Derivatives Written On A Power Of The Stock Price

This paper deals with the pricing of di%0Berent types of power options which are of practical relevance. First of all, a general valuation technique is developed that can be applied to several non-negative payo%0Bs being a function of the terminal stock price only. The key trick in the derivation is to forget about a choice of numeraires in the first place, but to use an appropriate density to change the measure. This allows for an elegant pricing equation in terms of artificial probabilities similar to the Black Scholes formula. However, the probabilities are not necessarily martingale measures. We gain economic intuition in this change of measure when we distinguish between complete and incomplete markets. In a complete market we derive that the measure corresponds to a traded numeraire portfolio, which means that it is a martingale measure whereas in incomplete markets this is in general no longer true. Furthermore, the method is applied to two examples. First, the standard Black Scholes framework is considered as an example for a complete market where the corresponding numeraire portfolio is computed explicitly. Second, we deal with stochastic volatility models as an example for an incomplete market. Closed-form solutions are derived for quot;power optionsquot; (written on a power of the underlying stock) with and without cap, and quot;powered optionsquot; (where the option payo%0B is raised to some power). There are two key contributions of this paper. First, we gain economic insight in the relationship between change of measure and change of numeraire in complete and incomplete markets. Second, we derive closed-form solutions for several types of power options under stochastic volatility.