A New Improvement Scheme on Approximation Methods for Probability Density Functions

Abstract

This paper develops a new scheme for improving an approximation method of a probability density function, which is inspired by the idea in best approximation in an inner product space. Moreover, we applies "Dykstra's cyclic projections algorithm" for its implementation. Numerical examples for application to an asymptotic expansion method in option pricing demonstrate the effectiveness of our scheme under Black-Scholes and SABR models.

Asymptotic expansion for some local volatility models arising in finance

“…Let us also note that, particularly during recent years, different asymptotic expansions approaches to other particular problems in mathematical finance have been developed, see, e.g., [Andersen et al(2013), Bayer et al(2014), Benarous et al(2013), Benhamou et al(2009), Breitung(1994, Cordoni et al(2015), Fouque et al (2009), Friz et al(2015, Fujii et al(2012), Gatheral et al(2012), Gulisashvili (2012), Kusuoka et al(2000), Lütkebohmert (2004), Shiraya et al(2017), Takahashi et al(2014), , Yoshida (2003)], see also [Albeverio et al(2012), Imkeller et al(2009), Peszat et al(2005)] for applications of similar expansion to other areas.…”