Minimum Action and Dealer Equilibrium in Stochastic-Volatility Option Markets

Abstract: We adapt a physics-inspired modeling idea to option markets: first specify a
principle that selects a system’s dynamics, then determine whether those dynamics
approach a stable statistical state. Dealers act under the physical probability measure
and balance the long-run cost of variance exposure against the cost of intervention.
Minimizing this expected action selects a feedback policy which, through the state
equation, determines the evolution of variance. Convergence to statistical equilibrium
is established separately rather than assumed in the objective.
In a solvable controlled-CIR benchmark, verification of the stabilizing control
yields restrictions linking action weights to mean reversion. The exact transition
law establishes convergence from every positive initial state to a Gamma invariant
distribution. In a specified finite-player dealer market, an invariant-law performance
identity verifies a stationary Nash equilibrium for inventory and variance controls.
With independent dealer shocks, average variance follows another CIR process.
Given the model’s declared order-flow rule, we construct a cross-maturity-consistent,
true-martingale stock-pricing density; a separate CARA risk bearer selects this
density for zero-volume option trades. Integrating the resulting affine transform over
the stationary Gamma distribution gives a semi-closed European marginal price.
For stationary-prior forward-at-the-money calls, the price rises with dealer count
and approaches a Black–Scholes limit at an explicit 1/N rate.
A distinct, restricted risk-allocation economy derives risk compensation from
customer endowment and market clearing. Simulations check the analytical calcu#2;
lations. The general interacting market and the model’s performance on observed
option data remain open to further testing.