An E-ARCH Model for the Term Structure of Implied Volatility of FX Options

this article, we contribute to the theoretical understanding of the volatility of option prices by studying empirically the dynamics of the term-structure of implied volatilities of currency options. We use a Principal Component Analysis (PCA) (Judge, 1988) combined with ARCH techniques to derive a statistical model for the evolution of the term structure of volatility. Thus, the present statistical analysis is not on the volatility of the underlying asset, as in traditional work (see Engle, 1994, 1995), but rather on the implied volatilities. The latter provide a "dimensionless" representation of the currency options market. Using historical data on the implied volatility of options on 13 currency pairs for the period Jan. 1, 1995 to July 30, 1996, we develop a three-factor term-structure model which appears to be applicable to all the studied currency pairs. A similar methodology was used by other authors in the study of term structure of interest-rates (Litterman and Scheinkman, 1991). There are, however, important structural differences between interest rates and implied volatilities. The main difference is that the term-structure of volatility is a stochastic process which is far from equilibrium. As an illustration of this, Figure 1 shows an "equilibrium" AR model fitted using least squares and maximum likelihood techniques compared against real data. One clearly observes more structure in the real data than in the equilibrium model, and unlike the latter, the real implied volatility data exhibits a trend. A formal test for non-stationarity or "trend" of time series is the unit root test (Enders, 1995). We apply this test to the term-structure of volatility, and the results 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 time 0 50 100 150 200 250 300 350 400..