Duality and Asset Pricing

The duality framework from Reflections on Duality maps onto asset pricing with very little friction. The payoff space is the primal space, the stochastic discount factor is the dual element, and the pricing formula is the bilinear pairing. Most of the central results in theoretical asset pricing — the Hansen-Jagannathan bound, the existence of risk-neutral measures, the non-uniqueness of the SDF in incomplete markets — are direct consequences of the Hilbert space machinery once the identification is made.

The Setup

Fix a probability space \((\Omega, \mathcal{F}, P)\) representing states of the world. The primal space is \(L^2(P)\), the space of square-integrable payoffs — random variables representing what securities deliver across states. This is a Hilbert space under the inner product

A pricing functional is a map \(\pi: X \to \mathbb{R}\) assigning a price to each payoff in some marketed subspace \(X \subseteq L^2\). No-arbitrage requires \(\pi\) to be linear and positive: non-negative payoffs get non-negative prices, strictly positive payoffs get strictly positive prices.

The SDF as Dual Element

Since \(L^2\) is a Hilbert space, \((L^2)^* \cong L^2\) via Riesz representation. Every continuous linear functional on \(L^2\) is represented by integration against some element of \(L^2\). Applied to the pricing functional:

for some \(m \in L^2\). This \(m\) is the stochastic discount factor (SDF) / pricing kernel / state price density. It is the Riesz representer of the pricing functional — the dual element whose inner product with any payoff produces its price.

The SDF is not an additional modeling construct. It is the weighting scheme that reweights each state \(\omega\) by \(m(\omega)\) before expectation aggregates. This is “weighing before summing” — exactly the structure identified in the general duality framework.

The Geometry

The pricing functional \(\pi\) has level sets \(\{x \in L^2 : \pi(x) = c\}\) for each price level \(c\). These are parallel hyperplanes in \(L^2\) — translates of \(\ker(\pi) = \{x : E[mx] = 0\}\). The SDF \(m\) is the normal vector to this family: the direction in \(L^2\) along which price increases most rapidly.

The zero-price hyperplane \(\ker(\pi)\) is the set of payoffs orthogonal to \(m\). For an excess return \(R^e\) (a difference of two returns on the same investment), the pricing condition \(\pi(R^e) = 0\) becomes \(E[mR^e] = 0\), i.e., \(R^e \perp m\). Excess returns lie in the kernel of the pricing functional — the hyperplane orthogonal to the SDF.

A terminological note: “pricing kernel” in finance refers to \(m\) (the normal vector, the representer), while “kernel” in linear algebra refers to \(\ker(\pi)\) (the null space, the hyperplane). These are orthogonal complements of each other. The finance usage follows the integral-kernel convention (\(m\) is the kernel of the integral \(\int m \cdot x \, dP\)), not the null-space convention.

What Each Duality Result Gives You

Riesz representation \(\to\) SDF existence. The fundamental theorem of asset pricing (no-arbitrage \(\to\) existence of a positive linear pricing functional) combined with Riesz representation gives the SDF. A continuous linear functional on a Hilbert space has a concrete representer. In asset pricing: prices are inner products with the SDF.

Cauchy-Schwarz \(\to\) Hansen-Jagannathan bound. For any excess return \(R^e\) with \(E[mR^e] = 0\):

Rearranging:

The Sharpe ratio of any excess return is bounded by the standard deviation of the SDF. This is Cauchy-Schwarz applied to the \(L^2\) inner product — a bound on the action of a linear functional in terms of the norm of its representer. The HJ bound is not an independent result; it is the operator-norm inequality read through the asset pricing dictionary.

Projection theorem \(\to\) minimum-variance SDF. If markets are incomplete, the pricing functional is defined only on the marketed subspace \(X \subset L^2\). Many elements of \(L^2\) can represent it on \(X\). Among all valid SDFs, the one with minimum \(L^2\)-norm (equivalently, minimum variance, since \(E[m]\) is pinned by the risk-free rate) is the orthogonal projection of any valid SDF onto \(X\). This is the unique Riesz representer within \(X\) — the minimum-norm solution. The Hilbert space projection theorem does the work.

Hahn-Banach \(\to\) multiple SDFs in incomplete markets. On the marketed subspace \(X\), the pricing functional is determined by data. Hahn-Banach guarantees extension to all of \(L^2\), but the extension is not unique. Each valid extension is a different SDF; each SDF corresponds to a different risk-neutral measure. The set of valid SDFs is an affine subspace: a translate of \(X^\perp\). In complete markets \(X = L^2\), so \(X^\perp = \{0\}\) and the SDF is unique.

Radon-Nikodym \(\to\) risk-neutral measure. Define a new measure \(Q\) by

where \(R_f = 1/E[m]\) is the gross risk-free rate. The pricing formula becomes

The risk-neutral measure \(Q\) has Radon-Nikodym derivative equal to the normalized SDF. This is the duality-theory connection in its measure-theoretic form: the SDF is a density in the Radon-Nikodym sense — it converts the physical measure \(P\) into the pricing measure \(Q\). No-arbitrage requires \(m > 0\) a.s., which gives \(Q \sim P\) (mutual absolute continuity): the two measures agree on which events are negligible. This is stronger than \(Q \ll P\) alone, and rules out states that the physical measure considers possible but the pricing measure ignores.

The Correspondence

The One Thing Duality Doesn’t Give You

Pure duality theory imposes no sign constraint on the representer. No-arbitrage does — it requires \(m > 0\) a.s. This positivity is an economic condition layered on top of the functional analysis. It is what ensures the change of measure is well-behaved (mutual absolute continuity, not just absolute continuity) and what makes the fundamental theorem of asset pricing a genuine theorem rather than a corollary of Riesz representation: the Riesz representer always exists; requiring it to be strictly positive is the financial content.