Option Pricing and Hedging with Regret Optimisation

This thesis focuses on the option pricing and hedging based on a regret optimisation problem in a discrete-time financial market model with proportional transaction costs. In such model, the no-arbitrage price interval can be very large. Such large interval makes it difficult for an investor to choose the “right” prices, which is a long standing difficulty in the field. We introduce an indifference pricing method based on minimising regret/disutility, and show that the spread between the buyer’s and seller’s prices can be much narrower than the no-arbitrage price interval. The regret optimisation problem allows possible fund injection/withdrawal at each time step, and in doing so it extends the classic utility maximisation problems in financial models. Moreover, by allowing the investor’s preference towards risk to be different at different time step, it also extends the optimal investment and consumption problem in financial market models with a finite horizon. In addition, the investor’s endowment that is considered in our setting is modelled by a portfolio flow which extends the notion of initial wealth. We prove that there exists a solution to the regret optimisation problem, and indifference prices are always within the no-arbitrage price interval. Under an exponential type regret function, we find a dynamic programming algorithm to construct a solution to a Lagrangian dual problem. By solving the dual problem, we can not only solve the regret optimisation problem but also calculate the option indifference prices. In binary models, we calculate the optimal injection/withdrawal strategy for various different values of given parameters, and also compute the indifference prices of various European options. The numerical results show that the bid-ask indifference price interval can be much narrower than the no-arbitrage price interval, and such smaller price interval can be used to guide the investor to choose the “right” prices.