Speaker : Professor Tai-Ho Wang (Baruch College, City University of New York)
Title : Variational most-likely-path approximation in local volatility models
Time : 2010-12-20 (Mon) 14:10 - 15:10
Place : 數學所 722 研討室 (台大院區)
Abstract:  We derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. The same methodology is also applied to address the problem of approximating the price of an Asian option under local volatility models. The application of the heat kernel expansion for the transition density between consecutive discrete sampling time points with the aid of Laplace asymptotic formula leads to a “path-integral” type expression for option prices. In the limit as the sampling time window approaches zero under this formalism, we end up with a constrained variational problem of finding an optimal path in time-price space. An approximation of the option price is obtained once the variation problem is solved. We conclude by presenting results of numerical tests of our approximations by a realistic S&P500 local volatility function. This talk is based on joint works with Jim Gatheral.