We choose the Andersen et al. (2007) and Lee and Mykland (2008) jump detection tests to detect intraday price jumps for ten foreign exchange rates and cojumps for six groups of two dollar exchange rates and one cross exchange rate at the one-minute frequency for five years from 2007 to 2011. We reject the null hypothesis that jumps are independent across rates as there are far more cojumps than predicted by independence for all rate combinations. We find that one dollar rate and the cross rate combination almost always has more cojumps than the two dollar rates combination. We also find some clustering of jumps and cojumps can be related to the macroeconomic news announcements affecting the exchange rates. The two selected jump detection tests find a similar number of jumps for ten foreign exchange rates.
We compare density forecasts for the prices of Dow Jones 30 stocks, obtained from 5-minute high-frequency returns and daily option prices for four horizons ranging from one day, one week, two weeks to one month. We use the Heston model which incorporates stochastic volatility to extract risk-neutral densities from option prices. From historical high-frequency returns, we use the HAR-RV model to calculate realised variances and lognormal price densities. We use a nonparametric transformation to transform risk-neutral densities into real-world densities and make comparisons based on log-likelihoods. For the sixty-eight combinations from seventeen stocks for four horizons, the transformed lognormal Black-Scholes model gives the highest log-likelihoods for fifty-nine combinations. The HAR-RV model and the Heston model have similar forecast accuracy for different horizons, either before or after applying a transformation which enhances the densities. The transformed real-world densities almost always pass the Kolmogorov-Smirnov and Berkowitz tests, while the untransformed risk-neutral densities almost always fail the diagnostic tests.