Call and put implied volatilities and the derivation of option implied trees

Abstract

Resting on the stylized fact that call and put prices imply different volatilities, the present paper proposes a methodology for the derivation of an arbitrage free implied tree that takes into account the information in both option classes. Specifically, we derive an implied tree that is characterised by interval values for the stock prices and we endogenously imply the corresponding artificial probabilities based on the risk neutral valuation argument. The implied tree obtained is then calibrated to market option prices by means of a non-linear optimisation routine. The methodology proposed is tested both in and out of sample using DAX index options data. Numerical results are benchmarked to the Derman and Kani’s approach. The comparison suggests that the methodology proposed in this paper, by taking into account the informational content of both call and put prices, highly improves both the in sample fitting and the out of sample performance.