NCFL in Rough Path Finance

• NCFL condition is a formal criterion ensuring absence of arbitrage by requiring that replicable trading gains have a zero mean under any equivalent measure.
• Its rough path formulation links admissible trading strategies with Gaussian unbiased integrators, effectively constraining price dynamics to semimartingales.
• Structural results using tools like Hermite polynomials demonstrate that maximal admissibility forces the market model to coincide with the Itô rough path lift of Brownian motion.

The No Controlled Free Lunch (NCFL) condition formalizes the absence of arbitrage opportunities in continuous-time, frictionless financial market models, particularly under generalizations provided by rough path theory. It ensures that, even when admissible trading strategies are highly flexible and controlled, markets retain an intrinsic constraint: replicable gains from trading must have mean zero under any equivalent measure. In its strongest formulation, the NCFL condition tightly characterizes the form of allowed asset price dynamics and directly connects fundamental no-arbitrage principles to probabilistic and analytic properties of stochastic integrators.

1. Rough Kreps–Yan Theorem

The Rough Kreps–Yan theorem generalizes the classical no-arbitrage–with–vanishing–risk concept to rough stochastic integrators. In classical asset pricing, the Kreps–Yan theorem states that the absence of arbitrage with vanishing risk is equivalent to the existence of an equivalent martingale measure. In the rough framework, this is extended as follows: a rough market model (the price process as a rough path $\mathbf{S}$) satisfies the NCFL condition if and only if there exists an equivalent measure $\mathbb{Q}$ such that

$$\frac{d\mathbb{Q}}{d\mathbb{P}} \in L^q_+ \quad \text{and} \quad \mathbb{E}_{\mathbb{Q}}\Big[\int_0^{\tau} Y_t\, d\mathbf{S}_t\Big] = 0\quad \forall\, Y \in \mathcal{H},\, \forall\ \tau \in \mathfrak{T}_T.$$

Here, $\mathcal{H}$ is the linear convex class of admissible controlled portfolios and $\mathfrak{T}_T$ is a collection of times (possibly stopping times). This links market no-arbitrage directly to an unbiasedness property: the rough integrator—when integrated against any admissible strategy—produces gain processes with zero mean under $\mathbb{Q}$.

2. Classification of Unbiased Rough Integrators

A central technical result is the complete classification of rough path lifts (integrators) that satisfy unbiasedness for various classes of controlled integrands.

• For Markovian-type integrands (polynomial functions and their derivatives), unbiasedness forces the integrator to be a Gaussian–Hermite rough path. In one dimension, this takes the form

$$\mathbf{X}_{s,t} = \big(1,\, X_{s,t},\, \mathbb{X}^2_{s,t},\, \dots,\, \mathbb{X}^{k}_{s,t}\big),$$

with higher-level increments specified by Hermite polynomials:

$\mathbb{X}^n_{s,t} = H_n(X_{s,t}, G_{s,t}),$

where $H_n$ is the $n$th Hermite polynomial and $$G_{s,t} = -\frac{1}{2}\big(\mathbb{E}[X_t^2] - \mathbb{E}[X_s^2]\big)$$.

• Enlarging the controlled strategy class to include signature-type portfolios (which encode pathwise historical information in terms of iterated integrals), the unbiasedness property forces the rough integrator to be infinitesimally close to the Itô rough path lift of a standard Brownian motion, possibly up to a time change. In the presence of all Markovian and signature-type strategies, the driver must satisfy additional Chen–Hermite balancing conditions, reproducing Brownian-increment structure.

3. Nature of Admissible Trading Strategies

Controlled portfolios are the admissible integrands in the rough integration framework, determined by Taylor-like local expansions and adapted to the price driver. The main classes considered are:

• Markovian strategies: Functions of current price state, e.g., $Y_t = (P(X_t), D P(X_t), \dots)$
• Signature-based strategies: Functions of path signatures, which are multilinear expressions dependent on the history of price increments.
• Enlarging the admissible class—e.g., by including simple strategies—imposes stronger restrictions on the rough integrator, eventually enforcing the classic semimartingale structure.

The theoretical results show that the richer the class of admissible trading strategies, the narrower the class of allowable integrators under the NCFL constraint.

4. Gaussianity as a Consequence of No-Arbitrage

A striking implication is that the NCFL condition does not merely assume Gaussianity of the noise process; rather, it derives Gaussianity as a necessary consequence. Specifically, the unbiasedness demand leads to a moment generating function of the price increments consistent with a normal law:

$$\mathbb{E}\left[e^{u X_{s,t} + G_{s,t} u^2}\right] = \exp(G_{s,t} u^2),$$

the exponentiated quadratic form characteristic of Gaussian random variables, with $X_{s,t} \sim \mathcal{N}(0, -2G_{s,t})$. In the rough path setting, this conclusion is robust under time changes and additional constraints arising from signature-type portfolios. In summary, Gaussianity is not a modeling assumption but a direct consequence of imposing the NCFL condition in frictionless continuous markets.

5. Constraints of Rough Path Theory and Collapse to Semimartingale Paradigm

Although rough path theory allows modeling with arbitrarily low regularity and broad classes of stochastic processes, maximal admissibility under the NCFL condition collapses permitted models to semimartingale price processes. That is:

• The only admissible rough integrators—when tested against the full span of Markovian and signature-based integrands—are those corresponding to the Itô rough path lift of a (possibly time-changed) Brownian motion.
• Integrators with non-Gaussian increments, non-Markovian memory, or nontrivial higher-level rough path corrections are ruled out by unbiasedness for all admissible strategies.

This result provides a definitive structural limitation on frictionless continuous trading models constructed via rough paths: only those that are (after appropriate transformations) semimartingales avoid arbitrage under maximal controlled strategy admissibility.

6. Technical Framework and Key Results

Several technical pillars underpin the NCFL condition in the rough path context:

• Rough market models: Defined via rough path lifts, $\mathbf{S} = (1, S, \mathbb{S}^2, \dots)$, with levels satisfying Chen’s relations.
• Controlled portfolios: Integrands $Y$ are jointly constructed with their Gubinelli derivative $Y'$ and exhibit expansions with vanishing remainders as $|t-s| \to 0$.
• Gain process: Interpreted as the limit of compensated Riemann sums over admissible partitions, producing the rough integral $\int_0^T Y_t d\mathbf{S}_t$.
• NCFL condition: Formally,

$$\overline{C}^p(\mathcal{H}, \mathfrak{T}_T) \cap L^p_+ = \{0\},$$

where $\overline{C}^p$ is the closure under the weak-star topology of $L^p$ of the cone of payoffs generated by zero-capital, admissible strategies, taken across all controlled portfolios and times.

• Functional-analytic separation: Use of Banach-space separation theorems ensures equivalence between no-arbitrage (NCFL) and the existence of equivalent measures under which all admissible gains are centered.
• Combinatorial classification: Complete Bell polynomials and Hermite polynomial representations are central tools, classifying the structure of admissible rough path drivers.

Overall Summary

Imposing the No Controlled Free Lunch condition in continuous-time, frictionless market models using rough path theory requires the price integrator to be unbiased for all admissible controlled strategies. This constraint leads to a precise algebraic and probabilistic classification: the only robust solutions are those whose increments are Gaussian (up to time reparametrizations) and whose rough lift coincides with the Itô lift of Brownian motion. Under maximal admissibility, rough market models revert to the semimartingale paradigm, precluding arbitrage and limiting rough path generality. This result conclusively delineates the ultimate capacity of the rough path framework in financial modeling and clarifies foundational relationships between pathwise stochastic calculus, no-arbitrage, and admissible trading.