Cosine Willow Tree Structure under Lévy Processes with Application to Pricing Variance Derivatives

Abstract

Lévy process models can capture the large price changes on sudden exogenous events and can better demonstrate the high peak and heavy tail characteristics of financial data. The Fourier transformation method is famous for pricing derivatives under the Lévy processes beause of its efficiency, how it separates models from payoff function, and how it handles models with characteristic functions, but it is criticized for its restriction on path dependency. In this article, we propose a unified cosine willow tree method, which inherits the merits of the transformation method but overcomes its shortcomings. Moreover, the hedging Greeks can be obtained as a by-product from the tree structure with minor extra cost. Some popular variance derivatives are also discussed to demonstrate the flexibility of the proposed method in handling path dependency. Finally, the theoretical convergence is analyzed for various Lévy process models.