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Instantaneous volatility of logarithmic return in the lognormal fractionalSABR model is driven by the exponentiation of a correlated fractional Brownianmotion. Due to the mixed nature of driving Brownian and fractional Brownianmotions, probability density for such a model is less studied in theliterature. We show in this paper a bridge representation for the joint densityof the lognormal fractional SABR model in a Fourier space. Evaluating thebridge representation along a properly chosen deterministic path yields a smalltime asymptotic expansion to the leading order for the probability density ofthe fractional SABR model. A direct generalization of the representation tojoint density at multiple times leads to a heuristic derivation of the largedeviations principle for the joint density in small time. Approximation ofimplied volatility is readily obtained by applying the Laplace asymptoticformula to the call or put prices and comparing coefficients.

Probability density of lognormal fractional SABR model