Power option pricing via Fast Fourier Transform

The basis of the option universe has been the European option, and much literature has been devoted to the extension of this option to create many new exotic options, including some with nonlinear payoffs. In this work, we study a European-style power option pricing, under a constant volatility dynamics, using the risk-neutral valuation within the Black-Scholes framework. Apart from applying the closed-form solution, we price the power option using the Fast Fourier Transform (FFT) technique which requires an analytical characteristic function of the power option. The resulting approximations are then compared with other numerical methods such as the Monte Carlo simulations, which show promising results and demonstrate the efficiency of the FFT technique as it can compute option prices for a whole range of strike prices. Besides, we show that there exists a relationship between the power call option and the power put option that is similar to the put-call parity relationship of vanilla options. We also find a transformation between the underlying asset and the power contract which enables us to obtain the pricing formulas of the power options from the vanilla options, as well as simplify the Greeks for power options. In addition to the Greeks derived from the closed-form solution, we present the Greeks using the pricing formula obtained from a characteristic function.