The article shows that the price of a European Call option with a higher strike can be seen as the price of a Call option on a Call option with a lower strike. This means the prices of these two contracts are the same. The researchers found new pricing functions for these options, including in the Black-Scholes model. They also discovered properties of the function that relates the price of the option to its underlying asset. Additionally, they explored implied volatility smiles for these options in the Black-Scholes model.