Researcher
Sharif Haason
Faculty Advisor
Dr. Ionut Florescu
Introduction
This research explores the Implied Willow Tree (IWT) methodology as an alternative framework for modeling asset price dynamics. The project adapts the IWT approach to the 10-Year Treasury Note futures market, extending previous work that primarily focused on equity options. Because Treasury futures prices behave differently from spot assets and their options are American-style contracts, the model required modifications to correctly account for these features.
The objective of the research was to evaluate whether the implied willow tree framework could accurately model the risk-neutral dynamics of Treasury futures prices and produce reliable prices for American options written on those futures.
Methodology
The implied willow tree constructs a probability tree that models the risk-neutral density of future asset prices using information extracted from market option prices.
Options data for 10-Year Treasury Note futures were collected from Bloomberg, including implied volatilities and trading volumes across multiple maturities. The model extracted the first four moments of the risk-neutral distribution—mean, variance, skewness, and kurtosis—using the Carr-Madan spanning formula, and applied a Johnson curve transformation to map these moments into the tree structure.
A calibration procedure then optimized node probabilities and transition probabilities so that the model’s option prices matched observed market prices while maintaining a smooth and realistic distribution.
Results & Analysis
The calibrated model successfully repriced the Treasury futures options used in the dataset, with most model prices falling within the market bid–ask spread, particularly for options near at-the-money strikes.
The research also evaluated the effect of the smoothing parameter used in calibration. Smaller values led to overfitting and irregular distributions, while larger values produced overly smooth distributions that failed to capture volatility skew. An intermediate value provided the best balance between accuracy and stability.
In addition, the model identified a potential mispricing in a specific strike, suggesting that the framework may also serve as a tool for detecting statistical arbitrage opportunities.
Conclusion
This research demonstrates that the Implied Willow Tree framework can be successfully adapted to Treasury futures markets, accurately modeling the risk-neutral dynamics of futures prices while accommodating American-style option features.
By capturing higher-order characteristics of asset return distributions, the model offers a flexible, data-driven alternative to traditional parametric pricing models and highlights opportunities for further research in derivative pricing and market modeling.