It has been demonstrated that European option premia computed with a binomial lattice, as first described by Cox, Ross, and Rubinstein (CRR, 1979), do not have a closed-form solution (Georgiadis, 2011). This stems from a lack of hypergeometricity, an artifact of Gosper's algorithm, and naturally...

It is well-known that the market prices of options produce implied volatilities that inexplicably vary by exercise price in a pattern often referred to as the volatility smile. This paper shows that not only do market prices produce volatility smiles, but so do model prices. This result occurs...