Improving the Willow Tree Method

An Enhanced Variant and an Empirical Comparison with the Binomial Tree Method

Abstract

This paper revisits the Willow Tree (WT) approach for pricing European call options under geometric Brownian motion and compares its numerical performance with the Black–Scholes benchmark and the Cox–Ross–Rubinstein (CRR) binomial tree. We implement two alternative WT sampling schemes, the first partial moment (FPM) and kurtosis matching (KM) methods, and examine the trade-off between pricing accuracy and computational cost. In low-volatility settings, a moderately refined WT discretization already delivers very small pricing errors at low marginal computational cost, with the FPM sampler providing the most stable performance across strikes. However, under long maturities and higher volatilities, the baseline WT exhibits a systematic negative bias. We show that this deterioration is closely associated with a mismatch in the discrete exponential-martingale condition induced by the bounded support of the terminal WT approximation. To mitigate this effect, we introduce an Esscher reweighting of the terminal WT probabilities that restores the martingale condition for the terminal approximation and substantially reduces pricing errors in the difficult regimes considered. In our experiments, the FPM sampler consistently outperforms KM in terms of residual pricing error, while Esscher reweighting yields the largest gains when maturity and volatility are high. Overall, the results suggest that WT with FPM performs well in benign regimes, whereas the Esscher-corrected version is more reliable when the baseline discretization becomes biased.