Parameters
Vkt denotes the variance for assetkat time pointt2f 1 ;...;Tg;
(^) kit denotes factor sensitivity (exposure) to factorifor assetkat time
pointt;
it denotes factor variance for factoriat time pointt;
ijt denotes the correlation between factoriand factorjat time pointt;
^2 sðkÞt denotes the asset-specific variance for assetkat time pointt.
In this case, as the factors are orthogonal, this simplifies to
Vkt¼
XF
i¼ 1(^) kit^2 ^2 itþ^2 sðkÞt: ð 13 : 1 Þ
These (asset) variances can be updated by considering the relationship between implied
volatility and factor model volatility. However, as diBartolomeo and Warrick (2005)
note, implied volatility is often a biased estimator of actual asset volatility. This is
because option-pricing methods, such as Black–Scholes (1973), are based on the
assumption that option positions can be hedged continuously at no cost. In the real
world, hedging is costly and positions are hedged periodically not continuously. Pos-
itions that are assumed to be risk-less under this pricing framework actually do carry
some risk. Traders compensate for this and bias their risk estimates upward. To avoid
this problem diBartolomeo and Warrick (2005) consider the relationship between
changes in implied volatility and changes in basic factor model volatility levels. We
can only study the relationship of those assets for which we have option-implied
volatility data. We can update our conditional estimate of asset variance at timet,
derived from the principal component model, to
V‘t¼V‘tM‘t ð 13 : 2 Þ
whereM‘tis an adjustment at timet, defined as
M‘t¼
hYw^1
r¼ 0
I‘tr
I‘tr 1
i
hwY^1
r¼ 0
V‘tr
V‘tr 1
i
ð 13 : 3 Þ
Sets and indices
‘2f 1 ;...;N 2 g denotes the assets for which option-implied volatility data are available.
Parameters
I‘t denotes option-implied variance observed for security‘at time pointt;
V‘t denotes the updated variance for security‘at time pointt;
w denotes the period considered for updating information.
Option-implied volatility (equivalently variance) levels update faster than factor model
estimates, so changes in this relationship should give us an improved estimate of future
risk. For each asset‘we have
V‘t¼
X
(^2) ‘itðitÞ^2 þ^2 sð‘Þt; ð 13 : 4 Þ
294 News and risk