Using an expansion of the transition density function of a 1-dimensional Brownian motion with drift, we give the first term in the asymptotics of European call option prices with respect to the time to the expiry T. We use this formula to calculate both the leading value of the implied volatility hat{sigma} and the first order deviation of hat{sigma} from its leading value. Some geometric interpretations will be discussed for these two terms. In particular, the leading value of the implied volatility could be interpreted as the Riemannian distance under the metric determined by the equation satisfied by the stock price S. This is a joint work with professor Elton Hsu.
I will give a quick survey of the background of the problem including some background knowledge in mathematical finance at the beginning of the talk. ( Slides)
Speaker: Cheng Ouyang, Northwestern University