A general Ornstein-Uhlenbeck stochastic volatility model ... · PDF fileA general Ornstein-Uhlenbeck stochastic volatility model with Lévy jumps Karl riedricFh Bannör Chair of Mathematical

Eine Einrichtung der TUM – gefördert von der

KPMG AG Wirtschaftsprüfungsgesellschaft

A general Ornstein-Uhlenbeck stochastic volatility modelwith Lévy jumps

Karl Friedrich Bannör

Chair of Mathematical Finance,Technische Universität München,Parkring 11, 85748 Garching-Hochbrück, Germany,email: karl.f.bannö[email protected],phone: +49 151 5800 3594.

Thorsten Schulz

Chair of Mathematical Finance,Technische Universität München,Parkring 11, 85748 Garching-Hochbrück, Germany,email: [email protected],phone: +49 89 289 18777.

Abstract

We present a general class of stochastic volatility models with jumps where thestochastic variance process follows a Lévy-driven Ornstein-Uhlenbeck (OU) processand the jumps in the log-price process follow a Lévy process. This �nancial marketmodel is a true extension of the Barndor�-Nielsen�Shephard (BNS) model class andcan establish a weak link between log-price jumps and volatility jumps. Further-more, we investigate the weak-link Γ-OU-BNS model as a special case, where wecalculate the characteristic function of the logarithmic price in closed form. More-over, we show that the classical Γ-OU-BNS model can be obtained as a limit ofweak-link Γ-OU-BNS models in the Skorokhod topology.

Keywords

�nancial market model, Barndor�-Nielsen�Shephard model, stochastic volatility, jump-di�usion model, time change, characteristic function, Lévy processes

2

1 Introduction

1 Introduction

Modeling �nancial assets is a task having craved for more and more sophistication dur-ing the last decades. Since complex products increasingly get into focus (i.e. derivativeson realized volatility or variance), more and more stylized facts of time series of assetprices are supposed to be captured. This naturally results into more sophisticated, butalso more complex models. Seeding in the groundbreaking works of Samuelson [1965]and Black and Scholes [1973], where the asset price follows a geometric Brownian mo-tion, many extensions and variants have been proposed as, e.g., introducing jumps inthe asset price as in Merton [1976]; Kou [2002], accounting for di�usion-style stochasticvolatility (e.g. Stein and Stein [1991]; Heston [1993]), combining both approaches (e.g.Bates [1996]), jump-style stochastic volatility models (e.g. Barndor�-Nielsen and Shep-hard [2001]), general pricing approaches as a�ne models (e.g. Du�e et al. [2000]). Onthe other hand, one would like the model to remain interpretable, i.e. being capable of�guring out which parameter accounts for which change in the asset price dynamics.

The model class of Barndor�-Nielsen and Shephard (BNS), which was introduced inBarndor�-Nielsen and Shephard [2001] and extended by Nicolato and Venardos [2003]to account for leverage e�ects, imposes a Lévy subordinator driven Ornstein�Uhlenbeckstructure for the squared volatility process. Furthermore, in the extended notion accord-ing to Nicolato and Venardos [2003], upward jumps in the squared volatility process arealways accompanied by downward jumps in the asset price. On the other hand, thereis empirical evidence (e.g. Jacod and Todorov [2010]) that asset prices and volatility donot always jump together, but there are separate jumps in both processes, which cannotbe captured by the BNS model class.

In this paper, we extend the BNS model class in a generic way, accounting for jumps inthe asset price as well as the squared volatility process that not necessarily jump together.Therefore, we employ a two-dimensional Lévy process to account for the jumps in thesquared volatility process and the asset price process, where the coordinate processes canhave any possible dependence structure. The classical BNS model (cf. Barndor�-Nielsenand Shephard [2001]) and the leverage e�ect BNS model (cf. Nicolato and Venardos[2003]) as well as the two-sided BNS model which can be used for modeling FX rates(cf. Bannör and Scherer [2013]) occur as special cases of this model framework. Wediscuss di�erent linking mechanisms to introduce dependence between asset price jumpsand squared volatility jumps and compute the characteristic function in a semi-closedform, for the general case as well as for special dependence structures. As a special case,we introduce the weak-link Γ-OU-BNS model, where the jumps are driven by coupledcompound Poisson processes employing the time change construction presented in Maiet al. [2014]. For this model, we compute the closed-form characteristic function andshow that the Γ-OU-BNS model can be regarded as a Skorokhod limit of the weak-linkΓ-OU-BNS model.

The remaining paper is organized as follows: In Section 2, we provide a short review ofthe BNS model class and introduce our extension afterwards. Section 3 deals with the

3

2 A general OU-driven stochastic volatility model class

di�erent possibilities of introducing dependence between Lévy processes. Afterwards, wecompute the characteristic function in Section 4 and analyze the limit behavior of theweak-link Γ-OU-BNS model.

2 A general OU-driven stochastic volatility model class

In this section, we discuss shortcomings of the classical BNS model class, motivating theconstruction of a general stochastic volatility model class with

• a stochastic variance process following a Lévy-subordinator-driven Ornstein-Uhlen-beck process,

• Lévy jumps in the log-price process, and

• �exible dependence between log-price process jumps and volatility process jumps.

2.1 A short review of the BNS model class

In the seminal paper Barndor�-Nielsen and Shephard [2001], a tractable stochasticvolatility model class was presented. In the BNS model class as described in Nico-lato and Venardos [2003], the dynamics of the log-price X = (Xt)t≥0 of a certain assetare governed by the SDEs

dXt = (µ+ βσ2t ) dt+ σt dWt + ρ dZt, (1)

dσ2t = −λσ2t dt+ dZt,

where W = (Wt)t≥0 is a Brownian motion, Z = (Zt)t≥0 a Lévy subordinator (indepen-dent of W ), µ, β ∈ R, ρ ≤ 0, and σ20, λ > 0. The BNS model incorporates stochasticvolatility by a Z-driven Ornstein-Uhlenbeck process, where volatility suddenly jumpsup by a jump in Z and calms down afterwards with exponential decay.1 Furthermore,upward jumps are always accompanied by downward jumps in the asset price process2,which accounts for modeling the leverage e�ect.

This model was, e.g., extended by Bannör and Scherer [2013] to incorporate two-sidedjumps in the asset price process to make it more suitable for FX rates modeling. But evenin case of equity modeling, the BNS model as described in (1) incorporates the leveragee�ect in a very strict manner. Every jump in the volatility process is accompaniedby a jump in the stock price and vice versa. Obviously, this strong link can seriouslybe doubted. On the one hand, a sudden jump in the stock price typically triggers

1In many formulations of BNS-type models, an additional time change t 7→ λt is employed to theprocess (Zt)t≥0, which is mainly for mathematical beauty. From a modeling point of view, theformulation without time change is equivalent.

2In the original paper Barndor�-Nielsen and Shephard [2001], the jump component of the asset priceprocess was nonexistent, which can be materialized by setting ρ = 0. The leverage e�ect wasemployed by Nicolato and Venardos [2003].

4

2.2 The OU-stochastic volatility model with jumps

several limit and stop orders. Hence, from an economic perspective, it makes sensethat rising volatility is a typical side e�ect of asset prices jumps. On the other hand,suddenly changing volatility can have manifold reasons; some reasons are presented in,e.g., Shiller [1989]. Furthermore, Jacod and Todorov [2010] scrutinize whether S&P500 jumps are typically associated with volatility jumps and obtain strong evidence forthe existence of separate and joint jumps. Thus, a model establishing di�erent levelsof dependence between asset price jumps and volatility jumps can be employed more�exibly and provides a more realistic behavior than the classical BNS model.


To mitigate the strong link between asset price jumps and volatility jumps that is pos-tulated in the BNS dynamics in (1), we therefore extend the BNS model by introducingtwo separate processes as jump drivers in the volatility and asset price process.

De�nition 2.1

We say that a positive asset price process S = (St)t≥0 follows an OU-stochastic volatil-ity model with Lévy jumps (OUSVLJ model), if the log-price process X = (Xt)t≥0 =(logSt)t≥0 has the dynamics

dXt = (µ+ βσ2t ) dt+ σt dWt + dYt, (2)


where W = (Wt)t≥0 is a Brownian motion, (Y, Z) = (Yt, Zt)t≥0 is a 2-dimensional Lévyprocess such that Z is a subordinator, µ, β ∈ R, and σ20, λ > 0.

Di�erent from the classical BNS dynamics in (1), the jumps of the asset price processoriginate from a separate Lévy process Y , which is possibly dependent of the Lévysubordinator Z driving the jumps in the volatility process. Some suggestions how toconstruct dependent Lévy processes are discussed in Section 3. In the remainder of thissection, we state some special cases of the introduced model class. First, we express the�classical� BNS model in the light of De�nition 2.1.

Example 2.2 (BNS model as a special case of De�nition 2.1)

By choosing Y such that Yt := ρZt, for all t ≥ 0, and for some ρ ≤ 0, the resulting modelwith negative linear dependence is exactly the BNS model with dynamics as described in(1).

Furthermore, the construction principle in the model described in De�nition 2.1 is very�exible and easily extends several models existing in the literature as, e.g., the jump-di�usion models described in Merton [1976]; Kou [2002]. We brie�y summarize specialcases of the OUSVLJ model class and their respective construction principles in Table 1.

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Model Choice of Y Choice of Z Restrictions

�Original� BNS 0 Z subordinator Z�Classical� BNS ρZ Z ρ ≤ 0, subordinator ZTwo-sided BNS ρ+Z

+ + ρ−Z− Z+ + Z− ρ+ ≥ 0 ≥ ρ−, subordinators Z+, Z−

OU-SV Merton model Y Z

subordinator Z,compound Poisson process Y

normal jumps

OU-SV Kou model Y Z

subordinator Z,compound Poisson process Ydouble exponential jumps

Weak-link Γ-OU-BNS −YT ZT independent subordinators Y,Z, T

Table 1 Several special cases of the OUSVLJ model that can either be found in the lit-erature or are simple extensions (OU-SV Merton and Kou model) incorporatingLévy-driven OU stochastic volatility.

A new model �tting in the OUSVLJ class, but not yet existing in the literature, isthe weak-link Γ-OU-BNS model, which employs the time change dependence structurepresented in Mai et al. [2014].

Example 2.3 (Weak-link Γ-OU-BNS model)

Let X = (Xt)t≥0 follow the dynamics

dXt = (µ+ βσ2t ) dt+ σt dWt − dYt,


with µ, β ∈ R, λ > 0, W = (Wt)t≥0 being Brownian motion, and Y = (Yt)t≥0,Z = (Zt)t≥0 being time-change dependent compound Poisson processes with exponentialjump sizes, i.e. there exist independent compound Poisson processes T = (Tt)t≥0, U =(Ut)t≥0, V = (Vt)t≥0 with respective intensitities cT , cY /(cT − cY ), cZ/(cT − cZ) ful�llingcT > 0, cY , cZ ∈ (0, cT ) and respective jump size distributions Exp(1),Exp(cT ηY /(cT −cY )),Exp(cT ηZ/(cT − cZ)), ηY , ηZ > 0, such that Y and Z can be represented as theT -time-change of the processes U and V , i.e. Yt := UTt and Zt := VTt a.s. for all t ≥ 0.

The time-change construction method in Example 2.3 yields a tractable and easy-to-simulate structure of dependency. A few properties of this construction are stated in thenext remark. For more information on this time-change construction as well as proofswe refer to Mai et al. [2014].

Remark 2.4 (Properties of the time-change construction in Example 2.3)

(i) The T -subordinated compound Poisson processes Y = UT , Z = VT are againcompound Poisson processes with intensities cY , cZ and jump size distributionsExp(ηY ), Exp(ηZ).

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(ii) For cmax := max {cY , cZ}, the correlation coe�cient of (Yt, Zt) is given by

Corr [Yt, Zt] =

√cY cZcT

= κ

√cY cZcmax

, for all t ∈ R,

with κ = cmax/cT ∈ (0, 1). We call κ the jump correlation parameter. In particu-lar, correlation coe�cients ranging from zero to

√cY cZ/cmax are possible, and the

correlation does not depend on the point in time t.

(iii) Due to the joint time change, the compound Poisson processes Y and Z are stochas-tically dependent. Moreover, it can be shown that the dependence structure of thetwo-dimensional process (Y, Z) is solely driven by the jump correlation parameterκ.

Figure 1 shows simulated paths for the classical BNS model and the weak-link Γ-OU-BNS model. The graphs in the �rst row show typical asset price paths of the two models.Corresponding to these paths, the graphs beneath exhibit the volatility process and thedaily log-returns. On the left side, the jump correlation parameter of the weak-link Γ-OU-BNS model is set to be 80%, on the right side, this parameter is 20%. Thus, we havea strong dependence between the asset price jumps and jumps in the volatility on theleft side and a weak dependence on the right side. For the sake of good comparability,the Brownian motions and the asset jump processes of both models coincide within onedependence con�guration. Therefore, the di�erence between the two models is deter-mined solely by the jumps in the volatility process. Moreover, the compound Poissonprocesses driving the volatility are identically distributed. One can easily see that thevolatility jumps are uncoupled from the asset price jumps in the weak-link Γ-OU-BNSmodel, i.e. there exist asset price jumps without simultaneous volatility jumps and, onthe other hand, there are volatility rises without negative asset price jumps. The higherthe jump dependence correlation parameter in the weak-link Γ-OU-BNS model is, thehigher seems the resemblance to the classical BNS model. This impression is con�rmedby a mathematical proof in Theorem 4.7.

Generalizing the two-sided BNS model presented in Bannör and Scherer [2013], one canalso establish a more complex dependence structure by employing linear dependence froman n-dimensional Lévy subordinator with independent coordinate processes. Hence, weformulate the linear dependence BNS model by employing matrix-vector notation.

Example 2.5 (Linear dependence BNS model)

Let Z = (Z1t , . . . , Z

nt )t≥0 be an n-dimensional Lévy subordinator with independent coor-

dinate processes and let ρ ∈ Rn. Furthermore, let ξ ∈ {0, 1}n with at least one ξj = 1,j = 1, . . . , n. Then the model following the dynamics

dXt = (µ+ βσ2t ) dt+ σt dWt + ρ′dZt,

dσ2t = −λσ2t dt+ ξ′dZt,

is called the general linear dependence BNS model.

7


0 0.5 1 1.5 2−0.15

−0.10

−0.05

0

0.05Asset log−returns

0 0.5 1 1.5 2−0.15

−0.10

−0.05

0


0 0.5 1 1.5 2−0.15

−0.10

−0.05

0


0 0.5 1 1.5 2−0.15

−0.10

−0.05

0


0 0.5 1 1.5 260

80

100

120Asset price

0 0.5 1 1.5 260

80

100

120Asset price

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4Volatility

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4Volatility

Weak−Link Γ−OU BNS Classical BNS

Figure 1 Sample paths of the asset price process, the volatility process, and the dailylog-returns for the classical BNS model and the weak-link Γ-OU-BNS model.Left: κ = 80%, right: κ = 20%.

8

3 The dependence structure between volatility and asset price jumps

Choosing n = 2, ξ = (1, 1)′, and ρ = (ρ−, ρ+) with ρ− ≤ 0 ≤ ρ+, the linear dependencemodel reduces to the two-sided BNS model of Bannör and Scherer [2013].

3 The dependence structure between volatility and asset price jumps

The main di�erence between the classical BNS model and the OUSVLJ model lies inthe relationship between volatility and asset jumps: In the classical BNS model, every(upward) volatility jump is accompanied by a downward jump in the asset price process,while the parameter ρ steers the magnitude of the asset price process jump. Conversely,in the general OU-stochastic volatility model, this close relationship is not anymorethe case: Similar to the development of the Cox-Ingersoll-Ross-type stochastic volatilitymodels from Heston [1993] over Bates [1996] to Du�e et al. [2000], the dependence ofvolatility and asset prices becomes more sophisticated, since we only assume to havesome dependence structure preserving the two-dimensional Lévy structure of Y and Z.In this section, we discuss di�erent possibilities of establishing dependence between Lévyprocesses.

3.1 Possibilities of introducing dependence between Lévy processes

There are several construction principles to obtain a two-dimensional Lévy processwith dependence from two one-dimensional Lévy processes. Deelstra and Petkovic[2010] neatly summarize three possibilities to construct dependent pure-jump Lévy pro-cesses3:

1. Linear combination of independent Lévy processes

2. Joint time change of independent Lévy processes by a Lévy subordinator

3. Linking the according Lévy measures by a Lévy copula

Obviously, other constructions are possible to create a dependent two-dimensional Lévyprocess (as, e.g., direct construction from a two-dimensional in�nitely divisible law),but the above construction principles provide �exible instruments, where one starts withindependent Lévy processes and results with dependent ones. A standard reference about�nancial modeling with Lévy processes with a focus on univariate Lévy processes is Contand Tankov [2004].

In most special cases presented in Table 1, one of the construction principles is applied.The classical and two-sided BNS model both stem from linear combinations of indepen-dent Lévy subordinators, resulting in a very immediate link between the processes: Inthese models, jumps in the volatility process are always accompanied by jumps in theasset price process. This property arises directly from introducing dependence via linear

3Obviously, dependence between the di�usion components is provided by using correlated Brownianmotions.

9

4 Pricing in the general OU stochastic volatility model

combination: Jumps in an independent building block are mapped by di�erent linearfunctions into both components.

When introducing dependence by joint time change between independent Lévy processes,the link between the jumps in the volatility and asset price process becomes weaker andmore blurry. Joint time change of two independent Lévy processes causes the probabilityof joint jumps to rise due to �common clocking�, but does not necessarily imply simul-taneous jumps. Hence, when introducing weak links between jumps, joint time changeis a tractable method. An example where a weak link between Γ-OU-BNS models is es-tablished, involving joint time-change, is discussed in Mai et al. [2014]: For independentcompound Poisson processes with exponential jump sizes, a time change with anotherexponential jump size compound Poisson process preserves the compound Poisson struc-ture. This construction can similarly be used to establish a weak link within a Γ-OU-BNSmodel, resulting in the weak-link Γ-OU-BNS model from Example 2.3. Here, we do notemploy a stochastic time change to model some kind of business time (as, e.g., in Lu-ciano and Schoutens [2006]), but we solely use the time change construction to establishsome kind of weak dependence between the Lévy processes that joint jumps occur in astochastic manner.

Linking the respective Lévy measures by Lévy copulas was promoted by Tankov [2004]and Kallsen and Tankov [2006]. Analoguously to linking marginal distributions by acopula (as described in Nelsen [2006]), one may link univariate, independent Lévy mea-sures by a Lévy copula. Lévy copulas are functions ful�lling some regularity conditionslinking the tail integrals w.r.t. the Lévy measures. Sklar's theorem for Lévy copulas (cf.[Kallsen and Tankov, 2006, Theorem 3.6]) states that this construction principle is auniversal one, i.e. every dependence structure in multidimensional Lévy processes canbe constructed from independent Lévy processes, linked by some suitable Lévy copula.For a pure mathematics point of view, the universal concept of Lévy copulas makesthe above mentioned constructions redundant. But for pricing purposes, a closed-formcharacteristic function of (integrated) variance and asset price process is typically help-ful. Furthermore, a tractable simulation scheme for Monte Carlo simulation is essential.With linear combination and joint time change of independent Lévy processes, the char-acteristic function of the factors can be calculated at least in a semi-closed form anda simulation scheme is immediately provided, while linking independent Lévy processeswith Lévy copulas typically exhibits cumbersomeness concerning these issues.


To ensure quick and convenient valuation of plain vanilla derivatives (i.e. for calibrationpurposes), many practitioners rely on Fourier pricing methods like FFT pricing (i.e. Carrand Madan [1999]; Raible [2000]) or the COS method described in Fang and Osterlee[2008]. Therefore, the knowledge of the characteristic function in a (semi-)closed form isessential. In this section, we compute the characteristic functions of several (sub-)types

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of the general OU stochastic volatility model. Furthermore, we investigate the limitbehavior of the weak-link Γ-OU-BNS model introduced in Example 2.3 and obtain theclassical BNS model as a Skorokhod limit.

To avoid confusion about the terminology, we de�ne the Laplace exponent of a d-dimensional Lévy process L = (Lt)t≥0, d ∈ N, by ψL(u) := logE[exp(u′L1)] for u ∈ Cdprovided the expected value exists.

We start with the computation of the �nite-dimensional distribution of the log-priceprocess in Theorem 4.1. This is done by calculating the joint characteristic function ofthe log-price process at �nitely many points in time.

Theorem 4.1 (Finite-dimensional distribution of the log-price process)

Let the logarithmic price process (Xt)t≥0 follow the SDE according to equation (2). Setn ∈ N, 0 = t0 ≤ t1 < · · · < tn, and u1, . . . , un ∈ R. De�ne uj :=

∑nk=j uk for all

1 ≤ j ≤ n. Then,

E

exp

n∑j=1

iujXtj

= exp

iu1X0 +

n∑j=1

iµujtj + f(uj)ε(tj−1, tj)e−λtj−1σ20 +

tj∫tj−1

ψ(Y,Z) (iuj , aj(t)) dt

,

with

f(u) :=1

λ

(iuβ − u2

2

),

ε(s, t) := 1− eλ(s−t),

aj(t) := f(uj)ε(t, tj) +n∑

k=j+1

f(uk)ε(tk−1, tk)eλ(t−tk−1) .

Proof

First, we state two simple calculations, which are needed later in the proof.

(i) By the de�nition of the squared volatility process, we get

tj∫tj−1

σ2t dt =

tj∫tj−1

1− eλ(t−tj)

λdZt +

1− eλ(tj−1−tj)

λσ2tj−1

=

tj∫tj−1

1− eλ(t−tj)

λdZt +

1− eλ(tj−1−tj)

λ

e−λtj−1σ20 +

tj−1∫0

eλ(t−tj−1) dZt

.

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(ii) Simple rearranging of summands yields

n∑j=1

f(uj)

tj−1∫0

ε(tj−1, tj)eλ(t−tj−1)dZt =

n∑j=1

j−1∑k=1

tk∫tk−1

f(uj)ε(tj−1, tj)eλ(t−tj−1)dZt

=n−1∑k=1

tk∫tk−1

n∑j=k+1

f(uk)ε(tj−1, tj)eλ(t−tj−1)dZt.

Second, we start calculating the joint characteristic function,

E

exp

n∑j=1

iujXtj

= E

exp

iu1Xt0 +n∑j=1

iuj(Xtj −Xtj−1

)=E

exp

iu1Xt0 +n∑j=1

iuj

tj∫tj−1

(µ+ βσ2t

)dt+

tj∫tj−1

σt dWt + Ytj − Ytj−1

.

Conditioning on the trajectories of Y and Z yields

= exp

iu1Xt0 + iµn∑j=1

ujtj

E

exp

n∑j=1

(iβuj −

1

2u2j

) tj∫tj−1

σ2t dt+n∑j=1

iuj(Ytj − Ytj−1

) ,

and by (i), we obtain

=E

exp

n∑j=1

f(uj)

tj∫tj−1

ε(t, tj)dZt + ε(tj−1, tj)

tj−1∫0

eλ(t−tj−1) dZt

+ iuj(Ytj − Ytj−1

)

× exp

iu1Xt0 +n∑j=1

(iµujtj + f(uj)ε(tj−1, tj)e

−λtj−1σ20

) .

Using (ii) yields

=E

exp

n∑j=1

tj∫tj−1

f(uj)ε(t, tj) +n∑

k=j+1

f(uk)ε(tk−1, tk)eλ(t−tk−1)

dZt + iuj(Ytj − Ytj−1

)

× exp

iu1Xt0 +

n∑j=1


−λtj−1σ20

) .

12


Since the 2-dimensional Lévy process (Y,Z) has independent increments, we get

=

n∏j=1

E

exp

tj∫tj−1

f(uj)ε(t, tj) +

n∑k=j+1

f(uk)ε(tk−1, tk)eλ(t−tk−1)

dZt + iuj(Ytj − Ytj−1

)

× exp

iu1Xt0 +

n∑j=1


−λtj−1σ20

) .

To obtain the �nal step, we have to apply the moment-generating function for stochas-tic integrals w.r.t. Lévy integrators, i.e. a straightforward multivariate generalization ofEberlein and Raible [1999]

= exp

iu1X0 +n∑j=1

iµujtj + f(uj)ε(tj−1, tj)e−λtj−1σ20 +

tj∫tj−1

ψ(Y,Z) (iuj , aj(t)) dt

.�

As an immediate corollary, we obtain a semi-closed form for the characteristic functionof the logarithmic price.

Corollary 4.2 (Characteristic function of the logarithmic price process)

Let the logarithmic price process (Xt)t≥0 follow the SDE according to equation (2). De�nethe abbreviations

f(u) :=1

λ

(iuβ − u2

2

),

ε(s, t) := 1− eλ(s−t).

Then, the characteristic function of the logarithmic price Xt, t ≥ 0 is given by

φXt(u) = exp

iuX0 + iuµt+ f(u)ε(0, t)σ20 +

t∫0

ψ(Y,Z) (iu, f(u)ε(s, t)) ds

,

denoting by ψ(Y,Z) the Laplace exponent of the two-dimensional Lévy process (Yt, Zt)t≥0.

Since the joint Laplace exponent of a two-dimensional Lévy process (which appears in theexpressions in Theorem 4.1 and Corollary 4.2) may be a cumbersome object, we calculateit for the special case of dependence arising from joint time change. The correspondingcalculations for dependence arising from linear dependence are straightforward, therefore,we omit them here.

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Lemma 4.3 (Joint Laplace exponent in case of joint time change)

If the two-dimensional Lévy process (Y, Z) = (Yt, Zt)t≥0 is constructed by jointly time-changing two independent Lévy processes U = (Ut)t≥0, V = (Vt)t≥0, i.e. it exists a Lévysubordinator T = (Tt)t≥0 such that Yt = UTt and Zt = VTt a.s. for all t > 0, then thejoint characteristic function of (Yt, Zt) can be calculated as

E [exp (i (uUTt + vVTt))] = exp (tψT (ψU (iu) + ψV (iv))) ,

where ψ∗ denotes the Laplace exponent of the corresponding process.

Proof

The claim follows easily by conditioning on Tt,

E [exp (i (uUTt + vVTt))] =E [exp ((ψU (iu) + ψV (iv))Tt)]

= exp (tψT (ψU (iu) + ψV (iv))) �

For the weak-link Γ-OU-BNS model, we immediately obtain an explicit expression forthe joint Laplace exponent.

Remark 4.4 (Joint Laplace exponent for the weak-link Γ-OU-BNS model)

In the special case of U, V being compound Poisson processes with exponential jumps andjoint time change T as well being a compound Poisson process with exponential jumpsand an average jump height of 1, the joint Laplace exponent reduces to

ψ(Y,Z)(u, v) =ψT (ψU (u) + ψV (v)) = cTcU

uηU−u + cV

vηV −v

1−(cU

uηU−u + cV

vηV −v

) .Finally, by plugging in the values for the parameters cU , cV , cT , ηU and ηV (cf. Exam-ple 2.3), we get a closed-form expression for the joint Laplace exponent,

=cT

cY ucT ηY −u(cT−cY ) + cZv

cT ηZ−v(cT−cZ)

1−(

cY ucT ηY −u(cT−cY ) + cZv

cT ηZ−v(cT−cZ)

) (3)

Moreover, the corresponding integral appearing in Corollary 4.2 can be computed in aclosed-form expression. This is done in Theorem 4.6.

As a special case, we can consider a one-sided time-change construction, where the termsbecome slightly simpler.

Remark 4.5 (Joint Laplace exponent of a one-sided time-change construction)

If the two-dimensional Lévy process (Y,Z) = (Yt, UYt)t≥0 is constructed by two indepen-dent compound Poisson processes Y, U with intensities cY , cU and jump size distributionsExp(ηY ), Exp(ηU ), then the joint Laplace exponent is given by

ψ(Y,Z)(u, v) = ψY (u+ ψU (v))

= cYu(ηU − v) + cUv

(ηY − u)(ηU − v)− cUv.

14


This construction is slightly simpler, but less �exible than the weak-link construction.In particular, a one-sided time-change construction only allows for separate jumps inone component, while the jumps in the other component always occur jointly. Later, weshow that a model resulting from such a time change construction can be obtained as theSkorokhod limit of the weak-link Γ-OU-BNS model (cf. Theorem 4.7).

Having now gathered all ingredients, we now show that the characateristic function forthe weak-link Γ-OU-BNS model can be calculated in closed form.Theorem 4.6 (Characteristic function of the weak-link Γ-OU-BNS model)

Let X = (Xt)t≥0 follow a weak-link Γ-OU-BNS model (cf. Example 2.3). Then thecharacteristic function can be calculated in closed form and is given by

log φXt(u) = iu(X0 + µt) + f(u)ε(0, t)σ20 −cTλ

(α(u) + δ(u)) log (γ(u)) + cT δ(u)t,

with the following abbreviations

f(u) :=1

λ

(iuβ − u2

2

),

ε(s, t) := 1− eλ(s−t),g(u) := cT cZηY + iucY (cT − cZ) + iucZ(cT − cY ),

h(u) := c2T ηY + iu(c2T − cY cZ),

k(u) := iucT cY ηZ ,

l(u) := c2T ηZ(ηY + iu),

α(u) :=g(u)

h(u),

δ(u) :=f(u)g(u)− k(u)

l(u)− f(u)h(u),

γ(u) :=l(u)

l(u)− ε(0, t)f(u)h(u).

Proof

Note �rst that by Corollary 4.2, the only thing left to show is that

t∫0

ψ(−Y,Z) (iu, f(u)ε(s, t)) ds = −cTλ

(α(u) + δ(u)) log(γ(u)) + cT δ(u)t.

Plugging in the speci�c joint Lévy exponent ψ(Y,Z) (cf. Remark 4.4), we obtain

t∫0

ψ(−Y,Z) (iu, f(u)ε(s, t)) ds =

t∫0

ψ(Y,Z) (−iu, f(u)ε(s, t)) ds

=

t∫0

cT−f(u)g(u) exp(−λt) exp(λs) + f(u)g(u)− k(u)

f(u)h(u) exp(−λt) exp(λs)− h(u)f(u) + l(u)ds

15


after applying some algebraic transformations and substitutions and adopting the termi-nology of the theorem. To solve this integral, we remark that for arbitrary x, y, z, w ∈ Cwith z exp(λs) + w, z, w 6= 0 for all s ∈ [0, t], the derivative of the function

ζ(s) :=1

λ

(xz− y

w

)log(z exp(λs) + w) +

y

ws, s ∈ [0, t]

is exactly

ζ ′(s) =x exp(λs) + y

z exp(λs) + w

for all s ∈ [0, t]. Hence, de�ning

x := −f(u)g(u) exp(−λt), y := f(u)g(u)− k(u)

z := f(u)h(u) exp(−λt), w := −h(u)f(u) + l(u),

yields

t∫0

ψ(−Y,Z) (iu, f(u)ε(s, t)) ds

=cT (ζ(t)− ζ(0)) =cTλ

(xz− y

w

)log

(z exp(λt) + w

z + w

)+cT yt

w,

which results in the proclaimed characteristic function. �

As described in Remark 2.4, the dependence structure between the squared volatilityprocess and the asset price process in the weak-link Γ-OU-BNS model can be completelydescribed by the time change intensity cT , or, alternatively, by the jump correlationparameter κ = max {cY , cZ} /cT , which �oats in the open unit interval (0, 1) and maytherefore be more interpretable than the time change intensity cT .

Obviously, the time-change construction in the weak-link Γ-OU-BNS model always estab-lishes nonlinear dependence between the asset price and the squared volatility process.Therefore, the weak-link Γ-OU-BNS model is not a true extension of the classical Γ-OU-BNS model. But we can show that the classical Γ-OU-BNS model occurs as a limitmodel in the Skorokhod topology as motivated in Figure 1. Thus, the weak-link Γ-OU-BNS model can be considered as an extension where the Γ-OU-BNS model occurs as alimiting case. Regarding the limit κ→ 1, the dependence structure is simpli�ed to lineardependence compared to the time change induced dependence in the weak-link Γ-OU-BNS model. Furthermore, dependent on the setting, a dependence structure resultingfrom a one-sided time change construction (cf. Remark 4.5) occurs as a limit behavior.Theorem 4.7 investigates the limit behavior of the weak-link Γ-OU-BNS model.

Theorem 4.7 (Skorokhod limit of the weak-link Γ-OU-BNS model)

Let the log-price process Xκ be given by a weak-link Γ-OU-BNS model (cf. Example 2.3),with κ being the respective jump correlation parameter as de�ned in Remark 2.4. Then,

16

5 Conclusion

the process Xκ converges to a process X for κ ↗ 1 in the Skorokhod topology. Thestructure of the limiting process X depends on the intensities cZ and cY in the followingway:

• cY > cZ :X can be represented by a construction as described in Remark 4.5, i.e. by the two-dimensional Lévy process (−Yt, Zt)t≥0 = (−Yt, UYt)t≥0, where Y , U are independentcompound Poisson processes with intensities cY , cZηY /(cY − cZ) and jump sizedistribution Exp(ηY ),Exp(cY ηZ/(cY − cZ)).

• cZ > cY :Again, X is given by a construction as described in Remark 4.5, i.e. by the two-dimensional Lévy process (−Yt, Zt)t≥0 = (−UZt , Zt)t≥0, where Z, U are indepen-dent compound Poisson processes with intensities cZ , cY ηZ/(cZ−cY ) and jump sizedistribution Exp(ηZ),Exp(cZηY /(cZ − cY )).

• cY = cZ :X is given by a classical BNS model, as described in Section 2.1, i.e. by the two-dimensional Lévy process (ρZt, Zt)t≥0, where ρ = −ηY /ηZ and Z is a compoundPoisson processes with intensity cZ and jump size distribution Exp(ηZ).

Proof

First, we note that the distribution of the semimartingale characteristics of Xκ does notdepend on the jump correlation parameter κ. Therefore, by [Jacod and Shiryaev, 2003,Theorem IV.4.18,Theorem IV.3.18], it su�ces to show that the �nite-dimensional distri-bution of Xκ converges to the �nite-dimensional distribution of X. Using Theorem 4.1,the problem can be reduced to show that the Laplace exponent of (Yκ, Zκ) converges point-wise to the Laplace exponent of (Y , Z).Consider the case, where cY > cZ , then by Remark 4.4,

limκ↗1

ψ(Yκ,Zκ)(u, v) =cY

uηY

+ cZvcY ηZ−v(cY −cZ)

1−(uηY

+ cZvcY ηZ−v(cY −cZ)

) = cYu(cY ηZcY −cZ − v

)+ cZηY

cy−cZ v

(ηY − u)(cY ηZcY −cZ − v

)− cZηY

cy−cZ v,

which is the claimed formula as in Remark 4.5. In case of cY < cZ , we get the resultanalogously.Now assume cY = cZ , then

limκ↗1

ψ(Yκ,Zκ)(u, v) = cZu+ v ηYηZ

ηY − u− v ηYηZ,

which coincides with the joint characteristic function of Z and ρZ. �

5 Conclusion

In this paper, we have extended the jump-di�usion type Barndor�-Nielsen�Shephardmodel class of Barndor�-Nielsen and Shephard [2001] to the more general OUSVLJ model

17

References

class, where the squared volatility structure follows a Lévy subordinator driven Ornstein�Uhlenbeck process, but the strong link between the jumps in the squared volatilityprocess and the asset price process is ameliorated by introducing weaker dependencestructures, capturing more realistic behavior of the asset price process. As a tractablemember, we have introduced the weak-link Γ-OU-BNS model, relying on a time changeconstruction. We have shown that the weak-link Γ-OU-BNS model has a closed-formcharacteristic function and the classical Γ-OU-BNS model results as a limit model.

Acknowledgements

We thank the KPMG Center of Excellence in Risk Management and the TUM GraduateSchool for supporting this work.

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A general Ornstein-Uhlenbeck stochastic volatility model ... · PDF fileA general Ornstein-Uhlenbeck stochastic volatility model with Lévy jumps Karl riedricFh Bannör Chair of Mathematical