Determining Implied Volatilities of American Options Using the Willow Tree Method

An option’s implied volatility is a measure of the option’s expected price fluctuation. It is a forward-looking, market-based estimate of volatility and is determined by the market price of the option. Implied volatility is a key ingredient in options pricing models such as the Black-Scholes model. It can be used to gauge investor sentiment and is affected by a number of factors, including the underlying asset’s price, time to expiration, interest rates, and dividend yields.

The usual approach for determining the implied volatility from the option’s price is the bisection method, which is a numerical root-finding technique. The implied volatility is the value of the volatility that, when input into the options pricing model, produces a model price that is equal to the observed market price of the option. The drawbacks of this method are that it is slow and it can be applied to European options only.

Reference [1] proposed a new approach for calculating implied volatilities of American options. It utilized the Willow tree method [2] that is more computationally efficient,

One is to increase the efficiency of the implied volatility calibration thanks to no need of reevaluation pn ij at each iteration. That is, the transition probabilities pnij just needs to be calculated once in the calibration. When the volatility is updated, we just need to update the tree nodes in (3.9) and (3.10) by the new value of the volatility, and perform a backward induction for evaluating the option price. It can save about 90% computational time in the option evaluation during the implied volatility calibration. The other is the tree structure can be constructed offline. That is, a basic willow tree structure can be formed in advance and stored in a database. Once the front desk requires to calibrate the implied volatility in real time, they can extract the basic tree structure from the database, adjust the tree nodes to satisfy a specific set of market parameters and calibrate the implied volatilities on the adjusted tree structure in a very short time.

Another advantage of the Willow tree method is that it can be easily extended to two- or three-factor models,

Our method is extendable to the two or three-factor models in [42]. The willow tree method manages to price vanilla and exotic options under various stochastic volatility models, see [34]. However, its efficiency may decrease as the more factors are introduced into the model. One of our future works is to explore the special structure in the two-factor or three-factor in [42] to construct an efficient method for commodity futures option pricing and implied volatilities determination.


[1]  Wei Xu, Aleksandar Šević, Željko Šević, Implied Volatility Surface Construction for Commodity Futures Options Traded in China, Research in International Business and Finance, 2022,

[2] M. Curran, Willow Power: Optimizing Derivative Pricing Trees, ALGO RESEARCH QUARTERLY, Vol. 4, No. 4, December 2001.

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