Option prices from operational-time reaction-boundary lattices

We consider the role of a continuum operational time u and its mapping to calendar time t and how these relate to event time for option pricing problems. We derive option-pricing equations from an operational-time Markov lattice rather than from a calendar-time diffusion. The primitive model is a nearest-neighbour log-price lattice with state- and time-dependent transition probabilities. Its Chapman-Kolmogorov decomposition yields discrete forward and backward equations, which converge under local finite-variance scaling to the usual continuum adjoint pair. In price variables, the backward equation gives a generalized European pricing PDE and reduces to Black-Scholes-Merton under the risk-neutral drift restriction and constant volatility. Interpreted as a reaction-boundary model for limit-order-book mid-prices, the construction identifies local volatility with an activity-rescaled risk-neutral bid-ask reaction-boundary variance. The framework separates the operational kernel, calendar-time projection, and pricing-measure choice, to clarify how unspanned clock, jump, or renewal risks can lead to incomplete-market pricing.