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Behavioral analysis of long-term implied volatilities

Min Xu (Scotiabank, Toronto, Canada)
Hong Xie (Manulife Financial Corp, Toronto, Canada)
Yuehua Wu (Department of Mathematics and Statistics, York University, Toronto, Canada)

Studies in Economics and Finance

ISSN: 1086-7376

Article publication date: 15 July 2019

Issue publication date: 27 July 2021




The purpose of this paper is to analyze different behaviors between long-term options’ implied volatilities and realized volatilities.


This paper uses a widely adopted short interest rate model that describes a stochastic process of the short interest rate to capture interest rate risk. Price a long-term option by a system of two stochastic processes to capture both underlying asset and interest rate volatilities. Model capital charges according to the Basel III regulatory specified approach. S&P 500 index and relevant data are used to illustrate how the proposed model works. Coup with the low interest rate scenario by first choosing an optimal time segment obtained by a multiple change-point detection method, and then using the data from the chosen time segment to estimate the CIR model parameters, and finally obtaining the final option price by incorporating the capital charge costs.


Monotonic increase in long-term option implied volatility can be explained mainly by interest rate risk, and the level of implied volatility can be explained by various valuation adjustments, particularly risk capital costs, which differ from existing published literatures that typically explained the differences in behaviors of long-term implied volatilities by the volatility of volatility or risk premium. The empirical results well explain long-term volatility behaviors.

Research limitations/implications

The authors only consider the market risk capital in this paper for demonstration purpose. Dealers may price the long-term options with the credit risk. It appears that other than the market risks such as underlying asset volatility and interest rate volatility, the market risk capital is a main nonmarket risk factor that significantly affects the long-term option prices.

Practical implications

Analysis helps readers and/or users of long-term options to understand why long-term option implied equity volatilities are much higher than observed. The framework offered in the paper provides some guidance if one would like to check if a long-term option is priced reasonable.


It is the first time to analyze mathematically long-term options’ volatility behavior in comparison with historically observed volatility.



The study was supported by the Natural Sciences and Engineering Research Council of Canada.


Xu, M., Xie, H. and Wu, Y. (2021), "Behavioral analysis of long-term implied volatilities", Studies in Economics and Finance, Vol. 38 No. 3, pp. 583-600.



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