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).

[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).

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

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].

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:

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 . By applying this: ts0 ∫∫00ss∑ i i=1

22 2 ∂∂ln SSttln σ St∂ln St d ln Stt=+dt (αλ−k)S dt + 2 dt ∂∂tStt2 ∂S

∂ ln St ++σ SdttB[ln ytSt−ln St] ∂St 22 11σ St ⎛⎞ 1 dSln tt=−(αλk)S dt+ ⎜⎟−2 dt+σStdBt+[ln yt+ln St−ln St] SStt2 ⎝⎠ St σ 2 dSln =−(αλk)dt− dt+σdB+ln y tt2 tt 2 σ N ln SS−=ln (αλ−−k)(t−0) +σ(B−B) +t ln y tt002 t∑i=1 i 2 σ N ln SS=+ln (αλ−−k)t+σB+t ln y tt0 2 t∑i=1 i

2 ⎧⎫σ N exp ln SS=+exp ln (αλ−−k)t+σB+t ln y ()tt⎨⎬0 t∑i=1 i ⎩⎭2 2 ⎪⎪⎧⎫⎛⎞σ Nt SS=−exp αλ−kt+σBexp ln y tt0 ⎨⎬⎜⎟t()∑i=1 i ⎩⎭⎪⎪⎝⎠2 2 σ Nt SS=−exp[(αλ−k)t+σB] y, tt0 Π i 2 i=1 or alternatively as:

σ 2 Nt SStt=−0 exp[(αλ−k)t+σB+∑ln yi]. 2 i=1

Using the previous definition of the log price (return) jump size ln(yYtt) ≡ :

σ 2 Nt SSt=−0 exp[(αλ−k)t+σBt+∑Yi] . (6) 2 i=1

This means that the asset price process {Stt ;0 ≤ ≤ T} is modeled as an exponential Lévy model of the form:

Lt SSt = 0 e ,

where X t is a Lévy process which is categorized as a Brownian motion with drift (continuous part) plus a compound Poisson process (jump part) such that:

σ 2 Nt Lktt=−()αλ− t+σB+∑Yi. 2 i=1 S In other words, 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

Nt Note that the compound Poisson jump process y =1 (in absolute price scale) or Π i i=1

NNtt ∑ln yi=∑Yi=0 (in log price scale) if Nt = 0 (i.e. no jumps between time 0 and t ) or ii==11 positive and negative jumps cancel each other out.

In the Black-Scholes case, log return ln(SSt / 0 ) is normally distributed:

σ 2 SS=−exp[(ασ)t+B] tt0 2 S σ 2 ln( t ) ∼ Normal[(α − )t,σ 2t]. S0 2

Nt But in MJD case, the existence of compound Poisson jump process ∑Yi makes log i=1 return non-normal. In Merton’s case the simple distributional assumption about the log 2 return jump size ()YNi ∼ (µ,δ ) enables the probability density of log return

xt= ln(SSt/ 0 ) to be obtained as a quickly converging series of the following form:

∞ PP()xtt∈=AN∑ (=i)P(xt∈ANt=i) i=0 ∞ −λti 2 et()λσ 22 P()xtt=−∑ Nx(;(α −λµk)t+i ,σt+iδ) (7) i=0 i!2

σ 2 where Nx(;(α −−λµk)t+i ,σ22t+iδ) t 2 2 ⎡ ⎪⎧⎛⎞σ 2 ⎪⎫⎤ ⎢xkt −−⎨⎜⎟αλ−t+iµ⎬⎥ 1 ⎣⎢ ⎩⎭⎪⎝⎠2 ⎪⎦⎥ =−exp[ 22 ] . 2(πσ22ti+ δ ) 2(σδti+ )

e−λti(λt) The term P()Ni== is the probability that the asset price jumps i times during t i! σ 2 the time interval of lengtht . And P()x ∈=AN i =N(x;(α −−λµk)t+i ,σ2t+iδ2) tt t2 is the Black-Scholes normal density of log-return assuming that the asset price jumps i times in the time interval of t . Therefore, the log-return density in the MJD model can be interpreted as the weighted average of the Black-Scholes normal density by the probability that the asset price jumps i times.

By Fourier transforming the Merton log-return density function with FT parameters (,ab,)= (1,1), its characteristic function is calculated as:

∞ φω()= exp()ixω P(x)dx ∫−∞ ttt

⎡⎤⎧⎫1122 =−exp ⎢⎥λωtiexp ⎨⎬()2 µδω−λt()1+iωk−tω{}−2iα+σ()i+ω. ⎣⎦⎩⎭22

After simplification:

φ()ωψ= exp[t (ω)] with the characteristic exponent (cumulant generating function):

⎪⎪⎧⎫⎛⎞δ 22ω⎛σ2 ⎞σ2ω2 ψ ()ωλ=−⎨⎬exp⎜⎟iiωµ −1+ω⎜α−−λk⎟−, (8) ⎩⎭⎪⎪⎝⎠22⎝ ⎠2

1 µδ+ 2 where ke≡2 −1. The characteristic exponent (8) can be alternatively obtained by substituting the Lévy measure of the MJD model:

2 λ ⎪⎪⎧⎫()dx − µ ()dx =−exp⎨⎬=λ f (dx) 2 2δ 2 2πδ ⎩⎭⎪⎪ fd()x∼ N(µ,δ 2 ) into the Lévy-Khinchin representation of the finite variation type (read Matsuda (2004)):

22 σω ∞ ψ (ωω) =−ib +{}exp(iωx) −1 (dx) 2 ∫−∞ 22 σω ∞ ψ (ωω) =−ib +{}exp(iωx) −1 λf (dx) 2 ∫−∞ 22 σω ∞ ψ (ωω) =−ib +λ{}exp(iωx) −1 f (dx) 2 ∫−∞ 22 σω ∞∞ ψ ()ωω=−ib +λeixω f (dx)−f (dx) 2 {∫∫−∞ −∞ }

∞ Note that efixω (dx) is the characteristic function of fd()x: ∫−∞

22 ∞ ⎛⎞δ ω efixω ()edx=−xpiµω . ∫−∞ ⎜⎟ ⎝⎠2

Therefore:

σω22 ⎪⎧ ⎛⎞δω22 ⎪⎫ ψ ()ωω=−ib +λ⎨exp⎜⎟iµω− −1⎬ , 22⎩⎭⎪ ⎝⎠⎪

σ 2 where b=−α −λk. This corresponds to (8). Characteristic exponent (8) generates 2 cumulants as follows:

σ 2 cumulant =−α −λk +λµ, 1 2 222 cumulant2 =+σ λδ +λµ , 23 cumulant3 =+λ(3δµ µ) , 4224 cumulant4 =+λ(3δµ6 δ+µ) .

Annualized (per unit of time) mean, variance, skewness, and excess kurtosis of the log- return density P(xt ) are computed from above cumulants as follows:

2 1 ⎛⎞µδ+ 2 σ 2 E[]xt ==cumulant1 α −−λλ⎜⎟e −1 +µ 2 ⎝⎠ 222 Variance[xt ] ==cumulant2 σ +λδ +λµ 23 cumulant3 λδ(3 µ+ µ) Skewness[]xt ==3/2 3/2 222 ()cumulant2 ()σλ++δλµ 4224 cumulant4 λ(3δµ+6 δ+µ) Excess Kurtosis[]xt ==4/2 2 . (9) 222 ()cumulant2 ()σλ++δλµ

We can observe several interesting properties of Merton’s log-return density P(xt ) .

Firstly, the sign of µ which is the expected log-return jump size, EY[ t ] = µ , determines the sign of skewness. The log-return density P(xt ) is negatively skewed if µ < 0 and it is symmetric if µ = 0 as illustrated in Figure 1.

3.5 3 y 2.5 t µ=−0.5 i 2 s

n 1.5 µ=

e 0

D 1 0.5 µ=0.5 0 -0.75 -0.25 0 0.25 0.75 log −return xt

Figure 1: Merton’s Log-Return Density for Different Values of µ . µ =−0.5 in blue, µ = 0 in red, and µ = 0.5 in green. Parameters fixed are τ = 0.25, α = 0.03, σ = 0.2 , λ =1, and δ = 0.1.

Table 1 Annualized Moments of Merton’s Log-Return Density in Figure 1

Model Mean Standard Deviation Skewness Excess Kurtosis

µ =−0.5 -0.0996 0.548 -0.852 0.864 µ = 0 0.005 0.3742 0 0.12 µ = 0.5 -0.147 0.5477 0.852 0.864

Secondly, larger value of intensity λ (which means that jumps are expected to occur more frequently) makes the density fatter-tailed as illustrated in Figure 2. Note that the excess kurtosis in the case λ =100 is much smaller than in the case λ =1 or λ =10 . This is because excess kurtosis is a standardized measure (by standard deviation).

3.5 3

y 2.5 t λ=1 i 2 s

n 1.5 λ=

e 10

D 1 0.5 λ=100 0 -0.75 -0.25 0 0.25 0.75 log −return xt

Figure 2: Merton’s Log-Return Density for Different Values of Intensity λ . λ =1 in blue, λ =10 in red, and λ =100 in green. Parameters fixed areτ = 0.25,α = 0.03, σ = 0.2 , µ = 0 , and δ = 0.1. Table 2 Annualized Moments of Merton’s Log-Return Density in Figure 2

Model Mean Standard Deviation Skewness Excess Kurtosis

λ =1 0.00499 0.2236 0 0.12 λ =10 -0.04012 0.3742 0 0.1531 λ =100 -0.49125 1.0198 0 0.0277

Also note that Merton’s log-return density has higher peak and fatter tails (more leptokurtic) when matched to the Black-Scholes normal counterpart as illustrated in Figure 9.3.

3 2.5 y

t 2 i

s 1.5 n µ=−0.5 e

D 1 0.5 BS 0 -1 -0.5 0 0.5 1 log −return xt

Figure 3: Merton Log-Return Density vs. Black-Scholes Log-Return Density (Normal). Parameters fixed for the Merton (in blue) are τ = 0.25, α = 0.03, σ = 0.2 , λ =1, µ =−0.5 , and δ = 0.1. Black-Scholes normal log-return density is plotted (in red) by matching the mean and variance to the Merton.

Table 3 Annualized Moments of Merton vs. Black-Scholes Log-Return Density in Figure 3

Model Mean Standard Deviation Skewness Excess Kurtosis

Merton with µ =−0.5 -0.0996 0.548 -0.852 0.864

Black-Scholes -0.0996 0.548 0 0

[3] Log Stock Price Process for Merton Jump-Diffusion Model

Log stock price dynamics can be obtained from the equation (6) as:

⎛⎞σ 2 Nt ln SStt=+ln 0 ⎜⎟αλ−−kt+σB+∑Yi. (10) ⎝⎠2 i=1

Probability density of log stock price ln St is obtained as a quickly converging series of the following form (i.e. conditionally normal):

∞ PP(ln SAtt∈=) ∑ (N=i)P(ln SAt∈Nt=i) i=0 ∞ et−λti()λσ⎛⎞⎛⎞2 P(ln SN) =+ln S;ln Sα −−λµkt+i,σ2t+iδ2, (11) tt∑ ⎜⎟0 ⎜⎟ i=0 i!2⎝⎠⎝⎠ where:

⎛⎞⎛⎞σ 2 NSln ;ln S+−α −λµkt+i,σ22t+iδ ⎜⎟t 0 ⎜⎟ ⎝⎠⎝⎠2 2 ⎡ ⎧ ⎛⎞⎛⎞2 ⎫ ⎤ ⎢ ⎪ σ ⎪ ⎥ ⎨ln SSt −+⎜⎟ln 0 ⎜⎟αλ−−kt+iµ⎬ 1 ⎢ ⎪ ⎝⎝2 ⎠⎠⎪ ⎥ =−exp ⎩⎭. ⎢ 22 ⎥ 2πσ()22ti+⎢δ 2()σδti+ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥

By Fourier transforming (11) with FT parameters (,ab,)= (1,1), its characteristic function is calculated as:

∞ φω()= exp()iSωln P(lnS)dlnS ∫−∞ ttt ⎡⎤⎛⎞δω22 ⎛σ2 ⎞σ2ω2 =−exp ⎢⎥λµti⎜⎟exp( ω ) −1 +iω⎜ln S0 +(α−−λk)t⎟−t, (12) ⎣⎦⎝⎠22⎝ ⎠2

δ 2 µ+ where ke=−2 1.

[4] Lévy Measure for Merton Jump-Diffusion Model

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).

The Lévy measure (dx) represents the arrival rate (i.e. total intensity) of jumps of sizes [,x xd+ x]. In other words, we can interpret the Lévy measure (dx) of a compound Poisson process as the measure of the average number of jumps per unit of time. Lévy measure is a positive measure on R , but it is not a probability measure since its total mass λ (in the compound Poisson case) does not have to equal1:

∫ ()dx =∈λ R+ .

A Poisson process and a compound Poisson process (i.e. a piecewise constant Lévy process) are called finite activity Lévy processes since their Lévy measures (dx) are finite (i.e. the average number of jumps per unit time is finite):

∞ ()dx < ∞ . ∫−∞

2 In Merton jump-diffusion case, the log-return jump size is (dxi ) ∼ i..i d. Normal(µ,δ ):

2 1 ()dxi − µ fd( xi ) =exp{−2 }. 2πδ 2 2δ

Therefore, the Lévy measure (dx) for Merton case can be expressed as:

λ(dx − µ)2 (dx) ==λ f (dx) exp{−2 }. (13) 2πδ 2 2δ

An example of Lévy measure (dx) for the log-return xt= ln(SSt/ 0 ) in MJD model is plotted in Figure 5. Each Lévy measure is symmetric (i.e. µ = 0 is used) with total mass 1, 2, and 4 respectively.

15 12.5 y

t 10 λ=1 i

s 7.5 n λ=

e 2

D 5 2.5 λ=4 0 -0.4 -0.2 0 0.2 0.4 log −return xt

Figure 5: Lévy Measures (dx) for the Log-Return xt= ln(SSt/ 0 ) in MJD Model for Different Values of Intensity λ . Parameters used are µ = 0 and δ = 0.1.

[5] Option Pricing: PDE Approach by Hedging

Consider a portfolio P of the one long option position VS(,t) on the underlying asset S written at time t and a short position of the underlying asset in quantity ∆ to derive option pricing functions in the presence of jumps:

PVtt= (S,t) −∆St. (14)

Portfolio value changes by in a very short period of time:

dPtt= dV (S ,t) −∆dSt. (15)

MJD dynamics of an asset price is given by equation (1) in the differential form as:

dSt =−()αλkdt+σdBtt+(y−1)dNt, St

dStt=−(α λσk)Sdt+SdtBt+(yt−1)SdtNt. (16)

Itô formula for the jump-diffusion process is given as (Cont and Tankov (2004)):

∂∂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 . Apply this to our case of ts0 ∫∫00ss∑ i i=1 option price functionVS(,t):

22 2 ∂∂VVσ St ∂V∂V dV (,Sttt)=+dt (αλ−k)S dt + 2 dt +σStdBt ∂∂tStt2 ∂S∂St

+−[Vy( ttS,t) V(St,t)]dNt. (17)

The term [Vy( ttS,t) −V(St,t)]dNt describes the difference in the option value when a jump occurs. Now the change in the portfolio value can be expressed as by substituting (16) and (17) into (15):

dPtt=−dV (S ,t) ∆dSt 22 2 ∂∂VVσ St ∂V∂V dPtt=+dt ()αλ−k S dt + 2 dt +σStdBt+[V (ytSt,t)−V (St,t)]dNt ∂∂tStt2 ∂S∂St

−∆{(α −λσkS) ttdt+ SdBt+ (yt−1)StdNt} 22 2 ⎧⎫∂∂VVσ St ∂V ⎛∂V⎞ dPtt=⎨⎬+()αλ− k S + 2 −∆()αλ− k Stdt +⎜⎟σSt−∆σStdBt ⎩⎭∂∂tStt2 ∂S ⎝∂St⎠

+−{Vy(,ttSt)V(St,t)−∆(1yt−)St}dNt. (18)

If there is no jump between time 0 and t (i.e. dNt = 0 ), the problem reduces to Black-

Scholes case in which setting ∆=∂V/ ∂St makes the portfolio risk-free leading to the following (i.e. the randomness dBt has been eliminated):

22 2 ⎧⎫∂∂VVσ St ∂V∂V ⎛∂V∂V⎞ dPtt=+⎨⎬()αλ−k S + 2 −()αλ−k Stdt +⎜⎟σSt−σStdBt ⎩⎭∂∂tStt2 ∂S∂St ⎝∂St∂St⎠ 22 2 ⎧⎫∂∂VVσ St dPt =+⎨⎬2 dt . ⎩⎭∂∂tS2 t

This in turn means that if there is a jump between time 0 and t (i.e. dNt ≠ 0 ), setting

∆=∂V/ ∂St does not eliminate the risk. Suppose we decided to hedge the randomness caused by diffusion part dBt in the underlying asset price (which are always present) and

not to hedge the randomness caused by jumps dNt (which occur infrequently) by setting

∆=∂V/ ∂St . Then, the change in the value of the portfolio is given by from equation (9.18):

22 2 ⎧⎫∂∂VVσ St ∂V∂V ⎛∂V∂V⎞ dPtt=+⎨⎬()αλ−k S + 2 −()αλ−k Stdt +⎜⎟σSt−σStdBt ⎩⎭∂∂tStt2 ∂S∂St ⎝∂St∂St⎠ ⎧⎫∂V +−⎨⎬Vy(,ttSt)V(St,t)−(yt−1)St dNt, ⎩⎭∂St 22 2 ⎧⎫∂∂VVσ St ⎧ ∂V⎫ dPtt=+⎨⎬2 dt +⎨V (,y Stt)−V (St,t)−(1yt−)St⎬dNt. (19) ⎩⎭∂∂tS2 tt⎩ ∂S⎭

Merton argues that the jump component ( dNt ) of the asset price process St is uncorrelated with the market as a whole. Then, the risk of jump is diversifiable (non- systematic) and it should earn no risk premium. Therefore, the portfolio is expected to grow at the risk-free interest rate r :

Ed[ Pt] = rPtdt. (20)

After substitution of (14) and (19) into (20) by setting ∆ =∂VS/ ∂ t :

22 2 ⎧⎫∂∂VVσ St ⎧ ∂V⎫ E[⎨⎬++2 dt ⎨V (yttS ,t)−V (,Stt)−(yt−1)St⎬dNt]=r{V (,Stt)−∆St}dt ⎩⎭∂∂tS2 tt⎩ ∂S⎭ 22 2 ∂∂VVσ St ∂V ∂V ( ++2 )dt E[V (yttS ,t)−V (,Stt)−(yt−1)St]E[dNt]=r{V (,Stt)−St}dt ∂∂tS2 tt∂S ∂St 22 2 ∂∂VVσ St ∂V ∂V ( ++2 )dt E[V (yttS ,t)−V (,Stt)−(yt−1)St]λdt =r{V (,Stt)−St}dt ∂∂tS2 tt∂S ∂St 22 2 ∂∂VVσ St ∂V ∂V ++2 λE[V(yttS,t)−V(,Stt )−(yt−1)St]=r{V(,Stt )−St} ∂∂tS2 tt∂S ∂St

Thus, the MJD counterpart of Black-Scholes PDE is:

22 2 ∂∂VVσ St ∂V ∂V ++2 rStt−rV +λE[(V y St,t)−V (St,t)]−λStE[yt−1]=0. (21) ∂∂tS2 tt∂S ∂St

where the term EV[ (yttS,t) −V(St,t)] involves the expectation operator and 1 µδ+ 2 2 Ey[1t −=]e −1≡k (which is the mean of relative asset price jump size). Obviously, if jump is not expected to occur (i.e.λ = 0 ), this reduces to Black-Scholes PDE:2

22 2 ∂∂VVσ St ∂V ++2 rSt −rV =0. ∂∂tS2 tt∂S

Merton’s simple assumption that the absolute price jump size is lognormally distributed 2 (i.e. the log-return jump size is normally distributed, Yytt≡ ln( ) ∼ N(µ,δ ) ) makes it possible to solve the jump-diffusion PDE to obtain the following price function of European vanilla options as a quickly converging series of the form:

∞ e−λτ ()λτ i ∑ VSBS (,t τ=−Tt,σi ,ri ), (22) i=0 i!

1 µ+ δ 2 where λλ=+(1 ke) =λ 2 , iδ 2 σσ22=+ , i τ

1 2 1 2 i()µ + δ ikln(1+ ) µδ+ rr=−λλk+ =r−(1e2 −)+ 2 , i ττ and VBS is the Black-Scholes price without jumps.

Thus, MJD option price can be interpreted as the weighted average of the Black-Scholes price conditional on that the underlying asset price jumps i times to the expiry with weights being the probability that the underlying jumps i times to the expiry.

[6] Option Pricing: Martingale Approach

Let {Bt ;0 ≤≤tT} be a standard Brownian motion process on a space (,Ω F ,P). Under actual probability measure P , the dynamics of MJD asset price process is given by equation (6) in the integral form:

σ 2 Nt SStt=−0 exp[(αλ−k)t+σB+∑Yk]. 2 k =1

2 This equation not only contains local derivatives but also links together option values at discontinuous values in S. This is called non-local nature.

Nt Nt We changed the index from ∑Yi to ∑Yk . This is trivial but readers will find the reason i=1 k=1 soon. MJD model is an example of an incomplete model because there are many equivalent martingale risk-neutral measures Q∼ P under which the discounted asset −rt price process {;eSt 0≤≤tT} becomes a martingale. Merton finds his equivalent martingale risk-neutral measure QM ∼ P by changing the drift of the Brownian motion process while keeping the other parts (most important is the jump measure, i.e. the distribution of jump times and jump sizes) unchanged:

2 Nt σ QM SSt=−0 exp[(r −λσk)t+Bt+∑Yk] under QM . (23) 2 k=1

QM Note that Bt is a standard Brownian motion process on (Ω,F ,QM ) and the process −rt Merton {;eSt 0≤≤tT} is a martingale under QM . Then, a European option price Vt(,St ) with payoff function HS( T ) is calculated as:

Merton −−r ()T t QM Vt(,St)= eE[H(ST)Ft]. (24)

Standard assumption is Ft= St, thus:

2 NTt− Merton −−r()T t QQM σ M Vt( , Stt) =−eE[H(Sexp[(r−λσk)(T−t) +BT−t+∑Yk]) St] 2 k=1

2 NTt− Merton −−r()T t QQM σ M Vt( , Stt) =−eE[H(Sexp[(r−λσk)(T−t) +BT−t+∑Yk])] . (25) 2 k=1

Poisson counter is (we would like to use index i for the number of jumps):

NiTt− = 0,1, 2,... ≡ .

And the compound Poisson process is distributed as:

NTt− 2 ∑YNk ∼ ormal(,iµ iδ ). k=1

Merton Thus, V(,tSt ) can be expressed as from equation (25) (i.e. by conditioning oni ):

Merton Vt(,St ) 2 i −−rT()t QQM σ M ==eN∑∑QMT( −−ti)E[H(Stexp[(r−−λσk)(T−t) +BTt+Yk])] . i≥0 2 k =1

Use τ =Tt−:

Merton Vt(,St ) −λτ i 2 i ⎡⎤⎛⎞2 −+rτµe ()λτ QQM σ δ/2 M =−eE∑∑⎢⎥H⎜⎟Stkexp[{r−λτ(e−1)} +σBτ +Y] . (26) i≥0 i!2⎣⎦⎝⎠k =1

Inside the exponential function is normally distributed:

2 i 2 σ µδ+ /2 QM {(re−−λτ−1)}+σBτ +∑Yk 2 k=1 2 ⎛⎞σ µδ+ 2 /2 2 2 ∼ Normal ⎜⎟{(r −−λ e −1)}τµ+i ,στ+iδ. ⎝⎠2

Rewrite it so that its distribution remains the same:

222 σσ2 τ+ iδ {(re−−λτµδ+ /2 −1)}+iµ+ BQM 2 τ τ 2 ⎛⎞σ µδ+ 2 /2 2 2 ∼ Normal ⎜⎟{(r −−λ e −1)}τµ+i ,στ+iδ. ⎝⎠2

Now we can rewrite equation (24) as (we can do this operation because a normal density is uniquely determined by only two parameters: its mean and variance):

Merton Vt(,St )

−λτ i 222 ⎡⎤⎛⎞2 −+rτµei()λτ QM σ δ/2 σ τ + δ QM =−eE∑ ⎢⎥H⎜⎟St exp[{r−λτ(e−1)} +iµ+Bτ ] i≥0 i!2⎢⎥⎜⎟τ ⎣⎦⎝⎠.

iiδδ22 We can always add (−)=0 inside the exponential function: 22ττ

−λτ i Merton −rτ e ()λτ Vt(,St )=×e ∑ i≥0 i! ⎡⎤⎛⎞222 2 Q σδiiδ 2 iδQ EHM ⎢⎥⎜⎟Sexp[{r−+( −) −λτ(eµδ+ /2 −1)} +iµ+σ2 +BM ] ⎜⎟t 22ττ2 ττ ⎣⎦⎢⎥⎝⎠ −λτ i Merton −rτ e ()λτ Vt(,St )=×e ∑ i≥0 i!

⎡⎤⎛⎞22 2 Q 1 iiδδ 2 iδQ EHM ⎢⎥⎜⎟Sexp[{r−+(σλ2/) +−(eµδ+ 2−1)}τ+iµ+σ2+BM ] ⎜⎟t 22ττ ττ ⎣⎦⎢⎥⎝⎠

iδ 2 Set σσ22=+ and rearrange: i τ

Merton Vt(,St ) −λτ i 2 ⎡⎤⎛⎞2 −rτ ei()λτ QM 12/δ µδ+ 2 QM =−eE∑ ⎢⎥H⎜⎟Stiexp[{rσλ+−(e−1)}τ+iµ+σiBτ ] i≥0 i!2⎣⎦⎝⎠2τ −λτ i 2 ⎡⎤⎛⎞2 −rτµei()λτ QM δ +δ/2 12 QM =+eE∑ ⎢⎥H⎜⎟Stiexp{iµλ−(e−1)τ}exp{(r−σ)τ+σiBτ } . i≥0 i!2⎣⎦⎝⎠2

Black-Scholes price can be expressed as:

1 VTBS (τσ=−t, S; ) =e−rτ EQQBS [H{Sexp(r−σ2 )τ+σBBS }] . tt2 τ

Finally, MJD pricing formula can be obtained as a weighted average of Black-Scholes price conditioned on the number of jumpsi :

Merton Vt(,St ) −λτ i 22 ei()λτδBS µδ+ 2 /2 2 iδ =≡∑ VS(τµ, itSexp{i+−λ(e−1)τ};σi≡σ+) . (27) i≥0 i!2 τ

Alternatively:

Merton Vt(,St )

−λτ i 2 ⎡⎤⎧⎫⎛⎞2 −+rτµei()λτ QM ⎪⎪δ/2 µ + iδ /212 QM =−eE∑ ⎢⎥H⎨⎬Stiexp{⎜⎟rλσ(e−1) +−τ+σiBτ } i≥0 i!2⎣⎦⎢⎥⎩⎭⎪⎪⎝⎠τ −λτ i 22 ei()λτ BS 2δ µδ+ 2 /2iµ + iδ /2 =≡∑ VS(,τσti; σ+,ri≡r−λ(e−1)+). (28) i≥0 i! ττ

1 µ+ δ 2 where λλ=+(1 ke) =λ 2 . As you might notice, this is the same result as the option pricing formula derived from solving a PDE by forming a risk-free portfolio in equation (22). PDE approach and Martingale approach are different approaches but they are related and give the same result.

[7] Option Pricing Example of Merton Jump-Diffusion Model

In this section the equation (22) is used to price hypothetical plain vanilla options: current stock price St = 50, risk-free interest rate r = 0.05, continuously compounded dividend yield q = 0.02, time to maturity τ = 0.25 years.

We need to be careful about volatilityσ . In the Black-Scholes case, the t -period standard deviation of log-return xt is:

Standard Deviation BS (xt) = σ BSt . (29)

Equation (10) tells that the t -period standard deviation of log-return xt in the Merton model is given as:

222 Standard DeviationMerton (xtM) =+(σ ertonλδ +λµ )t . (30)

This means that if we set σ BS = σ Merton , MJD prices are always greater (or equal to) than Black-Scholes prices because of the extra source of volatility λ (intensity), µ (mean log-return jump size), δ (standard deviation of log-return jump) (i.e. larger volatility is translated to larger option price):

Standard DeviationBS (xtt) ≤ Standard DeviationMerton (x ) 222 σ BS tt≤+()σ Merton λδ +λµ .

This very obvious point is illustrated in Figure 6 where diffusion volatility is setσ BS =σ Merton =0.2 . Note the followings: (1) In all four panels MJD price is always greater (or equal to) than BS price. (2) When Merton parameters ( λ , µ , and δ ) are small in Panel A, the difference between these two prices is small. (3) As intensity λ increases (i.e. increased expected number of jumps per unit of time), the t -period Merton standard deviation of log-return xt increases (equation (30)) leading to the larger difference between Merton price and BS price as illustrated in Panel B. (4) As Merton mean log-return jump size µ increases, the t -period Merton standard deviation of log- return xt increases (equation (30)) leading to the larger difference between Merton price and BS price as illustrated in Panel C. (5) As Merton standard deviation of log-return jump sizeδ increases, the t -period Merton standard deviation of log-return xt increases (equation (30)) leading to the larger difference between Merton price and BS price as illustrated in Panel D.

c 15 i r P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

A) Merton parameters: λ = 1, µ = -0.1, and δ = 0.1.

c 15 i r P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

B) Merton parameters: λ = 5, µ = -0.1, and δ = 0.1.

c 15 i r P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

C) Merton parameters: λ = 1, µ = -0.5, and δ = 0.1.

c 15 i r P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

D) Merton parameters: λ = 1, µ = -0.1, and δ = 0.5.

Figure 6: MJD Call Price vs. BS Call Price When Diffusion Volatility σ is same.

Parameters and variables used are St = 50, r = 0.05, q = 0.02, τ = 0.25, and

σ BS ==σ Merton 0.2 .

Next we consider a more delicate case where we restrict diffusion volatilitiesσ BS and

σ Merton such that standard deviations of MJD and BS log-return densities are the same:

Standard DeviationBS (xtt) = Standard DeviationMerton (x ) , 222 σ BS tt=+()σ Merton λδ +λµ .

Using the Merton parameters λ = 1, µ = -0.1, and δ = 0.1 and BS volatilityσ BS = 0.2 ,

Merton diffusion volatility is calculated asσ Merton = 0.141421. In this same standard deviation case, call price function is plotted in Figure 7 and put price function is plotted in Figure 8. It seems that MJD model overestimates in-the-money call and underestimates out-of-money call when compared to BS model. And MJD model overestimates out-of- money put and underestimates in-the-money put when compared to BS model.

c 15 i r P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

A) Range 30 to 70.

8 7 c

i 6 r

P 5 Merton l 4 l

a 3 Ce 2 BS 1 42 44 46 48 50 52 Strike K

B) Range 42 to 52.

Figure 7: MJD Call Price vs. BS Call Price When Restricting Merton Diffusion

Volatilityσ Merton . We set σ BS = 0.2 and σ Merton = 0.141421. Parameters and variables used are St = 50, r = 0.05, q = 0.02, τ = 0.25.

c 15 i r

P 10 Merton l l

a 5 Ce BS 0 30 40 50 60 70 Strike K

A) Range 30 to 70.

2.5 c

i 2 r

P 1.5 Merton l

l 1 a Ce 0.5 BS 0 42 44 46 48 50 52 Strike K

B) Range 42 to 52.

Figure 8: MJD Put Price vs. BS Put Price When Restricting Merton Diffusion

Volatilityσ Merton . We set σ BS = 0.2 and σ Merton = 0.141421. Parameters and variables used are St = 50, r = 0.05, q = 0.02, τ = 0.25.

References

Black, F. and Scholes, M., 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 3.

Carr, P., Madan, D. B., 1998, “Option Valuation Using the Fast Fourier Transform,” Journal of Computational Finance 2, 61-73.

Cont, R. and Tankov, P., 2004, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series.

Matsuda, K., 2004, Introduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models, Working Paper, Graduate School and University Center of the City University of New York.

Merton, R. C., 1971, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, 3, 373-413.

Merton, R. C., 1976, “Option Pricing When Underlying Stock Returns Are Continuous,” Journal of Financial Economics, 3, 125-144.

Wilmott, P., 1998, Derivatives, John Wiley & Sons