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Article

Implied Binomial Trees with Cubic Spline Smoothing

Yisong S. Tian
The Journal of Derivatives Spring 2015, 22 (3) 40-55; DOI: https://doi.org/10.3905/jod.2015.22.3.040
Yisong S. Tian
is a professor of finance at York University’s Schulich School of Business in Toronto, ON, Canada.
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  • For correspondence: ytian@schulich.yorku.ca
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Abstract

The binomial model is a workhorse of numerical option pricing. But the plain vanilla model produces a risk-neutral probability density for the stock price at expiration that becomes lognormal in the limit. This is consistent with Black–Scholes’ assumptions but not with actual option prices in the market. An alternative is an implied tree, which begins with the risk-neutral density extracted from the market prices for options with the desired maturity. The procedure constructs a lattice that is consistent with the desired distribution. An N-step tree has N + 1 terminal nodes, so fitting the tree to the market prices can be quite time consuming. The author proposes a simplification that greatly increases efficiency with negligible cost in accuracy. The trick is to calibrate only a subset of the terminal nodes to prices in the options market and fill in the other nodes by cubic spline interpolation.

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The Journal of Derivatives: 22 (3)
The Journal of Derivatives
Vol. 22, Issue 3
Spring 2015
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Implied Binomial Trees with Cubic Spline Smoothing
Yisong S. Tian
The Journal of Derivatives Feb 2015, 22 (3) 40-55; DOI: 10.3905/jod.2015.22.3.040

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Implied Binomial Trees with Cubic Spline Smoothing
Yisong S. Tian
The Journal of Derivatives Feb 2015, 22 (3) 40-55; DOI: 10.3905/jod.2015.22.3.040
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  • Article
    • Abstract
    • IMPLIED BINOMIAL TREES WITH CUBIC SPLINE SMOOTHING
    • NUMERICAL PERFORMANCE
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