Quadratic hedging strategies in affine models

76 Pages·2017·0.55 MB·English

MSc Stochastics and Financial Mathematics
Master Thesis
Quadratic hedging strategies in affine
models
Author: Supervisor:
Dafni Mitkidou dr. Asma Khedher
Examination date:
September 26, 2017
Korteweg-de Vries Institute for
Mathematics
Abstract
This thesis studies the problem of hedging a contingent claim in an incomplete mar-
ket. To approach this problem we use the method of quadratic hedging. The locally
risk-minimization and the mean-variance hedging are the two main quadratic hedging
approaches which are discussed in this context. We begin by giving an overview of
results and developments in these two areas. We then apply the theory to two affine
stochastic volatility models, namely the Heston model and the BNS model, and we
obtain semiexplicit formulas for the optimal hedging strategies.
Title: Quadratic hedging strategies in affine models
Author: Dafni Mitkidou, [email protected], 11137665
Supervisor: dr. Asma Khedher,
Second Examiner: dr. Peter Spreij
Examination date: September 26, 2017
Korteweg-de Vries Institute for Mathematics
University of Amsterdam
Science Park 105-107, 1098 XG Amsterdam
http://kdvi.uva.nl
2
Contents
Introduction 5
1 Basic Concepts 7
1.1 Semimartingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 L´evy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.3 Stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Financial market set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Locally risk-minimizing strategy 21
2.1 Risk-minimizing strategy: the stock price as a martingale . . . . . . . . . 21
2.2 Locally risk-minimizing strategy: the stock price as semimartingale . . . . 25
3 Mean-variance hedging strategy 32
3.1 The stock price as a martingale . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 The stock price as a semimartingale . . . . . . . . . . . . . . . . . . . . . 33
4 Quadratic hedging in affine stochastic volatility models 38
4.1 Introduction to affine stochastic volatility models . . . . . . . . . . . . . . 39
4.2 Locally risk-minimization in affine stochastic volatility models . . . . . . . 41
4.2.1 Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2 BNS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Mean-variance hedging in affine stochastic volatility models . . . . . . . . 55
4.3.1 Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 BNS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Conclusion 70
6 Future research 71
Popular summary 72
3
List of abbreviations
BNS Barndorff-Nielson and Shephard
E(L)MM equivalent (local) martingale measure
FS F¨ollmer-Schweizer
GKW Galtchouk-Kunita-Watanabe
LRM locally risk-minimizing
MMM minimal martingale measure
MVH mean-variance hedging
MVT mean-variance tradeoff process
RM risk-minimizing
SC structure condition
The following classes of processes are often used in the thesis.
V adapted processes with finite variation (p. 8)
A processes with integrable variation (p. 8)
A+ integrable increasing processes (p. 15)
S semimartingales (p. 8)
S2 square integrable semimartingales (p. 21)
M2 square integrable maringales (p.26)
If C is a class of processes, then C denotes the localized class. If we want to de-
loc
note the measure explicitly, then we use e.g. C(P).
For the following notation we refer to the page where the notation is introduced.
Q p. 19
G (µ) p. 15
loc
L(X) p. 27
L2(S) p. 22
4
Introduction
Onecentralprobleminfinancialmathematicsisthehedgingofcontingentclaims. Banks
and other financial institutions face the risk of losing money from trading various finan-
cial products. Therefore it is particularly important for them to form good hedging
strategies which minimize this risk as much as possible. The importance of hedging
has emerged after the recent subprime crisis of severe losses due to supposedly hedged
positions.
One of the most celebrated models which is used for pricing and hedging is the Black-
Scholes model (Black and Scholes (1973)). This model allows to obtain perfect hedging
strategies by assuming market completeness. In complete markets, where every contin-
gent claim is attainable, the risk involved in a portfolio can be eliminated. However, if
we would like to take a more realistic view of the the financial world, we should also con-
sider models which contain stochastic volatility effects or jumps. Markets which include
such models are incomplete. In incomplete markets not every contingent claim can be
replicated by a self-financing strategy and therefore perfect hedging strategies do not
exist.
Hedging in incomplete markets is a problem that has been studied extensively in the
past years. The existing literature provides several different methods for hedging a con-
tingent claim in incomplete markets. Suprhedging, utility maximization and quadratic
hedging are among the most commonly used approaches. A short description of these
methods, based on Chapter 10 of Cont and Tankov (2004), can be found below.
A superhedging strategy is a self-financing strategy such that at maturity the value of
thestrategyissurelygreaterthanthevalueofthederivative. Thecostofsuperhedgingis
defined as the cost of the cheapest superhedging strategy. One drawback of this method
is that it is often too expensive. Furthermore, it gives equal importance to hedging in
all scenarios which can occur, regardless of the actual loss in a given scenario. For more
details,wereferthereadertoElKarouiandQuenez(1995)andJouiniandKallal(1995).
On the other hand, utility hedging (see Hu et al. (2005)) is a more flexible approach
which involves weighting scenarios according to the losses incurred and minimizing this
weighted average loss. This idea is formalized using the notion of expected utility: For
every utility function U, which should be concave and increasing, we can determine the
related utility hedge by maximizing the function:
E[U(Z)]
over different payoffs Z. Risk is therefore measured by expected utility. In the general
5
case, this method provides rarely explicit solutions to the hedging problem. Quadratic
hedging is a special case of utility hedging. It can be defined as the choice of a strategy
which minimizes the hedging error in mean square sense. Hence risk in this case can be
seen as variance. In this approach the hedging strategies can be sometimes computed
explicitly. Thisfactmotivatedustostudyquadratichedginginthecontextofthisthesis.
There are two main quadratic hedging approaches, the locally risk-minimization and
the mean-variance hedging. In the theory of locally risk-minimization, we insist on the
replicating condition and the goal is to minimize the variance of the cost process at
any time. This method was developed first by F¨ollmer and Sondermann (1986) and
later on it was studied extensively by Schweizer (1991) and by F¨ollmer and Schweizer
(1991). On the other hand, the mean-variance hedging strategy aims to minimize in L2-
sensethedifferencebetweentheclaimatmaturityandtheportfolioatthattime, usinga
self-financingstrategy. ThisapproachwasintroducedbyBouleauandLamberton(1989).
Inthisthesiswefocusourattentionontheclassofaffineprocesses. Affineprocessesare
appealingbecausetheyhavepowerfulpropertieswhichmakethemanalyticallytractable.
Ifaprocessisaffine,itscharacteristicfunctionhasaclosed-formrepresentationandhence
one can obtain often explicit or semiexplicit formulas for pricing and hedging various
contingent claims. Duffie et al. (2003) give a rigorous mathematical foundation to the
theory of affine processes.
The purpose of this thesis is to exploit the rich structural properties of affine models
in order to obtain semiexplicit representations for the quadratic hedging strategies. In
a theoretical level, we study the results and developments in the areas of locally risk-
minimization and mean-variance hedging. On the computational side, we aim to apply
theseresultstoconcreteaffinestochasticvolatilitymodels, suchasHesonmodel(seeHe-
ston (1993)) and BNS model (see Barndorff-Nielsen and Shephard (2001)). These two
models feature important market characteristics such as jumps and stochastic volatility
and furthermore, they are well-known models that are often used in practice.
The thesis is structured as follows. Chapter 1 recalls preliminaries from the semi-
martingaletheoryandprovidesadescriptionofourfinancialmarket. Chapter2describes
the main results in the area of locally risk-minimization. The mean-variance hedging
strategy is discussed in Chapter 3. Chapter 4 introduces the class of affine processes and
includes the application of quadratic hedging theory to concrete affine stochastic volatil-
ity models. Finally, in Chapter 5 we sum up the results of the thesis and in Chapter 6
we recommend some points for future research. Chapters 2-4 begin with with a review
of the existing literature. Notice that in Chapters 1-3 most of the proofs are omitted
and the reader is referred to the related work where the proofs appear. On the other
hand, in Chapter 4, which constitutes our main contribution, we provide to the reader
all the necessary details.
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1 Basic Concepts
1.1 Semimartingale
1.1.1 General Theory
In this subsection wegivean overview ofthe semimartingaletheory. Wefocusmainly on
concepts which are important for the following chapters. Most of the results presented
in this section are based on the book of Jacod and Shiryaev (2003), unless it is stated
otherwise.
Let(Ω,F,P)beaprobabilityspace,T ∈ (0,∞)beourtimehorizonandF = (F )
t 0≤t≤T
be our filtration. Intuitively, F describes the information available at time t. Note that
t
all the random variables that are introduced throughout the whole thesis are defined on
this space.
Now let us define the set V as the set of all real-valued processes A with A(0) = 0
that are c`adl`ag, adapted and for which each path t → A(t,ω) has finite variation over
each finite interval [0,t]. The subset of processes from V that have integrable variation
is denoted by A and the localized class by A . To say that a process X ∈ V has locally
loc
integrable variation means that there is a sequence of stopping times τ increasing to
n
infinity, and such that the variations (cid:82)τn |dX| are all integrable. This is equivalent to X
0
being a locally integrable process with finite variation. A semimartingale is then defined
as below.
Definitions 1.1.
• A semimartingale X is a process of the form
X = X(0)+M +B,
with X(0) finite-valued and F -measurable, M a local martingale and B ∈ V.
0
• A special semimartingale X is a semimartingale which admits a decomposition
X = X(0)+M +B,
with B predictable.
The set of all semimartingales is denoted by S.
7
Example 1.2. Consider an Itˆo process
dX(t) = µ(t)dt+σ(t)dW(t),
where W is a Brownian motion and µ,σ adapted processes such that
(cid:90) t (cid:90) t
∀t > 0, |µ(s)|ds < ∞ and σ2(s)ds < ∞ a.s.
0 0
Then X is a special semimartingale.
We continue by introducing the concepts of quadratic covariation, compensator and
predictable quadratic covariation. These concepts appear often in this thesis and will be
used extensively in the computations of quadratic hedging strategies.
Definitions 1.3.
• The quadratic covariation of two semimartingales X and Y is defined as
(cid:90) · (cid:90) ·
[X,Y] = XY −X(0)Y(0)− X(t−)dY(t)− Y(t−)dX(t).
0 0
The quadratic covariation is characterized in Chapter I, Theorem 4.47 of Jacod
and Shiryaev (2003). For more details regarding the construction of stochastic
integrals w.r.t. a semimartingale see Chapter I, section 4d Jacod and Shiryaev
(2003).
• For processes X ∈ A we can define the unique process Xp, called the compen-
loc
sator under P, which is the predictable process in A such that X − Xp is a
loc
P-martingale.
• The predictable quadratic covariation of two semimartingales X,Y is the
compensator of the quadratic covariation [X,Y]. It is denoted by (cid:104)X,Y(cid:105) and
therefore also called the angle bracket of X and Y. The short hand notation (cid:104)X(cid:105)
will be used for the angle bracket (cid:104)X,X(cid:105).
In the following example we determine the quadratic covariation and the predictable
quadratic covariation of two Itˆo processes.
Example 1.4. Let
dX(t) = µ dt+σ dW (t),
1 1 1
dY(t) = µ dt+σ dW (t),
2 2 2
where W ,W are two Brownian motions with correlation ρ and µ ,µ ,σ ,σ constants.
1 2 1 2 1 2
8
By definition of quadratic covariation we obtain
(cid:90) t (cid:90) t
[µ t,µ t] = µ µ t2− µ µ sds− µ µ sds = µ µ t2−µ µ t2 = 0.
1 2 1 2 1 2 1 2 1 2 1 2
0 0
Similarly we can prove that [µ t] = [µ t] = 0. Furthermore, we have
1 2
(cid:90) t (cid:90) t
[µ t,σ W (t)] = µ σ W (t)− µ σ W (s)ds− µ σ sdW (s).
1 1 1 1 1 1 1 1 1 1 1 1
0 0
Application of Itˆo’s formula for the function f(x,t) = µ σ tx leads
1 1
(cid:90) t (cid:90) t
µ σ W (t) = µ σ W (s)ds+ µ σ sdW (s).
1 1 1 1 1 1 1 1 1
0 0
Therefore,
[µ t,σ W (t)] = 0.
1 1 1
Similarlywecanprovethat[µ t,σ W (t)] = [µ t,σ W (t)] = [µ t,σ W (t)] = 0. Finally,
1 2 2 2 1 1 2 2 2
it is known that [W (t),W (t)] = ρt. Hence, using linearity of quadratic covariation, we
1 2
get
[X,Y] = [σ W (t),σ W (t)] = σ σ ρt = (cid:104)X,Y(cid:105).
1 1 2 2 1 2
We remark that the compensator can be equivalently defined for processes X ∈ V. How-
ever, the additional condition X ∈ A guarantees the existence of the copmpensator
loc
(see Chapter I Theorems 3.17 and 3.18 Jacod and Shiryaev (2003)). Furthermore, the
compensator of a process is measure dependent, since the martingale property is not
invarient under measure changes. Hence the predictable quadratic covariation is also
measure dependent. On the other hand, the quadratic covariation is independent of the
measure we work with.
Proposition 1.5. If X is a local martingale and Y ∈ V, then also [X,Y] is a local
martingale and hence (cid:104)X,Y(cid:105) = 0.
Proof. This is proved in Chapter I, Proposition 4.49 Jacod and Shiryaev (2003).
The notion of quadratic covariation can be used now to define the concept of orthogonal
semimartingales.
Definition 1.6. TwoP-semimartingalesX andY arecalledorthogonalundermeasure
P if [X,Y] is a local martingale under P. Hence the angle bracket (cid:104)X,Y(cid:105) = 0.
Another important concept and a standard tool for this thesis is the characteristics of a
semimartingale. Assumewehavead-dimensionalsemimartingaleX withdecomposition
X = X(0)+M +B. Then the local martingale M has a unique decomposition in a
continuous local martingale Mc and a purely discontinuous local martingale Md:
X = X(0)+Mc+Md+B
9
The continous local martingale part of a semimartingale X is denoted by Xc = Mc,
while the discontinuous local martingale part is denoted by Xd = Md.
In the following definition we restrict ourselves to the specific case that X is a special
semimartingale.
Definition 1.7. Let X be an Rd-valued special semimartingale. Suppose that C is a
predictableRd×d-valuedprocesswhosevaluesarenon-negativesymmetricmatrices,both
with components of finite variation, and ν a predictable random measure on R ×Rd
+
(i.e. a family (ν(ω;∆)) ω ∈ Ω of measures on R × Rd with a certain predictability
+
property, see Jacod and Shiryaev (2003) for details). Then the triplet (B,C,ν) is called
characteristics of X if and only if eiλtrX −(cid:82)·eiλtrX(t−)dΨ(t,iλ) is a local martingale
0
for any λ ∈ Rd, where
1 (cid:90) t(cid:90)
Ψ(t,u) = utrB(t)+ utrC(t)u+ (eutrx−1−utrh(x))ν(ds,dx)
2
0 Rd
and h(x) denotes a fixed truncation function.
This integral version of the characteristics can alternatively be written in differential
form. Morespecifically, thereexistanincreasingpredictableprocessAandapredictable
triplet (b,c,F) such that
(cid:90) t
B(t) := b(s)dA(s),
0
(cid:90) t
C(t) := c(s)dA(s),
0
(cid:90) t
ν([0,t]×G) := F(s,G)dA(s) for t ∈ [0,T],G ∈ Bd.
0
In most applications the characteristics (B,C,ν) are actually absolutely continuous,
which means that one may choose A(t) = t. In this case we call the triplet (b,c,F) local
or differential characteristics of X. Intuitively, b denotes the local drift rate of X,
c the local covariance matrix of the continuous part and F the local L´evy measure of
jumps.
Example 1.8. For Itˆo processes as defined in Example 1.2, the local characteristics are
given by
b(t) = µ(t),
c(t) = σ2(t),
F = 0.
Usingthecharacteristictriplet(B,C,ν), everyRd-valuedsemimartingaleX canbewrit-
ten in the following form.
(cid:90) t(cid:90)
X(t) =X(0)+B(t)+Xc(t)+ h(x)(µ−ν)(ds,dx)
0 Rd
(cid:90) t(cid:90)
+ (x−h(x))(µ−ν)(ds,dx),
0 Rd
10