Rainbow Options in Discrete Time, II

This chapter is central for our exposition. It contains the main results describing the applications of the basic risk-neutral evaluation formula for games, developed in the previous chapter, to pricing rainbow (or colored) options under various market conditions. The first section sets the stage by defining the game of an investor with Nature leading to the basic game-theoretic expression for the hedging price in the simplest case of a standard European (rainbow) option without transaction costs. We then evaluate this expression using our risk-neutral evaluation formula, yielding various multidimensional extensions of the classic Cox–Ross–Rubinstein formula for a binomial model. The next sections show the essential simplifications that become available for submodular payoffs. In particular, even a unique risk-neutral selector can be specified sometimes, say, in the case of two-color options (for a still incomplete market). This is of key importance, as the major examples of real-life rainbow payoffs turn out to be submodular. Finally, transaction costs are fitted to our model.

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