On the Pricing of Power and Other Polynomial Options

Abstract

There are limitless possibilities for new kinds of options, with payoff patterns of all shapes and contingencies related to any number of events. The simplest of these payoffs, like those for a call spread or a strangle, for example, can be constructed out of a straightforward combination of plain vanilla puts and calls. This leads to very easy formulas for pricing and hedging them, but the possible payoff patterns invariably consist of a collection of linear segments defined by the intervals between the strikes of the options in the set. This article provides a way to significantly extend the achievable payoff patterns to any polynomial function of the final asset price. The building block is no longer a plain vanilla option, but a power option, the payoff of which is a function of (Sa - X) for some exponent a. The authors first show how to price such options under several important price processes, including Heston-style stochastic volatility and a general Lévy process. They then derive the formula for an option with a general polynomial payoff, as a function of power options. Finally, they offer some examples to illustrate the technique, including parabolic payoff call options.