The article presents a new way to calculate implied volatility for models with changing volatility over time. By using a mathematical method called the heat kernel expansion on a specific type of surface, the researchers found a more accurate approximation for certain financial models. They applied this method to a popular model called SABR and showed that their formula works better than the traditional one in certain cases. Additionally, they were able to solve the SABR model for specific values of its parameters. This new approach could also be useful for other financial models in the future.