Entropic Dynamics of Jump-Diffusion Option Pricing

Standard models of stock price dynamics and option valuation usually begin by postulating stochastic processes. This paper develops an entropic inference framework that derives these processes from information constraints. The key symmetry is that markets reward returns rather than price levels, which selects log price as the dynamical variable. Price changes are represented by two channels. The continuous channel carries constraints of continuity and directionality. The jump channel carries the arrival rate and the first two moments of jump size. Since these constraints apply to disjoint parts of the microstate, the channels factorize. The resulting dynamics is the Merton jump diffusion, with Geometric Brownian Motion as the no jump limit. The log price density satisfies the Kolmogorov Feller equation, whose no jump limit is the Fokker Planck equation. The same inferential principle, with no arbitrage imposed through the mean log return, selects the Esscher transform from the many martingale measures available in an incomplete market. The option price then satisfies Mertons partial integro differential equation, and the risk neutral mixture of lognormal distributions generates the implied volatility smile. The Black Scholes results are recovered when jumps vanish. What changes from one model to another is not the inference, but the information supplied to it.