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Introduction to Merton Jump Diffusion Model

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Introduction to Merton Jump Diffusion Model © 2004 Kazuhisa Matsuda All rights reserved. Introduction to Merton Jump Diffusion Model Kazuhisa Matsuda Department of Economics The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016-4309 Email: [email protected] http://www.maxmatsuda.com/ December 2004 Abstract This paper presents everything you need to know about Merton jump diffusion (we call it MJD) model. MJD model is one of the first beyond Black-Scholes model in the sense that it tries to capture the negative skewness and excess kurtosis of the log stock price density P(ln(SST / 0 )) by a simple addition of a compound Possion jump process. Introduction of this jump process adds three extra parameters λ , µ , and δ (to the original BS model) which give the users to control skewness and excess kurtosis of the P(ln(SST / 0 )) . Merton’s original approach for pricing is to use the conditional normality of MJD model and expresses the option price as conditional Black-Scholes type solution. But modern approach of its pricing is to use the Fourier transform method by Carr and Madan (1999) which is disccused in Matsuda (2004). © 2004 Kazuhisa Matsuda All rights reserved. 1 © 2004 Kazuhisa Matsuda All rights reserved. [1] Model Type In this section the basic structure of MJD model is described without the derivation of the model which will be done in the next section. MJD model is an exponential Lévy model of the form: Lt SSt = 0 e , where the stock price process {Stt ;0 ≤ ≤ T} is modeled as an exponential of a Lévy process {Ltt ;0 ≤≤T}. Merton’s choice of the Lévy process is a Brownian motion with drift (continuous diffusion process) plus a compound Poisson process (discontinuous jump process) such that: σ 2 Nt Lktt=−()αλ− t+σB+∑Yi, 2 i=1 where {Bt ;0 ≤≤tT} is a standard Brownian motion process. The term 2 σ N (α −−λkt)+σB is a Brownian motion with drift process and the term t Y is a 2 t ∑i=1 i compound Poisson jump process. The only difference between the Black-Scholes and the N N MJD is the addition of the term t Y . A compound Poisson jump process t Y ∑i=1 i ∑i=1 i contains two sources of randomness. The first is the Poisson process dNt with intensity (i.e. average number of jumps per unit of time) λ which causes the asset price to jump randomly (i.e. random timing). Once the asset price jumps, how much it jumps is also modeled random (i.e. random jump size). Merton assumes that log stock price jump size 2 follows normal distribution, ()dxi ∼ i..i d. Normal(µ,δ ): 2 1 ()dxi − µ fd( xi ) =exp{−2 }. 2πδ 2 2δ It is assumed that these two sources of randomness are independent of each other. By introducing three extra parameters λ , µ , and δ to the original BS model, Merton JD model tries to capture the (negative) skewness and excess kurtosis of the log return density P (ln (SSt / 0 )) which deviates from the BS normal log return density. Lévy measure (dx) of a compound Poisson process is given by the multiplication of the intensity and the jump size density fd()x: ()dx = λ f (dx). 2 © 2004 Kazuhisa Matsuda All rights reserved. A compound Poisson process (i.e. a piecewise constant Lévy process) is called finite activity Lévy process since its Lévy measure (dx) is finite (i.e. the average number of jumps per unit time is finite): ∞ ()dx = λ <∞. ∫−∞ The fact that an asset price St is modeled as an exponential of Lévy process Lt means S that its log-return ln( t ) is modeled as a Lévy process such that: S0 2 Nt St σ ln( ) ==Lktt(αλ− −)t+σB+∑Yi. S0 2 i=1 Let’s derive the model. [2] Model Derivation In MJD model, changes in the asset price consist of normal (continuous diffusion) component that is modeled by a Brownian motion with drift process and abnormal (discontinuous, i.e. jump) component that is modeled by a compound Poisson process. Asset price jumps are assumed to occur independently and identically. The probability that an asset price jumps during a small time interval dt can be written using a Poisson process dNt as: Pr {an asset price jumps once in dt } = Pr{ dNt = 1} ≅ λdt , Pr {an asset price jumps more than once in dt } = Pr{ dNt ≥ 2 } ≅ 0 , Pr {an asset price does not jump in dt } = Pr{ dNt = 0 } ≅ 1− λdt , where the parameter λ ∈ R+ is the intensity of the jump process (the mean number of jumps per unit of time) which is independent of time t . Suppose in the small time interval dt the asset price jumps from St to yStt (we call yt as absolute price jump size). So the relative price jump size (i.e. percentage change in the asset price caused by the jump) is: dStty St− St = =yt −1, SStt 3 © 2004 Kazuhisa Matsuda All rights reserved. where Merton assumes that the absolute price jump size yt is a nonnegative random 2 variables drawn from lognormal distribution, i.e. ln( yit ) ∼ .i.d.N(µ,δ ) . This in turn 1 2 µ+ δ 22 2 22µδ+ δ implies that Ey[]t = e and Ey[( tt− E[y]) ] =e (e −1) . This is because if 1 2 ab+ 22 ln x ∼ Na( ,b) , then x ∼ Lognormal(,e 2 e2ab+ (eb −1)). MJD dynamics of asset price which incorporates the above properties takes the SDE of the form: dSt =−()αλkdt+σdBtt+(y−1)dNt, (1) St where α is the instantaneous expected return on the asset, σ is the instantaneous volatility of the asset return conditional on that jump does not occur, Bt is a standard Brownian motion process, and Nt is an Poisson process with intensity λ . Standard assumption is that (Bt ) , (Nt ) , and (yt ) are independent. The relative price jump size 1 µδ+ 2 2 of St , yt −1, is lognormally distributed with the mean Ey[1t − ]=e −1≡k and the 22µδ+ 22δ 1 variance Ey[( tt−−1 E[ y−1]) ] =e (e −1) . This may be confusing to some readers, so we will repeat it again. Merton assumes that the absolute price jump size yt is a lognormal randon variable such that: 1 2 µδ+ 22 2 2µδ+ δ ()yt ∼ i.i.d.Lognormal(e ,e (e −1)). (2) This is equivalent to saying that Merton assumes that the relative price jump size yt −1 is a lognormal random variable such that: 1 2 µδ+ 22 2 2µδ+ δ (1yt −≡)∼ i.i.d.Lognormal(k e −1,e (e −1)). (3) This is equivalent to saying that Merton assumes that the log price jump size ln yt≡ Yt is a normal random variable such that: 2 ln( yit ) ∼ .i.d.Normal(µ,δ ) . (4) yS This is equivalent to saying that Merton assumes that the log-return jump size ln( tt) is St a normal random variable such that: 1For random variable x , Variance[1x − ]= Variance[x]. 4 © 2004 Kazuhisa Matsuda All rights reserved. yStt 2 ln( ) =≡ln(yYtt) ∼ i.i.d.Normal(µ,δ ) . (5) St It is extremely important to note: 1 µδ+ 2 2 Ey[ tt−=1] e −1 ≡k ≠ E[ln(y)] =µ , because ln Ey[ tt−≠1] E[ln(y−1)] =E[ln(yt)]. The expected relative price change Ed[ St/ St] from the jump part dNt in the time interval dt is λkdt since E[(ytt−=1)dN ] E[yt−1]E[dNt] =kλdt . This is the predictable part of the jump. This is why the instantaneous expected return on the asset αdt is adjusted by −λkdt in the drift term of the jump-diffusion process to make the jump part an unpredictable innovation: dSt E[]=−Ek[(αλ)dt]+E[σdBtt]+E[(y−1)dNt] St dS E[]t =−(α λλk)dt +0+kdt =αdt . St Some researchers include this adjustment term for predictable part of the jump −λkdt in the drift term of the Brownian motion process leading to the following simpler (?) specification: dSt =+ασdt dBtt+(1y −)dNt St λkt B ∼ Normal(,− t) t σ dS λkdt E[]t =+αdt σ(− )+λkdt =αdt . St σ But we choose to explicitely subtract λkdt from the instantsneous expected return αdt because we prefer to keep Bt as a standard (zero-drift) Brownian motion process. Realize that there are two sources of randomness in MJD process. The first source is the Poisson Process dNt which causes the asset price to jump randomly. Once the asset price jumps, how much it jumps (the jump size) is also random. It is assumed that these two sources of randomness are independent of each other. If the asset price does not jump in small time interval dt (i.e. dNt = 0 ), then the jump- diffusion process is simply a Brownian motion motion with drift process: 5 © 2004 Kazuhisa Matsuda All rights reserved. dSt =−()αλkdt+σdBt . St If the asset price jumps in dt (dNt = 1): dSt =−()αλkdt+σdBtt+(y−1), St the relative price jump size is yt −1. Suppose that the lognormal random drawing yt is 0.8, the asset price falls by 20%. Let’s solve SDE of (1). From (1), MJD dynamics of an asset price is: dStt=−(α λσk)Sdt+SdtBt+(yt−1)SdtNt. Cont and Tankov (2004) give the Itô formula for the jump-diffusion process as: ∂∂f (,Xt) f(X,t) σ 22∂f(,Xt) df (,X t)=+ttdt b dt +ttdt tt∂∂tx2 ∂x2 ∂fX(,t) ++σ t dB [(f X +∆X )−f (X )], tt∂x t−−tt where bt corresponds to the drift term and σ t corresponds to the volatility term of a N tt t jump-diffusion process X =+X b ds +σ dB +∆X .
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