Implied Binomial Trees with Cubic Spline Smoothing

Posted: 20 May 2019

See all articles by Yisong S. Tian

Yisong S. Tian

York University - Schulich School of Business

Date Written: November 5, 2014


Implied binomial trees are typically constructed by fitting a risk-neutral density (in the form of ending nodal probabilities) to observed option prices (e.g., Rubinstein [1994]). This commonly used approach requires the solution of a high dimensional quadratic program with the number of unknowns proportional to the number of binomial periods. In this paper, we propose a more efficient implementation of implied binomial trees by incorporating cubic spline smoothing in the quadratic program. Only a selected subset of ending nodal probabilities is treated as unknowns while the remainder is interpolated using cubic splines. The reduction in dimensionality of the quadratic program can substantially improve the efficiency of implied binomial trees without any loss in numerical accuracy. More importantly, our smoothing method can overcome the overfitting problem in the implied binomial tree scheme and minimize distortions to the extracted risk-neutral density even when option prices are observed with error.

Keywords: Implied binomial trees, Cubic splines, Risk-neutral density, Risk-neutral moments, Numerical efficiency

JEL Classification: G13

Suggested Citation

Tian, Yisong Sam, Implied Binomial Trees with Cubic Spline Smoothing (November 5, 2014). Journal of Derivatives, Forthcoming,, Available at SSRN: or

Yisong Sam Tian (Contact Author)

York University - Schulich School of Business ( email )

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