An Orthogonal Series Expansions Method to Hedge and Price European-Type Options

Closed-form pricing formulae and option Greeks are obtained for European-type options using an orthogonal polynomial series -- complex Fourier series. We assume that risky assets are driven by exponential Lévy processes and stochastic volatility models. We provide a succinct error analysis to demonstrate that we can achieve an exponential convergence rate in the pricing method in many cases as long as we choose the correct definite computational interval. As a novel pricing method, we also numerically demonstrate that complex Fourier series are more accurate and computationally effective when implemented to price in/at/out-of-the-money options or options with complicated contingent claims