This paper represents the first attempts to distinguish between long-and short-memory (with level shifts) in volatility of Peruvian stock and Forex rate returns.We utilise the approach of Perron and Qu (2010). The daily data span the period 3 January, 1990 to 13 June, 2013 for the stock market returns, and from 3 January, 1997 to 24 June, 2013 for the Forex rate returns. The analysis of the ACF, the periodogram and the fractional parameter estimates for both volatilities suggests that the theoretical predictions of the simple mixture model of Perron and Qu (2010) are correct. The results are more conclusive for the stock market volatility. The application of one statistic suggests rejection of the long-memory hypothesis for both volatilities. Other two statistics provide weak evidence against the null hypothesis, above all for the Forex rate market. To reinforce the findings, some results associated with other investigations are presented.