| Using an expansion of the transition density function of a 1-dimensional Brownian motion with drift, we give the corresponding first term in the asymptotics of European call option prices with respect to the time to the expiry. We use this formula to calculate both the leading value of the model-based implied volatility and the first order deviation of the model-based implied volatility from its leading value. Some geometric interpretations will be discussed for these two terms. In particular, the leading value of the model based implied volatility could be interpreted as the Riemannian distance under the metric determined by the equation satisfied by the stock price process. |
