introduction to merton jump diffusion model - …：默顿跳跃扩散模型的介绍—….pdf

? 2004 Kazuhisa Matsuda All rights reserved. Introduction t o Merton Jump Di ffusion Model Kazuhisa Matsuda Department of Econom ics The Graduate Cent e r, The City University of New York, 365 Fifth Avenue, New York, NY 10016-4309 Email: m a xmatsuda@ m / Decem ber 2004 Abstract This paper presents everything you need to know about Merton jum p diffusion (we call it MJD) m odel . MJD m odel is one of the first be yond Black-Scholes m odel in the sense that it trie s to ca pture the ne gativ e skewness and excess kurto sis of the log stock price density ( ) 0 ln ( / ) T SS P by a sim p le addition of pound Po ssion jump process. Introduction of this jum p process adds three ex tra p a ram e ters λ, μ , and δ (to the original BS model) which give the users to control sk ewness and excess kurto sis of the ( ) 0 ln ( / ) T SS P . Merton ’s original app r oach for pricin g is to use th e conditional norm a lity of MJD m o del and express e s the op tion price as con d ition a l Bla c k-Scholes type solution . But m odern approach of its pricing is to use the Fourier transform m e thod by Carr and Madan (1999) which is used in Matsuda (2004). ? 2004 Kazuhisa Matsuda All rights reserved. 1 ? 2004 Kazuhisa Matsuda All rights reserved. [1] Model T y pe In this section the basic stru cture of MJD m odel is describe d without the de rivation of the model which will be do ne in the n e xt section. MJD m odel is an exponentia l Lévy model of the form : 0 e L t SS = , where the s t ock price process { ; 0 } t St T ≤≤ is m o deled as an exponential of a Lévy p r o c e s s { . Merton’s choice of the Lévy pro cess is a Brownian motion with drift (continuous diffusion process) plus pound Poisson process (discontinuous jum p process) such that: ; 0 } t Lt T ≤≤ 2 1 () 2 t N tt i Lk t B σαλσ= =??+ + i Y ∑, w h e r e { ; 0 } t B t T ≤≤ is a standard Brownian m o tion process. T h e term 2 ( ) 2 t kt B σαλσ??+ is a Bro