An E-ARCH Model For the Term Structure of Implied Volatility of FX
Options
Marco Avellaneda and Yingzi Zhu
Applied Mathematical Finance, 1997
Abstract
We construct a statistical model for the term-structure of implied volatilities
of currency options based on daily historical data for 13 currency paris
over a 19-month period. We examine the joint evolution of 1-month, 2-months,
3-months, 6-months, and 1-year at-the-money (50 delta) options in all
the currency pairs. We show that there exist three uncorrelated state variables
(principal components) which account for the parallel movement, slope oscillation
and curvature of the term structure and which explain, on average, the movements
of the term-structure of volatility to more than 95 % in all cases.
We test and construct an exponential ARCH , or E-ARCH, model for each state
variable. One of the applications of this model is to produce confidence
bands for the term-structure of volatility.