Option prices from operational-time reaction-boundary lattices

Battle Count: 86

The paper is primarily theoretical and does not provide directly implementable trading strategies or numerical algorithms. However, it offers important conceptual clarity for quantitative practitioners: (1) it clarifies why local volatility surfaces cannot separately identify operational variance from clock activity, which affects model calibration; (2) it explains when market completeness fails (unspanned clock, jump, or renewal risk), which is critical for hedging path-dependent derivatives; (3) it shows that finite-mesh effects can cause materially different prices for discretely monitored claims even when European prices agree; (4) it provides a unified framework for understanding when BSM assumptions break down. The practical relevance is indirect but foundational for model risk management and derivative pricing.

The theoretical framework requires advanced knowledge of stochastic process theory (Markov generators, martingale problems, weak convergence), partial differential equations (forward/backward parabolic equations, Fokker-Planck equations), probability theory (Chapman-Kolmogorov equations, Levy processes, subordination), and mathematical finance (risk-neutral pricing, Dupire inversion, market completeness). The paper itself does not provide code or numerical algorithms, but implementing the hierarchy would require solving PDEs/PIDEs, handling clock projections, and managing multiple model branches. The mathematical sophistication is high, though the core derivation (Proposition 1) is presented in an accessible manner.

The paper provides complete self-contained mathematical derivations from first principles without invoking Ito calculus. All propositions, lemmas, and theorems are stated with explicit assumptions and proofs. Appendices provide process-level convergence conditions (Appendix C), deterministic clock projection proofs (Appendix D), clock-variance equivalence results (Appendix E), finite-mesh examples (Appendix F), and unspanned clock risk constructions (Appendix G). However, the paper is purely theoretical with no numerical experiments or code, so reproducibility is limited to verifying the mathematical arguments. The companion paper [2] (arXiv:2607.05011) is referenced for the full reaction-boundary variance derivation.