ABSTRACT
Abstract Pricing and Hedging derivatives products is essentially a problem of portfolio optimization. Once a measure of risk has been chosen, the price can be defined as the mean value of the profit and loss (P&L) and the best hedging strategy is the optimal control which minimizes the risk. In the Black-Scholes model, the only source of risk is the spot process and the optimal control is the delta-strategy which cancels the risk. However, under the introduction of stochastic volatility, the market model becomes incomplete. The resulting risk is finite and the delta-strategy is not optimal. A portfolio optimization problem appears also naturally if we assume that the market is illiquid and the trading strategy affects the price movements. In the following, we will focus on these optimal control problems when the market is incomplete and the market is illiquid. Our study involves the use of perturbation methods for non-linear PDEs.
