Valuation of exotic options under shortselling constraints as a singular stochastic control problem

Option trading in financial markets is increasing worldwide. The valuation and hedging of options on equities, currencies, commodities is now well understood. However, there are certain exotic options which cause substantial hedging difficulties for the practitioner and which the standard valuation and hedging formulas do not capture. To include such hedging difficulties in the price of the option we impose constraints on the hedging portfolio.; To motivate this concept we consider a prototype of such a dangerous option: The up-and-out call option that can knock out in the money. The usual delta-hedge suggested to the seller can lead to unbounded short positions of the underlying asset. To prevent this undesirable situation, we impose a short-selling constraint and discuss its effect on the value of the option. Furthermore we present an analytical solution to this constrained valuation problem in a constant coefficient geometric Brownian motion model.; In 1993, Cvitanic and Karatzas showed in a general Brownian motion driven market that constrained valuation of any path-dependent option under convex constraints is equivalent to solving a certain stochastic control problem. Unfortunately this is difficult to solve in generality.; In 1997, Broadie, Cvitanic and Soner developed a procedure for valuation of path-independent options under convex constraints. The final value of the option is a function of the underlying asset, and this function is replaced by a smoother function, a procedure we call "face-lifting". The option with the face-lifted payoff is then priced in an unconstrained market.; Our new result helps pricing path-dependent options in a way similar to the face-lifting procedure. We restrict our attention to the following special case of a convex constraint: we put a lower bound on the gearing (the ratio of value of holdings of underlying asset to value of the option). We will show that in a one-dimensional constant coefficient Brownian motion model the Cvitanic-Karatzas stochastic control problem can be viewed as a singular stochastic control problem, which often admits a solution by inspection. In particular we will be able to analyze a variety of barrier options even with time dependent barriers, look-back options, Asian options and the example of a book of two up-and-out call options.