Instantaneous volatility
of logarithmic return in the lognormal fractional SABR model is driven by the
exponentiation of a correlated fractional Brownian motion. Due to the mixed
nature of driving Brownian and fractional Brownian motions, probability
density for such a model is less studied in the literature. We show in this
paper a bridge representation for the joint density of the lognormal
fractional SABR model in a Fourier space. Evaluating the bridge
representation along a properly chosen deterministic path yields a small time
asymptotic expansion to the leading order for the probability density of the fractional
SABR model. A direct generalization of the representation to joint density at
multiple times leads to a heuristic derivation of the large deviations
principle for the joint density in small time. Approximation of implied
volatility is readily obtained by applying the Laplace asymptotic formula to
the call or put prices and comparing coefficients. This is a joint work
with Jiro Akahori and Tai-Ho Wang. |