You have to modify the code source to do that. Roughly, in the Catalog controller, you can change the HomePageProducts action to call PrepareProductOverviewModel with the preparePriceModel to true and in the view use the ProductBox partial view instead of the SmallProductBox.
However, the site will be slower because computing the price model takes a significant amount of time especially if you have complex discounts.
In my implementation, I cached the ProductModels (with appropriate variations for displaying the correct price depending on tax settings, preferences, currency, language, etc) for each product, making things significantly faster. The cache gets refreshed automatically whenever a product, product variant, category or discount is modified.