Instantaneous and Averaged Volatility in Two-Side Filtration Model of Financial Asset Pricing

A two-factor approach to asset pricing based on averaged historical and instantaneous volatility defined by a marginal investor’s beliefs and herding behaviour is proposed. For the two-side filtration, backward SDE-defined stochastic dynamics under the risk-neutral probability measure are determined by a target price distribution at given horizon with parameters averaged over a subset of active market agents. For the current price at market equilibrium and instantaneous volatility, the distribution of acceptable price of risk is obtained. The found implied volatility dependencies on strike and maturity are corresponding to the historical data for options by Carr and Wu (2016). The liquidity discount for bonds and options is derived. A generalized solution for the FBSDE and a partial solution for the stochastic terminal conditions are found. The developed two-factor approach is well-suited to deep learning pricing algorithms.