The Nobel-prize-winning Black-Scholes model is the most celebrated work on option pricing and hedging. It presumes a complete market without arbitrage. It is also an example of a theoretically developed mathematical model which turned out to have important implications for daily operations and thinking about real financial decisions and other real-world decisions. Current research in derivative pricing examines conditions more complicated than those addressed by Black-Sholes, including incomplete markets where: the modelling of price processes involves jumps (market crashes) and stochastic volatilities; where trading opportunities are more restricted; and where market information is incomplete. There is need for a transparent, strong and applicable pricing and hedging theory that can address incomplete market models. A standard approach under such circumstance is utility based derivative pricing. While intellectually interesting, there are some limitations that would inhibit widespread adoption of this approach. The basic assumption that an institution's preferences are captured by a utility function is simply not valid in most situations.
Motivated by the idea of utility-based pricing, this project will develop a theory of option pricing and hedging that that maintains the central importance of optimal hedging to the theory of pricing but replaces the criterion of maximizing utility with minimization of risk exposure. In essence, the financial intermediary is assumed to buy or sell an option for an amount such that with active hedging, her risk exposure will not increase at expiration. This would be the minimal condition for her to be willing to enter the deal in the first place, therefore the theory will provide a set of reservation prices. The option-pricing method will be implemented by means of an abstract convex risk measure. All the definitions and properties of pricing and hedging strategies derived are related to the general properties of the risk measure and the admissibility of the hedging processes, and are quite independent of the specifics of the underlying processes. Solving the existence problem of optimal hedging strategies and optimal derivative designs mathematically involve such probabilistic methods as stochastic optimal control and duality theory. Numerical solutions will be sought when closed-form solutions are not available.
With the explosion of both research and practice in risk management after recent events of financial crises, this approach potentially provides a much more practical solution to pricing and hedging issues through risk measures. The concept can also be adapted to general valuation questions of uncertain quantities where there are risk limits and the possibility to trade in the underlying products to hedge the risk exposure.
