M. This is the geometric content of linear regression. Second, we establish the Cauchy-Schwarz inequality, which bounds the correlation between any two payoffs and underlies the Hansen-Jagannathan volatility bound. Third, we prove the Riesz representation theorem: every linear pricing functional on the payoff space can be written as an inner product with a unique random variable. That random variable is the stochastic discount factor.

Probability Structure

Uncertainty is represented by a finite set \mathcal{S} = \{1, \ldots, S\} of states, defining a finite probability space (\mathcal{S}, q). The set of all random variables defined in \mathcal{S} is denoted by L and is called the payoff space. Thus, for any x \in L we have that x = \begin{bmatrix} x(1) \\ x(2) \\ \vdots \\ x(S) \end{bmatrix} \in \mathbb{R}^{S} defines all the possible payoffs in each state, and the probability of getting a payoff in a particular state is given by \operatorname{P}(x = x(s)) = q(s) for all s \in \mathcal{S}. We assume throughout that q(s) > 0 for all s \in \mathcal{S}, that is, we will not consider possible outcomes that happen with probability zero.

The payoff space is clearly a linear vector space since for any x, y \in L and \alpha, \beta \in \mathbb{R} we have that \alpha x + \beta y \in L. We endow the payoff space with an inner product \langle{\cdot, \cdot}\rangle: L \times L \rightarrow \mathbb{R} defined such that for any x, y \in L, we have that \langle{x, y}\rangle = \operatorname{E}(xy) = \sum_{s = 1}^{S} q(s)x(s) y(s). In finite-dimensional spaces, we can use the inner product to define the Euclidean norm \lVert{\cdot}\rVert: L \rightarrow \mathbb{R}^{+} for all x \in L as \lVert{x}\rVert = \sqrt{\langle{x, x}\rangle}. Clearly, \lVert{x}\rVert = 0 \Leftrightarrow x = 0. The second moment of x, defined as \lVert{x}\rVert^{2} = \operatorname{E}(x^{2}), measures the mean squared payoff across states and serves as the natural notion of distance between payoffs: \lVert{y - z}\rVert is small when y and z are close in every state on average.

Note that the inner product can be related to standard statistical moments. For any x, y \in L, we have \langle{x, y}\rangle = \operatorname{E}(xy) = \operatorname{Cov}(x, y) + \operatorname{E}(x)\operatorname{E}(y), where \operatorname{Cov}(x, y) = \operatorname{E}[(x - \operatorname{E}(x))(y - \operatorname{E}(y))] denotes the covariance between x and y. In particular, \lVert{x}\rVert^{2} = \operatorname{V}(x) + [\operatorname{E}(x)]^{2}, where \operatorname{V}(x) = \operatorname{Cov}(x, x) is the variance of x.

Since L consists of all random variables on \mathcal{S}, it is a vector space of dimension S. A natural basis is provided by the Arrow-Debreu securities \{e_{1}, e_{2}, \ldots, e_{S}\}, where e_{s}(i) = \begin{cases} 1 & \text{if } i = s, \\ 0 & \text{otherwise,} \end{cases} for each s \in \mathcal{S}. The security e_{s} pays one unit in state s and zero in every other state. Any payoff x \in L can be expressed in this basis as x = \sum_{s=1}^{S} x(s)\, e_{s}, since in state i the right-hand side evaluates to x(i). Under the inner product defined above, the Arrow-Debreu securities are mutually orthogonal: for s \neq t, the product e_{s} e_{t} is zero in every state, so \langle{e_{s}, e_{t}}\rangle = \operatorname{E}(e_{s} e_{t}) = 0. Moreover, \langle{e_{s}, e_{s}}\rangle = \operatorname{E}(e_{s}^{2}) = q(s), so the Arrow-Debreu securities form an orthogonal—though not orthonormal—basis for L.

Projections

Given x, y \in L, consider the vectors y_{x} = \alpha x and z = y - y_{x}. We say that y_{x} is the projection of y on the subspace generated by \{x\} if the norm of z is minimal. To obtain the projection, we need to compute the \alpha that minimizes \lVert{z}\rVert^{2} = \lVert{y - \alpha x}\rVert^{2} = \operatorname{E}[(y - \alpha x)^{2}]. The first-order condition of this problem is: 0 = \operatorname{E}[(y - \alpha x) x] = \langle{y - \alpha x, x}\rangle = \langle{z, x}\rangle, which implies that \alpha = \dfrac{\langle{x, y}\rangle}{\langle{x, x}\rangle} and \langle{z, y_{x}}\rangle = 0. Statistically, \alpha is the population regression coefficient of y on x with no intercept: it is the unique scalar that makes the residual z = y - \alpha x uncorrelated with x.

We say that two vectors x, y \in L are orthogonal if their inner product is equal to zero. Thus, we have that y_{x} \mathrel\bot z, implying that the vector y can be decomposed into two orthogonal components. Indeed, we have that \lVert{y}\rVert^{2} = \lVert{z + y_{x}}\rVert^{2} = \lVert{z}\rVert^{2} + 2 \langle{z, y_{x}}\rangle + \lVert{y_{x}}\rVert^{2} = \lVert{z}\rVert^{2} + \lVert{y_{x}}\rVert^{2}, which is a generalization of the classical Pythagorean theorem.

Property 1 (Orthogonal Decomposition) Given x, y \in L, the projection of y on the subspace generated by \{x\} is given by y_{x} = \dfrac{\langle{x, y}\rangle}{\langle{x, x}\rangle} x. The vector z = y - y_{x} is orthogonal to y_{x}, implying that \lVert{y}\rVert^{2} = \lVert{z}\rVert^{2} + \lVert{y_{x}}\rVert^{2}. \tag{1}

Equation (1) implies that \Vert y \Vert^{2} \geq \Vert y_{x} \Vert^{2}, with equality occurring whenever y is proportional to x. Therefore, we have that \lVert{y}\rVert^{2} \geq \lVert{y_{x}}\rVert^{2} = \left\lVert \frac{\langle{x, y}\rangle}{\langle{x, x}\rangle} x \right\rVert^{2} = \frac{\langle{x, y}\rangle^{2}}{\lVert{x}\rVert^{2}}. The previous expression is known as the Cauchy-Schwartz inequality and is fundamental in the study of Euclidean vector spaces.

Property 2 (Cauchy-Schwartz Inequality) Given x, y \in L we have that |\langle{x, y}\rangle| \leq \lVert{x}\rVert \lVert{y}\rVert. \tag{2}

The Cauchy-Schwartz inequality implies that, for any two nonzero x, y \in L, the ratio \dfrac{\langle{x, y}\rangle}{\lVert{x}\rVert\lVert{y}\rVert} lies in [-1, 1] and can be interpreted as the cosine of the angle \theta \in [0, \pi] between x and y. Specifically, |\langle{x, y}\rangle| = \lVert{x}\rVert\lVert{y}\rVert if and only if y is proportional to x, i.e., the two vectors are collinear. When \operatorname{E}(x) = \operatorname{E}(y) = 0, the inner product reduces to \langle{x, y}\rangle = \operatorname{Cov}(x, y), and the ratio \dfrac{\langle{x, y}\rangle}{\lVert{x}\rVert\lVert{y}\rVert} coincides with the correlation coefficient between x and y.

Projection onto a Subspace

The projection result established in Property 1 extends naturally to higher-dimensional subspaces. Let M \subseteq L be a subspace with linearly independent basis \{x_{1}, x_{2}, \ldots, x_{N}\}, and collect the basis elements into the random vector \mathbf{x} = (x_{1}, \ldots, x_{N})'. We say that y_{M} \in M is the projection of y \in L onto M if it minimizes \lVert{y - z}\rVert over all z \in M.

Since any z \in M takes the form \mathbf{a}'\mathbf{x} for some \mathbf{a} \in \mathbb{R}^{N}, the projection solves \min_{\mathbf{a} \in \mathbb{R}^{N}} \lVert{y - \mathbf{a}'\mathbf{x}}\rVert^{2} = \min_{\mathbf{a} \in \mathbb{R}^{N}} \operatorname{E}\left[(y - \mathbf{a}'\mathbf{x})^{2}\right]. The first-order condition with respect to \mathbf{a} yields \operatorname{E}\left[(y - \mathbf{a}'\mathbf{x})\mathbf{x}\right] = \operatorname{E}(y\mathbf{x}) - \operatorname{E}(\mathbf{x}\mathbf{x}')\mathbf{a} = \mathbf{0}. The Gram matrix G = \operatorname{E}(\mathbf{x}\mathbf{x}') has (i,j) entry \operatorname{E}(x_i x_j), the second cross-moment of the i-th and j-th basis payoffs; it is positive definite whenever the basis payoffs are linearly independent. Provided G is invertible, the unique minimizer is \mathbf{a} = G^{-1}\operatorname{E}(y\mathbf{x}), so y_{M} = \mathbf{a}'\mathbf{x} = \operatorname{E}(y\mathbf{x})'G^{-1}\mathbf{x}. The residual z = y - y_{M} satisfies \langle{z, x_{i}}\rangle = 0 for every i = 1, \ldots, N. By linearity, this extends to \langle{z, x}\rangle = 0 for every x \in M, and the Pythagorean identity becomes \lVert{y}\rVert^{2} = \lVert{y_{M}}\rVert^{2} + \lVert{z}\rVert^{2}.

Property 3 (Projection onto a Subspace) Let M \subseteq L be a subspace with linearly independent basis \{x_{1}, \ldots, x_{N}\}, let \mathbf{x} = (x_{1}, \ldots, x_{N})', and assume the Gram matrix G = \operatorname{E}(\mathbf{x}\mathbf{x}') is invertible. For any y \in L, the projection onto M is y_{M} = \operatorname{E}(y\mathbf{x})'G^{-1}\mathbf{x}. The residual z = y - y_{M} satisfies \langle{z, x}\rangle = 0 for all x \in M, and \lVert{y}\rVert^{2} = \lVert{y_{M}}\rVert^{2} + \lVert{z}\rVert^{2}.

The set of all payoffs in L that are orthogonal to M is called the orthogonal complement of M: M^{\mathrel\bot} = \{z \in L : \langle{z, x}\rangle = 0 \text{ for all } x \in M\}. Since M^{\mathrel\bot} is closed under addition and scalar multiplication, it is itself a subspace of L. The projection theorem implies that every y \in L admits the unique decomposition y = y_{M} + z with y_{M} \in M and z \in M^{\mathrel\bot}, so M \cap M^{\mathrel\bot} = \{0\} and L = M \oplus M^{\mathrel\bot}. This orthogonal direct sum decomposition is the key geometric fact underlying the structure of stochastic discount factors: as we will see in the next notebook, any valid SDF m decomposes as m = x^{*} + e, where x^{*} \in X is the projection of m onto the subspace of traded payoffs and e \in X^{\mathrel\bot} is the orthogonal residual.

Linear Functionals

A central object in asset pricing is a pricing functional that assigns a price to every traded payoff. Under the law of one price, such a functional must be linear: the price of a portfolio equals the sum of the prices of its components. Understanding when and how linear functionals can be represented will therefore be essential for characterizing stochastic discount factors.

Given x, y \in L and \alpha, \beta \in \mathbb{R}, a linear functional f: L \rightarrow \mathbb{R} satisfies f(\alpha x + \beta y) = \alpha f(x) + \beta f(y).

We say that the linear functional f : L \rightarrow \mathbb{R} is bounded if |f(x)| \leq M \lVert{x}\rVert for all x \in L. In other words, the absolute value of the functional cannot grow infinitely for a finite x. A bounded linear functional is also called a continuous linear functional. The smallest M for which this inequality remains true is called the norm of f, i.e., \lVert{f}\rVert = \inf \{M: |f(x)| \leq M \lVert{x}\rVert, \text{ for all } x \in L\}.

For a given m \in L and any x \in L, the functional f(x) = \langle{m, x}\rangle = \operatorname{E}(m x) = \sum_{s = 1}^{S} q(s)m(s) x(s) is linear since f(\alpha x + \beta y) = \operatorname{E}(m (\alpha x + \beta y)) = \alpha \operatorname{E}(m x) + \beta \operatorname{E}(m y) = \alpha f(x) + \beta f(y). Furthermore, the Cauchy-Schwartz inequality implies that |f(x)| = |\langle{m, x}\rangle| \leq \lVert{m}\rVert \lVert{x}\rVert, showing that the linear functional f is bounded and hence continuous. Since the previous inequality is an equality whenever x is proportional to m, we have that \lVert{m}\rVert is the smallest bound of f, showing that \lVert{f}\rVert = \lVert{m}\rVert.

Conversely, consider a linear functional f: L \rightarrow \mathbb{R}. Its kernel K = \{x \in L: f(x) = 0\} is a subspace of codimension one, so K^{\mathrel\bot} is one-dimensional: it is spanned by some nonzero z satisfying \langle{x, z}\rangle = 0 for all x \in K. Without loss of generality, assume that z has been appropriately scaled so that f(z) = 1.

Given any x \in L, we have that x - f(x) z \in K since f(x - f(x) z) = f(x) - f(x) f(z) = 0. Moreover, z \mathrel\bot K, implying that 0 = \langle{x - f(x) z, z}\rangle = \langle{x, z}\rangle - f(x) \langle{z, z}\rangle. The previous expression implies that f(x) = \frac{\langle{x, z}\rangle}{\langle{z, z}\rangle} = \langle{x, m}\rangle, where m = \dfrac{z}{\lVert{z}\rVert^{2}}. The previous analysis is an important result known as the Riesz representation theorem.

Property 4 (Riesz Representation Theorem) If f: L \rightarrow \mathbb{R} is a bounded linear functional, there exists a unique vector m \in L such that for all x \in L, f(x) = \langle{m, x}\rangle. Furthermore, we have \lVert{f}\rVert = \lVert{m}\rVert and every m determines a unique bounded linear functional.

The Riesz representation theorem has a direct and important consequence for asset pricing. Suppose that p: L \rightarrow \mathbb{R} is a linear pricing functional satisfying the law of one price. Then the theorem guarantees the existence of a unique m \in L such that p(x) = \langle{m, x}\rangle = \operatorname{E}(mx) for every payoff x \in L. The random variable m is called a stochastic discount factor, and characterizing its properties—positivity, uniqueness, and variance bounds—is the central task of asset pricing theory.