An Efficient Convergent Willow Tree Method for American and Exotic Option Pricing under Stochastic Volatility Models

Abstract

Stochastic volatility models can describe the evolution of financial assets, such as stocks, currencies, and commodities, better than the classic Black–Scholes model. Some strategic decision-making problems also involve path-dependent and American-style options. In this article, the authors propose a novel, efficient, accurate, and unified two-factor willow tree method to price exotic and American options under the stochastic volatility models, such as the Heston, 3/2, 4/2, Hull–White, Stein–Stein, and a-Hypergeometric models. They also present the convergence analysis of their proposed tree method. They then apply the tree method to price European and American options, and the expected present value and survival rate in a dividend-and-ruin problem. Numerical results demonstrate the efficiency, accuracy, and convergence of their method.

TOPICS: Options, volatility measures, factor-based models, analysis of individual factors/risk premia

Key Findings

• • The authors propose an efficient and unified two-dimensional willow tree structure for various stochastic volatility models.

• • The convergence rate of the two-dimensional willow tree method is O(Δt).

• • The authors apply the willow tree to evaluate the present firm value and survival rate of a dividend-and-ruin problem, which embeds the lookback, the reflecting and absorbing barrier, and the stopping time features.