Correction to 'Leverage and volatility feedback effects in high-frequency data' [J. Financial Econometrics 4 (2006) 353--384]

where B t and W t are correlated Brownian motions with corr(dB t , dW t ) = ρ. For simplicity, it is assumed that µ = c = 0. If the continuously compounded returns from time t to time t + ∆ are Bollerslev et al. (2006) it is proved that, for n = 0, 1, 2, . . .,

where a ∆ = (1 -e -κ∆ )/κ.

In order to obtain (2) these authors use the following distribution of the squared returns

Observe, however, that R 2 t,t+∆ = (p t+∆ -p t ) 2 cannot depend on the values of R u,u+∆ with u ∈ [t, t + ∆], that is, on returns which are posterior to t + ∆. Here we will obtain the correct expression for the distribution of the squared returns and check that, under this new distribution, result (2) in Bollerslev et al. (2006) still holds.

In order to obtain the distribution of the squared returns observe that R 2 t,t+∆ = p 2 t+∆ + p 2 t -2 p t p t+∆ . By Itô’s Lemma we have that

Using ( 1) and ( 4) we have that

(5)