A Rough Theory of Markets

This paper develops a structural market theory in which roughness, systemic synchronization, crash-like episodes, and incomplete visible pricing and hedging arise from one common mechanism. The setting is an N-asset Gaussian Volterra market with directed latent contagion kernels, asymmetric information flow, and an aggregate market-volatility functional. The admissibility condition is internalized at market scale: active contagion channels must remain source-screened visible and admit a genuinely shrinking projected normalization, and the admissible class is characterized rather than stipulated. Within that class, smooth directional contagion is locally degenerate at high frequency, so roughness becomes the only stable observable contagion phase. The same dominant rough edges define a screened contagion operator, and above its Perron threshold latent stress synchronizes along a common mode and becomes macroscopically amplifying. Under threshold-triggered activation, a coarse observer can see a stochastic jump limit even though the primitive market remains continuous. The paper then shows that visible prices and visible hedges are compressions of latent market-variance risk: compression to visible pricing summaries leaves a lower-bounded convexity gap, and visible hedging of variance-linked claims leaves an explicit residual-risk lower bound whenever the synchronized latent mode retains conditional variance. Together, these results yield a unified market theory in which asymmetric latent contagion forces roughness, roughness organizes synchronization, synchronized activation can generate crash-like episodes, and visible pricing and hedging remain phase-dependent compressions of latent market-variance risk.