A Rough Theory of Markets

Markets can look rough, crash-like, and only partially hedgeable even when their microscopic volatility channels remain continuous. This paper gives a unified structural explanation through two governing principles. First, markets have phases. Second, visible prices are compressions of a higher-dimensional latent contagion network. The paper studies an N-asset Gaussian Volterra market with directed latent contagion kernels, asymmetric information flow, and an aggregate market-volatility functional M. The key admissibility condition builds on the projected observability and minimax logic of [15] and is internalized at market scale. Active contagion channels must remain source-screened visible and must admit a genuinely shrinking local normalization in the projected experiment. The admissible class is characterized rather than stipulated. For active off-diagonal channels, source-screened visibility, nonzero market loading, and an exponent H_ij < 1/2 are exactly the conditions for a nondegenerate shrinking projected experiment. Within that class, smooth directional contagion is locally degenerate at high frequency. The only stable observable contagion phase is rough. Once active market-dominant channels remain projected-minimax admissible and aggregate without leading-order cancellation, the aggregate market-volatility exponent is pinned down by the roughest active source-screened edge and is strictly below one half. Those same dominant edges define a screened contagion operator on the N × N network. Above its Perron threshold, latent stress synchronizes along a common mode and becomes macroscopically amplifying. If threshold-triggered activation makes the synchronized bursts contract faster than observation resolution, a coarse observer sees a stochastic jump limit even though the primitive market remains continuous. The same synchronized mode governs pricing and hedging. For the integrated market variance Q_M = ∫₀ᵀ M(t)² dt, compression to any visible pricing summary leaves a lower-bounded latent convexity gap, and visible hedges for variance-linked claims satisfy an explicit lower bound on residual risk whenever the synchronized latent mode retains conditional variance. In particular, leading-order short-maturity skew statistics of the type identified in [16] are visible Q-summaries but remain incomplete relative to latent market-variance convexity. The paper therefore develops one coherent market theory in which asymmetric latent contagion forces roughness, roughness organizes synchronization, synchronization produces crash-like market episodes, and visible pricing and hedging remain phase-dependent compressions of latent market-variance risk.