Market with Markov jump volatility I: Price of risk monitoring as an optimal filtering problem

The first part of series is devoted to investigating the market price of risk in a financial system. It contains riskless bank deposits, risky base assets, and their derivatives. The model of the underlying price evolution represents a stochastic differential system with stochastic volatility which is a hidden Markov jump process. The investigated market is incomplete and has no arbitrage possibilities. The market price of risk, which corresponds to a prevailing martingale measure, can be characterized via the hidden Markov jump process but can not be restored precisely. However, it can be estimated optimally using the observations of both the derivative and underlying prices. Using the concept of the prevailing martingale measure existence, one can derive a system of the partial differential equations which describes an evolution of the derivative prices and represents some analog of the classic Black–Sholes equation. Then, one can convert the calculation problem for the market price of risk to the optimal state filtering in a differential stochastic observation system. The paper also discusses various aspects of the numerical realization for the stated estimation problem.