Sequential Modeling of Dependent Jump Processes

02/20/2014 03:35 PM54-63_Wilm_Scherer_TP_March_2014_Final.indd 54

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Sequential Modeling of Dependent Jump ProcessesJan-Frederik MaiXAIA Investment GmbH, MünchenMatthias SchererTechnische Universität München, GarchingTh orsten SchulzTechnische Universität München, Garching, e-mail: [email protected]

AbstractTwo multivariate models for asset returns are introduced. Both generalize popular univariate models without altering their marginal laws; a very convenient feature, e.g., for a sequential calibration of the model’s parameters to market quotes. The first is a double exponential jump-diffusion model with constant volatility as pre-sented in the univariate case by Kou (2002). The second is a generalization of the stochastic volatility model of Gamma–Ornstein–Uhlenbeck type as first presented by Barndorff-Nielsen and Shephard (2001). Univariate processes of these kinds are driven by a Brownian motion as well as a compound Poisson process. Unlike other existing multivariate extensions of these univariate models, one important advantage of the presented approach is to keep the parameters that govern the dependence structure between the assets separate from the parameters that determine the mar-ginal distributions of the assets. This is achieved by introducing dependence to the univariate compound Poisson processes by means of a bespoke common stochastic time change.

Keywords: jump-diffusion process, Lévy subordinator, multivariate Barndorff–Nielsen–Shephard model, multivariate Kou model, sequential calibration, time change

1 Introd uctionUnivariate jump-diffusion processes are a common tool in modeling price proc-esses of individual financial assets; see, e.g., Cont and Tankov (2004) for an overview. On the contrary, the literature on multivariate jump-diffusion models is quite rare. Multivariate models are, however, important to price multi-asset derivatives and to investigate the risk within a portfolio.

In the present article, we present a new methodology to generalize univariate models to the multivariate case. Jump-diffusion models, whose jump part is driven by a compound Poisson process with exponentially distributed jumps, are considered. Our multivariate extension is based on a tricky construction of dependent Poisson processes. This construction is applied to yield multivariate generalizations of two univariate jump-diffusion models: Kou’s model (Kou, 2002), a model with constant

volatility and two-sided exponentially distributed jumps, and the BNS-�-OU model (Barndorff-Nielsen and Shephard, 2001), a model with stochastic volatility of Ornstein–Uhlenbeck type and one-sided exponentially distributed jumps. Unlike existing multivariate extensions of these univariate models (see, e.g., for a multivari-ate BNS model, Pigorsch and Stelzer (2008) or Muhle-Karbe et al. (2012) and for a multivariate Kou model, Huang and Kou (2006)), we use a bottom-up approach. That means, we start with d univariate models and merge these to one multivariate model by introducing a certain dependence structure. The most appealing feature of our ansatz is the separability of the marginal distributions from the dependence structure, rendering our multivariate models quite handy. We can divide the model parameters into two sets: the parameters determining the marginal distribution of each one-dimensional model and the parameters determining the dependence structure. This separation feature provides nice effects in terms of practical issues. For example, a calibration can be carried out in two subsequent steps: first, the univariate models can be calibrated to market quotes of options on single assets and second, one can set the dependence parameters without affecting the already fixed marginal distributions. Moreover, the dependence of the jump parts in our d-dimensional models can be con-trolled by one single parameter �Ë(0,1), independently of the dimension d.

This article is structured as follows: Section 2 presents the construction of dependent compound Poisson processes, which provides the crucial separation property. This construction is the basis for our multivariate extensions of the two mentioned univariate models, which are introduced in Section 3. Section 4 provides some algorithms to simulate paths in our multivariate models. In Section 5 we cali-brate our models to market quotes and price a multi-asset option. Distributional properties of the construction in Section 2 are investigated in Section 6. Finally, Section 7 concludes.

2 The constru ction of dependent compound Poisson processesThe following construction introduces dependence to d compound Poisson pro-cesses with intensities c1,…, cd > 0 and exponentially distributed jumps with means


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by Lemma 2.1. Moreover, in Remark 6.4 we show that this construction is invariant under the choice of �0. Therefore, dependence can be controlled solely by the intensity c0, respectively by the parameter �. Figure 1 illustrates this construction.

3 Multivariate models of BNS and Kou typeThe construction of dependent compound Poisson processes from Remark 2.2 is now applied to build multivariate jump-diffusion models. This approach applies to all univariate jump-diffusion models, whose jump part is driven by a compound Poisson process with exponentially distributed jump sizes. In the following we con-sider two jump-diffusion models of that kind: one with a constant and deterministic volatility process, Kou’s model (cf. Kou (2002)) and one with a stochastic volatility process, the BNS-�-OU model (cf. Barndorff-Nielsen and Shephard (2001)). In both models the characteristic function of the log price process is known in closed form. Therefore, we are able to compute option prices via Fourier methods (cf. Carr and Madan (1999), Raible (2000)).

Our multivariate generalization supports a flexible dependence structure, even though the dependence of the jump parts is driven by only one parameter. Individual jumps of one asset, joint jumps of a part of the portfolio, as well as simultaneous jumps of all asset values are possible. The jump sizes at common jump times are dependent, which is investigated in Section 6. Moreover, we calculate the joint char-acteristic function of the log price processes within these models. This is needed later on for pricing multi-asset options by means of a multivariate extension of the one-dimensional Fourier method (cf. Eberlein et al. (2010)).

Univariate Kou modelKou’s model is an exponential jump-diffusion model with constant and deterministic volatility. It supports positive and negative jumps, driven by independent compound Poisson processes with exponentially distributed jump sizes. That is, the asset value process S = {St}t g 0 is given by St = S0 exp(Xt), where

Xt = 𝜇 t + 𝜎Wt + Z+t − Z−

t ,

with S0 > 0 and � > 0. Here, Z+ = { Z t + }t g 0 and Z– = { Z t

– }t g 0 are compound Poisson proc-esses with intensity c+, with Exp(�+) -distributed jump sizes and intensity c–, with Exp(�–) -distributed jump sizes, respectively. W = {Wt}t g 0 is a standard Brownian motion. All processes are mutually independent. Under an equivalent martingale measure, the drift has to satisfy

𝜇 = r− 𝜎2

2− c+

𝜂+ − 1+ c−

𝜂− + 1,

where r denotes the constant risk-free interest rate. Relevant for the pricing of options via fast Fourier transform methods is that the characteristic function of Xt is known:

𝔼[eiu Xt]= exp

(t(iu𝜇 − 1

2𝜎2 u2 + c+ i u

𝜂+− i u− c− i u

𝜂− + iu

)), t ≥ 0. (1)

Multivariate Kou modelWe model a portfolio of d assets, each represented by a one-dimensional Kou model. The dependence between the diffusion components is treated as in the standard mar-ket models living in a Brownian world, i.e., we consider d correlated Brownian motions W(1),…, W(d). The jumps are subdivided into positive and negative components, mod-eled by two independent compound Poisson processes with exponentially distributed jumps. Jumps of different assets are made dependent via the tricky construction from Section 2, allowing two or more assets to jump simultaneously. The induced dependence between the jump components is determined by the parameters �+Ë(0,1)

1/�1,…, 1/�d > 0. Starting with d independent compound Poisson processes, they are linked by means of a special stochastic time change, parameterized by one parameter �Ë(0,1), which controls the strength of dependence. This approach is appealing, since the parameters (c1,�1),…, (cd,�d) remain invariant under the presented time change. Thus, it is possible to obtain the parameters (c1,�1),…, (cd,�d) from univariate asset prices and to specify the dependence parameter � in a second step. Lemma 2.1 states that a suitable subordination of two independent compound Poisson processes with exponentially distributed jump sizes produces again a compound Poisson proc-ess of this kind. A proof can be found in the Appendix.

Lemma 2.1 (Underlying construction) Suppose one is given �0 > 0, c0 > 0, � > 0, and c Ë (0, c0). Let Y = {Yt}t g 0 be a compound Poisson process with intensity c�0/(c0 – c) and Exp(c0�/(c0 – c)) -distributed jumps and let T = {Tt}t g 0 be a compound Poisson process with intensity c0 and Exp(�0)-distributed jumps. Assume that Y and T are independent and define the process Z = {Zt}t g 0 … {YTt

}t g 0. It follows that Z is a compound Poisson process with intensity c and Exp(�) -distributed jumps.

Remark 2.2 (Idea of the multivariate construction) Dependence is introduced to d compound Poisson processes Z(1),…, Z(d) with intensities c1,…, cd and jump size distributions Exp(�1),…, Exp(�d) by taking the same underlying process T = {Tt}t g 0 as joint time transformation. Let T be a compound Poisson process with jump size distribution Exp(�0) and intensity c0 = max 1 f i f d{ci}/�, where �Ë(0,1). Construct for each 1 f i f d the process Z(i) by { Z t

(i) }t g 0 … { Y Tt (i) }t g 0, where Y(i) is a compound Poisson

process with intensity ci�0/(c0 – ci) and Exp(c0�i/(c0 – ci)) -distributed jumps. The proc-esses Y(1),…, Y(d), T are independent. Then, the processes Z(1),…, Z(d) are dependent compound Poisson processes, which have the desired parameters c1,…, cd and �1,…, �d

Figure 1: The first graph shows paths of three independent compound Poisson processes. The second graph shows the subordinator T, which serves as joint time transformation. The third graph gives the resulting processes in our construction. We observe that these often jump at the same point in time. Moreover, the jump magnitudes at such events are dependent.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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and �–Ë(0,1). Independently of the choice of �+ and �–, the marginal distributions – which are equivalent to those in the univariate Kou model – remain the same. We are thus able to describe the portfolio model by two separated sets of parameters:

1. The parameters determining the marginal distributions of the assets – one parameter for the diffusion volatility, as well as two parameters for the inten-sities of the jumps and two parameters determining the average jump sizes.

2. One set of parameters for the dependence structure of the assets – a correlation matrix � for the diffusion parts and the coefficients �+ and �– for the jump parts.

The construction works as follows. We consider a probability space (Ω,, ℙ) , on which we define the following processes:

(a) For i =1,…, d, the processes �it + �i W t (i) , where W = (W(1),…, W(d)) is a d

-dimensional standard Brownian motion with correlation matrix �, �i denotes the risk-neutral drift, and �i represents the volatility of the diffusion part.

(b) Independently of the processes in (a), we define independent Poisson pro-cesses N(1),…, N(d) with intensities c1 � 0

+ /( c 0 + – c1),…, cd � 0

+ /( c 0 + – cd) and N(–1),…,

N(–d) with intensities c–1 � 0 – /( c 0

– – c–1),…, c–d � 0 – /( c 0

– – c–d). Moreover, for each i =1,…, d, we let { J j

(i) }jËN and { J j (–i) }jËN be sequences of i.i.d. random variables

with J(i)1 ∼ Exp(c+0 𝜂i∕(c

+0 − ci)

) and J(−i)1 ∼ Exp

(c−0 𝜂−i∕(c

−0 − c−i)

), inde-

pendently of the previous processes. Here, we suppose we have given intensi-ties c1,…, cd > 0 and c–1,…, c–d > 0, and we set c 0

+ … max1 f i f d {ci}/�+ and c 0 – …

max1 f i f d{c–i}/�–, where �+, �–Ë(0,1). (c) Independently of the processes in (a) and (b), we let T+ = { T t

+ }t g 0 and T– = { T t

– }t g 0 be compound Poisson processes with intensities c 0 + and c 0

– and jump size distributions Exp( � 0

+ ) and Exp( � 0 – ).

Having defined these processes on our probability space, for each i =1,…, d we describe asset i as S 0

(i) times the exponential of the process

X(i)t = 𝜇i t + 𝜎i W

(i)t +Z(i)

t − Z(−i)t ,

with Z(i)t =

N (i)T+t∑j=1

J(i)j and Z(−i)t =

N (−i)T−t∑j=1

J(−i)j .

(2)

For pricing multi-asset options by Fourier methods, we need the joint characteristic function of X(1),…, X(d). The following theorem presents a closed-form expression of this joint characteristic function in our multivariate Kou model. The derivation can be found in the Appendix.

Theorem 3.1 (Joint characteristic function in the multivariate Kou model) For all u = (u1,…, ud) Ë Cd, define

𝛼+(u) ∶=d∑

k=1

ck i ukc+0 𝜂k − iuk

(c+0 − ck

) and 𝛼−(u) ∶=d∑

k=1

c−k i ukc−0 𝜂−k + i uk

(c−0 − c−k

) .

Then, the joint characteristic function of X t (1) ,…, X t

(d) is given by

𝜑d(u) ∶= 𝔼

[d∏

k=1eiuk X

(k)t

]

= exp(t(i u⊤ 𝜇 − 1

2u⊤ Σ̂u +

c+0 𝛼+(u)

1 − 𝛼+(u)−

c−0 𝛼−(u)

1 + 𝛼−(u)

)),

where �̂ denotes the covariance matrix of (�1 W 1 (1) ,…, �d W 1

(d) ).

Univariate BNS-�-OU modelIn the BNS-�-OU model, the stochastic variance process �2 = { � t

2 }t g 0 is given by a �-OU process. The asset process S = {St}t g 0 is S0 times the exponential of X = {Xt}t g 0, where

dXt = 𝜇t dt+ 𝜎t dWt + 𝜌 dZt, X0 = 0,

d𝜎2t = −𝜆 𝜎2

t dt+ dZt, (3)

with S0 > 0, �0 > 0, � > 0, and � < 0. Here, Z = {Zt}t g 0 is a compound Poisson process with intensity c and with Exp(�) -distributed jump size and W = {Wt}t g 0 is a standard Brownian motion. Under an equivalent martingale measure, the drift has to satisfy

𝜇t = r −c 𝜌

𝜂 − 𝜌− 1

2𝜎2t ,

where r denotes the constant risk-free interest rate. For more details on the choice of the risk-neutral measure within this model setup, we refer to Nicolato and Venardos (2003). In the BNS-�-OU model the squared volatility process �2 increases by jumps and declines exponentially between two consecutive jumps. The rate of decay is set by the slow-down parameter �. A solution of Equation (3) is given by the OU pro-cess:

𝜎2t = e−𝜆 t 𝜎2

0 + e−𝜆 t ∫t

0e𝜆 s dZs. (4)

Owing to the fact that the so-called leverage parameter � is negative, positive jumps in the stock return process are not supported. Figure 2 illustrates the joint behavior of the asset price process and the volatility process.

One important fact about the BNS-�-OU model is that the characteristic func-tion of Xt can be represented in a closed-form expression (cf. Barndorff-Nielsen et al. (2002)):

𝔼[eiu Xt]= exp

(iu t(r − c 𝜌

𝜂 − 𝜌

)−g h 𝜎2

0 +c

𝜂 − f2

(𝜂

𝜆log

𝜂 − f1𝜂 − i u 𝜌

+ f2 t))

, (5)

Figure 2: A negative jump in the asset value process coincides with an increase in the volatility, which is a quite realistic stylized fact.

0 0.5 1−0.1

−0.05

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with

g ∶= u2 + i u

2, h ∶=

1 − exp(−𝜆 t)𝜆

, f1 ∶= i u 𝜌− g h, f2 ∶= i u 𝜌−g𝜆.

Multivariate BNS-�-OU modelWe model a portfolio of d assets, each represented as a one-dimensional BNS-�-OU model. The dependence between the diffusion components is treated as in the standard market models living in a Brownian world. The jump components driving the volatility processes, however, are made dependent via the tricky con-struction of Section 2, making it possible for two or more assets to jump simul-taneously, and introducing dependence to the stochastic volatility processes. The induced dependence between the jump components is determined by the parameter �. Independently of the choice of �, the marginal distributions remain the same. We are thus able to describe the portfolio model by two separate sets of parameters:

1. The parameters determining the marginal distributions of the assets. A �-OU process with leverage under an equivalent martingale measure is determined by five parameters – one parameter for the jump times, one parameter for the jump sizes, one slow-down parameter for the stochastic volatility, one lever-age parameter, and one initial value for the volatility process.

2. One set of parameters for the dependence structure of the assets. A correla-tion matrix � for the Brownian parts and the coefficient � for the jump parts.

The construction works as follows. We consider a probability space (Ω,, ℙ), on which we define the following processes:

(a) The process W = (W(1),…, W(d)), which is a d -dimensional standard Brownian motion with correlation matrix �.

(b) Independently of the process in (a), we define the independent Poisson pro-cesses N(1),…, N(d) with intensities c1�0/(c0 – c1),…, cd�0/(c0 – cd). Moreover, for each i =1,…, d we let { J j

(i) }jËN be a sequence of i.i.d. random variables with J 1 (i) í

Exp(c0�i/(c0 – ci)), independently of the previous processes. Here, we suppose the intensities c1,…, cd > 0 to be given and we set c0 … max1 f i f d{ci}/�, where �Ë(0,1).

(c) Independently of the processes in (a) and (b), we let T = {Tt}t g 0 be a com-pound Poisson process with intensity c0 and jump size distribution Exp(�0).

Having defined these processes on our probability space, for each i =1,…, d we describe asset i by a one-dimensional BNS-�-OU model. That is S t

(i) = S 0 (i) exp ( X t

(i) ), where

dX(i)

t =(r −

ci 𝜌i𝜂i − 𝜌i

− 12

(𝜎 (i)t

)2)dt + 𝜎 (i)

t dW(i)t + 𝜌i dZ

(i)t , (6)

d(𝜎(i)t

)2= −𝜆i

(𝜎(i)t

)2dt+ dZ(i)

t . (7)

The process Z(i) = { Z t (i) }t g 0 is defined by

Z(i)t =

N (i)Tt∑

j=1J(i)j .

For pricing multi-asset options by Fourier methods, the joint characteristic function of (X(1),…, X(d)) is useful. In our multivariate BNS-�-OU model we can at least calcu-late it in closed form within the special case of independent Brownian motions. This is done in the next theorem. A proof can be found in the Appendix.

Theorem 3.2 (Joint characteristic function in the multivariate BNS model) Assume that W(1),…, W(d) are independent. Define for all 0 f s f t, 1 f k f d, and fixed u1,…, ud Ë C,

fk(s; uk) =i uk + u2k2𝜆k

(e𝜆k (s−t) − 1

)+ iuk 𝜌k.

Then, the joint characteristic function of ( X t (1) ,…, X t

(d) ) is given by

𝔼

[ d∏k=1

eiuk X(k)t

]= exp

( d∑k=1

iuk t(r −

ck 𝜌k𝜂k − 𝜌k

)+iuk + u2k2 𝜆k

(e−𝜆k t − 1

) (𝜎 (k)0

)2)

exp⎛⎜⎜⎝

t

∫0

c0⎛⎜⎜⎝ 11 −∑d

k=1ck fk (s;uk )

c0 𝜂k−fk (s;uk )(c0 −ck)− 1⎞⎟⎟⎠ ds⎞⎟⎟⎠ .

4 Efficient a sset value simulationIn this section we present efficient simulation schemes for the asset value processes. For that, we need Algorithm 1, which implements the construction of dependent compound Poisson processes Z(1),…, Z(d) from Section 2. W.l.o.g. we set the jump size parameter �0 of T equal to 1. As shown in Remark 6.4, this does not affect the distri-bution of the asset values.

Algorithm 1 (Construction of dependent compound Poisson processes) Suppose the following parameters to be given: parameters for the univariate processes Z(1),…, Z(d), i.e. c1,…, cd and �1,…, �d, the parameter � for the dependence, and the maturity t*>0.

1. Simulate N í Poisson (c0t*), which equals the number of jumps of the subordi-nator T until time t*. Here, c0 … max1 f i f d{ci}/�.

2. Draw N independent and Uniform[0, t*]-distributed random variables and compute their order statistic. This serves as the jump times of T and, moreover, the possible jump times of Z(1),…, Z(d) (see Sato (1999, Proposition 3.4)).

3. Draw N independent and Exp (1)-distributed random variables E1,…, EN. Ei represents the jump size of T at the i-th jump time.

4. For each 1 f i f d, draw N Poisson-distributed random variables M 1 (i) ,…, M N (i)

with parameters E1ci/(c0 – ci),…, ENci/(c0 – ci). 5. For each 1 f i f d, 1 f j f N, draw an Erlang ( M j

(i) , c0�i/(c0 – ci))-distributed random variable, which is the jump size of Z(i) at the j-th possible jump time.

Algorithms 2 and 3 simulate paths of d asset value processes within the presented multivariate models.

Algorithm 2 (Paths of the asset values in the multivariate Kou model) Suppose the following parameters to be given: the initial values S 0

(1) ,…, S 0 (d) for each asset, the

constant volatilities �1,…, �d for each asset, parameters for the jump processes Z(1),…, Z(d), i.e. c1,…, cd and �1,…, �d, parameters for the jump processes Z(–1),…, Z(–d), i.e. c–1,…, c–d and �–1,…, �–d, the maturity t*, the correlation matrix � of the d -dimensional Brownian motion (W(1),…, W(d)), the dependence parameters �+,�– for the jump parts, and the constant interest rate r.

1. Perform Algorithm 1 and get dependent processes Z(1),…, Z(d) . 2. Perform Algorithm 1 and get dependent processes Z(–1),…, Z(–d). 3. Introduce a grid on [0,t*] and apply for each 1 f i f d an Euler discretization of

Equation (2). It has to be checked additionally at each grid point whether one has to add jumps of Z(–i), Z(i) or not.

4. Return at each grid point S 0 (i) exp(X(i)) for all 1 f i f d.


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Algorithm 3 (Paths of the asset values in the multivariate BNS model) Suppose the following parameters to be given: the initial values for each asset, i.e. S 0

(1) ,…, S 0 (d) and

( � 0 (1) )2 ,…, ( � 0

(d) )2, parameters for the processes Z(1),…, Z(d), i.e. c1,…, cd and �1,…, �d, the slow-down parameters of the volatility processes �1,…, �d , the leverage parameters �1,…, �d , the maturity t*, the correlation matrix � of the d-dimensional Brownian motion (W(1),…, W(d)), the dependence parameter � for the jump parts, and the con-stant interest rate r.

1. Perform Algorithm 1 and get dependent processes Z(1),…, Z(d). 2. Introduce a grid on [0,t*] and apply for each 1 f i f d an Euler discretization of

the SDEs (6) and (7). It has to be checked additionally at each grid point wheth-er one has to add a jump of Z(i) or not.

3. Return at each grid point S 0 (i) exp(X(i)) for all 1 f i f d.

Algorithms 4 and 5 do not simulate the whole path. They can be used to more efficiently perform a Monte Carlo simulation that simulates the final value of d asset price processes at the fixed time t*. This is useful for the pricing of multi-asset European options. Note that this simulation of the final value is unbiased.

Algorithm 4 (Final asset values in the multivariate Kou model) Suppose the same parameters to be given as in Algorithm 2.

1. Perform Algorithm 1 and get Z(1) ,…, Z(d). 2. Perform Algorithm 1 and get Z(–1) ,…, Z(–d). 3. For each 1 f i f d draw a random variable

Ri ∼

(t∗(r −

𝜎2i

2−

ci𝜂i − 1

+c−i

𝜂−i + 1

), t∗ 𝜎2

i

).

The correlations of R1,…, Rd are given by the correlation matrix �. 4. Return for each 1 f i f d: S 0

(i) exp( X t* (i) ) = S 0

(i) exp (Ri + Z t* (i) – Z t*

(–i) ).

Algorithm 5 (Final asset values in the multivariate BNS model) Suppose the same parameters to be given as in Algorithm 3. The idea is to use the jump times of T = {Tt}t g 0 as grid points for the simulation.

1. Perform Algorithm 1 and get dependent processes Z(1),…, Z(d). 2. The volatility process (Equation (4)) is deterministic between two consecutive

jump times. Therefore, we use the jump times as a grid for the simulation of the paths. To account for the change in the asset value process between two consecu-tive jump times t1 and t2 of Z(i), one has to add a normal-distributed random variable Ri with

Ri ∼ ((

r−ci 𝜌i𝜂i − 𝜌i

)(t2 − t1) −

12𝜈(i)t1 ,t2

, 𝜈(i)t1,t2

), (8)

where 𝜈 (i)t1 ,t2 =

t2−t1

∫0

(𝜎 (i)t1

)2e−𝜆i t dt = 1

𝜆i

(𝜎 (i)t1

)2 (1 − e−𝜆i (t2−t1 )

).

Left to determine is the correlation of (Ri)1 f i f d, which is done in Lemma 4.1. All combined, the asset prices at time t2 are given as

X(i)t2

= X(i)t1

+ Ri+ 𝜌(Z(i)t2− Z(i)

t1

).

Iteratively, we get the final values X t* (1) ,…, X t*

(d) . 3. Return S 0

(i) exp( X t* (i) ) for all 1 f i f d.

Lemma 4.1 Let Ri be defined as in (8) and assume that Corr ( W t (i) , W t

(j) )= ij, i≠j, then

Corr (Ri , Rj) = 2 𝜃ij

√𝜆i 𝜆j

𝜆i + 𝜆j

1 − e−12 (𝜆i+𝜆j) (t2−t1 )√(

1 − e−𝜆i (t2−t1 )) (

1 − e−𝜆j (t2−t1)) .

A proof can be found in the Appendix.

5 Model calibration and the pricing of multi-asset optionsA calibration of the presented multivariate models can be carried out in two separate steps. Owing to the fact that the marginal distributions can be separated from the dependence structure within our models, it is possible to keep the parameters gov-erning the dependence separate from the parameters governing the marginal distri-butions. Therefore, in a first step we calibrate independently each univariate model and in a second step we calibrate the parameters driving the dependence structure. In doing so, the fixed univariate parameters are not affected by the second step. Since there is little market data for multi-asset options, this two-step method is very appealing: we can disintegrate one big calibration problem into two smaller ones. The univariate models can be calibrated to prices of plain vanilla options, which is a well-known task.

In this section, we consider two-dimensional models of two indexes, the DAX and the ESTX 50 Net Return. We aim at pricing European options depending on both indexes. Thus, we calibrate at first the univariate models of Kou and BNS type. To do so, we use market data for European call and put options on the indexes. All market quotes are closing prices as of March 30, 2012. Actually, implied volatilities of bid and ask prices of put and call options are given.1 The expiry dates of these options range from 1 month to 1 year. For each maturity, we consider various strikes. Option prices with a wide bid–ask spread are withdrawn. If there is a put option and a call option with the same strike and maturity, we use the option having a smaller bid–ask spread, which is usually the out-of-the-money option.

After thinning out the implied volatility quotes,2 we calibrate the univariate mod-els to the mid implied volatilities via minimizing the absolute distance of the model implied volatilities to market implied volatilities, with equal weights on every option. Here, option prices in the univariate Kou models and in the univariate BNS models are obtained via Fourier inversion (cf. Carr and Madan (1999)) by means of the char-acteristic function of the log prices (see Equations (1) and (5)).

The calibration of the parameters governing the dependence could be done in a quite similar way. We calibrate the multivariate model with already fixed univariate parameters to market quotes of European multi-asset options, for example best-of-two options. Again, prices in the multivariate Kou model can be obtained via Fourier methods. Here we have to use a multi-dimensional extension of the one-dimensional Fourier method (cf. Eberlein et al. (2010)). Prices in the multivariate BNS model have to be computed via Monte Carlo simulation, because we do not know the joint characteristic function of the log prices. In the bivariate case one has to calibrate only two dependence parameters: the correlation of the Brownian motions and the parameter � driving the dependence of the jump parts.3 Tables 1 and 2 show the results of the calibration of the univariate models. Figures 3 and 4 show simulated

Table 1: Calibrated parameters in the univariate Kou modelsS0

� c – c + � – DAX 6946.8 0.1673 0.2729 0 3.8953ESTX 4210 0.1816 0.1641 0 2.8379


^TECHNICAL PAPER

WILMOTT magazine 59

Table 2: Calibrated parameters in the univariate BNS models S0 �0 c � � �

DAX 6946.8 0.16 1.2426 7.0068 2.8025 –0.5398ESTX 4210 0.1755 0.6506 4.2776 1.7224 –0.4620

0 1 2 3 4 5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

DAX log returns

0 1 2 3 4 5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

ESTX log returns

0 1 2 3 4 55000

6000

7000

8000

9000

10000DAX

0 1 2 3 4 53000

4000

5000

6000

7000ESTX

Figure 3: The left graphs show simulated paths of the DAX and ESTX with calibrated parameters in the multivariate Kou model with a 5-year time horizon. The right graphs show the corresponding daily log returns. We observe one joint jump within this time interval. Here, the correlation of the Brownian motions is 0.5 and � = 0.7.

0 1 2 3 4 53000

4000

5000

6000

7000ESTX

0 1 2 3 4 5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

ESTX log returns

0 1 2 3 4 55000

6000

7000

8000

9000

10000DAX

0 1 2 3 4 5

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

DAX log returns

Figure 4: The left graphs show simulated paths of the DAX and ESTX with calibrated parameters in the multivariate BNS model. The right graphs show the corresponding daily log returns. We observe one joint jump within this time interval, as well as some individual jumps. Here, the correlation of the Brownian motions is 0.5 and � = 0.7.

paths of the bivariate models with calibrated univariate parameters and fixed dependence parameters.

In the absence of positive jumps in the calibrated bivariate Kou model (c+ = 0), dependence is driven by only two parameters, similar to the multivariate BNS model. In the following we price a multi-asset option with a payoff after 1 year of


60 WILMOTT magazine

max{K − max

{eXDAX

1 , eXESTX1

}, 0}

.

That is, we consider a put option with strike K > 0 on the maximum of the two nor-malized indexes. This option can be used as an insurance against a global market crash, because one gets a payoff if the relative performance of both indexes is smaller than K. Here, XDAX and XESTX represent the log price processes and we set K = 0.9. Figure 5 shows model prices of this put option as a function of the dependence parameters.

6 Distributional pr opertiesIn this section, we further investigate the distributional properties of our multivari-ate construction of compound Poisson processes. Of special interest is the implied dependence structure. As a first step we compute the correlation of two compound Poisson processes in our setup.

Theorem 6.1 (Correlation of the compound Poisson processes) Let { Z t (1) }t g 0 =

{ Y Tt (1) }t g 0 and { Z t

(2) }t g 0 = { Y Tt (2) }t g 0 be two dependent compound Poisson processes, con-

structed by the approach presented in Section 2. Set cmax … max1 f i f d {ci}, then

Corr

(Z(1)t , Z(2)

t

)= 𝜅

√c1 c2cmax

.

Proof Wald’s formula (see Klenke (2008, Theorem 5.5)) implies that for square inte-grable i.i.d. random variables J1, J2,… and an independent square integrable random variable N ËN0 we have

𝔼

[ N∑k=1

Jk

]= 𝔼 [N] 𝔼

[J1], Var

[ N∑k=1

Jk

]= 𝔼 [N] Var

[J1]+ Var[N]

(𝔼[J1] )2

. (9)

Using the equalities in (9), we get

𝔼

[Z(1)t Z(2)

t

]= 𝔼

[𝔼

[Z(1)t|||Tt

]𝔼

[Z(2)t|||Tt

]]=

c1 c2 𝜂20𝜂1 𝜂2 c20

𝔼[T2t

]=

c1 c2𝜂1 𝜂2

(2 tc0

+ t2). (10)

Equality (10) and the equalities in (9) imply the claim. In particular, correlation coefficients ranging from 0 to

√c1 c2 /cmax are possible,

and the correlation is independent of t. This gives rise to the guess that our construc-tion of dependent compound Poisson processes is not affected by the actual choice of �0. This is confirmed in Remark 6.4.Let {Zt}t g 0 = {YTt

}t g 0 be constructed as in Lemma 2.1 with parameters (c,�). That is, let J1, J2,… be independent Exp(c0�/(c0 – c))-distributed random variables and let N be an independent Poisson process with intensity c�0/(c0 – c), such that

Zt = YTt =

NTt∑k=1

Jk.

This representation of Z implies that Z jumps at time t if and only if NT jumps at time t, and NT jumps at time t only if T jumps at time t. The following theorem establishes the jump size distribution of NT.

Theorem 6.2 (Link to Geometric law) Let … inf {t g 0|Tt ≠ 0}. Then, the random variable NT

follows a Geometric distribution on N0 with parameter 1 – c/c0, i.e.

ℙ(NT𝜏 = k

)=(

cc0

)k (1 − c

c0

), k ∈ ℕ0.

Proof For k g 0 it holds that

ℙ(NT𝜏 = k

)=

∞

∫0

ℙ

(NT𝜏 = k|||T𝜏 = x

)𝜂0 e

−𝜂0 x dx

=∞

∫0

1k!

(x c 𝜂0c0 − c

)k

e−x c𝜂0c0−c 𝜂0 e−𝜂0 x dx

=

∞

∫0

1k!

ck 𝜂k+10

(c0 − c)kxk e−x

c0 𝜂0c0−c dx.

–1

–0.5

0

0.5

1

0.25

0.5

0.75

0

50

100

150

200

250

300

350

Correlation

Bivariate Kou model

κ

Option p

rice in b

p

–1

–0.5

0

0.5

1

0.25

0.5

0.75

0

50

100

150

200

250

300

350

Correlation

Bivariate BNS model

κ

Option p

rice in b

p

Figure 5: The left graph shows option prices in the bivariate Kou model, the right graph shows option prices in the bivariate BNS model. The prices are given in basis points as a function of the two dependence parameters: the correlation of the Brownian motions and the parameter � for the jump dependence.


^TECHNICAL PAPER

WILMOTT magazine 61

Iterative application of partial integration yields

ℙ(NT𝜏

= k)=

∞

∫0

(cc0

)k

𝜂0 e−x c0 𝜂0

c0 −c dx =(

cc0

)k (1 − c

c0

).

Theorem 6.2 motivates an intuitive representation of the presented construction: potential jump times of Z(1) ,…, Z(d) are given by a Poisson process with intensity c0. The jump size of Z(i) is a sum of independent Exp(c0�i/(c0 – ci))-distributed random variables. The number of added variables is given by the jump size of N T (i) , which is Geometric (1 – ci/c0)-distributed. Note that P( N y

(i) = 0) = 1 – ci /c0 > 0, and, hence, a potential jump time needs not lead to an actual jump of Z(i). Moreover, the jump sizes of N T (1) ,…, N T (d) are dependent; the joint distribution is investigated in the next theorem.

Theorem 6.3 (Link to a multivariate Geometric law) Let … inf {t g 0|Tt ≠ 0}. The joint distribution of the random variables N T

(1) ,…, N T (d) is given by

ℙ(N (1)

T𝜏= k1,… ,N (d)

T𝜏= kn)= k!∏d

i=1 ki!

(1 +

d∑i=1

cic0 − ci

)−k−1 n∏i=1

(ci

c0 − ci

)ki,

where k =n∑i=1

ki .

Proof Straightforward adaptation of the proof of Theorem 6.2. The distribution in Theorem 6.3 is a multivariate Geometric distribution in the sense of Srivastava and Bagchi (1985), which was further investigated by Sreehari and Vasudeva (2012).

Remark 6.4 (Parameter driving the dependence structure) The joint distribution of N T

(1) ,…, N T (d) does not depend on the choice of the jump size parameter �0 of T = {Tt}t g 0.

Thus, the construction presented in Section 2 is independent of the actual choice of �0. The dependence in our construction can be controlled by just one parameter, namely the intensity c0. Denote by pd … P( N T

(1) g 1,…, N T (d) g 1) the probability that the first jump of T leads

to a joint jump of Z(1),…, Z(d). This probability can be obtained by Theorem 6.3. The number of joint jumps follows a Poisson process with parameter c0 pd, due to the thinning property of a Poisson process. In the two-dimensional case, we get by using Theorem 6.3:

p2 = 1 + ℙ

(N (1)

T𝜏= 0 , N (2)

T𝜏= 0)−ℙ(N (1)

T𝜏= 0)− ℙ(N (2)

T𝜏= 0)

=

c1 c2 (2 c0 − c1 − c2)c30 − c0 c1 c2

. (11)

Set cmax … max{c1, c2}, cmin … min{c1, c2}, and c0 … cmax /�, where � Ë (0,1). Then

lim𝜅↘0

p2 = 0 and lim𝜅↗1

p2 =cmin

cmax,

which is a very intuitive result: as � tends to 0, the probability of a common jump is 0, and, hence, there is no dependence. On the contrary, maximal dependence can be obtained when � tends to 1. Then, the compound Poisson process with higher inten-sity jumps at almost every potential jump time given by the jump times of T. And thus, almost every jump of the process with lower intensity is a common jump.

7 ConclusionWe have presented two multivariate generalizations of jump-diffusion models w ith exponentially distributed jumps. These models generalize the univariate BNS

model and the Kou model to the multivariate case. The generalizations are based on a construction of dependent compound Poisson processes supporting a separa-tion of marginal distributions from dependence. This is achieved by a common time shift. Owing to that separation property, the resulting multivariate models are quite tractable. Calibration can be done in two steps: firstly, we derive univari-ate models by calibrating the model parameters to market quotes of plain vanilla options; secondly, we set the parameters driving the dependence, which is a much easier task than calibrating the whole set of parameters at once. Moreover, we have presented some efficient simulation schemes and priced multi-asset options within both models.

AcknowledgmentsWe thank the KPMG Center of Excellence in Risk Management for making this work possible. Furthermore, we thank the TUM Graduate School for supporting these studies.

AppendixTheorem A.1(Sato, 1999, Theorem 30.4) Let Y and T be two independent Lévy sub-ordinators with Laplace exp onents �Y, �T. Then, the process Z = {Zt}t g 0 … {YTt

}t g 0 is a Lévy subordinator with Laplace exponent �Z given by

ΨZ(x) = ΨT(ΨY(x)

), x ≥ 0.

Proof (of Lemma 2.1) The Laplace exponent of a compound Poisson process L = {Lt}t g 0 with intensity c and Exp(�)-distributed jump size is given by

ΨL(x) = − log𝔼

[e−x L1

]= c x

𝜂 + x, x ≥ 0.

By Theorem A.1, the process Z is again a Lévy subordinator with Laplace exponent

ΨZ(x) = ΨT(ΨY(x)

)= c0

c𝜂0c0−c

xc0 𝜂

c0 −c+x

𝜂0 +c 𝜂0c0−c

xc0 𝜂

c0−c+x

= c x𝜂 + x

.

Proof (of Theorem 3.1)

𝔼

[ d∏k=1

eiuk X(k)t

]= 𝔼

[ d∏k=1

eiuk(𝜇k t+𝜎k W (k)

t +Z(k)t −Z(−k)t

)]

= exp

( d∑k=1

i uk 𝜇k t

)𝔼

[ d∏k=1

eiuk 𝜎k W(k)t

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

=∶A

𝔼

[d∏

k=1eiuk Z

(k)t

]⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

=∶B

𝔼

[d∏

k=1e−iuk Z

(−k)t

]⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

=∶C

.

Term A is the characteristic function of a multivariate normal-distributed random variable, and, hence

A = exp(− 12u⊤ Σ̂u t

).

By conditioning on T t + , Z T (1) ,…, Z T (d) become independent and Z t

(i) | T t + is a compound

Poisson distributed random variable with intensity T t + ci � 0

+ /( c 0 + – ci) and jump size dis-

tribution Exp( c 0 + �i/( c 0

+ – ci)). Furthermore, Re(�+(u))<0, which yields


62 WILMOTT magazine

As T has independent increments, we obtain

=

N∏k=0

𝔼

[exp

(d∑i=1

ci 𝜂0 fi (tk)c0 𝜂i − fi (tk)

(c0 − ci

) (Ttk+1 − Ttk

))].

Under the assumption of 1 > Re

(d∑i=1

ci fi (tk)c0 𝜂i − fi(tk )

), it holds by the stationarity of

the increments of T that

= exp⎛⎜⎜⎝

N∑k=0

c0∑d

i=1ci fi (tk)

c0 𝜂i−fi (tk )(c0 −ci)

1 −∑d

i=1ci fi (tk )

c0 𝜂i−fi (tk )(c0−ci)

(tk+1 − tk

)⎞⎟⎟⎠ . If the mesh of the partition goes to zero, the last term converges to

exp⎛⎜⎜⎝

t

∫0

c0⎛⎜⎜⎝ 1

1 −∑d

i=1ci fi (s)

c0 𝜂i−fi (s)(c0 −ci)− 1⎞⎟⎟⎠ ds⎞⎟⎟⎠ .

Finally, we can show that

exp

(d∑i=1

N∑k=0

fi (tk)(Z(i)tk+1

−Z(i)tk

)) L1−→ exp

⎛⎜⎜⎝d∑i=1

t

∫0

fi (s) dZ(i)s

⎞⎟⎟⎠ . The last argument works in a similar way to the proof of Lemma 3.1 in Eberlein and Raible (1999). Proof (of Theorem 3.2)

𝔼

[d∏

k=1eiuk X

(k)t

]= 𝔼

[ d∏k=1

exp(i uk

(r t −


t − 12 ∫

t

0

(𝜎 (k)s

)2 ds+∫

t

0𝜎 (k)s dW(k)

s + 𝜌k Z(k)t

))].

Conditioning on the trajectories of Z(1),…, Z(d) yields

= 𝔼

[ d∏k=1

exp(i uk

(− 12 ∫

t

0

(𝜎 (k)s

)2 ds + 𝜌k Z(k)t

))

𝔼

[exp(∫

t

0iuk 𝜎

(k)s dW(k)

s

)||||| (Z(k)u

)u≤t

]]exp

(d∑

k=1iuk t

(r −


))

= 𝔼

[ d∏k=1

exp(− 12(i uk + u2k

)∫

t

0

(𝜎 (k)s)2 ds+ i uk𝜌k Z

(k)t

)]

exp

( d∑k=1

iuk t(r−


)),

and we get by Equation (7),

=𝔼

[d∏

k=1exp

(i uk + u2k2𝜆k

((𝜎 (k)t

)2−(𝜎 (k)0

)2− Z(k)

t

)+ iuk 𝜌k Z

(k)t

)]

exp

(d∑

k=1i uk t

(r −


)).

Equation (4) gives

B = 𝔼

[d∏

k=1𝔼[ei uk Z

(k)t|||T+

t

]]= 𝔼

[eT

+t 𝜂+0 𝛼+ (u)

]= exp

( c+0 𝛼+(u)

1 − 𝛼+(u) t).

In a quite similar way we get

C = exp

(−

c−0 𝛼−(u)

1+ 𝛼−(u)t).

The following lemma is a multivariate extension of the key formula in Eberlein and Raible (1999) for the special case of our construction of dependent Poisson proc-esses. This is a useful tool for calculating the joint characteristic function in the BNS setup.

Lemma A.2 Let Z(1),…, Z(d) be dependent compound Poisson processes constructed as presented in Section 2. Fix some t>0 and let, for all 1 f i f d, fi: [0,t]™C be left continu-ous functions with limits from the right, such that

Re(fi (s)) <

c0 𝜂ic0 − ci

and Re

( d∑i=1

ci fi(s)c0 𝜂i − fi (s)

)< 1, ∀s ∈ [0, t].

Then,

𝔼

⎡⎢⎢⎣exp⎛⎜⎜⎝

d∑i=1

t

∫0

fi (s) dZ(i)s

⎞⎟⎟⎠⎤⎥⎥⎦ = exp

⎛⎜⎜⎝t

∫0

c0⎛⎜⎜⎝ 11 −∑d

i=1ci fi (s)

c0 𝜂i−fi (s)(c0−ci)− 1⎞⎟⎟⎠ ds⎞⎟⎟⎠ .

Proof For any partition 0 = t0 < … <tN+1 = t of the interval [0,t] we get

𝔼

[exp

(d∑i=1

N∑k=0

fi (tk)(Z(i)tk+1

− Z(i)tk

))]

= 𝔼

[𝔼

[exp

(N∑k=0

d∑i=1

fi (tk)(Z(i)tk+1

− Z(i)tk

))||||||Tt1−Tt0

]].

Note that d∑i=1

fi(tk )(Z(i)tk+1 −Z(i)

tk

) is independent of (Tt1

–Tt0) for all k > 0, hence

= 𝔼

[𝔼

[exp

(N∑k=1

d∑i=1

fi (tk)(Z(i)tk+1

− Z(i)tk

))]

𝔼

[exp

(d∑i=1

fi(t0)(Z(i)t1− Z(i)

t0

))||||||Tt1 − Tt0

]].

Iteratively, we obtain

= 𝔼

[N∏k=0

d∏i=1

𝔼

[exp(fi(tk)

(Z(i)tk+1

− Z(i)tk

))||||Ttk+1− Ttk

]].

Z tk+1 (i) – Z tk

(i) |Ttk+1 – Ttk

has the same distribution as V 1 (i) , where { V t

(i) }tg0 is a compound Poisson process with intensity (Ttk+1

– Ttk)ci�0/(c0 – ci) and jump size distribution

Exp(c0�i/(c0 – ci)). Since c0�i/(c0 – ci) > Re( fi(tk)) , we get

= 𝔼

⎡⎢⎢⎢⎣N∏k=0

d∏i=1

exp

⎛⎜⎜⎜⎝ci 𝜂0c0 −ci

(Ttk+1 − Ttk

)fi(tk )

c0 𝜂ic0−ci

− fi (tk)

⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦ .


TECHNICAL PAPER

WILMOTT magazine 63

W

=𝔼

[ d∏k=1

exp

(∫

t

0

(iuk + u2

k

2 𝜆k

(e𝜆k (s−t) − 1

)+ i uk 𝜌k

)dZ(k)

s

)]

exp

( d∑k=1

i uk t(r−


)+

iuk + u2k2𝜆k

(e−𝜆k t − 1

) (𝜎(k)0

)2).

Define for all 0 f s f t, 1 f k f d,

fk(s; uk) =

i uk + u2k2𝜆k

(e𝜆k (s−t) − 1

)+ iuk 𝜌k,

and note that

Re(fk (s; uk)) < 0 and Re

(d∑

k=1

ck fk(s; uk)c0 𝜂k − fk (s; uk)

)< 0, ∀s ∈ [0, t].

Hence, Lemma A.2 yields the statement. Proof (of Lemma 4.1)

Corr(Ri , Rj) = Corr⎛⎜⎜⎝

t2

∫t1

𝜎 (i)t dW(i)

t ,

t2

∫t1

𝜎(j)t dW(j)

t

⎞⎟⎟⎠= Corr

⎛⎜⎜⎝t2−t1

∫0

𝜎 (i)t1e−

12 𝜆i t dW(i)

t ,

t2−t1

∫0

𝜎 (j)t1e−

12 𝜆j t dW(j)

t

⎞⎟⎟⎠= 𝜃ij

∫ t2−t10 𝜎 (i)

t1𝜎 (j)t1e−

12 (𝜆i+𝜆j ) t dt√

𝜈(i)t1 ,t2 𝜈

(j)t1 ,t2

= 2 𝜃ij

√𝜆i 𝜆j

𝜆i + 𝜆j

1 − e−12 (𝜆i+𝜆j) (t2−t1 )√(

1 − e−𝜆i (t2−t1 )) (

1 − e−𝜆j (t2−t1)) .

Jan-Frederik Mai studied Mathematics and Economics at the universities of Ulm (Germany) and Syracuse, NY (USA), earning a PhD in Mathematical Finance at Technische Universität München (Germany) in 2010. Since then, he has been a quantitative analyst at XAIA Investment GmbH (formerly Assenagon Credit Management GmbH), with a focus on credit and credit–equity strategies in the high-yield market. In addition, he has co-authored several papers and a book focusing on dependence modeling, in particular with a focus on applications in Mathematical Finance.

Matthias Scherer is a Professor of Financial Mathematics at Technische Universität München. His research interests include the pricing and risk management of financial derivatives, probability theory, statistics, and efficient numerical tools. He is particularly interested in dependence concepts and copula models. He is co-author of the book Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications (World Scientific, 2012).

ENDNOTES1. Initially, implied volatilities of the ESTX 50 (price index) are given. Therefore, we transform the strike prices to the ESTX 50 Net Return (performance index) and assume the implied volatilities of these indexes to be equal.2. We use 187 mid implied volatilities for ESTX, 328 for DAX.3. Unfortunately, there is not enough market data for multi-asset options to get sen-sible calibration results. Time series analysis of historical index series may put things right here. This is, however, beyond the scope of this article.

REFERENCESBarndorff-Nielsen, O.E. and Shephard, N. 2001. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63, 167–241.Barndorff-Nielsen, O.E., Nicolato, E., and Shephard, N. 2002. Some recent develop-ments in stochastic volatility modeling. Quantitative Finance, 2, 11–23.Carr, P. and Madan, D. 1999. 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. London: Chapman & Hall/CRC Financial Mathematics Series.Eberlein, E. and Raible, S. 1999. Term structure models driven by general Lévy proc-esses. Mathematical Finance, 9(1), 31–53.Eberlein, E., Glau, K., and Papapantoleon, A. 2010. Analysis of Fourier transform valua-tion formulas and applications. Applied Mathematical Finance, 17(3), 211–240.Huang, Z. and Kou, S.G. 2006. First passage times and analytical solutions for options on two assets with jump risk. Preprint. Available at http://www.learningace.com/doc/2225569/3636b950bc73a152c54d23c2f4f671ba/kou_05_06.Klenke, A. 2008. Probability Theory. Berlin: Springer-Verlag.Kou, S.G. 2002. A jump-diffusion model for option pricing. Management Science, 48(8), 1086–1101.Muhle-Karbe, J., Pfaffel, O., and Stelzer, R. 2012. Option pricing in multivariate stochas-tic volatility models of OU type. SIAM Journal on Financial Mathematics, 3, 66–94.Nicolato, E. and Venardos, E. 2003. Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Mathematical Finance, 13(4), 445–466.Pigorsch, C. and Stelzer, R. 2008. A multivariate generalization of the Ornstein–Uhlenbeck stochastic volatility model. Working Paper.Raible, S. 2000. Lévy processes in finance: Theory, numerics, and empirical facts. PhD thesis, Albert-Ludwigs-Universität Freiburg.Sato, K.-I. 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press.Sreehari, M. and Vasudeva, R. 2012. Characterizations of multivariate geometric distri-butions in terms of conditional distributions. Metrika, 75, 271–286.Srivastava, R.C. and Bagchi, K.S.N. 1985. On some characterizations of the univariate and multivariate geometric distributions. Journal of the Indian Statistical Association, 23, 27–33.

Thorsten Schulz finished his Diploma in Mathematics at Technische Universität München in September 2011. For a period of 6 months he worked as a student trainee at MEAG. From October 2011 to February 2012 he was employed as a research assistant to the Chair of Mathematical Finance. Since March 2012, he has been working on his PhD at the KPMG Center of Excellence in Risk Management with a focus on dependence modeling.

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Sequential Modeling of Dependent Jump Processes