An Approximation of the Cumulant Generating Functions of Diffusion Models and the Pseudo-likelihood Estimation Method

Abstract

Di ffusi on i s a basi c mathemati cal tool for modern fi nanci al engi neeri ng. The theory of the esti mati on methods for di ffusi on model s i s an i mportant topi c of the fi nanci al engi neeri ng. Many researches have been tri ed to appl y the l i kel i hood esti mati on method for esti mati ng di ffusi on model s. However, the l i kel i hood esti mati on method for di ffusi on i s compl i cated and needs much amount of computi ng.In thi s paper we devel op the esti mati on methods whi ch are si mpl e enough to be compared to the Eul er approxi mati on method, and effi ci ent enough stati sti cal l y to be compared to the l i kel i hood esti mati on method. We devi se pseudo-l i kel i hood and propose the maxi mum pseudo-l i kel i hood esti mati on methods. The pseudo-l i kel i hoods are obtai ned by approxi mati ng the transi ti on densi ty wi th normal di stri buti ons. The means and the vari ances of the di stri buti ons are obtai ned from the del ta expansi on suggested by Lee, Song and Lee (2012).We compare the newl y suggested esti mators wi th other exi sti ng esti mators by si mul ati on study. From the si mul ati on study we fi nd the maxi mum pseudo-l i kel i hood esti mator has very si mi l ar properti es wi th the maxi mum l i kel i hood esti mator. Al so the maxi mum pseudo-l i kel i hood esti mator i s easy to appl y to general di ffusi on model s, and can be obtai ned by si mpl e numeri cal steps.Keyword：Di ffusi on Model s, Transi ti on Densi ty, Del ta Expansi on, The Maxi mum Li kel i hood Esti mator 한국경영과학회지 제38권 제1호