Option hedging for high frequency data models

Hedging strategies for contingent claims are studied in a general model for high frequency data. The dynamics of the risky asset price is described through a marked point process Y , whose local characteristics depend on some hidden state variable X. The two processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. Since the market considered is incomplete one has to choose some approach to hedging derivatives. We choose the local risk-minimization criterion. When the price of the risky asset is a general semimartingale, if an optimal strategy exists, the value of the portfolio is computed in the terms of the so-called minimal martingale measure and may be interpreted as a possible arbitrage-free price. In the case where the price of the risky asset is modeled directly under a martingale measure, the computation of the risk-minimizing hedging strategy is given. By using a projection result, we also obtain the risk-minimizing hedging strategy under partial information when the hedger is restricted to observing only the past asset prices and not the exogenous process X which drives their dynamics.