Gram-Charlier expansions and option pricing - Knipe, Joshua (Department of Finance and Tax, 2022)
Gram-Charlier expansions provide a tractable way of fitting risk-neutral distributions to asset prices. This allows the model to capture skewness, excess kurtosis and higher moments in observed asset returns. Schlogl (2013) proposes a calibration method to ensure the fitted densities are valid and arbitrage free. This method is implemented with standard foreign exchange options and gives an exact fit when enough moments are included in the calibration process. GramCharlier expansions also result in analytic solutions for many exotic option prices through an extremely general framework. This relies on representing an option as a portfolio of the M-binaries defined by Skipper and Buchen (2003). Geometric Asian options are priced using this approach and compared to the corresponding Black-Scholes prices. Numerical examples highlight the effect skewness and excess kurtosis can have on these option prices, particularly for options that are out-the-money. Gram-Charlier distributions are also combined with Monte Carlo simulations to estimate option prices for calls and geometric Asian options. The results show convergence to the analytical solutions for all cases. Additionally, Gram-Charlier estimates for arithmetic Asian options are calculated and compared to Black-Scholes estimates.