---
title: Fundamental Theorem of Asset Pricing
url: https://www.emergentmind.com/topics/fundamental-theorem-of-asset-pricing
type: topic
---

# Fundamental Theorem of Asset Pricing

The Fundamental Theorem of Asset Pricing (FTAP) characterizes the absence of arbitrage in financial markets by the existence of a suitable family of probability measures under which discounted asset prices are martingales. The FTAP provides the conceptual and mathematical foundation for risk-neutral pricing, super-hedging duality, and the analysis of market completeness versus incompleteness. The theorem’s structural form, technical hypotheses, and duality results have been generalized to accommodate model uncertainty, bid-ask spreads, and static hedging in options markets. In modern treatments, robustness to model ambiguity and the presence of trading frictions are systematically addressed.

## 1. Market Framework and Model Uncertainty

Consider a discrete-time financial market on a measurable space \((\Omega,\mathcal{F})\) with time indices \(t = 0,\ldots,T\). The primary risky assets are modeled as a Borel-measurable \(\mathbb{R}^d\)-valued process \(S = (S^1_t, \ldots, S^d_t)_{t=0}^T\). 

A finite family of European-style options \(g = (g^1,\ldots,g^e)\), each \(g^i:\Omega\to\mathbb{R}\), are available for static trading at initial time, with associated bid (\(\underline{g}\)) and ask (\(\overline{g}\)) prices:
\[
\underline{g} = (\underline{g}^1,\ldots,\underline{g}^e), \qquad \overline{g} = (\overline{g}^1,\ldots,\overline{g}^e), \qquad \underline{g}\leq \overline{g} \text{ (componentwise)}.
\]
Dynamic trading occurs in the stocks, with the set \(\mathcal{H}\) of predictable, \(\mathbb{R}^d\)-valued portfolios \(H = (H_t)_{t=0}^{T-1}\). The cumulative trading gain is
\[
(H\cdot S)_T = \sum_{t=0}^{T-1} H_t\cdot(S_{t+1} - S_t).
\]
Static option positions \(h = (h^1,\ldots,h^e) \in \mathbb{R}^e\) enter at time zero, with asymmetric initial costs due to bid-ask spreads, and yield payoffs
\[
h^+(g - \overline{g}) - h^-(g - \underline{g}), \quad \text{where } h^+, h^-\text{ are the positive and negative parts of } h.
\]
The overall terminal wealth for the semi-static strategy \((H,h)\) with zero initial capital is
\[
(H\cdot S)_T + h^+(g - \overline{g}) - h^-(g - \underline{g}).
\]
Model uncertainty is represented by a convex, possibly nondominated collection \(\mathcal{P}\) of probability measures on \((\Omega,\mathcal{F})\). Properties are said to hold \(\mathcal{P}\)-quasi surely (q.s.) if they fail only on a \(\mathcal{P}\)-polar set, i.e., a set that is null under every \(P\in\mathcal{P}\) [1309.2728].

## 2. Robust No-Arbitrage under Bid-Ask Spreads

**Classical No-Arbitrage** (\(\mathrm{NA}(\mathcal{P})\)):  
No-arbitrage is defined as follows: any semi-static strategy \((H,h)\) that is non-negative \(\mathcal{P}\)-q.s. in terminal payoff must in fact be exactly zero \(\mathcal{P}\)-q.s.:
\[
(H\cdot S)_T + h^+(g-\overline{g}) - h^-(g-\underline{g}) \geq 0 \ \mathcal{P}\text{-q.s.} \implies
(H\cdot S)_T + h^+(g-\overline{g}) - h^-(g-\underline{g}) = 0 \ \mathcal{P}\text{-q.s.}
\]

**Robust No-Arbitrage** (\(\mathrm{NA}^r(\mathcal{P})\)):  
A more stringent notion, required under bid-ask spreads, states that there exist strictly tighter quotes
\[
[\underline{g}', \overline{g}'] \subset \operatorname{ri}[\underline{g},\overline{g}]
\]
(with strict inequalities in nontrivial cases), such that \(\mathrm{NA}(\mathcal{P})\) holds for these perturbed spreads.

When all bid and ask prices coincide, \(\mathrm{NA}^r(\mathcal{P}) \iff \mathrm{NA}(\mathcal{P})\). However, when spreads are present, this robustification eliminates "arbitrage at the boundary" [1309.2728].

**Non-Redundancy Assumption:**  
If a hedging option with nonzero spread is not replicable by stock trading together with static positions in the other options (\(g^i\) is non-redundant), then \(\mathrm{NA}(\mathcal{P})\) alone implies \(\mathrm{NA}^r(\mathcal{P})\).

## 3. Fundamental Theorem of Asset Pricing: Main Statement

**Fundamental Theorem (Robust version):**  
Suppose all payoffs \(|g^i|\leq\varphi\) for some Borel \(\varphi\geq1\). Define the set of martingale measures calibrated to the (bid–ask) quotes:
\[
M := \left\{ Q:\; S \text{ is a }Q\text{-martingale},\; E_Q[g^i]\in[\underline{g}^i,\overline{g}^i],\ i=1,\ldots,e \right\}.
\]
Then:
\[
\mathrm{NA}^r(\mathcal{P}) \iff M\neq\emptyset,
\]
i.e., robust no-arbitrage holds if and only if there exists a probability measure \(Q\) under which the stocks are martingales and all hedging options are correctly priced inside their bid-ask intervals [1309.2728].

## 4. Super-Hedging Duality and Attainable Claims

**Super-Hedging Price:**  
Given a (Borel) contingent claim \(f\) with \(|f|\le\varphi\),
\[
\pi^*(f) := \inf\left\{ x\in \mathbb{R}: \exists (H,h)\in\mathcal{H}\times\mathbb{R}^e,\, x + (H\cdot S)_T + h^+(g-\overline{g}) - h^-(g-\underline{g}) \geq f \ \mathcal{P}\text{-q.s.} \right\}.
\]

**Duality Theorem:**  
Under \(\mathrm{NA}^r(\mathcal{P})\),
\[
\pi^*(f) = \sup_{Q\in M} E_Q[f],
\]
and the supremum is also equal to the supremum over all \(Q\) for which \(E_Q[g]\in \operatorname{ri} [\underline{g}, \overline{g}]\). There exists an optimal semi-static strategy attaining the infimum.

**Closedness of the Attainable Set:**  
Let
\[
\mathcal{C} =
\left\{
(H\cdot S)_T + h^+(g - \overline{g}) - h^-(g - \underline{g}) : (H,h)\in\mathcal{H}\times\mathbb{R}^e
\right\} 
- \mathcal{L}_+^0,
\]
where \(\mathcal{L}_+^0\) is the set of non-negative Borel functions. Under \(\mathrm{NA}^r(\mathcal{P})\), \(\mathcal{C}\) is closed in the topology of \(\mathcal{P}\)-q.s. convergence.

The closedness is nontrivial because static options with bid-ask spreads create a non-linear, asymmetric set of payoffs, preventing the use of classical Komlós-type arguments applicable to linear cones.

## 5. Key Proof Techniques

**Equivalence in the FTAP:**  
- If \(M\neq\emptyset\) (existence of a calibrated martingale measure), any semi-static strategy with non-negative \(\mathcal{P}\)-q.s. payoff cannot yield an arbitrage since every such \(Q\) precludes it by linearity and positivity of expectation.
- Conversely, under robust no-arbitrage, separating hyperplane techniques (via convex analysis) are used to construct an admissible \(Q\); the strict "robustness" of NA^r ensures that one can select \(Q\) with expectations inside the bid-ask intervals, preventing approximate replication of spread options.

**Super-Hedging Duality:**  
- The closedness of \(\mathcal{C}\) ensures existence of an optimal hedging strategy.
- A duality argument (separation of convex sets) gives equality between the super-hedging price and the supremum over calibrated martingale measures.

**Measurability and Convexity:**  
- The framework relies on Borel measurability of all processes and claims, and on the availability of universally measurable selectors for extracting convergent subsequences.
- The convexity of the set \(M\) of martingale measures is essential, as is direct convex separation (Krein–Milman type arguments are avoided due to non-dominated \(\mathcal{P}\)).

## 6. Generalizations and Context

The robust FTAP in [1309.2728] encompasses both frictionless and bid-ask markets, multi-asset settings, and non-dominated model uncertainty. Non-redundancy of spread options restores equivalence between classical and robust no-arbitrage, ensuring that no artificial arbitrage can arise from price boundary effects.

The duality results and closedness properties enable a comprehensive analysis of super-hedging under minimal assumptions. These advances connect to a broader literature on robust pricing, model-free approaches, and super-replication in incomplete and uncertain markets. The use of semi-static hedging—mixing dynamic trading of the underlying with static option positions—models realistic markets with transaction costs and a wide menu of traded derivatives.

This robust FTAP framework serves as a canonical reference in the literature of model uncertainty and transaction costs, both for theoretical extensions (including dynamic or pathwise approaches) and for applications to numerical and empirical asset price systems [1309.2728].

Source: https://www.emergentmind.com/topics/fundamental-theorem-of-asset-pricing