Bridging Asset Pricing and Market Microstructure: Option Valuation in Roll’s Framework

We introduce a binary tree for pricing contingent claims when the underlying security prices exhibit history dependence. We apply the model to the specific cases of moving-average and autoregressive behavior that are characteristic of price histories induced by market microstructure behavior. Our model is market-complete and arbitrage-free. When passing to the risk-neutral measure, the model preserves all parameters governing the natural-world price dynamics, including the instantaneous mean of the asset return and the instantaneous probabilities for the direction of asset price movement. This preservation holds for arbitrarily small, but non-zero, time increments characteristic of market microstructure transactions. In the (unrealistic) limit of continuous trading, the model reduces to continuous diffusion price processes, with the concomitant loss of the microstructure information.