A volatility smile is a common graph shape that results from plotting the strike price and implied volatility of a group of options with the same expiration date. The volatility smile is so named because it looks like a person smiling. The implied volatility is derived from the Black-Scholes model, and the volatility adjusts according to the option's maturity and the extent to which it is in-the-money (moneyness).
The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, during the time span between the valuation date and the option's expiration date. The model reduces possibilities of price changes, and removes the possibility for arbitrage.