Abstract
We compare density forecasts for the prices of Dow Jones 30 stocks, obtained from 5-minute high-frequency returns and daily option prices. We use the Black-Scholes and Heston models to extract risk-neutral densities from option prices. From historical high-frequency returns, we use the HAR-RV model to calculate realized variances and lognormal price densities. We use a nonparametric transformation to transform risk-neutral and historical densities into real-world densities and make comparisons based on log-likelihoods. The Black-Scholes model gives the highest log-likelihoods for all four horizons evaluated, ranging from one day to one month, both before and after applying transformations. The HAR-RV model and the Heston model give similar log-likelihoods for all four horizons.
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