Equity premium puzzle: The equity premium puzzle is thought to be one of the most important outstanding questions in neoclassical economics. It is founded on the basis that over the last one hundred years or so the average real return to stocks in the US has been substantially higher than that of bonds. The puzzle lies in explaining the causes behind this equity premium. While there are a number of different theories regarding the puzzle, there still exists no definitive agreement on its cause.
So solved in this context means consensus, not as in something mathematically rigorous. I think this is the problem with a lot of these unsolved problems. When I think of something being solved, it means that it absolves doubts.
Improved Black–Scholes and binomial options pricing models: The Black–Scholes model and the more general binomial options pricing models are a collection of equations that seek to model and price equity and call options. While the models are widely used, they have many significant limitations. Chief among them are the model's inability to account for historical market movements and their frequent overpricing of options, with the overpricing increasing with the time to maturity. The development of a model that can properly account for the pricing of call options on an asset with stochastic volatility is considered an open problem in financial economics.
This is already done. There are a plethora of models to account for everything. It's not an open problem. It's just messy mathematically and does not have the elegant solution like the Black Scholes option pricing model does.