January 5th, 2011, 10:44 am
"back bone" is a naive concept for capturing the ATM dynamics.But, "back bone" does not capture the ATM dynamics. if beta equal to 1 or \rho \lambda is large. Here, \lambda is the strength of vol of vol compared to the local volatility, which is Eq.(3.1b) in the original Hagan paper.As Bartletts shows, change of underlying accompanines the following change of alpha:\Delta \alpha = \rho \nu / f^\beta \Delta f Then, approximated ATM vol ( \alpha / f^(1-\beta)) changes as:\Delta ATM = \alpha f^(\beta-2) ( (\beta -1) + \rho \lambda) \Delta fThe first term is the "back bone".The second term is due to the change of volatility.Using Barlett's delta, the change of ATM is also hedged by the underlying.If the SAMR model is right, Bartlet's delta is appropriate.