Fundamental Theorem of Asset Pricing

• The Fundamental Theorem of Asset Pricing is a key result that defines arbitrage absence via the existence of calibrated martingale measures under model uncertainty.
• It employs duality techniques and convex analysis to validate super-hedging strategies even in markets with bid-ask spreads and trading frictions.
• Robust generalizations extend classical pricing methods to incomplete markets by integrating static option trading with dynamic stock portfolios.

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 $P\in\mathcal{P}$ (Bayraktar et al., 2013).

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" (Bayraktar et al., 2013).

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 (Bayraktar et al., 2013).

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 (Bayraktar et al., 2013) 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 (Bayraktar et al., 2013).