Can anyone solve the smile problem?

volatility models were the local volatility models1. They inferred a volatility dependent on the stock price level and time that accommodates the market price of vanillas within the Black-Scholes framework (Dupire (1994), Derman & Kani (1994), Rubinstein (1994)). Indeed, local volatility models postulate that the underlying follows a lognormal diffusion process equation dS S = π(t)dt + σ (S, t) dW