variance-optimal hedging in general affine stochastic volatility models

VARIANCE-OPTIMAL HEDGING IN GENERAL AFFINE STOCHASTIC VOLATILITY MODELSAuthor(s): JAN KALLSEN and ARND PAUWELSSource: Advances in Applied Probability, Vol. 42, No. 1 (MARCH 2010), pp. 83-105Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/25683808 .

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Adv. Appl. Prob. 42, 83-105 (2010) Printed in Northern Ireland

?AppliedProbability Trust 2010

VARIANCE-OPTIMAL HEDGING IN GENERAL AFFINE STOCHASTIC VOLATILITY MODELS

JAN KALLSEN,* Christian-Albrechts-Universitat zu Kiel

ARND PAUWELS,** MEAG AssetManagement GmbH

Abstract

We consider variance-optimal hedging in general continuous-time affine stochastic

volatility models. The optimal hedge and the associated hedging error are determined

semiexplicitly in the case that the stock price follows a martingale. The integral

representation of the solution opens the door to efficient numerical computation. The

setup includes models with jumps in the stock price and in the activity process. It also

allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

Keywords: Variance-optimal hedging; stochastic volatility; Galtchouk-Kunita-Watanabe

decomposition; affine process; Laplace transform

2000 Mathematics Subject Classification: Primary 91B28; 60H05; 60G48; 93E20

1. Introduction

Often observed so-called stylized features of stock returns include semiheavy tails, volatility clustering, and the leverage effect (i.e. negative correlation between changes in volatility and stock prices). Stochastic volatility (SV) models account for these observations. Examples in the literature include the Heston model [14] and the Levy-driven stochastic volatility models

put forward in [2]. Other SV models are based on time-changed Levy processes as in [7]. All these examples are affine in the sense of [10]. Further instances of such affine stochastic

volatility models are discussed in [18] and [33]. Stochastic volatility typically leads to an incomplete market, i.e. perfect hedging strategies

do not exist for many contingent claims. As a way out, we may try to minimize the expected squared hedging error

E[(v + # ST -

H)2]

over all initial endowments v e R and all admissible hedging strategies &. Here, H denotes the discounted payoff at time T of a European-style contingent claim and S denotes the discounted

price process of the underlying stock. The stochastic integral # S stands for the gains from

dynamic trading in the stock according to strategy This problem and in particular its general structure have been extensively studied in the

literature; cf., e.g. [1], [8], [29], [35], and the references therein. If S is a martingale, the solution is determined in [11] based on the Galtchouk-Kunita-Watanabe decomposition. To be more specific, let Vt := E[H | !Ft] denote the martingale generated by H. Then i;* := Vb is

Received 18 May 2009; revision received 21 October 2009. * Postal address: Mathematisches Seminar, Christian-Albrechts-Universitat zu Kiel, Christian-Albrechts-Platz 4,

24098 Kiel, Germany. Email address: [email protected] ** Postal address: MEAG AssetManagement GmbH, Abteilung Risikocontrolling, Oskar-von-Miller-Ring 18, 80333

Miinchen, Germany. Email address: [email protected]

83



84 J. KALLSEN AND A. PAUWELS

the optimal initial endowment in the above hedging problem and the optimal hedging strategy can be written as

' d(S,S),'

(L1)

where ( , ) denotes the predictable covariation process. The corresponding hedging error equals

E[(v* + #* ST - H)2] = E[(V, V)T - (#* S)Tl (1.2)

Alternative representations of the solution are provided in [5] using the carre du champ operator and in [4] based on Malliavin derivatives. All these general characterisations do not

immediately lead to numerical results in concrete models. For example, it may not be obvious how to evaluate (1.1) and (1.2) because analytical expressions for V are typically not available.

Numerical approaches are discussed in, e.g. [9], [13], and [16]. Heath et al. [13] used

partial differential equation methods for specific continuous stochastic volatility models. Cont et al. [9] and similarly Hubalek and Sgarra [15] considered an SV model involving jumps. A partial integro-differential equation is solved by finite-difference schemes in order to obtain the process V above. The hedging error is computed by Monte Carlo simulation. This approach is applied to exotic contingent claims and it allows for options as hedging instruments.

In this paper we study variance-optimal hedging in a general affine stochastic volatility model. The objective is to determine semiexplicit expressions for the optimal hedging strategy and the hedging error which can be evaluated without implementing involved numerical schemes or computer-intensive Monte Carlo simulations. They are obtained with the help of integral transform techniques which are used widely in option pricing (cf., e.g. [6] and [31]) and in [16] for the mean-variance hedging problem without stochastic volatility.

We focus on the case that S is a martingale. Firstly, this allows us to cover a broader class of volatility structures than without this restriction, e.g. those involving a leverage term.

Secondly, we believe that the excess drift of asset prices is of secondary importance for the hedging problem. Finally, quadratic hedging appears as an auxiliary problem in a first order approximation to utility-based derivative pricing and hedging; cf. [3], [19], [21], [24], [25], and [26]. Here the variance-optimal hedge must be determined under some equivalent

martingale measure, i.e. S is by default a martingale under the relevant measure. For a treatment of the more involved non-martingale case, we refer the reader to [23], which generalizes some

of the present results. In such context a measure change to the?generally signed?variance optimal martingale measure plays a major role implicitly or explicitly. However, the setup in [23] allows for leverage only in very special cases. Moreover, it is written on a less rigorous

mathematical level. The structure of the paper is as follows. In Section 2 we introduce the general affine stochastic

volatility model. Subsequently, we discuss an integral representation of the contingent claim

that is to be hedged. Section 4 contains the solution to the hedging problem in this setup. For

numerical results in concrete parametric models, we refer to the forthcoming paper [20]. Proofs of the main results are to be found in the final section.

Unexplained notation is used as in [17]. Superscripts generally refer to components of a

vector or vector-valued process rather than powers. The few exceptions should be obvious from the context. As a key tool, we need the notion of semimartingale characteristics (J5, C, v). For a summary of important results, we refer the reader to Appendix A. Here h = (h\,h2)

generally denotes a componentwise truncation function on R2, i.e. h(x\, xi) = (h(x\), h(x2)), where h: R ?> R is a one-dimensional truncation function, which can, e.g. be chosen of the



Variance-optimal hedging in general affine stochastic volatility models 85

form 10 ifxk = 0,

,1 ,n .u (1-3) (1 a |jcfc|)- otherwise.

1**1 All characteristics and Levy-Khintchine triplets on R and R2, respectively, are expressed relative to truncation functions h and h, respectively. For ease of exposition, we occasionally use the particular function (1.3) in the proofs, but the results hold for arbitrary h.

2. The general affine stochastic volatility model

We denote the discounted price process of a univariate stock by S = Sbexp(Z), i.e. Z stands for logarithmic returns. Moreover, we consider a positive activity process y leading to randomly changing volatility in our general setup. We assume that the bivariate process (y, Z) is an affine semimartingale in the sense of [10]. More specifically, we suppose that the characteristics (By,z, Cy ,z, vy,z) of the R+ x R-valued semimartingale (y, Z) are of the form

*t'Z= f G8(0)+i8(i))v-)<k, (2.1) Jo

CJ,Z = / (Y(0) + Y(i)ys-)ds, Jo

v*z([0, t]xG)= f (p(0)(G) + <P(i)(G)ys-) ds, (2.2) Jo

for all G e 3$2 and t e [0, T], where Ofyy), Y(j), <P(j)), j =0, 1, are Levy-Khintchine triplets on R2 which are admissible in the sense of [10, Definition 2.6] or [18, Definition 3.1], i.e.

fa) e R2, y(j) is a symmetric, nonnegative matrix in R2x2, and <p(7) is a a-finite measure on R2 \ {0} satisfying /(1a \x\2)<p(j)(dx) < oo,

1,1 1,2 2,1 n no>

= no)

= no>

= ?

<p(0)((R+ x R)c) = <p(i)((R+ x R)c) = 0, i.e. <p(0) and are actually Levy measures

on R+ x R,

f hi(x)(p(0)(dx) < oo and/^0)

- /hi(x)(p(0)(dx)

> 0,

/x$p($(dx) < oo. This additional condition prevents explosion in finite time; cf. [10, Lemma 9.2].

We refer to this setup as a general affine stochastic volatility model. We provide a few examples of popular affine SV models in the literature.

Example 2.1. The Heston [14] model can be written as

dZ, = (/x + Syt) dt + y/yi&w}, dyt = (k - Xyt) dt + cr^dW?, where /c > 0, /x, 8, A, and cr denote constants, and Wl and W2 are Wiener processes with constant correlation ?. The bivariate process (y, Z) is affine in the sense of (2.1)-(2.2) with

(^(o),K(0)^(0)) = (q'0'0)'

ca,..?... ?.?-((?) (? 7)4




Example 2.2. The so-called BNS model of [2] is of the form

dZt = (ji + 8yt-) dt + y/y^dWt +odzt, dyt = -Xyt- dt + dzf,

where /x, 8, X, and q are constants and W is a Wiener process. Here z denotes a subordinator, i.e. an increasing Levy process whose Levy-Khintchine triplet we write as (bz, 0, Fz). The

process (y, Z) is affine in the sense of (2.1)-(2.2) with

/ OO

^(0)(G) = / lG(x, Qx)Fz(dx) for all G e ?2, Jo

and

<*?.*,?.,)?).?> cf. [18, Section 4.3]. In order to obtain more realistic autocorrelation patterns in volatility, BarndorfT-Nielsen and Shephard replaced the Levy-driven Ornstein-Uhlenbeck process yt by a linear combination of such processes. This extension has a multivariate affine structure

according to [18, Section 4.3]. It can be treated along similar lines as the simpler BNS model above. We stick to the bivariate case in this paper in order not to confuse the reader with heavy notation.

Example 2.3. Carr et al. [7] considered stochastic volatility models which are based on time

changed Levy processes, namely

Zt ? lit + xyt + QZt, dYt = yt- dt, dyt = -Xyt? dt + dzt,

where /x, q, and X are constants, and z and x denote independent Levy processes with triplets (bz, 0, Fz) and (bx, cx, Fx), respectively. Here z is supposed to be increasing. The model is of the form (2.1)-(2.2) with

^(0) = + Qbz + f^ihiQx)

~ Qh(x))Fz(dx)j

' Y(0) = ?'

poo

<p{0)(G) = / Qx)Fz(dx) for all G e ?2, Jo

Ad = (^).

w) = (o cx)'

^(1)(G) = fR

1g(0' *)fX^

for all G e iB2; cf. [18, Section 4.4]. If we choose x to be a Brownian motion with drift, we

obtain the dynamics of the BNS model above. Another possible choice of the time change rate y is a square-root process:

Zt = fit + xyt + q(yt -

yo), dYt = yt dt, dyt = (k -

Xyt) dt + a^dWt,

where k > 0, /x, ?, a, and <j are constants, w denotes a Wiener process, and x is an independent




Levy process with triplet (bx, cx, Fx). This model is of the form (2.1)-(2.2) with

*? = = *V + c*)

<G> = IlG(0'x)FX(dx)

for all G g S2\ cf. [18, Section 4.4]. If we choose X to be a Brownian motion with drift, we recover the dynamics of the Heston model?up to a rescaling of the volatility process y.

The Levy-Khintchine triplets /(;), (P(j)), j = 0, 1, can be associated to corresponding Levy exponents

fjQi) := uTpw + ^uTyU)u +

j(ef*

- 1 - uTh(x))<pU)(dx). (2.3)

The functions V^', y = 0, 1, are defined on

Uj := \ u eC2: / exp(Re(w)T;c)<p(;-)(d;t) < oo}. I J{\x\>\} J

Assumption 2.1. order for S to be a locally square-integrable martingale, we assume that

(0, 2) g t/o fl U\ and

^o(0,l) = ih(0,l) = 0. (2.4)

Moreover, we suppose that

^0(0,2)^0 or Vi (0,2)^0,

w order to avoid the degenerate case, Z = 0.

Proposition 2.1. Assumption 2.1 implies that S g M^c.

Proof. The differential characteristics (bz, cz, Fz) of Z can be easily calculated from

(2.1)-(2.2); cf. Proposition A. 1 or A.2. They are of the form

^ = Pf0) + 0<V'- ct = n2o')2 + y?i)y<->

Ftz(G) = j

lG(x2)<P(0)(dx) + J

lG(x2)^(i)(dx)yf_

for all G g <?. Using Propositions A.l and A.2, we obtain, for the stochastic logarithm X := X(S) = 1/S_ S,

*>? = bf + l-cf + j(h(ex

- 1) - /k*))F,z(dx),

FX(G) = j

lG(ex - l)Ftz(dx) for all G e ?

for the differential characteristics (Z?x, cx, Fx) of X. In particular, we have

f x2Ftx(dx) =

J(e*2

- l)2^(0)(dx) + (e*2 -

l)2Wl)(dx))y,_.




In view of Assumption 2.1 and [17, Proposition II.2.29], X is a locally square-integrable semimartingale. Moreover, (2.4) implies that

bf + j(x

- h(x))Ftx(dx) = V(o)(0, 1) + V(i)(0, Dy,_ = 0,

which implies that X is actually a local martingale. Since S(X)- is locally bounded, S =

SoS(X) = So(l + ?(X)- X) is a locally square-integrable martingale as well.

Owing to the results of [10], the characteristic function of the bivariate affine process (y, Z) is known explicitly.

Proposition 2.2. The conditional characteristic function of(y, Z) is of the form

E[exp(wiy,+5 + u2Zt+s) | Ff] = exp(*I>0(s, u\, u2) + ^\(s, u\,u2)yt + u2Zt),

u e C_ x iR, where ty\: R+ x C_ x iR ? C_ solves the initial value problem

d ? ty\(t, U\,U2) = ^iOPi(f, "1, u2), u2), ^l(0, Ml, w2) = mi, (2.5)

and ^I>0: R+ x C_ x it -> C w g/vera by

Vo(t,u\, u2) = f 1ro(Vi(s,ui,U2),U2)ds. (2.6) Jo

Here C_ w defined as C_ := (zgC: Re(z) < 0}.

Proo/ This is a special case of [18, Theorem 3.2].

3. European options

The hedging problem cannot be solved in closed form even for geometric Levy processes without stochastic volatility. In order to obtain at least semiexplicit solutions, we consider

European-style claims whose discounted payoff at time T is of the form H = f(Sj). More

specifically, we assume that the function /: (0, oo) ? R can be written in integral form as

f(s) = f

szTl(dz)

with some finite complex measure II on a strip := {z C: R' < Re(z)_< R}, where

R\ R e R. The measure n is supposed to be symmetric in the sense that 11(A) = 11(A) for

A e <S(C) and ~A := {z e C: z e A}. In most cases we can choose Rf = R and the measure

n has a density; cf. [16]. For example, we have

j /?/?+ioo K^~z

(s-K)+ = ? sz?--dz

2*1 Jr-ioo z(! -

z>

with R > 1 for a European call with strike K, which means that

1 Kx~z n(ck) = ? ??-dz

2ttiz(1 -z)




on Sf := R + iR. The put f(s) = (K ?

s)+ corresponds to the same formula but with R < 0.

The integral representation of many other payoffs can be found in [16]. For the derivation of formulae in the next section, some moment conditions are needed. We

phrase them here in terms of analytical properties of the characteristic exponents in Proposi tion 2.2.

Assumption 3.1. We assume that the following conditions hold.

(i) For some s > 0, the mappings (u \, u2) h-> ̂o(t, u\, u2), (/, w i, u2) have an analytic extension on

S:={ueC2: (Re(wi), Re(w2)) e V?(0)}

for all t e [0, T], where

M0 :=sup{2î(f,0,r): r e [Rf A 0, R v 0], t e [0, T]},

and

Vg(a) := (-oo, (M0 v 0) + s) x ({IR' A 0) ? ?, (2R v a) + e)

for a e R+. These extensions are again denoted by *I>o cmd ̂i, respectively.

(ii) The mappings t i?ô(t, u\, ui) and t i-> ̂\(t,u\, u2) are continuous on [0, T]for any (u\, u2) e S.

(iii) Ve(2) C UoC)U\ . This is satisfied if the mappings iR ?> C, w \j/j(u)for j = 0, 1 /zave

analytic extensions to Ve(2), m which case representation (2.3) holds for this extension. Note that this implies the integrability condition (0,2) e UoDU\ in Assumption 2.1.

Remark 3.1. (i) From [36, Theorem III.13.XI], it follows that the analytic extensions *I>o> solve (2.5) and (2.6) as well. Moreover, *I>o, *I>i, I^ô* and D2^\ are continuous on [0, T] x S. For later use, we also note that the mapping t ty$t,0, z) is twice continuously differentiable on [0, T] for all z eSf.

(ii) From [10, Theorem 2.16(h)], it follows with Assumption 3.1(i) that

E[exp(wiy,+5 + u2Zt+s) \ Ft] = exp(vI>o(s, u\,u2) + ^\(s, u\,u2)yt + u2Zt) (3.1)

for all u e S. It is easy to see that ̂j(t,u\, u2) is real-valued for u e R2 H S, j = 0, 1.

(iii) By the first remark we have V0(t, 0, z) = fQ i/o(^\(s, 0, z), z)dsforz e S. Equation (3.1) and Jensen's inequality yield Re(î(^, 0, z)) < ^\(t, 0, Re(z)) and, hence, (V\(t, 0, z), z) e S for all f e [0, T] and all z e 4/.

Lemma 3.1. Under Assumption 3.1(i), is a square-integrable random variable for any z Sf. The same is true for H.

Proof. In view of Remark 3.1(h) we have

E[|4|2] < exp(2(|/?/| v \R$\logS0\)E[e2R'ZT +z2rzt] < oo.




Applying Holder's inequality we have

E[|#|2]<E \SzT\\n\(dz)^

<in|(^)E^ I4i2|n|(dz)

< |n|(^/)2exp(2(|/?/| v |/e|)|log^0|)E[e2/?/^ +e2*zH < oo.

Here |11| indicates the total variation measure of n in the sense of [32, Section 6.1].

A key role in the hedging problem is played by the square-integrable martingale V generated by H, i.e.

Vt :=E[H | Ft], te[0,Tl

We call it an option price process because it could be used as such without introducing arbitrage to the market. But note that we do not assume H to be traded, let alone with price process V.

Proposition 3.1. Under Assumption 3.1(i), we have

Vt = j

V(z),II(dz), te[0,T], (3.2)

where the square-integrable martingale V(z)forz ? &f is defined as V(z)t := E[Sj | !Ft]for t e [0, Tl

Proof. For all t e [0, T] and ze^/,we have the estimate

j E[|V(z),|]|n|(dz)<

j E[|V(z)r|]|n|(dz)

< j

E[eRe(z)los5nin|(dz)

<E[e*'IogSr H-e^logS7,]|n|(4/) < oo

from Remark 3.1(ii) and the finiteness of n. An application of Fubini's theorem yields

E f V(z),n(dz)lcj =E^E[SZT

| <Ft]icnm

= E^f

SzTU(dz) lc = mvt ic]

for all C ? 5^. This implies the assertion.

The process V(z) will be determined in Theorem 4.1, below, as a by-product. Via (3.2) this

leads to an integral representation of the option price process which is of interest in itself. It

extends similar formulae in [7] and [33] to the more general class of processes considered in

this paper.




4. Variance-optimal hedging

We turn now to the hedging problem itself. We generally assume that Assumptions 2.1 and 3.1 hold. We define the set of admissible trading strategies as

0 := {# predictable process: E[|#|2 (S, S)T] < oo}.

We call an initial capital v* e R and an admissible strategy #* variance-optimal (hedge) if

they minimize

E[(v + # ST- H)2]

over all such pairs (u,i?)elx0. The residue

j0 := E[(v* + #* ST -

H)2]

is referred to as the minimal hedging error. The following characterization of the variance

optimal hedge constitutes the first main result of this paper.

Theorem 4.1. The variance-optimal initial capital v* and the variance-optimal hedging strat

egy #* are given by

= j

V(z)0n(dz),

,._r^3*^?aM?zn(*). (4.1)

where the process V(z) satisfies

V(z)t = Sf exp(M/0(T

- t, 0, z) + *i(T - t, 0, z)yt), ze*f, (4.2)

So, <5i e R, and the functions /co, /ci: [0, T] x ? C are defined as

Kj(t, z) := - r, 0, z), z + 1) -

VyOW -

*, 0, z), z), (4.3)

5, 1=^(0,2), 7 =0,1. (4.4)

As is well known, the variance-optimal hedging strategy #* given by (4.1) also yields the solution to

min E[(v + & ST -

H)2l (4.5)

where iJ e R denotes a given initial capital instead of the optimizer from the previous theorem. Our second main result concerns the hedging error. This quantity gives an idea of the remaining risk. The seller of the option may take it into account in order to decide what risk premium to

charge for the claim.

Theorem 4.2. The minimal hedging error is given by

fff* Mt, zi,z2)drn(dzi)n(dz2) */S0^0,Si ?0,

Jo=' fff

^2(f,zi,Z2)d/n(dzi)n(dz2) ifS0 = 0,

jjj^ J3(f,zi,z2)d/n(dzi)n(dz2) i/5i=o.




The integrals over % f have to be understood in the sense of the Cauchy principal value (cf the

following remark). The integrands Jk: [0, T] x Sf2

-> C, k = 1, 2, 3, in these expressions are defined as

J\ (t,Z\,Zl)

= SZI+Z2^? ^exp(*0(f,

?1,^1+ zi) + ?1, zi + z2)yo)

X ^(D2Vl/0(f,

?!, zi + Z2) + 02*l(*. ?l, Zl + Z2)V0) + ̂

~2^Q)

i o e

X Jo (^o

+ eXP(^lS

+ *?(''

1?8(5') + Zl + Z2)

+ *! ^, ̂

log(s) + ?is, zi + Z2^)yoj ds^,

$11+12 ^ h(t, z\,zi) = -exploit, ?i, zi + Z2) + *i(Mi, zi + z2)yo)

X + (?>2*o(Ml,Zl +Z2) + 02*l(',?l,Zl +Z2)W)??2),

zi, z2) = ? ? exp((^(0, zi) + <Ao(0, z2))0T - 0 + iAo(0, zi + zi)t).

77*e constants <$o and 8 \ are defined as in (4.4). The remaining variables are specified as follows:

aj =aj(t, z\,zi) := fj(S$t, zi, z2), zi + z2)

- fj(^\(T

- t, 0, zi), zi) -

fj(^\(T - t, 0, z2), zi),

no =rio(t, zi, zi) '= &o<*o(t, zi, zi) -

k0(t, z$Ko(t, zi),

m = m(f,z\,zi)

:=8oa$t, zi, zi) + 8\ao(t, z\, zi) -

K\(t, z$K0(t, zi) -

K$t, zi)K0(t, zi),

m = rji(t,zuzi) \= 8\a\(t,z\,zi) -

k\(t, z$k\(t, zi),

$j =

?,(*, zuzi) := Vj(T -

t, 0, zi) + Vj(T -

t, 0, z2), j = 0, 1,

with kq and k\ from (4.3). For ease of notation, we dropped the arguments of some functions in

the formulae above. The mappings no, n\,rji, ao, a\, fn> o,nd ?i are defined on [0, T] x %j2.

Remark 4.1. (i) The integrals in the previous theorem are to be understood in the sense that

70=lim f f f Jk(t,zuzi)dtU(dzi)Tl(dzi), ct?o J0

where

Scf := {z eC: R' < Re(z) < R, \lm(z)\ < c}.

It is not obvious whether integrability holds on all of Sf.

(ii) Condition 8\ = 0 actually implies that Z is a Levy process. The integral of J3 relative to t can easily be evaluated in closed form, which is done in [16, Theorem 3.2].




(iii) The minimal hedging error in (4.5) for fixed initial endowment v instead of the optimal v equals

J0 + (v-v)2.

For concrete models and numerical results, we refer the reader to the companion paper [20].

5. Proofs of the main results

This section is devoted to the proofs of Theorems 4.1 and 4.2. We start by analyzing the

martingales V(z) in Proposition 3.1.

Lemma 5.1. V(z) in Proposition 3.1 is of the form (4.2).

Proof The proof follows immediately from (3.1).

The Galtchouk-Kunita-Watanabe (GKW) decomposition of V(z) is determined in the

following lemma.

Lemma 5.2. Fix z Sf. If we set

V(z)t-Ko(t,z) + Ki(t,z)yt- /C1. Hz)t := -t;-r?T-, (5.1)

St- 8o + 8\yt L(z) :=V(z)-V(z)o-#(z)-S,

then $(z) E ? (here liberally extended to complex-valued processes), L(z) is a square integrable martingale, and LS is a local martingale. Here Kj(t, z) and 8j, j = 0,1, are

defined as in (4.3) and (4.4), respectively.

Proof The denominator in (5.1) is positive because the constants 8$,8\ e R+ do not both vanish (cf. Assumption 2.1). The differential characteristics (bs, cs, Fs) of S can be obtained from (2.1)-(2.2) and Proposition A.2. We have

cf = s,2_(y(o')2 + rlifyt-) and F?(G) =

j lG(St-(eX2 - l)Mo)(dx) +

j lG(St-(eX2 - l)Mi)(dx)yf_

for all G e By [17, Proposition II.2.29b], this implies that

(S,S)t= f(css + [ x2Fss(dx))ds= f S2_(8o + $iys-)ds, (5.2) Jo \ Jr / Jo

where the second equality follows from

yfj) + /(e"2

~ D^U)^)

= 2> - 2^(0, 1) = ^-(0, 2) = Sj.

Another application of Proposition A.2 allows us to compute the differential characteris tics (bv^s, cViz)>s, Fv^s) of (V(z), S) = g(I, y, Z), where It = t denotes the identity process and

g(t xx xi) = (S? exP(*o(r " ?' z) + *l V ~ r' ?' z^ + ZX*A U

\ S0exp(x2) J'




Again, using [17, Proposition II.2.29b], this leads to

(V{z), S)t = ?(^cV(z),Srtl

+ jT2

Jcijc2/^v(z)'5(d(jci, jc2?)

ds

= f V(z)s-Ss-(K0(s,z) + Ki(s,z)ys-)ds. (5.3) Jo

We conclude that

?(z)'(S,S) = (V(z),S). (5.4)

Hence, (L, S) = 0, which implies that LS is a local martingale. Equation (5.4) also yields

<L(z), L(i)> + |#(z)|2 (5, S> = <L(z), ~Uz)) + <S, S> = <V(z), Vfe)>. Since V(z) is a square-integrable martingale, we have E[(V(z), V(z))t] < oo, which implies that^(z) 0 and L(z) e M2.

For later use, we need a technical result.

Lemma 5.3. Let rn := inf {t > 0: yt $ [l/n, n] or St ? [l/n, n]} A T. For any n e N, there exists a constant c(n) < oo such that

/ E[l[[0,rn]] \V(z)t\2(a0(t, z, z) + ai(f, z, z)y,)]df < c(n) (5.5) Jo

holds for all z G The functions cto and a\ are defined as in Theorem 4.2.

Proof Fix n N. In this proof c denotes a generic constant that does not depend on t e [0, T] or z ? 4/\ It may vary from line to line. Setting q := max{|/?'|, \R\} implies that l[[0,rn[[QRe{z)lo^St) <nQ <c. Since ^7(zT,

= ^7(21,^2), we have *I>i(f, zi, Z2) =

*I>1 (t, zi, Z2) for zi, Z2 ̂ iR and, hence, for zi, Z2 4/ by uniqueness of analytic continuations. We obtain

?,-(*, z, z) =

V;(2ReOI>i(r -

f, 0, z)), 2Re(z)) -

2Re(^(î(T - t, 0, z), z))

> (Re(*i(r - f, 0, z)), Re(z))y0)(Re(vl/1(r

- f, 0, z)), Re(z))T

+ dm(*i(r - f, 0, z)), Im(z))y0)(lm(vl/1(r

- *, 0, z)), Im(z))T

+ J(exp(Re(vI/!(r

- t, 0, z))*i + Re(z)*2) - \)2<p<j)(&c) > 0

for j ? 0, 1 because is positive semi-definite. Hence,

K\(z) := / E[1[[0,T?]] l{Re(*i(T-/,0,z))<0}

Jo

x exp(2Re(vI/0(r -

f, 0, z)) + 2Re(*i(r -

t, 0, z))y,)S2Re(z) x (a0(f,z,z) + a\(t,z,z)yt)]&t

< exp^ReOW

- t, 0, z)) + ^Re(î(r

- t, 0, z))^c

x (Vr0(2Re(*i(r - t, 0, z)), 2Re(z))

- 2Rt(î(T

- t, 0, z), z))

+ n(î(2Re(î(r - t, 0, z)), 2Re(z))

-2Re(iAi(î(r-r,0, z),z))))dr.




Using (2.5) and (2.6), we obtain

Ki(z) <c |exp^2Re(*I>0(F

- t, 0, z)) + ^Re(*i(7

- t, 0, z))^

x ^0(2Re(î(r - t, 0, z)), 2Re(z)) dr

+ c |exp^2Re(*0(r

- t, 0, z)) + ^Re(*i(T

- t, 0, z))^

x tfri(2Re(î(r - f, 0, z)), 2Re(z)) dr

+ jt (2Re(Vo(T

- t, 0, z)) + ^Re(î(7

- r, 0, z)))

x exp^2Re(*0(r

- *, 0, z)) + ^Re(vI>i(T

- f, 0, z))^

dr

=: c(K2(z) + K3(z) + \K4(z)\).

Equation (3.1) and Jensen's inequality yield

2Re(*7-(f, 0, z)) < */(f, 0, 2Re(z)) for all (r, z) [0, T]x4f.

By continuity of the mappings *I/0 and (cf. Assumption 3.1(h)) we have

\K4(z)\ = exp^Re(z))

- exp^ReOW,

0, z)) + ^Re(*i(7\

0, z)))

< expQRe(z)^)

+ exp^*0(7\

0, 2Re(z)) + 0, 2Re(z))^

< c.

Similarly, we obtain estimates

g(t, z) := 2Re(î(r -

t, 0, z)) < *i(r - f, 0, 2Re(z)) < c,

2Re(^0(r - r, 0, z)) < *oCT

- t, 0, 2Re(z)) < c.

For j = 0, 1, we have

ie?(,'z)/"^(g(^),2Re(z)) n

"2 < -d^l + elyJ^Dc + (l/JÎ + 2clyif|)|g?(f, z)| +

^\y^\\g?(t, z)\2 eg?(t<z)

+ Ie??(',Z) j

leg(t,z)x,+2Re(z)x2 _ l _

^ z)^(xi) _

2Rfi(Z)A(x2)|^0.)((Lc)J

where g?(t, z) g(t, z)/n. Boundedness of g implies that we have

egn(t'z)\gn(t,z)\k <c




for k = 0, 1, 2 and some constant c. Furthermore, we have

Ie*?(f.z) [ |eg(^)*i+2Re(z)*2 _

i-g(t,zMxi)-2Re(z)h(x2)\<P(j)(dx) n J

< Ie*n(*.z)(

f eM^+2R^nj)(dx) + f eM^+2R,X2(p{j)(dx) n \h\x\>\) h\x\>\)

+ (\+2Q + \g{t,z)\)<pU){{\x\ > 1}))

+ lQ8n(t,z) ( |e*(^)*i+2Re(z)*2

_ 1 _ ^ z)jC]

_ 2Re(z)jC2|p(,/)(dJC). n h\x\<\)

Since <pq) is a Levy measure, <p(;)({|Jt| > 1}) is finite. The first two integrals on the right-hand side are bounded in view of Assumption 3.1 (iii). Note that

|eg(f,*)*i+2Re(z)*2 _ 1 _

z)xi _

2Re(z)x2| < c{x\+xl)

for all |*| < 1. Since f^^ix2

+ x\)<p(j)(dx) < oo, we obtain

Ieg?(f.z) j

|e*(f,z)*i+2Re(z)*2 _ j _

^ z)fc(jCl) _

2Re(z)A(jc2)|p0-)(dJc) < c.

Altogether, we conclude that

-e8(t>z)/nifj(g(t, z), 2Re(z)) < c < oo ft

uniformly in (f, z) g [0, T] x <&f, j = 0, 1. This in turn yields #2(z) < c, K^(z) < c,

and, hence,

0 < Ki(z) < c < oo (5.6)

for all z Sf. Analogously, we show that

K5(z) := I E[l[[o,Tn]]l{Re(*i(r-fAz))>0} Jo

x exp(2Re(^0(r - r, 0, z)) + 2Re(*i(r

- *, 0, z))y,-)S,2Re(z) x (a0(?,z,z) +<xi(t9z,z)yt)]dt

< c. (5.7)

Assertion (5.5) follows now from (5.6) and (5.7) because

|V(z)r|2 = sfRe(z) exp(2Re(v!/0(r - t, 0, z)) + 2Re(^i(r

- f, 0, z))yt)

due to Lemma 5.1.

Corollary 5.1. Using the notation of Lemma 5.3, we have

B[(/ ̂ (Z)|2|n|(dZ)) # (5' S)*? < ??' (5'8)

j E[(L(z),Ife))TJ|n|(dz) < oo. (5.9)




Proof. By (5.1) and (5.2) we have

K-=t[(f W(z)\2\n$dz)^ ' (S, S)Tn

=e iv(Z)fi2i^^t:(f")jf'2'n'^d; Uo J <50 + <$iy.

Similar to the derivation of (5.3) we derive

(V(zi), V(zi))t =

j V(zi)sV(z2)s(<xo(s, zuz2) + ai(s, zu z2)ys) ds

for zi, z2 ? Sf. Equations (5.2) and (5.3) imply that

Hz2)'(V(zi),S)t = ?(zi)*(S,V(z2))t = (Hzi)&(z2))*(S, S)t

= / V(zi)sV(z2)s-r?-ds. Jo 8o + 8iys

This leads to

(L(zi),L(z2))t= f V(zi)sV(z2)s Jo

x (ao(s,zuZ2)

+a[(s,z\,Z2)ys

(k0(s,z$ + k$s, z$ys)(Ko(s, Z2) + k\(s,Z2)ys)\ A 1A, -i

- }ds (5A?)

for zi, z2 ? Sf. The angle bracket (L(z), L(z)) = (L(z), L(z)) is a nonnegative increasing process. Hence,

ko(f,z) + Ki(r,z)y,|2 ^ . , . 0 < -?-7- <

<xo(t,z,z)+ai(t,z,z)yt (5.11) <$0 +<$iyr

for all t e [0, T] because Kj(t, z) = (r, z) for y = 0, 1. With the help of Fubini's theorem we deduce that

K<ff E[l[[0,Tn]]|^

Since n is a finite measure on Sf, estimate (5.8) follows from Lemma 5.3. In view of (5.10) and (5.11) we have

(L(z),L(z))Tn < [

" \V(z)t\\ao(t,z9z)+ai(t9z9z)yt)dt. (5.12) Jo

Hence, (5.9) follows from Lemma 5.3 as well.

We can now determine the GKW decomposition of V.




Lemma 5.4. A real-valued admissible trading strategy is defined by

#* := j

0(z)n(dz).

Moreover,

L := j

L(z)U(dz)

is a real-valued square-integrable martingale orthogonal to S (i.e. such that LS is a local

martingale). Finally, we have V = V0 + #*'S + L.

Proof. Using Holder's inequality and (5.8), we obtain

E ?

\&?\2d(S,S)t <E |0(z),||n|(dz))

d(S,S)t

<\n\(4f)E |^u)|2|n|(dz)^-(5,5>rn < 00

for the stopping times xn from Lemma 5.3. Hence, #* e L20C(S). Lemma 5.2 implies that

SL(z) e M\oc for all z Sf. Let (crn)n>\ denote a localizing sequence for SGn e J?^c. Since

L(z) e M2, it follows that (SL(z))?n eM\cf. [17, Theorem 1.4.2]. Now we set rn := xn A an. Since (L(z), L(z)) is an increasing process, Jensen's inequality yields

(E[|L(z)T|])2 <E[|L(z)r|2] <E[(L(z),LU))r] (5.13)

for any stopping time r. In Lemma 5.3 we have shown that

/ E[l[[0,fn]] \V(z)t\2(ao(t, z, z) + ai(r, z, z)y,)]dr Jo

is uniformly bounded in z e Sf. Owing to STn e M2, we have

K := sup{E[(S^)2]: t e [0, T]} < oo.

In view of the Cauchy-Schwarz inequality, (5.12), and (5.13), we obtain

f E[|^L(z)f"|]|n|(dz)

< f ^j\[lmrn]]\V(z)t\Ha0?

|n|(dz) < oo

and, similarly,

e[| |L(z)rnArlin|(dz)J =

j E[|L(z)rT"|]|n|(dz) < oo

for t e [0, T]. In particular, the integral in the definition of L is finite. An application of




Fubini's theorem yields

E[(S?L]n -

SJ?LJ?) \a\ = j

E[(Sj"L(z?tn -

S]nL(z)]n) lA]II(<k) = 0

for A e !FS and s < t. Hence, SL e M\oc and, therefore, (S, L) = 0. A similar Fubini-type argument shows that LTn e M because L(z) e M2 for all z e Sf and M2 is stable under

stopping. Furthermore, (5.9) leads to

sup E[\LTt?\2]=E[\LTn\2] te[0,T]

< jj

E[|L(zi)T||L(z2)rJ]|n|((kl)|n|(dZ2)

< |n|(^) f

E[|L(z)rj2]|n|(dz)

< \n\(sf) JE[(L(z),I(i)>Tj|n|(dz)

< oo

and, hence, L e Obviously, L starts in zero. From Proposition 3.1 and Lemma 5.2, we know that

V = Vb + f

S)n(dz) + L.

In view of (5.8), Fubini's theorem for stochastic integrals [30, Theorem IV.65] yields

j (Hz) S)Tl(dz) =

(^f 0(z)n(dz)^ =

From

(#* -

&*) (5, S) = (V, S) -

(V, S) = 0,

it follows that #* and hence also L is real valued. The admissibility of #* and the square integrability of L follow as in Lemma 5.2 from V e M2.

The GKW decomposition of V leads to the variance-optimal hedge.

Proof of Theorem 4.1. The assertion follows from Lemma 5.4 combined with [34, Theo rem 3 and Corollary 10].

Finally, we turn to the proof of the formula for the hedging error. Observe that the superscript c in the following does not refer to the continuous martingale part of processes.

5.1. Proof of Theorem 4.2

Let us consider the truncated payoff function

fc(s) := j

^lT(dz),

where c e R+, Tlc(A) := Tl(<8cf

n A) for all A e ?(Sf), and

Scf := {z e C: R' < Re(z) < R, \lm(z)\ < c}.




Observe that fc is real valued because n is symmetric. We define Hc := fc(Sr) by the

corresponding contingent claim. Its price process Vc is defined by Vtc := E[HC | Ft] as in Section 3. By Proposition 3.1 we have

Vtc = f V(z)tnc(dz) = f V(z)tU(dz).

Holder's inequality yields

\Vt-Vt\2<^c |V(z)rlin|(dz)) < \n\(sf)

f |V(z)r|2|n|(dz)

< (\T\\(*f))2(e2RlozST + e2*'10^).

Since the expression on the right-hand side is square integrable, we have

Hc = V? VT = H as c -> oo

by dominated convergence. If we set

:= j

#(z)Uc(dz) and Lc := f

L(z)Uc(dz),

then Vc = Vc(0) + #*-S + L

is the GKW decomposition of Vc in the sense of Lemma 5.4. Theorem 3.8 of [27] shows that

LCT converges to Lt in L2. Hence, we have

7o = E[L2T] = lim E[(L^)2] = lim E[(LC, Lc)Tl (5.14) c?>oo c-*oo

In order to finish the proof, we need the following Fubini-type theorem for the predictable covariation.

Lemma 5.5. We have

(Lc, Lc) = jf

<L(zi), L(z2))nc(dz!)nc(dz2),

where (L(zi), L(z2)) is given by (5.10).

Proof. This is shown similarly as in the proof of [16, Theorem 3.2]; cf. [28, Lemma 3.26] for details.

We can now complete the proof of Theorem 4.2. From (5.14), it follows with Lemma 5.5

that

E[L2]= lim e\ f [ (L(z1),L(z2)>rn(dzi)n(dz2)l. (5.15)




We focus on the case in which So, 8\ ̂ 0. The others follow along the same lines. Note that

(X, X) is an increasing and integrable process for all C-valued square-integrable martingales X. From (5.10), it follows that

E[(L(zi), L(Z2))t] = I* {y- E[V(zi), V(z2)tyt] + mh~~m&*

E[V(zi), V(z2)t] jo \0\ 6\

m^-m^oh + mSl\V(zi)tV(z2)tJ\. +?--2-

E - dr, (5.16)

where 770, ?7i, and r?2 defined as in the assertion depend on f, zx, and z2. In view of Assump tion 3.1 (i) and Remark 3.1(h), we obtain

E[V(zi)tV(z2)t] = ^?SZi+Z2 exp(vl/0(r, ?!, zx + z2) + ?1, zi + z2)y0), (5.17)

where ?0 and ?1 defined as in the assertion depend on t, zx, and z2. For (jci, jc2) ^, we set

go(xx,x2) := exp(x\yt + x2Zt) and *2) := E[g0(*i, *2)L Moreover, let e) :=

{z C: |z ?

?11 < s] denote the ball around ?1 with radius s. Choose s := e/2 for s as in

Assumption 3.1(i). We have

esup{|Z>i*0?,zi +z2)|: ? < e(Mo+2^'(e2*z' + e2*'2').

The right-hand side of this inequality has finite expectation by Assumption 3.1(i) and Remark 3.1(h). Since

go(?i + +Z2)-go<S\,zx H-z2) ~. flri . , d ~m - < s + sup{|Digo(?, zi + 22)1: ? e e)} n

for sufficiently small |n|, dominated convergence yields

E[V(zx)tV(z2)tyt] =

^Sz0^Z2E[Dxgo(^uzx+Z2)] =

^?Sz0l+Z2Dxgx(l;uzx+Z2) = E[V(zx)tV(z2)t](D2Vo(t, ?1, zx + Z2) + 02*1 fr, ?1, zx + z2)yo). (5.18)

Moreover,

^-X^*-jr(^M(? **-)?>)* with r: [0, 1] -+ C, s i-> (Si/So) log(s) + f is. It follows that

ErV(zi)fVfa)r' [ <5o + <5iyf .

, e-<5ofi/<Si -e A0

^

x ^

E jf exp^^

log(i) + fts^y,

+ j-hs + (Zl + zi)Z^

ds

+ I1E jf sexp^^log(s)

+ |i^y,

+ ^|i5

+ (Zi+z2)Z^ds ̂




By 80, 8\ > 0 we have

exp((^ log(j) +

%is^yt +

j-Sis + (zi +

z2)Zt^j < Q80(M0v0)/Si (e(M0vO)yt+2RZt + e(Af0vO)yf+2i?'Z, ^ (5.19)

for all s e (0, 1]. Since the right-hand side of this inequality has finite expectation (cf.

Assumption 3.1(i) and Remark 3.1(H)), Fubini's theorem and, once more, Remark 3.1(ii) yield

E\V(zi),V(z2)t-\_c,nsZI+Z2c-^ L So + Siyt J 0 <5i

x exp^*I>0^, ^

log(5) + f is, z\ + zi^

+ vj/, (t, ̂ log(s) + fis, zi +

z2)yo) ds. (5.20)

From (5.17)-(5.19) and from the continuity of *o, *i, D2*o,andD2*i on[0, T]x S, we obtain

E[|V(z),|2] = E[V(z),V(z)(] <*i(c),

E[|V{z),\2y,} = E[V(z)tV(z)tyt] < k2(c),

JM1=EM<,3(C), 180 + <5iyr J L 5o + 8\yt _

for all (r, z) [0, T] x ?cf

and some finite numbers &i(c), fc2(c), and ̂ (c) that may depend on c, but not on t or z. Consequently, (5.16) yields

E[(L(z), L(z))t] = E[(L(z), L(z)>r]

< jf ̂2(r,

z, z)|^

+ (si\m(t, z, z)| + 8o\n2(t, z, z)|)^

+ (8\\m(t, z, z)\ + Mi\m(t, z, z)\+ , z, z)l)^)

d*

< *4(c)

for all z ?f

and a positive constant k^ic) depending only on c because the mappings

(f,z) h> mi*, z,z), T]i(t, z,z), r)2(t, z,z)

are continuous on [0, T] x ^ by Assumption 3.1(h). Hence, Fubini's theorem yields

E[4]=limf f E[(L(zi),L(z2)>r]n(dzi)n(dz2)

in view of (5.15). The assertion follows now from (5.16)?(5.18) and (5.20).




Appendix A

The dynamic behaviour of a semimartingale can be described in terms of predictable charac teristics (B, C, v) (cf. [17] for a definition and further background). For most continuous-time

processes in applications, the characteristics factorize as follows.

Definition A.l. Let X be an R^-valued semimartingale with characteristics (#, C, v) relative to some truncation function h: Rd -> M.d. If there are some R^-valued predictable process b, some predictable Rdxd-valued process c whose values are nonnegative, symmetric matrices, and some transition kernel F from (Q x R+, P) into (RJ, &d) such that

Bt = f bs ds, Ct = f cs ds, Jo Jo

v([0, t] x G) = f Fs(G)ds for* e [0, T], G e &d, Jo

we call (b,c, F) differential characteristics of X.

We can interpret bt or rather bt+f(x?h(x))Ft (dx) as a drift rate, ct as a diffusion coefficient, and Ft as a local jump measure. For a Levy process, the triplet (b, c, F) is deterministic and does not depend on t. In this case it coincides with the Levy-Khintchine triplet appearing in the

Levy-Khintchine formula. In a sense the differential characteristics generalize this triplet for more general semimartingales. They are typically derived from other 'local' representations of the process, e.g. in terms of a stochastic differential equation.

Proposition A.l. Let X be an Rd-valued semimartingale with differential characteristics

(b,c, F), and let % be an Rnxd-valued predictable process whose components j =

1, ..., n, are integrable with respect to X. Then the W1-valued integral process ? X :=

(?7' X)j-\fm^n has differential characteristics (b, c, F) of the form

bt = $tbt + j(h&x)

- $Mx))Ft(dx),

ct = $tCtHj,

Ft(G) = j

\G^tx)Ft(dx) for all G e &n with 0 i G.

Here, h and h denote truncation functions on Rd and W1, respectively.

Proof. Cf. [17, IX.5.3] and [22, Lemma 3].

Ito's formula can be expressed in terms of characteristics as follows.

Proposition A.2. Let X denote an Rd-valued semimartingale with differential characteristics (b,c, F). Suppose that g: U -> R" is twice continuously differentiate on some open subset

U C Rd such that X and X_ are U-valued. Then the W1-valued semimartingale g(X) has

differential characteristics (b, c, F) of the form d

1 d

b\ = YJDkgi(Xt-)bkt + -

D*V(X,-)c*'' k=l k,l=l

+ J (hi(g(Xt-+x)-g(X,-))-

J^ Dkg'iXt^hHx^Ftidx),




d

c\J = ]T Dkg^X^c^DigHXt-), kj=i

Ft(G) = J

lG(g(Xt- + jc) - g(Xt-))Ft(dx) for all G e ?n with 0 i G,

1,7 = 1, ..., n. Here, etc. denote partial derivatives and h and h truncation functions on

Rd and W1, respectively.

Proof Cf. [12, Corollary A.6].

Acknowledgements

The first author gratefully acknowledges partial support through Sachbeihilfe KA1682/2-1 of the Deutsche Forschungsgemeinschaft. Thanks are also due to Richard Vierthauer for valuable discussions.

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