|Speaker :||1.Prof. Tai-Ho Wang (Baruch College, The City University of New York) 2. Prof. Jiro Akahori (Ritsumeikan University)|
|Title :||1.Probability density of lognormal fractional SABR model 2.A modification of Malliavin-Mancino's Fourier estimation method|
|Time :||2017-06-29 (Thu) 15:00 -|
|Place :||Seminar Room 722, Institute of Mathematics (NTU Campus)|
1.Instantaneous volatility of logarithmic return in 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 models are less known in the literature. We present in this talk 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 an expansion of the probability density for the fractional SABR model in Fourier space. 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. The presentation is based on a joint work with Jiro Akahori and Xiaoming Song.
2.Among so-called high-frequency statistics, the Fourier method introduced by P. Malliavin and M. Mancino is very special in that the method estimates differentiation of the volatility(spot volatility). I will talk on general feature of the method as well as a modification made by N.L. Liu, Mancino, Y. Yasuda, and myself. The consistency of the modification is also discussed. Further generalization, based on Malliavin's idea of "Ito atlas", is an optional topic if I have enough time.