Operations Research Letters 42 (2014) 27–33

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Operations Research Letters

journal homepage: www.elsevier.com/locate/orl

Option pricing under jump-diffusion models with mean-revertingbivariate jumpsDaniel Wei-Chung Miao a,b,∗, Xenos Chang-Shuo Lin c, Wan-Ling Chao a

a National Taiwan University of Science and Technology, Taipei, 106, Taiwanb Imperial College Business School, South Kensington Campus, London SW7 2AZ, UKc Aletheia University, No. 32, Zhenli St., Danshui Dist., New Taipei City 25103, Taiwan

a r t i c l e i n f o

Article history:Received 10 March 2013Received in revised form11 November 2013Accepted 11 November 2013Available online 16 November 2013

Keywords:Options pricingJump-diffusion modelsMean-revertingBivariate jumpsDiscrete Ornstein–Uhlenbeck processImplied volatility smiles

a b s t r a c t

We propose a jump-diffusion model where the bivariate jumps are serially correlated with a mean-reverting structure. Mathematical analysis of the jump accumulation process is given, and the Europeancall option price is derived in analytical form. The model and analysis are further extended to allow formore general jump sizes. Numerical examples are provided to investigate the effects ofmean-reversion injumps on the risk-neutral return distributions, option prices, hedging parameters, and implied volatilitysmiles.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In options pricing, the jump-diffusionmodel is a popular exten-sion from the diffusion model for the underlying stock price pro-cess. Earlierwork can be dated back toMerton’s classicalmodel [7],and there have been a few variations proposedmore recently, suchas [1,5,6]. In these models, the risk-neutral dynamics of the stockprice St is assumed to follow

dStSt

= (r − q − λk)dt + σdWt + dJt , (1)

where r, q are risk-free rate and dividend yield, Wt is a Brownianmotion, Jt =

N(t)j=1 (Yj − 1) is a compound Poisson process with

i.i.d. random jump ratio Yj and k = E[Yj]−1 is the mean jump size.The Poisson processN(t)with jump arrival rateλ records the num-ber of jumps happening in the time interval [0, t]. The stock price

at time t can also expressed as St = S0e(r−q− σ22 −λk)t+σWt+Xt , where

Xt =N(t)

j=1 ln Yj is another compound Poisson process (as opposedto Jt ). The variations in the above-mentioned studies differ in theirassumptions on the distribution of ln Yj. For example, in Merton’s

∗ Corresponding author at: National Taiwan University of Science and Technol-ogy, Taipei, 106, Taiwan.

E-mail address:miao@mail.ntust.edu.tw (D.W.-C. Miao).

0167-6377/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.orl.2013.11.004

model [7], ln Yj follows a normal distribution. In the model consid-ered by Amin [1] and Gukhal [5], ln Yj follows a bivariate distribu-tionmeaning that it is either+δ or−δ. Kou [6] proposed to assumethat ln Yj follows adouble exponential distribution to reflect its lep-tokurtic feature. In these main stream jump-diffusion models, thejump process Jt or Xt is time homogeneous and has no serial corre-lation because both the underlying Poisson process and the jumpsize distribution are time homogeneous (with a fixed arrival rate λand a fixed distribution in ln Yj).

In view of this lack, this paper intends to consider an extensionof the bivariate jump-diffusion (BJD) model of [1,5] such that thejump process is serially correlated. The serial correlation is intro-duced by assuming that it is less likely for the process to see thenext jump going toward the same direction as the previous jump.If a positive (negative) jump has happened, then the probability ofseeing a further positive (negative) jumpwill become smaller. Thisis motivated by the empirical observations that markets tend tooverreact to unexpected events and the effect from a jump-causingevent tends to become weaker as time goes by (see the empiricalstudies such as [3,2,4]). This makes the probability of seeing a se-quence of positive (negative) jumps smaller than the correspond-ing probability in a timehomogeneousmodelwhere the successivejumps follow i.i.d. distributions.

To this end, we construct a model termed mean-reverting bi-variate jump-diffusion (MR-BJD)model which nests the BJDmodelas a special case. The mean-reverting property can be seen in the

28 D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33

jump accumulation process Xt in that it tends to be pulled backto its mean value when it moves away. We investigate the mathe-matical properties of the jumpaccumulationprocess in theMR-BJDmodel, and link it to the discrete OU process [8] so that wemay usethe results of the latter process to conduct our analysis.

The analysis of the jump accumulation process Xt yields someanalytical results that are useful in the subsequent derivations ofthe option pricing formulas for the proposedMR-BJDmodel. Theseformulas are obtained in analytical form which makes their calcu-lations a simple task. To incorporate more general jump size dis-tribution, we consider an extended version of the MR-BJD model,termed theMR-BNJDmodel, andprovide its full analysis. A series ofnumerical examples are provided to examine the effects from themean-reverting property on the risk-neutral distributions, optionprices and hedging parameters. In addition, our numerical exam-ples also demonstrate how themean-reversion in jumps affects theshape of volatility smiles.

2. Modeling mean-reverting bivariate jumps

In this section we generalize the bivariate jump-diffusion (BJD)model such that the jumps are serially correlated with a mean-reverting structure. In the original BJD model as proposed in [1,5],the successive jump sizes are assumed to follow a bivariate distri-bution as below

ln Yj =

+δ, with probability p,−δ, with probability 1 − p.

The mean jump size (mean return caused by a jump) is k = E[Y ]−

1 = peδ+ (1 − p)e−δ

− 1. The compound Poisson process Xt =N(t)j=1 ln Yj measures the accumulated contribution from the jumps

occurring up to time t . Clearly, Xt takes values from {. . . , −2δ, −δ,0, +δ, +2δ, . . .}. The characteristic function of Xt can be obtainedeasily as E[eiαXt ] = eλt[peiαδ

+(1−p)e−iαδ−1].

With k defined above, it is not difficult to check that in the fol-lowing BJD model

St = S0e

r−q− σ2

2

t−λkt+σWt+Xt

, (2)

the martingale condition E[St ] = S0e(r−q)t holds and this ensuresthe absence of arbitrage (since E[eXt ] = eλkt ). The call option pricecan be obtained by the conditioning technique (conditional onN(t)and the number of up jumps; see [1,5]) as

CBJD =

∞n=0

ni=0

e−λt (λt)

n

n!

ni

pi(1 − p)n−i

× CBS(S0e−λkt+(2i−n)δ, K , r, q, σ , t)

, (3)

where CBS(S0, K , r, q, σ , t) = S0e−qtN(d1) − Ke−rtN(d2), d1 =

ln

S0K

+(r−q+ σ2

2 )t

σ√t

, d2 = d1 − σ√t is the Black–Scholes formula for

call option price.In this BJD model, since {ln Yj} is an i.i.d. random sequence, the

jump accumulation process Xt is actually an asymmetric randomwalk. In the MR-BJD model, we intend to introduce serial cor-relation to {ln Yj} in order to reflect the decaying momentum ofsuccessive jumps. Suppose that the jth jump is an up jump withprobability p and a down jump with probability 1 − p. The prob-ability of seeing an up or down jump in the (j + 1)-th jump is nolonger the same, but depends on the outcome of the jth jump inthe following way:

If ln Yj = ±δ, then

ln Yj+1 =

+δ, with probability p ∓ ξ ;−δ, with probability 1 − p ± ξ . (4)

In other words, if the jth jump turns out to be an up (down) jump,then the probability of seeing a further up (down) jump will bedecreased by ξ . It is clear that, with the above type of serial corre-lation, the jump accumulation process Xt tends to be pulled backtoward a central (mean) value when it goes far away. The MR-BJDmodel is the stock pricemodel (2) where Xt has such type ofmean-reverting property, and the additional parameter ξ measures thestrength of mean-reversion.

We now provide an analysis on the jump accumulation processXt because it serves as the basis of the MR-BJD model. Let ps bethe starting up jump probability (the first jump will be an up jumpwith probability ps). Apparently, the successive up jump probabil-ities will be p = ps + iξ where i is an integer. For convenience, weassume the up and down jump probabilities take values from thefollowing (identical) setsup jump probability p ∈ {1, 1 − ξ, . . . , 0.5, . . . , ξ , 0},down jump probability 1 − p ∈ {0, ξ , . . . , 0.5, . . . , 1 − ξ, 1},and ps belongs to these sets. Note that 1 and 0 also belong to thesesets, and this ensures the legitimacy of the model (avoid yielding anegative or greater-than-one probability). These up (down) jumpprobabilities form a decreasing (increasing) arithmetic sequence,starting with 1 (0) and ending with 0 (1). In addition, 0.5 also be-longs to these sets so that there is a casewhere up and down jumpsare equally likely to happen.

Let the up probability takes a general form (ℓ is the index of the2m + 1 values)pℓ = 1 − ℓξ, ℓ = 0, . . . , 2m,

where ξ =12m . The starting probability ps is one of them, i.e. ps =

1−ℓξ for some ℓ. The jump accumulation process Xt =N(t)

j=1 ln Yjis then a process of finite states taking values from X = {x0, . . . ,xm, . . . , x2m} = {θ − mδ, . . . , θ, . . . , θ + mδ}, orxℓ = θ − (m − ℓ)δ, ℓ = 0, . . . , 2m.

The central state value θ is the mean value of X corresponding tothe case where p = 1 − p = 0.5. Note that xℓ = 0 for some ℓbecause Xt starts with X0 = 0 ∈ X. Each time when a jump hap-pens,Xt moves up or downby δ. One significant difference from theBJD case is that the domain of the Xt in BJD is unbounded (infinitestates) but the domain of the Xt inMR-BJD is bounded (finite states,because the probability of jumping beyond [x0, x2m] is zero). Thiscan also be seen from the number of states 2m+1 =

1ξ+1 → ∞ as

ξ → 0. The mean-reverting property is observed from the greaterprobabilities of pulling Xt back toward θ when it moves fartheraway from it.

Fig. 1 shows two possible scenarios of the process Xt . We seethat ξ represents the strength of mean-reversion. Starting withthe same probability set (ps, 1 − ps) = (0.6, 0.4), (a) shows thestronger mean-reversion case with ξ = 0.1 while (b) shows therelatively weaker case with ξ = 0.05. The mean value θ dependson ξ by the following formula

θ =(ps − 0.5)δ

ξ

which equals δ in (a) and 2δ in (b).Note that the jump accumulation process Xt is a continuous-

time Markov chain (CTMC). At each state value xℓ ∈ X, it may ei-ther jump up to state xℓ+1 with a transition rate uℓ or jump downto state xℓ−1 with rate dℓ. Because the jumps are driven by the Pois-son processN(t)whose arrival rate isλ, the up anddown transitionrates for Xt = xℓ are given byuℓ = λpℓ = λ(1 − ℓξ), dℓ = λ(1 − pℓ) = λ(ℓξ). (5)The process Xt with parameter set (λ, δ, ps, ξ) is now well definedas a CTMC.

It is worth discussing the analogue relation between the CTMCXt and its corresponding diffusion process Xt . In the original BJDcase, the jump accumulation process Xt has a fixed tendency to

D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33 29

Fig. 1. The jump accumulation process Xt with successive upward (solid) and downward (dashed) jumps occurring at τj, j = 1, 2, . . . . The starting up and down jumpprobabilities are ps = 0.6 and 1 − ps = 0.4. (a) stronger mean-reversion with ξ = 0.1 (m = 5, θ = δ); (b) weaker mean-reversion with ξ = 0.05 (m = 10, θ = 2δ).

move upward or downward. Therefore, it can be seen as a discreteversion of the Brownian motion with drift Xt which solves dXt =

βdt + γ dWt . The symmetric case p = 0.5 in Xt corresponds toβ = 0 in Xt . In the asymmetric case, both processes tend to moveupward if p > 0.5 and β > 0 and tend to move downward ifp < 0.5 and β < 0.

To link the jump accumulation process Xt in the MR-BJD modelto the corresponding diffusion process, we consider the followingmean-reverting process:

dXt = κ(θ − Xt)dt + γ dWt . (6)This is the well-known Ornstein–Uhlenbeck (OU) process whichwas originally proposed in physical literature in 1930s [9]. Theparameters κ, θ, γ are the speed ofmean reversion, long runmean,and volatility. A discrete version of this OU process is discussedin [8]with some analytical results provided.Wewould like to showthat the Xt in theMR-BJD is indeed the discrete OU process definedin [8] so that we may conduct this study based on those results.

In [8], the discrete version (whichwill also be denoted as Xt andintended to be linked to our jump accumulation process) of theOU process Xt defined in (6) is constructed by matching their firsttwo infinitesimal moments, i.e. E[Xn

t+1t |Xt = x] = E[Xnt+1t |Xt =

x], ∀x, n = 1, 2, 1t → 0. Let this discrete OU process Xt be param-eterized by (κ, θ, γ , δ) where δ is the step size. We intend for thisXt to take values from the same finite setX = {xℓ = θ − (m−ℓ)δ,ℓ = 0, . . . , 2m}. Then Xt is again a CTMC with up and down tran-sition rates given by

uℓ =γ 2

+ δκ(θ − xℓ)

2δ2, dℓ =

γ 2− δκ(θ − xℓ)

2δ2. (7)

We need to ensure that this CTMC will not move beyond [x0, x2m]

and all the transition rates are legitimate. As seen from [8], this canbe achieved by setting u2m = d0 = 0 to get m =

γ 2

κδ2or δ =

γ√mκ

.With this relation, the discrete OU process Xt is constructed as awell-defined CTMC.

The following proposition provides an analytical result of thediscrete OU process which will be used later.

Proposition 2.1. For the discrete OU process Xt parameterized by(κ, θ, γ , δ), its characteristic function is given by

φ(α) = E[eiαXt ] = eiαθ−2θδ

tanh−1e−κt tanh

iαδ2

×

cos

αδ

2

1 + e−2κt tan2

αδ

2

2γ 2

κδ2

. (8)

Proof. Note that the moment generating function of Xt , denotedE[eαXt ], is derived in [8, Proposition 5, Eq. (16)]. The claimed char-acteristic function is obtained by replacingα with iα in the formula

of E[eαXt ] together with the following relation

cosh iαδ

2

cosh

tanh−1 e−κt tanh

iαδ2

= cos

αδ

2

1 + e−2κt tan2

αδ

2

. �

The next proposition relates the jump accumulation process inthe MR-BJD model to the discrete OU process defined in [8].

Proposition 2.2. The jump accumulation process in the MR-BJD, pa-rameterized by (λ, δ, ps, ξ), and the discrete OU process, parame-terized by (κ, θ, γ , δ), are the same continuous-time Markov chaindefined on X. Their parameters are connected by

κ = 2λξ, θ =(ps − 0.5)δ

ξ, γ =

√λδ. (9)

Proof. With both processes defined on the same X, it remains tocheck that the up and down jump rates at each state xℓ = θ −(m−

ℓ)δ, ℓ = 0, . . . , 2m have an exact match. To see this, we first notefrom (5) and (7) that uℓ +dℓ = λ for the first process and uℓ +dℓ =

γ 2

δ2for the second process. (Note that uℓ+dℓ does not depend on ℓ.)

This indicates that the relation λ =γ 2

δ2or γ =

√λδ should hold.

Also note that m =12ξ in the first process and m =

γ 2

κδ2=

λκin

the second process, which implies κ = 2λξ . By further using the θformula (as seen earlier),we see that the uℓ and dℓ in these twopro-cesses are matched exactly for all ℓ = 0, . . . , 2m. This completesthe proof. �

Now that the two specifications (λ, δ, ps, ξ) and (κ, θ, γ , δ) canbe used interchangeably for Xt , wemay discuss further on its prop-erties by using the existing results. In the following proposition,we study the autocorrelation function of the jump accumulationprocess in the MR-BJDmodel, and examine the convergence to thelimiting process, i.e. the jump accumulation process in the originalBJD model.

Proposition 2.3. Consider the jump accumulation process Xt in theMR-BJDwith parameter set (λ, δ, ps, ξ). The autocorrelation functionof Xt at steady state is given by

ρ(τ) = e−2λξτ , τ > 0.

As ξ → 0, the serial correlation disappears and Xt converges in dis-tribution to XBJD

t , the counterpart process for the BJD.Proof. Consider the corresponding discrete OU process with pa-rameter set (κ, θ, γ , δ). When the original (continuous) OU pro-cess is at steady state, Xt ∼ N(θ,

γ 2

2κ ), and E[Xt+τ |Xt ] = Xte−κτ+

30 D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33

–1

1

0.5

–0.5

0

0–π π–π/2 π/2

–1

1

0.5

–0.5

0

0–π π–π/2 π/2

Fig. 2. The integrand function in Eq. (11) as a function of α ∈ [−π, π]: (a) real part; (b) imaginary part. The two curves are for δ = 1 and 0.5, where the δ = 1 curve makesa full cycle but the δ = 0.5 curve only makes a half cycle. Other parameters are λ = 2, ps = 0.6, ξ = 0.1, t = 1, ℓ = 8.

θ(1−e−κτ ). As shown in [8], themean and variance of Xt and Xt areidentical for all t , i.e. E[Xt ] = θ,Var[Xt ] =

γ 2

2κ , and E[Xt+τ |Xt ] =

Xte−κτ+ θ(1 − e−κτ ) (recall that the first two infinitesimal mo-

ments are matched). Therefore, E[XtXt+τ ] = E[E[XtXt+τ |Xt ]] =

E[X2t e

−κτ+ Xtθ(1 − e−κτ )] = (

γ 2

2κ + θ2)e−κτ+ θ2(1 − e−κτ ) =

γ 2

2κ e−κτ

+ θ2. The autocorrelation function is obtained as

ρ(τ) =E[(Xt − E[Xt ])(Xt+τ − E[Xt+τ ])]

Var[Xt ]

=E[XtXt+τ ] − θ2

Var[Xt ]= e−κτ

= e−2λξτ

which goes to 1 for all τ > 0 as ξ → 0.The convergence is distribution Xt → XBJD

t can be proved bythe convergence of their characteristic functions. Let φ(ξ) (hereξ is stressed but α is deemphasized) represent the characteristicfunction (8) with parameters (κ, θ, γ , δ) converted to (λ, δ, ps, ξ).We need to show that, as ξ → 0, φ(ξ) → eλt[pseiαδ

+(1−ps)e−iαδ−1],

the characteristic function of XBJDt . To this end, we write lnφ(ξ) =

a(ξ)/ξ where

a(ξ) = (2ps − 1)iαδ

2− tanh−1

e−2λξ t tanh

iαδ

2

+ ln

cos

αδ

2

1 + e−4λξ t tan2

αδ

2

.

In order to apply l’Hôpital’s rule, we calculate

a′(ξ) =2λt(2ps − 1)e−2λξ t tanh

iαδ2

1 − e−4λξ t tanh2 iαδ

2

−2λte−4λξ t tan2

αδ2

1 + e−4λξ t tan2

αδ2

.Therefore, we obtain

limξ→0

lnφ(ξ) = limξ→0

a′(ξ) = a′(0)

=2λt(2ps − 1) tanh

iαδ2

1 − tanh2 iαδ

2

−2λt tan2

αδ2

1 + tan2

αδ2

= λt[(2ps − 1) sinh(iαδ) + cos(αδ) − 1]

= λt[pseiαδ+ (1 − ps)e−iαδ

− 1]

which implies the desired result. �

We end this section by giving the probability mass function ofXt which is useful in the derivation of option pricing formula.

Proposition 2.4. For the jump accumulation process Xt parameter-ized by (λ, δ, ps, ξ), the probability mass function of Xt is indepen-dent of δ and given by

g(t, xℓ) =12π

π

−π

eiα(m−ℓ)−2ps−1

ξtanh−1

e−2λξ t tanh

iα2

×

cos

α

2

1 + e−4λξ t tan2

α

2

1ξ

dα, (10)

where ℓ = 0, . . . , 2m.

Proof. The probability mass function of Xt can be obtained fromthe characteristic function φ(α) given in (8) by using the inversionformula:

g(t, xℓ) = P(Xt = xℓ)

= limω→∞

12ω

ω

−ω

e−iαxℓφ(α)dα, xℓ ∈ X. (11)

In (11), the step size δ (appearing in φ(α)) is actually irrelevant inthe probability distribution of Xt , because the state probability dis-tribution of a CTMC is determined by the transition rates amongstates. Under the specification (λ, δ, ps, ξ), once λ, ps, ξ are given,the transition rates uℓ and dℓ are fully determined by (5) and thusg(t, xℓ) is fixed regardless of δ. Without loss of generality, we mayset δ = 1 in (11). In addition, it can be checked that the integrandfunction in (11) is a periodic function, which makes a cycle overα = [−

πδ, π

δ]. (See Fig. 2 for typical plots of its real and imaginary

partswith δ = 1 and0.5. Also note that the imaginary part is an oddfunction which integrates to zero.) With the chosen δ = 1, in (11)we only need to integrate over α ∈ [−π, π]. Together with κ =

2λξ, θ =(ps−0.5)δ

ξ, we obtain the claimed result. �

3. Option pricing formula under the MR-BJD model

In this sectionwederive the analytical formula for the Europeancall option price under the proposed MR-BJD model. Let t denotethematurity time of the call option. Generalized from (2), the stockprice under the risk-neutral measure is assumed to follow

St = S0e

r−q− σ2

2

t−η+σWt+Xt

(12)

where Xt =N(t)

j=1 ln Yj is the jump accumulation process as de-fined in Section 2. The constant η in (12) is introduced for thisMR-BJD model to be free of arbitrage, i.e. η is chosen such that the

D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33 31

martingale condition E[St ] = S0e(r−q)t is satisfied. It is clear that inthe special case of BJD, η = λkt (see (2)). To seek η in the general-ized MR-BJD model, we note that eη

= E[eXt ] which can be foundfrom (8). Therefore, for a given t , we have

η =2ps − 1

ξ

δ

2− tanh−1

e−2λξ t tanh

δ

2

+1ξln

cosh

δ2

cosh

tanh−1 e−2λξ t tanh

δ2

. (13)

The following proposition ensures its correctness by showing theexpected convergence relation from MR-BJD to BJD.

Proposition 3.1. The following convergence relation holds:

limξ→0

η = λkt.

Proof. The proof is similar to that of Proposition 2.3 in spirit. Toapply l’Hôpital’s rule, we rewrite (13) as η = b(ξ)/ξ (note thathere b(ξ) is a real function but a(ξ) in the proof of Proposition 2.3is a complex function), and calculate

b′(ξ) =2λte−2λξ t tanh

δ2

2ps − 1 + e−2λξ t tanh

δ2

1 −

e−2λξ t tanh

δ2

2 .

Therefore, the limit is obtained by

limξ→0

η = limξ→0

b′(ξ)

=2λt tanh

δ2

2ps − 1 + tanh

δ2

1 − tanh2 δ

2

= λt[pseδ

+ (1 − ps)e−δ− 1]

as claimed. �

Under the MR-BJD model (12), the log return over the time pe-riod [0, t] is Rt = ln(St/S0) = (r − q −

σ 2

2 )t − η + σWt + Xt .It is useful to derive its probability density function (pdf) and themain statistics. Note that, conditional on Xt = xℓ, Rt is normallydistributed with mean (r − q −

σ 2

2 )t − η + xℓ and variance σ 2t .Therefore, the pdf of Rt can be calculated by unconditioning as

fRt (y) =1

√2πσ 2t

2mℓ=0

g(t, xℓ)

× exp

−

y −

r − q −

σ 2

2

t − η + xℓ

22σ 2t

. (14)

Because theMGFs ofWt and Xt are both known, so is theMGF of Rt .Themoments can be obtained fromsuccessive differentiations, andthe main statistics including mean, variance, skewness and kurto-sis are obtained as

E[Rt ] =

r − q −

σ 2

2

t − η + (2ps − 1)

1 − e−2λξ t

2ξ

δ,

Var(Rt) = σ 2t +

1 − e−4λξ t

4ξ

δ2,

Skew(Rt) =

(2ps − 1)

(1−e−4λξ t )e−2λξ t

4ξ

δ3

Var(Rt)32

,

Kurt(Rt) = 3 +

(3e−4λξ t

−1)(1−e−4λξ t )8ξ

δ4

Var(Rt)2.

Since (Rt |Xt = xℓ) follows a normal distribution as in the Black–Scholesmodel, the optionpricing formula under theMR-BJDmodel

should have amixture formof the Black–Scholes formulaswith pa-rameters slightly adjusted (like the formula (3) for the BJD model).The results are presented as follows.

Proposition 3.2. For the MR-BJD model (12), the call option price isgiven by

CMR-BJD =

2mℓ=0

g(t, xℓ)CBS(S0e−η+xℓ , K , r, q, σ , t), (15)

where g(t, xℓ) is defined in (10).

Proof. By conditioning on Xt = xℓ, the call option price is calcu-lated as

CMR-BJD = e−rtE[(St − K)+]

=

2mℓ=0

P(Xt = xℓ)e−rtE[(St − K)+|Xt = xℓ].

Because (St |Xt = xℓ) = (S0e−η+xℓ)e(r−q− σ22 )t+σWt is log-normally

distributed, the term e−rtE[(St − K)+|Xt = xℓ] is actually the calloption price under a Black–Scholes model with initial stock pricechanged to S0e−η+xℓ . This yields the claimed formula (15). �

4. An extension to the MR-BNJD Model

The bivariate distribution of jump size is somewhat limited be-cause it only takes two values. In this section we extend the mean-reverting jumps to amore general jump size distribution. Based on(4), we assume ln Yj follows a bivariate normal distribution, i.e.

If ln Yj ∼ N(±δ, ν2), then

ln Yj+1 ∼

N(+δ, ν2), with probability p ∓ ξ ;N(−δ, ν2), with probability 1 − p ± ξ .

(16)

Here N(a, b) stands for a normal distributionwithmean a and vari-ance b. A jump-diffusionmodel with the above jump size structureis termed as the mean-reverting bivariate normal jump-diffusion(MR-BNJD) model which nests the MR-BJD as a special case. (It re-duces to MR-BJD if ν = 0, and further reduces to Amin’s BJD ifξ = 0.) In the absence of mean-reversion, i.e. ξ = 0, the jump sizefollows a bivariate normal (BN) distribution which is a mixture oftwo normal distributions. The mean jump size for this BNJDmodel

is obtained as k = E[Y ]−1 = eν22 [peδ

+(1−p)e−δ]−1which takes

a similar form to the k in the BJD model. Note that it also containsMerton’s jump-diffusion model [7] as a special case (when p = 1or 0).

Generalized from (12), the stock price process under the MR-

BNJD is assumed to follow St = S0e(r−q− σ22 )t−η+σWt+Zt where

Zt =N(t)

j=1 ln Yj. To keep notations consistent with those used inthe MR-BJD, we let Xt = δ

N(t)j=1 [1{ln Yj∼N(+δ,ν2)} − 1{ln Yj∼N(−δ,ν2)}]

where 1{A} stands for an indicator random variable of event A.Namely, Xt captures the accumulated contribution from the up-ward and downward jumps (each of them contributes +δ or −δto Xt ), and we have Xt ∈ {. . . , −δ, 0, +δ, . . .} as in the MR-BJD.Clearly, Zt requires the knowledge of the numbers of up and downjumps during [0, t]. In view of this, we write Zt(n, ℓ) = (Zt |N(t) =

n, Xt = xℓ) ∼ N(xℓ, nν2). For later unconditioning,we need P(N(t)= n) and P(Xt = xℓ|N(t) = n). The former is obtained easily asN(t) follows a Poisson distribution. To find the latter conditionalprobability, we need to consider Xt at the jump epochs τn, n =

0, 1, 2 . . . as a discrete-time Markov chain (DTMC) and calculatethe state probability vector from π(n) = π(0)Pn. Here P = [pij]where 0 ≤ i, j ≤ 2m is the transition probability matrix withnonzero elements pℓ,ℓ+1 = 1 − ℓξ (ℓ ≤ 2m − 1) and pℓ,ℓ−1 = ℓξ

(ℓ ≥ 1). The 1 × (2m + 1) probability vector π(n) =πℓ(n)

32 D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33

Fig. 3. The risk-neutral return distributions (fRt (y) vs. y) under different values of ξ = 0 (BJD), 0.025, 0.05, 0.1: (a) pdf shown in normal scale; (b) pdf shown in log scale.Other parameters are r = 0.08, q = 0, σ = 0.25, t = 1, λ = 5, δ = 0.25, ps = 0.6.

Fig. 4. Call prices and deltas under different ξ = 0, 0.025, 0.05, 0.1: (a) C vs. S0; (b) ∆ vs. S0 . The strike price K = 100 and other parameters are identical to those in Fig. 3.

where πℓ(n) = P(Xt = xℓ|N(t) = n), 0 ≤ ℓ ≤ 2m gives the prob-ability distribution of Xt if n jumps have happened. It can be cal-culated from the initial probability vector π(0) =

πℓ(0)

where

πℓ(0) = 1 for ℓ = (1 − ps)/ξ and πℓ(0) = 0 otherwise.For the MR-BNJD model to be free of arbitrage, the martingale

condition must hold. This leads to

η = ln

∞n=0

2mℓ=0

e−λt(λt)n

n!πℓ(n) exp

xℓ +

12nν2

, (17)

which is a generalized version of (13). As in the MR-BJD, since(Rt |N(t) = n, Xt = xℓ) follows a normal distribution, the call op-tion price under the MR-BNJD model can be derived as a mixtureform of the Black–Scholes formulas as below

CMR-BNJD =

∞n=0

2mℓ=0

e−λt(λt)n

n!πℓ(n)

× e12 nν

2−η+xℓCBS(S0, K , r, q, σ , t) (18)

with two adjusted parameters r = r +12 nν

2−η+xℓt and σ 2

=

σ 2+

nν2t .

5. Numerical examples

This section provides numerical examples to investigate theeffects from themean-reverting jumps on the return distributions,option prices, and volatility smiles. The results are plotted underdifferent values of parameter ξ including the special case ξ = 0which corresponds to the original BJD.

5.1. Return distributions

The probability density functions of the return Rt (as given by(14)) are plotted in Fig. 3. It is observed that in the BJD case ξ = 0,the distribution is more widely spread than other mean-revertingcases with ξ = 0. As seen in (a), when ξ increases, the distribu-tion becomesmore andmore tightly distributed around the centervalue which also moves toward zero. This agrees with intuition inthat the presence of mean-reversion will make it more narrowlydistributed and symmetric. For the maximal ξ = 0.1, the distribu-tion has the smallest variance and the least heavy tails. The differ-ences in their tail-heaviness can be seen clearly in (b) where thedensities are shown in log scale.

5.2. Option prices and hedging parameters

Fig. 4 shows how the call option price and the hedging parame-ter (delta ∆ =

∂C∂S0

) vary with the initial stock price. Again we takeξ = 0 as the benchmark case. In (a), we see that as ξ gets larger,the call price becomes lower. This is because the stock return vari-ance (or stock volatility) becomes smaller and in turn makes theoption price lower. As shown in (b), the deviations in delta causedby a nonzero ξ are dependent on the option’s moneyness. Whenthe option is at-the-money (ATM, S0 = K ), there is little change indelta. But the delta becomes larger when it is in-the-money (ITM,S0 > K ) and becomes smaller when it is out-of-the-money (OTM,S0 < K ). This can bemore or less seen from the slope of the curve in(a). It also indicates that ignoring the presence of mean-reversionin jumps will introduce some hedging errors.

D.W.-C. Miao et al. / Operations Research Letters 42 (2014) 27–33 33

Fig. 5. Implied volatility (IV) smiles under different ξ = 0, 0.025, 0.05, 0.1: (a) ps = 0.4; (b) ps = 0.5; (c) ps = 0.6. Other parameters are r = 0.08, q = 0, σ = 0.25, t =

0.5, λ = 2, δ = 0.25.

5.3. Implied volatility smiles

Finally we look at the effects on the curves of volatility smiles.Fig. 5 shows the results under different starting probabilities ps =

0.4, 0.5, 0.6. (Note that only ps = 0.6 is considered in Figs. 3 and4.) The arrangement is intended to show how the asymmetry inreturn distribution influences the volatility smiles. It is easy to seethat ps = 0.5 is a special symmetric case (symmetric return dis-tribution), and the distribution will be skewed toward the right forps > 0.5 (as seen in Fig. 3 where ps = 0.6) and toward the leftfor ps < 0.5. These different types of asymmetry are reflected inthe volatility smiles as seen in (a), (b), (c) for ps = 0.4, 0.5, 0.6,respectively. In each plot, as ξ gets larger, the smile curve movesdownward because of the smaller stock return variance. In addi-tion, as ξ gets larger, the curves also become less curved and lessasymmetric compared to the ξ = 0 case.

References

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