Quadratic hedging strategies in affine models

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MSc Stochastics and Financial Mathematics

Master Thesis

Quadratic hedging strategies in affine

models

Author: Supervisor:

Dafni Mitkidou

dr. Asma Khedher

Examination date:

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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, dafnimitkidou@student.uva.nl, 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

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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 Cloc denotes the localized class. If we want to

de-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

Gloc(µ) p. 15

L(X) p. 27

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Introduction

One central problem in financial mathematics is the hedging of contingent claims. 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 Black-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 the strategy is surely greater than the value of the derivative. The cost of superhedging is 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, we refer the reader to El Karoui and Quenez (1995) and Jouini and Kallal (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)]

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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. This fact motivated us to study quadratic hedging in the context of this thesis. 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 -sense the difference between the claim at maturity and the portfolio at that time, using a self-financing strategy. This approach was introduced by Bouleau and Lamberton (1989). In this thesis we focus our attention on the class of affine processes. Affine processes are appealing because they have powerful properties which make them analytically tractable. If a process is affine, its characteristic function has a closed-form representation and hence 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 these results to concrete affine stochastic volatility models, such as Heson model (see He-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-martingale theory and provides a description of our financial market. Chapter 2 describes 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 we give an overview of the semimartingale theory. We focus mainly 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) be a probability space, T ∈ (0, ∞) be our time horizon and F = (Ft)0≤t≤T

be our filtration. Intuitively, Ft describes the information available at time t. Note that

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 Aloc. To say that a process X ∈ V has locally

integrable variation means that there is a sequence of stopping times τn increasing to

infinity, and such that the variationsRτn

0 |dX| are all integrable. This is equivalent to X

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 F0-measurable, M a local martingale and B ∈ V.

• A special semimartingale X is a semimartingale which admits a decomposition X = X(0) + M + B,

with B predictable.

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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 ∀t > 0, Z t 0 |µ(s)|ds < ∞ and Z t 0 σ2(s)ds < ∞ a.s. 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 [X, Y ] = XY − X(0)Y (0) − Z · 0 X(t−)dY (t) − Z · 0 Y (t−)dX(t).

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 ∈ Alocwe can define the unique process Xp, called the

compen-sator under P, which is the predictable process in Aloc such that X − Xp is a

P-martingale.

• The predictable quadratic covariation of two semimartingales X, Y is the compensator of the quadratic covariation [X, Y ]. It is denoted by hX, Y i and therefore also called the angle bracket of X and Y . The short hand notation hXi will be used for the angle bracket hX, Xi.

In the following example we determine the quadratic covariation and the predictable quadratic covariation of two Itˆo processes.

Example 1.4. Let

dX(t) = µ1dt + σ1dW1(t),

dY (t) = µ2dt + σ2dW2(t),

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By definition of quadratic covariation we obtain [µ1t, µ2t] = µ1µ2t2− Z t 0 µ1µ2sds − Z t 0 µ1µ2sds = µ1µ2t2− µ1µ2t2= 0.

Similarly we can prove that [µ1t] = [µ2t] = 0. Furthermore, we have

[µ1t, σ1W1(t)] = µ1σ1W1(t) − Z t 0 µ1σ1W1(s)ds − Z t 0 µ1σ1sdW1(s).

Application of Itˆo’s formula for the function f (x, t) = µ1σ1tx leads

µ1σ1W1(t) = Z t 0 µ1σ1W1(s)ds + Z t 0 µ1σ1sdW1(s). Therefore, [µ1t, σ1W1(t)] = 0.

Similarly we can prove that [µ1t, σ2W2(t)] = [µ2t, σ1W1(t)] = [µ2t, σ2W2(t)] = 0. Finally,

it is known that [W1(t), W2(t)] = ρt. Hence, using linearity of quadratic covariation, we

get

[X, Y ] = [σ1W1(t), σ2W2(t)] = σ1σ2ρt = hX, Y i.

We remark that the compensator can be equivalently defined for processes X ∈ V. How-ever, the additional condition X ∈ Aloc guarantees the existence of the copmpensator

(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 hX, Y i = 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. Two P-semimartingales X and Y are called orthogonal under measure P if [X, Y ] is a local martingale under P. Hence the angle bracket hX, Y i = 0.

Another important concept and a standard tool for this thesis is the characteristics of a semimartingale. Assume we have a d-dimensional semimartingale X with decomposition 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:

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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 predictable Rd×d-valued process whose values are non-negative symmetric matrices, 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 −R·

0e

iλtr_X(t−)

dΨ(t, iλ) is a local martingale for any λ ∈ Rd, where

Ψ(t, u) = utrB(t) +1 2u tr_{C(t)u +} Z t 0 Z Rd (eutrx− 1 − utrh(x))ν(ds, dx) and h(x) denotes a fixed truncation function.

This integral version of the characteristics can alternatively be written in differential form. More specifically, there exist an increasing predictable process A and a predictable triplet (b, c, F ) such that

B(t) := Z t 0 b(s)dA(s), C(t) := Z t 0 c(s)dA(s), ν([0, t] × G) := Z t 0 F (s, G)dA(s) for t ∈ [0, T ], G ∈ Bd.

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.

Using the characteristic triplet (B, C, ν), every Rd-valued semimartingale X can be writ-ten in the following form.

X(t) =X(0) + B(t) + Xc(t) + Z t 0 Z Rd h(x)(µ − ν)(ds, dx) + Z t 0 Z Rd (x − h(x))(µ − ν)(ds, dx),

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where X(0) is finite-valued and F0-measurable, µ is the random measure of jumps of X

and h the truncation function. The latter form of X is called canonical representa-tion.

We finish this subsection by stating some technical results, which will help us to compute the local charcteristics of a special semimartingale X in Chapter 4.

Definition 1.9. Let X be an Rd-valued semimartingale with local characteristics (b, c, F ). Then the Rd×d-valued predictable process

˜

c(t) := c(t) + Z

Rd

xxtrF (t, dx)

is called modified second characteristic of X if the integral exists.

This modified second characteristic appears in the context of predictable covariation processes.

Lemma 1.10. Let X be an Rd-valued semimartingale with characteristics (b, c, F ) and modified second characteristics ˜c. If the corresponding integral exists, we have

hX_i, Xji(t) =

Z t

0

˜ cij(s)ds

for the predictable covariation process of the components of X = (X1, · · · , Xd).

Proof. This is proved in Proposition 1.2 ˇCern´y and Kallsen (2007).

Proposition 1.11. Let X be an Rd-valued predictable process with local characteristics (b, c, F ) and H an Rn×d-valued predictable process which is integrable with respect to X. The local characteristics (ˆb, ˆc, ˆF ) of the Rn-valued integral process

Z · 0 H(t)dX(t) := Z · 0 Hj(t)dX(t) j=1,··· ,n

are of the form ˆ b(t) = H(t)b(t) + Z Rd (˜h(H(t)x) − H(t)h(x))F (t, dx), ˆ c(t) = H(t)c(t)H(t)tr, ˆ F (t, A) = Z Rd 1A(H(t)x)F (t, dx) ∀A ∈ Bn with 0 /∈ A.

Here ˜h : Rn→ Rn _{and h : R}d_{→ R}d _{denote the truncation functions which are used on}

Rn and on Rd respectively.

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The following lemma is a version of Itˆo’s formula for semimartingales.

Lemma 1.12. Let U be an open subset of Rd and X = (Xi)0≤i≤d an U -valued

semi-martingale such that X− is U -valued as well. Moreover, let f : U → R be twice

differentiable. Then f (X) is a semimartingale, and we have

f (X(t)) =f (X(0)) + d X i=1 Z t 0 ∂if (X(s−))dXi(s) + 1 2 d X i,j=1 Z t 0 ∂ijf (X(s−))dhXic, Xjci(s) + X 0≤s≤t f (X(s)) − f (X(s−)) − d X i=1 ∂if (X(s−))∆Xi(s) .

In terms of the characteristic triplet (B, C, ν) of a semimartingale X, Itˆo’s formula takes the following form

f (X(t)) =f (X(0)) + d X i=1 Z t 0 ∂if (X(s−))dXi(s) + 1 2 d X i,j=1 Z t 0 ∂ijf (X(s−))dC(s) + Z t 0 f (X(s−) + x) − f (X(s−)) − d X i=1 ∂if (X(s−))x µ(s, dx). Proof. This is proved in Theorem I.4.57 of Jacod and Shiryaev (2003).

A direct application of Itˆo’s formula provides the local characteristics of a semimartingale f (X), where f is a twice differentiable function.

Proposition 1.13. Let X be an Rd-valued semimartingale with local characteristics (b, c, F ). Suppose that f : U → Rn is twice differentiable on some open subset U ⊂ Rd such that X, X− are U -valued. Then the local characteristics (ˆb, ˆc, ˆF ) of the Rn-valued

semimartingale f (X) are given by ˆ bi(t) = d X k=1 ∂kfi(X(t−))bk(t) + 1 2 d X k,l=1 ∂kfi(X(t−))ckl(t) + Z Rd fi(X(t−) + x) − fi(X(t−)) − d X k=1 ∂kfi(X(t−))xk F (t, dx), ˆ cij(t) = d X k,l=1 ∂kfi(X(t−))ckl(t)∂lfj(X(t−)), ˆ F (t, A) = Z Rd 1A(f (X(t−) + x) − f (X(t−)))F (t, dx) ∀A ∈ Bn with 0 6∈ A.

Proof. This is proved in Corollary A.6 of Goll and Kallsen (2000).

In the following two definitions we introduce the stochastic exponential and logarithm which we will use in the context of change of measures.

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Definition 1.14. The stochastic exponential E (X) of a semimartingale (X(t))0≤t≤T is the solution Z to Z(t) = 1 + Z t 0 Z(s−)dX(s).

The converse of the stochastic exponential is the stochastic logarithm.

Definition 1.15. For a semimartingale X = (X(t))0≤t≤T , the stochastic logarithm

L(X) is the solution Y to Y (t) = Z t 0 1 X(s−)dX(s).

As a simple application of Itˆo’s formula for semimartingales, we will work out the dy-namics of the stochastic exponential and the stochastic logarithm of a semimartingale.

Example 1.16. We consider the function f : R+ → R with f(x) = log x. Then f

is twice differentiable with f0(x) = _x1 and f00(x) = −_x12. Applying Itˆo’s formula to

f (Z(t)) = log Z(t), we get log Z(t) = log Z(0) + Z t 0 1 Z(s−)dZ(s) − 1 2 Z t 0 1 Z2_(s−)dhZ c_i(s) + X 0≤s≤t log Z(s) − log Z(s−) − ∆Z(s) Z(s−) ⇔ log Z(t) = Z t 0 1 Z(s−)Z(s−)dX(s) − 1 2 Z t 0 1 Z2_(s−)Z 2_(s−)dhXc_i(s) + X 0≤s≤t log 1 +∆Z(s) Z(s−) −∆Z(t) Z(t−) ⇔ log Z(t) = X(t) − X(0) −1 2hX c_{i(t) +} X 0≤s≤t log(1 + ∆X(s)) − ∆X(s) . Exponentiating gives the solution

Z(t) = E (X)(t) = eX(t)−X(0)−12hX

c_i(t) Y

0≤s≤t

(1 + ∆X(s))e−∆X(s).

Example 1.17. Let X = (X(t))0≤t≤T be a positive semimartingale with X(0)=1. We

consider the function f : R+ → R with f(x) = log x, as in previous example. Applying

Itˆo’s formula to f (Z(t)) = log Z(t), yields log X(t) = log X(0) + Z t 0 1 X(s−)dX(s) − 1 2 Z t 0 1 X2_(s−)dhX c_i(s) + X 0≤s≤t log X(s) − log X(s−) − ∆X(s) X(s−)

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⇔ L(X) = Z t 0 1 X(s−)dX(s) = log X(t) + 1 2 Z t 0 1 X2_(s−)dhX c_i(s) − X 0≤s≤t log 1 +∆X(s) X(s−) −∆X(s) X(s−)

Proposition 1.20 below shows how we can derive the local characteristics of a semi-martingale X under a new measure Q, knowing the characteristics of X under the original measure P.

Proposition 1.18. Let X be an Rd_{-valued semimartingale with differential}

character-istics (b, c, F ). Suppose that Qloc P with density process

Z = E Z · 0 H(s)dXc(s) + Z · 0 Z Rd W (µX − νX)(ds, dx) for some H ∈ L(Xc_{), W ∈ G}

loc(µX)1, where Xcdenotes the continuous martingale part

of X and µX, νX the random measure of jumps of X and its compensator. Then the differential characteristics (ˆb, ˆc, ˆF ) of X relative to Q are given by

ˆ_{b(t) = b(t) + H}tr_{c(t) +} Z Rd W (t, x)h(x)F (t, dx) ˆ c(t) = c(t) ˆ F (t, A) = Z Rd 1A(x)(1 + W (t, x))F (t, dx) ∀A ∈ Bd.

Proof. A version of this theorem is proved in Chapter III, Theorem 3.24 of Jacod and Shiryaev (2003).

In applications the density process Z cannot always be expressed in the form of a stochastic exponential. An alternative way to derive the local characteristics of a semimartingale under a new measure is given below.

Proposition 1.19. Let X be an Rd_{-valued semimartingale and P}? _{∼ P a probability}

measure with density process Z. If (X, L(Z)) admits local characteristics (b, c, F ), the

1_{Let ˆ}_W t(ω) = R RdW (ω, t, x)ν(ω; {t} × dx) if R Rd|W (ω, t, x)|ν(ω; {t} × dx) < ∞ and +∞ otherwise.

We denote by Gloc(µX) the set of all ˜P-measurable (P = P × B(R˜ d)) real-valued functions W on ˜

Ω(= [0, t] × B(Rd)) such that the process ˜W = W (ω, t, ∆Xt(ω)1{∆Xt(ω)6=0}(ω, t) − ˆWt(ω) satisfies

X s≤· ( ˜Ws)2 1/2 ∈ A+ loc,

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P?-characteristics (b?, c?, F?) of (X, L(Z)) are given by b?(t) = bi(t) + ci,d+1(t) + Z Rd xixd+1F (t, dx), i = 1, · · · , d + 1, c?(t) = c(t), F?(t, A) = Z Rd 1A(x)(1 + xd+1)F (t, dx).

Proof. This is proved in Lemma 5.1 of Kallsen (2003).

The following result is needed for deriving the mean-variance hedging formulas in section 4.3.

Lemma 1.20 (Kallsen and Vierthauer (2013)). Let X = (X1, · · · , Xd) be an Rd-valued

semimartingale such that Xddoes not have jumps of size −1. If (b, c, F ) denote the local

characteristics of X, then the local characteristics (ˆb, ˆc, ˆF ) of the Rd+1-valued semi-martingale Y = (Y1, · · · , Yd, Yd+1) := X1, · · · , Xd−1, log |E (Xd)|, X s≤· 1{∆Xd<−1} are given by ˆ_b_i_{(t) = b}_i_(t), _{i = 1, · · · , d − 1,} ˆ_b_d_{(t) = b}_d_{(t) −}1 2cdd(t) + Z Rd (log |1 + xd| − xd)F (t, dx), ˆ_b_d+1₌Z Rd 1_{(−∞,−1)}(xd)F (t, dx), ˆ cij(t) = cij(t), i, j = 1, · · · , d, ˆ cd+1,i(t) = 0, i = 1, · · · , d + 1, ˆ F (t, A) = Z Rd 1A(x1, · · · , xd−1, log |1 + xd|, 1(−∞,−1)(xd))F (t, dx) ∀A ∈ Bd+1.

Proof. This is proved in Lemma A.6 of Kallsen and Vierthauer (2013).

1.1.2 L´evy processes

In this subsection we consider L´evy processes which constitute one subclass of semi-martingales. We give an overview of the most important results concerning L´evy pro-cesses. A formal definition is given below.

Definition 1.21. A stochastic process X = (X(t))t≥0 is said to be a L´evy process if

it satisfies the following properties: • X(0) = 0 a.s.

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• X has increments independent of the past: X(t) − X(s) is independent of F_s for all 0 ≤ s < t < ∞.

• X has stationary increments: X(t) − X(s) has the same distribution as X(t − s) for all 0 ≤ s < t < ∞.

• X(t) is continuous in probability: ∀ > 0, limh→∞P(|X(t + h) − X(t)| > ) = 0. The following proposition can be seen as an alternative definition of a L´evy process. Proposition 1.22. An Rd-valued semimartingale X with X(0) = 0 is a L´evy process if and only if it has a version (b, c, F ) of local characteristics which does not depend on (ω, t).

Proof. This is proved in Chapter II, Proposition 4.19 of Jacod and Shiryaev (2003). Two well-known examples of L´evy processes are the Brownian motion and the Pois-son process.

The distribution of a L´evy process is characterized by its characteristic function which is given by the L´evy-Khintchine formula

ΦX(t) := E(eiλ

tr_X(t)

) = etψ(iλ) where the L´evy exponent ψ is given by

ψ(u) = utrβ + 1 2u tr_{γu +} Z Rd (eutrx− 1 − utrh(x))φ(dx), u ∈ iRd (1.1) and h : Rd→ Rd _{denotes a fixed truncation function as e.g. h(x) = x1}

{|x|≤1}.

The triplet (β, γ, φ) is called L´evy-Khintchine triplet and denotes the local charac-teristics of a L´evy process.

Using the L´evy-Khintchine triplet (β, γ, φ) the L´evy process L can be decomposed as follows L(t) = βt +√γW (t) + Z t 0 Z |x|≥1 xµL(ds, dx) + Z t 0 Z |x|<1 x(µL− νL)(ds, dx),

where h(x) = x1{|x|≤1} is used as a truncation function, W denotes a Brownian motion,

µL is a Poisson process and νL(dt, dx) = φ(dx)dt is its compensator.

Example 1.23. One subclass of Lévy processes that is of special interest is the subor-dinators. Subordinators are increasing Lévy processes. The Lévy-Khintchine triplet of a subordinator Z is given by (bZ, 0, FZ), where bZ ≥ 0.The Lévy exponent ψZ(u) of Z is given by

ψZ(u) = ubZ+

Z ∞

0

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The canonical decomposition of the process Z w.r.t. the identity truncation function is given by Z(t) = bZt + Z t 0 Z ∞ 0 x(µZ− νZ)(ds, dx).

1.1.3 Stochastic volatility models

Stochastic volatility models can be seen as examples of semimartingales. In this sub-section we consider two stochastic volatility models, namely the Heston model and the BNS model, and we compute their local characteristics. These models will be studied extensively in the context of this thesis.

Example 1.24. The Heston model is one of the most well-known stochastic volatility models. It was introduced in Heston (1993). The dynamics of the model are given by

dS(t) S(t) = µυ(t)dt + p υ(t)dW1(t) (1.3) dυ(t) = (κ − λυ(t))dt + σpυ(t)[ρdW1(t) + p 1 − ρ2_dW 2(t)] (1.4)

where µ, κ ≥ 0, λ, σ and ρ are constants and W1_{, W}2 _{independent Brownian motions.}

In order to have a strictly positive solution for υ, we need to impose the additional condition

κ ≥ 1 2σ

2_,

(see Chapter XI of Revuz and Yor (1991)). The semimartingale (υ, S) has local charac-teristics (b, c, F ) where b =κ − λυ(t) µS(t)υ(t) , c = σ2υ(t) S(t)σρυ(t) S(t)σρυ(t) S2_(t)υ(t) , F (dx) = 0.

Example 1.25. The BNS model was introduced in a paper by Barndorff-Nielsen and Shephard (2001). The dynamics of the model are given by

dX(t) = (µ + βυ(t−))dt +pυ(t−)dW (t) + ρdZ(λt) (1.5)

dυ(t) = −λυ(t−)dt + dZ(λt) (1.6)

where X(t) = log S(t), X(0) = 0, υ(0) > 0, µ, β ∈ R, ρ ≤ 0 and λ > 0. W denotes a Wiener process and Z a subordinator with L´evy-Khintchine triplet (bZ, 0, FZ), bZ ≥ 0. We use h(x) = x as truncation function. The local characteristics of the semimartingale (X, υ) are given by the triplet (b, c, F ), where

b = λbZ_{− λυ(t−)} µ + ρλbZ+ βυ(t−) , c =0 0 0 υ(t−) , F (A) = λ Z ∞ 0 1A(x, ρx)FZ(dx), ∀A ∈ B2_.

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1.2 Financial market set-up

In this section we introduce our market model and we present some necessary results which appear in the context of quadratic hedging strategies.

We denote by ˜S = ( ˜Si(t))0≤t≤T , i∈{0,··· ,d} the d+1-dimensional random vector

repre-senting the nonnegative prices of some risky assets at time t. For simplicity we as-sume that ( ˜S0(t))0≤t≤T has strictly positive value. Then the discounted price process

S = (Si(t))0≤t≤T ,i∈{0,··· ,d} is defined by Si(t) = ˜ Si(t) ˜ S0(t) for i = 1, · · · d and 0 ≤ t ≤ T .

Note that the asset S0(t) = 1 for every 0 ≤ t ≤ T . We assume that the prices Si(t),

i ∈ {1, · · · d} are observed at time t and hence adapted to our filtration F.

Next we denote by φ = (θ, η) = (θ(t), η(t))0≤t≤T our dynamic portfolio strategies,

where θ is a d-dimensional predictable process and η is adapted. In such a strategy, θi(t) describes the number of units of assets i held at time t and η(t) is the amount

of money invested in asset zero. At any time t the value of the portfolio is given by Vφ(t) = θtr(t)S(t) + η(t).

Definition 1.26. A dynamic portfolio strategy is called self-financing strategy if V is given by

Vφ(t) = V (0) +

Z t 0

θ(s)dS(s),

where V (0) := Vφ(0) is the initial amount required to start the strategy.

Notice that θ is a predictable locally bounded process. Hence the above integral is well defined if we assume that S is a semimartingale (see Dol´eans-Dade and Meyer (1970)). Intuitively speaking, a portfolio is called self-financing if the purchase of a new asset is financed only by selling assets which are already in the portfolio.

Throughout the whole thesis we assume that the market is arbitrage free. A formal definition of the latter concept follows below.

Definition 1.27. A self-financing dynamic portfolio strategy φ = (θ, η) is an arbitrage opportunity over [0, T ] if V (0) ≤ 0, Vφ(T ) ≥ 0 P-a.s. and P(Vφ(T ) > 0) > 0. A market

is called arbitrage free if arbitrage opportunities don’t exist.

Definition 1.28. An equivalent (local) martingale measure (E(L)MM) Q ∼ P has the property that the discounted price processes Si are Q-(local) martingales for all

i ∈ {1, · · · , d}. We denote the convex set of all ELMMs with Q.

The theorem below is a version of the First Fundamental Theorem of Asset Pricing (FFTAP). A first version of this theorem was proved by Harrison and Kreps (1979).

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Theorem 1.29. A market is arbitrage free if and only if the set Q is nonempty. According to the FFTAP, our assumption of arbitrage free market implies that in our market there exists at least one ELMM Q.

The following three definitions are significant in the context of quadratic hedging strate-gies.

Definitions 1.30.

• A contingent claim H is a nonnegative FT-measurable random variable.

• A contingent claim H is called attainable if there exists a self-financing strategy that satisfies Vφ(T ) = H P-a.s. (replicating condition).

• An arbitrage free market is complete if every contingent claim is attainable. One example of contingent claim is the well-known European call (respectively put) option on an asset Si with maturity T and strike price K. In this case H = (Si(T )−K)+

(respectively H = (K − Si(T ))+), i = 1, · · · , d.

The statement below is known as the second Fundamental Theorem of Asset Pricing and it provides a necessary and sufficient condition for a market to be complete. Theorem 1.31. A market is complete if and only if there exists a unique ELMM. Proof. This is proved in Theorem 3.35 of Harrison and Pliska (1981).

In complete markets, where every contingent claim is attainable, the risk involved in a portfolio can be eliminated. However, completeness can be easily destroyed as soon as we consider even minor modification of a basic complete model. For example, if we consider in our model stochastic volatility, then the market becomes incomplete (see example 1.32 below). In incomplete markets not every contingent claim is attainable and therefore the portfolio cannot be protected against all losses, i.e. we cannot form self-financing hedging strategies which eliminate the risk. Quadratic hedging strategies is a method that is used in order to hedge contingent claims in incomplete markets. Example 1.32. Let us consider the Heston model with dynamics given by (1.3)-(1.4). For this example we assume that the risk-free rate r equals zero. We will prove that the contingent claim H = (S(T )−K)+_{is not attainable and hence the market is incomplete.}

We follow Frey (1997) to characterize all the equivalent martingale measures Q. M (T ) = dQ dP _F T = exp − Z T 0 γ(u)dW1(u) + Z T 0 ν(u)dW2(u) −1 2 Z T 0 γ2(u)du + Z T 0 ν2(u)du .

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By Girsanov’s theorem, under the change of measure M (T ) we have two independent Q-Brownian motions W1Q and W2Q defined by:

dWQ

1 (t) = dW1(t) + γ(t)dt

dWQ

2 (t) = dW2(t) + ν(t)dt

The dynamics of the asset price under the new measure Q are given by: dS(t) =pυ(t)S(t)dWQ 1 (t) =pυ(t)S(t)[dW1(t) + γ(t)dt] =pυ(t)γ(t)S(t)dt +pυ(t)S(t)dW1(t) Hence, γ(t) = µpυ(t).

The dynamcis of the stochastic volatility under the measure Q are given by: dυ(t) = [κ − λυ(t) − ρσpυ(t)γ(t) −p1 − ρ2_σp υ(t)ν(t)]dt + σpυ(t)dBQ_(t) where dBQ_{(t) = ρdW}Q 1 (t) + p 1 − ρ2_dWQ 2 (t)

We can observe that for every choice of {ν(t), 0 ≤ t ≤ T } S still has zero drift and remains martingale. Therefore ν is not uniquely determined, which means that also the equivalent martingale measure Q is not unique. Hence the payoff H will have different value for different measures.

In incomplete markets the set Q contains at least two ELMMs. Moreover, Q is convex and therefore by a property of convex sets Q contains infinitely many elements. That makes pricing a contingent claim in incomplete markets challenging, since there is not any more a unique arbitrage free price.

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2 Locally risk-minimizing strategy

The first quadratic approach we consider is called (locally) risk-minimization. In this approach one insists on the replicating condition, i.e. Vφ(T ) = H. Because in incomplete

markets these strategies φ cannot be self-financing, we aim to find a strategy φ∗ with a ”small” cost.

F¨ollmer and Sondermann (1986) were the first who studied this quadratic hedging approach. They focused their attention on the case that the underlying asset S is a lo-cal martingale and they proved the existence and the uniqueness of a ”risk-minimizing” strategy in this framework. Later their results were extended to the semimartingele case for the first time by Schweizer (1988). The foundation of the ”locally risk-minimizing” hedging strategies is described in Schweizer (1990), where the equivalence between the orthogonality of martingales and the risk-minimality under small petrubations is proved. In Schweizer (1991) the concept of locally risk-minimization is introduced as a method to hedge contingent claims when the underlying asset S is a semimartingale.

In this chapter we begin by giving an overview of the study of F¨ollmer and Sonder-mann (1986). We continue by describing the theory of locally risk-minimization based on the results of Schweizer (2001).

2.1 Risk-minimizing strategy: the stock price as a

martingale

Let us assume that S is a local martingale under the original measure P. We start by giving some important definitions and we continue by presenting the results of F¨ollmer and Sondermann (1986).

Definition 2.1. Assume S is a local martingale under the measure P. A couple φ = (θ, η) is called trading strategy if

• θ is predictable process • θ ∈ L2_{(S) with} L2(S) = θ ∈ Rd predictable E[ Z T 0

θ(u)d[S, S](u)θtr(u)] 1/2

< ∞

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• The value process V_φ= θtrS + η of the strategy φ has right-continuous paths and E[V_φ2(t)] < ∞ for every t ∈ [0, T ].

Definition 2.2. For any trading strategy φ, the cost process Cφ is defined by

Cφ(t) = Vφ(t) −

Z t

0

θ(u)dS(u), 0 ≤ t ≤ T

where C(t) is intuitively the difference between the value of the portfolio at time t and the gains/losses made from trading in the financial market up to time t.

The risk process Rφ of φ is defined by

Rφ(t) = E[(Cφ(T ) − Cφ(t))2|Ft], 0 ≤ t ≤ T.

In case that φ is a self-financing strategy, the cost process Cφ will be constant

over time and hence the risk process Rφ will be zero. But in incomplete markets the

strategy φ cannot be self-financing. Thus from all strategies that satisfy the replicating condition, we aim to find the one that minimizes the risk process. Such a strategy is called risk-minimizing. A definition follows below.

Definition 2.3. A trading strategy φ is called risk-minimizing (RM) if Vφ(T ) = H

P-a.s. and if for any trading strategy φ such that V˜ _φ˜(t) = H P-a.s., we have

Rφ(t) ≤ R_φ˜(t) P − a.s. for every t ∈ [0, T ].

As we have already mentioned, RM strategies are not in general self-financing. How-ever, according to Lemma 2.5 below they are mean-self-financing. A formal definition of the latter concept follows below.

Definition 2.4. A trading strategy φ is called mean-self-financing if its cost process Cφis a martingale under P.

Lemma 2.5. Any RM trading strategy φ is also mean-self-financing. Proof. Fix t0 ∈ [0, T ] and define ˜φ by setting ˜θ := θ and

˜ θtr(t)S(t) + ˜η(t) = V_φ˜(t) := Vφ(t)1[0,t0)(t) + E Vφ(T ) − Z T t θ(u)dS(u) Ft 1[t0,T ](t)

choosing a c´adl´ag version. Then ˜φ is a trading strategy with V_φ˜(T ) = Vφ(T ) and because

C_φ˜(T ) = Cφ(T ) and C_φ˜(t0) = E[C_φ˜(T )|Ft0],

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implies that

Rφ(t0) = R_φ˜(t0) + (Cφ(t0) − E[Cφ(T )|Ft0])

2_.

Because φ is risk-minimizing, we conclude that

Cφ(t0) = E[Cφ(T )|Ft0] P − a.s.

and since t0 is arbitrary, the assertion follows.

Our goal is to hedge a contingent claim H ∈ L2(P). According to F¨ollmer and Son-dermann (1986) the key to finding an RM strategy is the Galtchouk-Kunita-Watanabe (GKW) decomposition.

Definition 2.6. An FT-measurable random variable H has a

Galtchouk-Kunita-Watanabe decomposition with respect to the local martingale M if there exists a constant H(0), a process ξ ∈ Lloc(M ) and a local martingale L, such that [L, M ] is a

local martingale, and

H = H(0) + Z T

0

ξ(u)dM (u) + L(T ).

If H is squared integrable and S is a locally squared integrable martingale then the existence of the GKW decomposition is ensured. F¨ollmer and Sondermann (1986) defined the RM hedging strategy only in this setting. However, Schweizer (2001) proved that the conditions on S are too strong and that it is sufficient for S to be local martin-gale under P.

By using the GKW decomposition the contingent claim H can be uniquely written as H = E[H|F(0)] + Z T 0 ξP_{(u)dS(u) + L}P_{(T )} P − a.s. (2.1)

with ξP _{∈ L}2_{(S), L}P _{∈ M}2 _{with L}P_{(0) = 0 and P-orthogonal to S.}

Theorem 2.7. Let S be a local P-martingale. Then every contingent claim H ∈ L2(P) admits a unique RM strategy φ∗ with Vφ∗(T ) = H, P-a.s. In terms of the decomposition

(2.1), φ∗ is explicitely given by

Vφ∗(t) = E[H|F_t] =: V∗(t), 0 ≤ t ≤ T,

θ∗= ξP_{= (dhSi)}inv_dhV∗_{, Si,}

η∗ = V∗− (θ∗)trS, Cφ? = E[H|F_t] + LP.

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Proof. We start by proving that the strategy φ∗ is RM. Notice first that φ∗ is a trading strategy with Vφ∗(T ) = H. Now fix t ∈ [0, T ] and an arbitrary trading strategy ˜φ with

V_φ˜(T ) = H. The same argument as in the proof of Lemma 2.5 shows that we may

assume that C_φ˜(t) = E[C_φ˜(T )|Ft] and so by using (2.1) and the martingale property of

R _˜ θdS we get C_φ˜(T ) − C_φ˜(t) = H − Z T t ˜ θ(u)dS(u) − E[H|Ft] = LP_{(T ) − L}P_{(t) +} Z T t ξP_{(u) − ˜}_θ(u) dS(u). Because Cφ∗ = E[H|F_t] + LP and LP is orthogonal to S, we obtain

R_φ˜(t) = Rφ∗(t) + E " Z T t (ξP_{(u) − ˜}_θ(u))dS(u) 2 F_t # ≥ R_φ∗(t). Hence φ∗ is RM.

Next we prove the uniqueness of φ∗. Assume that there is some other strategy ˜φ which is also RM. Then C_φ˜ must be a martingale by Lemma 2.5. and then the same argument

as before gives for t = 0

R_φ˜(0) = Rφ∗(0) + E

" Z T

0

(ξP_{(u) − ˜}_θ(u))tr_d[S](u)(ξP_{(u) − ˜}_θ(u))

F0 # .

Because ˜φ is RM, this implies that ˜θ = ξP _{= θ}∗ _{and since C} ˜

φ is a martingale and

V_φ˜(T ) = Vφ∗(T ), we also obtain ˜φ = φ∗.

Finally we prove that ξP _{= (dhSi))}inv_dhV∗_{, Si. Using (2.1) and the martingale property}

we have V∗(t) = E[H|Ft] = E[H] + Z t 0 ξP_{(u)dS(u) + L}P_(t) ⇒ dV∗(t) = ξP_{(t)dS(t) + dL}P_(t). Hence

dhV∗, Si(t) = ξP_{(t)dhSi(t) + dhL}P_{, Si(t) ⇔ ξ}P_{(t) = (dhSi(t))}inv_dhV∗_{, Si(t)}

where by inv we denote the Moore–Penrose pseudoinverse, which is a generalization of the inverse matrix (for more details see Penrose (1955)).

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2.2 Locally risk-minimizing strategy: the stock price as

semimartingale

Now we present a generalization of the RM approach in case that our price process S is a semimartingale under P. However, this is not so simple. Schweizer (1988) proved by giving a counterexample that finding an RM strategy may be a problem without solution.

Proposition 2.8. If S is not a local P-martingale, a contingent claim H admits in general no RM strategy φ with Vφ(T ) = H.

Proof. This is proved in Proposition 3.1 of Schweizer (2001).

Since we can no longer find an RM strategy φ, Schweizer (1988) extended the theory of risk-minimization to locally risk-minimization. His basic idea was to control the riskness of a strategy as measured by its local cost fluctuations.

In order to continue with the description of LRM strategies, we set some assumptions on the price process S. Since the set of all ELMM Q is nonempty we know already that the price process S is a semimartingale under P. Next to that we impose the following conditions:

(C1) S ∈ S_loc2 (P), where S = S(0) + M + A

with M ∈ M2_loc(P) such that M (0) = 0 and A is an Rd-valued predictable process of finite variation such that A(0) = 0.

(C2) The structure condition (SC) is satisfied.

We say that the process S satisfies the (SC) if there exists an Rd-valued predictable process bλ such that A is absolutely continuous with respect to hM i in the sense that Ai(t) = Z t 0 dhM i(s)bλ(s) i = d X j=1 Z t 0 b λj(s)dhMi, Mji(s), 0 ≤ t ≤ T, i = 1, · · · , d

and such that the mean-variance tradeoff process (MVT)

b K(t) = Z t 0 b λtr(s)dhM i(s)bλ(s) = d X i,j=1 Z t 0 b λi(s)bλj(s)dhMi, Mji(s)

is finite P-a.s. for each t ∈ [0, T ].

Notice that the absence of arbitrage in our market guarantees according to Delbaen and Schachermayer (1994) that the condition (C2) is satisfied.

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Definition 2.9. Θ denotes the space of all processes θ ∈ L(S) for which the stochas-tic integral R θdS is in the space S2(P) of semimartingales. Equivalently, θ must be predictable with E " Z T 0 θtr(s)d[M ](s)θ(s) + Z T 0 |θtr(s)dA(s)| 2# < ∞.

In the case that S is a semimartingale the definition of trading strategy has to be adjusted.

Definition 2.10. Assume S = S(0) + M + A is a semimartingale under the measure P. A couple φ = (θ, η) is called a trading strategy if

• θ is a predictable process • θ belongs to the space Θ • η is adapted

• V_φ= θtrS + η has right-continuous paths and E[Vφ2(t)] < ∞ for every t ∈ [0, T ].

The next definition of small perturbation will help us to to define formally the concept of LRM strategy. However, from now on we focus on one-dimensional assets. Although similar results can be obtained for the multidimensional case (see Schweizer (2008)), in the context of this thesis we restrict our attention to the case d = 1.

Definition 2.11. A small perturbation is a trading strategy ∆ = (δ, ) such that δ is bounded, the variation of R δdA is bounded (uniformly in t and ω) and δT = T = 0.

For any subinterval (s, t] of [0, T ], we then define the small perturbation ∆|(s,t] = (δ1(s,t], 1(s,t])

Next we define partitions τ = (ti)0≤i≤N of the interval [0, T ]. A partition of [0, T ] is

a finite set τ = {t0, t1, · · · , tk} of time points with 0 = t0 < t1 < · · · < tk = T and

the mesh size of τ is |τ | = max

ti,ti+1∈τ

(ti+1− ti). A sequence (τn)n∈N is called increasing if

τn⊆ τn+1 for all n and it tends to the identity if lim

n→∞|τn| = 0.

Definition 2.12. For a trading strategy φ, a small petrurbation ∆ and a partition τ of [0, T ], we set

rτ(φ, ∆) = X

ti,ti+1∈τ

Rφ+∆|_(ti,ti+1](ti) − Rφ(ti)

E[hM i(ti+1) − hM i(ti)|F (ti)]

1_(t_i_,t_i+1_].

φ is called LRM strategy if lim inf

n→∞ r

τn_{(φ, ∆) ≥ 0} _{(P × hM i) − a.e. on Ω × [0, T ]}

for every small petrubation ∆ and every increasing sequence (τn)n∈Nof partitions tending

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Although the definition of LRM strategies is rather technical, they can be computed quite easily using the following results.

Lemma 2.13. Suppose that hM i is a P-a.s. strictly increasing. If a trading strategy is LRM, it is also mean self-financing.

Proof. This is proved in Lemma 2.1 of Schweizer (1991).

Theorem 2.14. Suppose that S satisfies the (SC), M is in M2(P), hM i is P-a.s. strictly increasing, A is P-a.s. continuous and E[ bK(T )] < ∞. Let H ∈ L2(P), be a contingent claim and φ a trading strategy with Vφ(T ) = H P-a.s. Then φ is LRM if and only if φ

is mean-self-financing and the martingale Cφ is orthogonal to M .

Proof. This is proved in Theorem 3.3 of Schweizer (2001). The following definition is true for the general case d ≥ 1.

Definition 2.15. Let H ∈ L2(P) be a contingent claim. A trading strategy φ with Vφ(T ) = H P-a.s. is called pseudo-locally risk-minimizing or pseudo-optimal for

H if φ is mean-self-financing and the martingale Cφ is orthogonal to M .

Notice that under the conditions of Theorem 2.14, finding an LRM strategy essentialy boils down to finding a pseudo-optimal strategy. Therefore instead of looking for LRM strategies we can focus on finding pseudo-optimal strategies. The key to finding such strategies is the so-called F¨ollmer-Schweizer (FS) decomposition.

Definition 2.16. An FT-measurable and square-integrable random variable H admits

a F¨ollmer-Schweizer (FS) decomposition if there exists a constant H(0), a (S-integrable) process ξF S,P ∈ Θ and a square-integrable martingale LF S,P_{, with L}F S,P_{(0) =}

0, such that [LF S,P, M ] is a local martingale, and

H = H(0) + Z T

0

ξF S,P(u)dS(u) + LF S,P(T ). (2.2) We remark that the FS decomposition is an extension of the GKW decomposition and it was first introduced in F¨ollmer and Schweizer (1991) but with the focus on the continuous case. The extension to the discontinuous case under certain assumptions was given by Ansel and Stricker (1992), who showed uniqueness and existence of the decomposition given that the one-dimensional semimartingale S is locally bounded. The strongest result concerning the existence and uniqueness of FS decomposition is given in Choulli et al. (1998).

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Proposition 2.17. A contingent claim H ∈ L2(P) admits a pseudo-optimal trading strategy φ with Vφ(T ) = H P-a.s. if and only if H can be written in the form of (2.2).

The strategy φ is then given by

θ(t) = ξF S,P(t), 0 ≤ t ≤ T and

Cφ(t) = H(0) + LF S,P(t), 0 ≤ t ≤ T ;

its value process is Vφ(t) = Cφ(t) + Z t 0 θ(u)dS(u) = H(0) + Z t 0 ξF S,P(u)dS(u) + LF S,P(t), 0 ≤ t ≤ T (2.3) so that η(t) = Vφ(t) − ξF S,P(t)S(t) is also determined by the above description.

Proof. (⇐) Assume that H can be written in the form of (2.2). Furthermore, we have that H = Vφ(T ) = Cφ(T ) + Z T 0 θ(u)dS(u) = Cφ(0) + Z T 0 θ(u)dS(u) + Cφ(T ) − Cφ(0). Then, Cφ(0) = H(0), θ(t) = ξF S,P(t), 0 ≤ t ≤ T, Cφ(t) − Cφ(0) = LF S,P(t), 0 ≤ t ≤ T.

This implies that Cφ is a square integrable martingale, orthogonal to M . Hence by

definition, φ is a pseudo-optimal strategy.

(⇒) Assume that the strategy φ is pseudo-optimal. We have that H = Vφ(T ) = Cφ(T ) + Z T 0 θ(u)dS(u) = Cφ(0) + Z T 0 θ(u)dS(u) + Cφ(T ) − Cφ(0),

where (by definition of pseudo-optimality) Cφis a square integrable martingale,

orthog-onal to M and Cφ(0) ∈ L2(P) F0-measurable. Therefore,

H = H(0) + Z T

0

ξF S,P(u)dS(u) + LF S,P(T ), with H(0) = Cφ(0), ξF S,P= θ and LF S,P = Cφ− Cφ(0).

The preceding result is very important because it allows us to obtain an LRM strategy. Therefore we continue by explaining how one can obtain formula (2.2) for a given contingent claim by considering a change of measure. The suitable measure is called minimal martingale measure and a formal definition is given below.

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Definition 2.18. The minimal martingale measure (MMM) bP related to the P-semimartingale S is the martingale measure such that any P-local martingale which is orthogonal to M under P remains a local martingale under bP.

We define the density of the MMM as follows:

b Z = E − Z b λdM ,

where bλ can be computed using the (SC) (see condition (C2)) and M denotes the mar-tingale part of S.

Proposition 2.19. Suppose that the conditions (C1) and (C2) are satisfied. If further-more the MVT defined in (C2) is uniformly bounded in t and ω then we have that bZ is a true P-martingale and square integrable under P. Moreover, if bZ is strictly positive then the minimal martingale density is given by bZ

dbP

dP := bZ(T ) ∈ L

2

(P), where the MMM bP ∈ Q.

Proof. In Theorem II.2 of Lépingle and Mémin (1978) it is proved that uniform bound-ness of bK in t and ω yields that bZ ∈ M2(P). Furthermore, in Theorem 3.5 of Föllmer and Schweizer (1991) it is proved that bZS is a P-martingale and therefore we have that b

P ∈ Q.

Notice that if bZ is not strictly positive, which is possible in discontinuous cases, we cannot link a probability measure with the density bZ. Hence it is not possible to define b

P. Choulli et al. (1998) introduced the concept of the E -martingales to overcome this problem.

In the following lemma, we compute the number of risky assets in a LRM strategy. Lemma 2.20. Suppose that bZ > 0 and that the contingent claim H admits an FS decomposition. Then

ξF S,P(t) = dhV

M_{, M i(t)}

dhM i(t) , 0 ≤ t ≤ T, (2.4)

where VM denotes the P-martingale part of Vφ.

Proof. Using (2.3) we have that

dVφ(t) = ξF S,P(t)dS(t) + dLF S,P(t)

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Hence

dhVφ, M i(t) = ξF S,P(t)dhM i(t) + ξF S,P(t)dhA, M i(t) + dhLF S,P, M i(t).

Using Proposition 1.7 leads that hA, M i(t) = 0 and by assumption hLF S,P_{, M i(t) = 0.}

Therefore

ξF S,P(t) = dhVφ, M i(t)

dhM i(t) , 0 ≤ t ≤ T.

If we denote by VM(t) = R₀tξF S,P(s)dM (s) + LF S,P(t) the P-martingale part of the process Vφ(t) we can see that

hVM_{, M i(t) = hV, M i(t),} _{0 ≤ t ≤ T.}

Therefore ξF S,P can be computed equivalently by

ξF S,P(t) = dhV

M_{, M i(t)}

dhM i(t) , 0 ≤ t ≤ T.

A question that might arise at this point is whether the FS decomposition of H under the original measure P and the GKW decomposition of H under the MMM bP coincide. In other words can we compute the LRM strategy using the GKW decomposition of H under bP instead of using the FS decomposition of H under P?

b

ZV (φ) is a P-martingale for a pseudo-optimal trading strategy φ and hence we get Vφ(t) = E H bZ(T ) b Z(t) |F (t) = Eb_P[H|F (t)] := bV (t), 0 ≤ t ≤ T (2.5)

• Since S is a local b_{P-martingale then the GKW decomposition of b}V under bP with respect to S is b V (t) = H(0) + Z t 0 b ξ(u)dS(u) + bL(t) with h bL, Sib_P = 0 (2.6)

where bL is a local bP-martingale. • The FS decomposition of b_{V under P is}

b

V (t) = H(0) + Z t

0

ξF S,P(u)dS(u) + LF S,P(t) with hLF S,P, M i = 0

where LF S,P is a local P-martingale.

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• LF S,P _{is b}

P-local martingale, which is always true by definition of MMM. • b_{L is a P-local martingale orthogonal to M .}

• LF S,P _{is orthogonal to S under b}

P, i.e. hLF S,P, Sib_P _{= 0.}

Schweizer (1991) proved that the latter two conditions are satisfied only if S is a continuous process. Hence in this case finding an LRM strategy essentially amounts to finding the Galtchouk-Kunita-Watanabe decomposition of H under the MMM bP. In what follows we determine the relationship between ξF S,P and bξ as in Section 3.3 of Vandaele (2010).

Proposition 2.21. Suppose that bZ > 0 and that H admits an FS decomposition. If moreover [ bL, M ] ∈ Aloc, then

ξF S,P(t) = bξ(t) +dh bL, M i(t) dhM i(t) , for t ∈ [0, T ].

Proof. We proved already that

b

ξ(t) = dh bV , Si

b P_(t)

dhSib_P(t)

(this is the case of one dimensional stock price process),

ξF S,P(t) = dhV

M_{, M i(t)}

dhM i(t) =

dh bV , M i(t) dhM i(t) . From the latter equation we get

ξF S,P(t)dhM i(t) = dh bV , M i(t).

Combining this with formula (2.5), we obtain the following relationship between ξF S,P and bξ.

ξF S,P(t)(dhM i)(t) = bξ(t)dhS, M i + dh bL, M i(t)

= bξ(t)(dhM i(t) + dhA, M i(t)) + dh bL, M i(t) = bξ(t)dhM i(t) + dh bL, M i(t),

where the latter equality follows directly from Proposition 1.5. Therefore we obtain ξF S,P(t) = bξ(t) +dh bL, M i(t)

dhM i(t) .

Notice that the assumption [ bL, M ] ∈ Aloc guarantees the existence of the predictable

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3 Mean-variance hedging strategy

The second quadratic approach we consider is called mean-variance hedging (hence-forth MVH). This approach insists on the self-financing constraint and tries to minimize the difference between the contingent claim and the portfolio at maturity T . The MVH strategy aims to minimize the risk globally, which on one hand is a big advantage, but on the other hand determining the strategy explicitly becomes harder.

This quadratic approach was first introduced by Bouleau and Lamberton (1989), who determined a self-financing portfolio minimizing the risk in case the underlying process is a martingale and when it is a function of a Markov process. Duffie and Richardson (1991) studied the MVH problem in a correlated Brownian motion setting, where a hedger is faced with a commitment in one asset and the opportunity to con-tinuously trade future contracts on the correlated other. An extension to more general type of claims is given in Schweizer (1992). In the years that followed many researchers studied the problem of MVH with the focus mainly on continuous processes. Schweizer (2001) and Pham (2000) give an overview of the main results in the area of MVH for continuous price processes.

An extension to the case of discontinuous price processes is given in Arai (2005). In particular Arai extended the results of Gourieroux et al. (1998) and Rheinländer and Schweizer (1997) to the case of the cádlág special semimartingale. Finally, ˇCerný and Kallsen (2007) provided a new characterization of MVH strategies in general semi-martingale markets. The key point in their paper is the introduction of a new probability measure, namely the opportunity neutral measure, which we define later in this chapter. The goal of this chapter is to present some significant notions and main theorems that will be applied in Chapter 4, rather than give an extensive overview of all the results in the area of MVH strategies. Similarly to Chapter 2, we begin by introducing the MVH for the martingale case and we continue by extending the results to the semimartingale case.

3.1 The stock price as a martingale

Let us assume that S is a martingale under the original measure P. The results we present are based on Bouleau and Lamberton (1989). As we mentioned above, an additional condition that S is a function of some Markov process is imposed on their paper. However, according to Schweizer (2001) their basic idea can also be applied to

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our more general framework.

As discussed above we focus on self-financing trading strategies. Definition 1.24 implies that these strategies are described by a pair φ0 = (V (0), θ) ∈ L2(P) × L2(S). Since the contingent claim H ∈ L2(P) is not in general attainable, then at maturity T we have a shortfall given by

H − Vφ0(T ) = H − V (0) −

Z T

0

θ(u)dS(u). (3.1)

Therefore we get a residual risk

J0(φ0) := E[(H − Vφ0(T ))2].

Our goal is to minimize this risk by suitably choosing a strategy φ0 = (V (0), ¯θ) among all self-financing trading strategies φ0. This can be achieved by computing the orthog-onal projection of H on the space spanned by L2(F0, P) and the stochastic integrals

RT

0 θ(u)dS(u). Using the GKW decomposition of H (2.1) we get

J0(φ0) = E[(E[H|F(0)] + Z T 0 ξP_{(u)dS(u) + L}P_{(T ) − V (0) −} Z T 0 θ(u)dS(u))2]. (3.2) Proposition 3.1 (Bouleau and Lamberton (1989)). Let S be a martingale. Then the MVH strategy φ0 = (V (0), ¯θ) that minimizes (3.2) is given by

V (0) = E[H|F0],

¯

θ(t) = ξP_{(t) = (dhSi(t))}inv_{dhV, Si(t),} _{0 ≤ t ≤ T.}

Notice that the number of risky assets invested using the MVH strategy coincides with the number invested using the RM hedging strategy. However, the amount invested in the riskless asset (S0) differs. The minimal residual risk of strategy φ0 is given by

J0(V (0), ¯θ) = E[(LP(T ))2] = Var[LP(T )].

3.2 The stock price as a semimartingale

If S fails to be a martingale under the original measure P then hedging the contin-gent claim H becomes more complicated. The problem remains the same: we aim to find a pair φ0 = (V0, θ) that minimizes the L2(P)-norm of the shortfall (3.1). The results

presented in this section rely on ˇCern´y and Kallsen (2007). In order to simplify some expressions we assume that the asset S is one-dimensional. However, the same theory applies to the multi-dimensional case.

To begin with, we first need to choose a reasonable set of trading strategies. This can be a problem because if this set is too large, arbitrage opportunities may exist and

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if it is too small we may not be able to find optimal strategies. In the literature different authors make different choices of such sets and this might lead to confusions. However, this problem seems to be fixed by ˇCern´y and Kallsen (2007) who chose their set Θ in-spired by Delbaen and Schachermayer (1994).

Before we define the set Θ, we need to explain the concept of simple strategies. An Rd-valued process θ is called simple if it is of the form θ = ξ1_]]τ₁_,τ₂_]]1with τ1 ≤ τ2

stopping times and ξ a bounded Fτ1-measurable random variable.

Definition 3.2. We call θ ∈ L(S) admissible strategy if there exists some sequence (θ(n))n∈N of simple strategies such that

Z t

0

θ(n)(u)dS(u) → Z t

0

θ(u)dS(u) in probability for any t ∈ [0, T ] and Z T 0 θ(n)(u)dS(u) → Z T 0 θ(u)dS(u) in L2(P).

Similarly, we call (V (0), θ) ∈ L0(Ω, F0, P) × L(S) admissible strategy pair if there exist

some sequences (V(n)(0))_n∈Nin L2(Ω, F0, P) and (θ(n))n∈N of simple strategies such that

V(n)(0) + Z t 0 θ(n)(u)dS(u) → V (0) + Z t 0

θ(u)dS(u) in probability for any t ∈ [0, T ] and V(n)(0) + Z T 0 θ(n)(u)dS(u) → V (0) + Z T 0 θ(u)dS(u) in L2(P). We set Θ := {θ ∈ L(S)|θ admissible}, L2_(F 0) × Θ := {(V (0), θ) ∈ L0(Ω, F0, P) × L(S)|(V (0), θ) admissible}.

We can now define formally our optimization problem.

Definition 3.3. We call an admissible strategy pair φ0 _{= (V (0), θ) optimal if φ}0 ₌

(V0, θ) = (V0, θ) minimizes the expected squared hedging error J0(φ0) over all admissible

strategy pairs φ0 = (V0, θ).

The first step on our way to determine the MVH strategy is to define the so-called opportunity process and adjustment process.

Definitions 3.4. We call opportunity process the process L that satisfies L(t) = inf θ E 1 − Z T 0 θ(s)dS(s) 2 Ft , (3.3) 1_]]τ

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for t ∈ [0, T ]. The adjustment process is related to the optimizer θ(t) in (3.3). The process a that satisfies

θ(t)(s) = 1 − Z s− t θ(t)(u)dS(u) a(s), for s ∈ [t, T ] is called adjustment process.

Intuitively, the adjustment process stands for the optimal number of shares per unit of wealth.

Both processes L and a play a crucial role in the determination of the MVH strategy. Theorem 3.5 below helps us to characterize them.

Theorem 3.5. The opportunity process is the unique semimartingale L such that: 1. L(t), L(t−) are (0, 1]-valued for 0 ≤ t ≤ T ,

2. L(T ) = 1,

3. The joint characteristics of (S, L) solve the equation bL(t) = L(t−)¯b(t) 2 ¯ c(t) where ¯ b(t) := bS(t) + c SL_(t) L(t−) + Z x1 x2 L(t−)F S,L_{(t, dx),} ¯ c(t) := cS(t) + Z x2₁ 1 + x2 L(t−) FS,L(t, dx) outside some P × A-null set.

4. aE Z −a1_{]]τ,T ]]}dS (t−)1_{]]τ,T ]]}∈ Θ for 0 ≤ t ≤ T, aE Z −a1_{]]τ,T ]]}dS (t−)L(t) is of class (D)2 for 0 ≤ t ≤ T

hold for a := ¯b(t)_c(t)_¯ for 0 ≤ t ≤ T and any stopping time τ . In this case a meets the requirement of an adjustment process.

Proof. This is proved in Theorem 3.25 of ˇCern´y and Kallsen (2007).

2

A stochastic process X = {Xt: t ∈ R+} is of class (D) if {Xτ : τ < ∞ is a stopping time} is uniformly

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A main ingredient in the paper of ˇCern´y and Kallsen (2007) is the introduction of a new measure, namely the the opportunity-neutral measure. The opportunity process L allows us to define it as follows.

Definition 3.6. We call the probability measure P?opportunity-neutral probability measure if P? ∼ P with density process

ZP? _:= L

L(0)AK,

where AK = exp R₀tbK(s)ds and bK denotes the drift part part in the local character-istics of the stochastic logarithm K := L(L) =R₀·_L(t−)1 dL(t).

We remark that the opportunity-neutral probability measure is typically not a mar-tingale measure. In some instances it actually equals P (for more details see Section 3.6

ˇ

Cern´y and Kallsen (2007)).

Definition 3.7. We call P?-minimal logarithm process the process N?that satisfies N? := −

Z ·

0

ˆ

adMS?(t)

where ˆa = (1 + ∆AK)a and MS? consists the martingale part of the semimartingale S w.r.t. measure P?.

Recall that Vφ(t) = EQ[H|Ft] for Q ∈ Q. However, P? is not necessarily a martingale

measure and therefore the preceding expression does not hold. Lemma 3.8 below gives another characterization for the process V .

Lemma 3.8. There is a unique semimartingale V satisfying Vφ(t) = EP?[HE (N

?_{− (N}?₎t_{)(T )|F}

t] := V?(t) for 0 ≤ t ≤ T

where (N?)t denotes the process N? stopped at t.

Proof. This is proved in Lemma 4.3 of ˇCern´y and Kallsen (2007).

Definition 3.9. We call the process V? from Lemma 3.8 mean-value process of the option.

We now come back to the hedging problem from Definition 3.2. The following theorem determines the MVH strategy.

Theorem 3.10. The optimal pair φ0 = (V (0), ¯θ) is given by V (0) = V?(0),

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where θ? is specified by its feedback equation θ?(t) = ξ?(t) − V (0)?+ Z t− 0 θ?(u)dS(u) − V?(t−) a(t) (3.4) and ξ?(t) = hS, V i P?_(t) hSiP?_(t) (3.5) for t ∈ [0, T ].

Proof. This is proved in Theorem 4.10 of ˇCern´y and Kallsen (2007).

To conclude, in order to determine the MVH strategy one can proceed as follows: • First, one determines the opportunity process L and the adjustment process a

using the characterization in Theorem 3.5.

• Then, with the help of these processes one can determine the opportunity neutral measure P? and the P?-minimal logarithm process N?.

• Finally, the mean-value process V?_{can be determined and this leads to the optimal}

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4 Quadratic hedging in affine stochastic

volatility models

In this chapter we apply the theory of quadratic hedging on affine stochastic volatil-ity models. The reason we choose this class of models is that they feature most market characteristics such as jumps and stochastic volatility and also most models used in prac-tice fall into this category (i.e. Heston (1993), exponential L´evy processes, Barndorff-Nielsen and Shephard (2001) etc.) Furthermore, the affine properties of this class of models allow us to obtain semiexplicit solutions to our problem.

Time-homogeneous affine processes have been characterized completely and inves-tigated thoroughly in a paper by Duffie et al. (2003). They focused on regular affine processes and they provided foundations for a wide range of financial applications. The extension to the time-inhomogeneous case was studied by Filipovi´c (2005). Kallsen (2006) reviewed the theory of Duffie et al. (2003) with the focus on affine stochastic volatility models. He studied this class of processes from the point of view of semi-martingales and by doing that he explained the intuition behind the semimartingale characteristics.

Now coming back to our main topic, many researchers applied the theory of quadratic hedging strategies on affine stochastic volatility models. In the area of locally risk-minimization Poulsen et al. (2009) studied the application of LRM theory on general stochastic volatility models, including affine models, like for example Heston, and Arai et al. (2017) determined the LRM strategy for BNS model using Malliavin calculus. However, in both papers the focus was not specifically on the affine structural prop-erties of the models and the hedging strategies are not given in closed-form . On the MVH problem, Kallsen and Pauwels (2010) considered general affine stochastic volatility models and they provided a semiexplicit solution in the case that the stock price is a martingale. Kallsen and Vierthauer (2013) extended this solution to the semimartingle case under specific conditions and they illustrated their approach numerically for a L´evy driven stochastic volatility model with jumps.

The goal of this chapter is to use the affine structure of some models in order to derive semiexplicit solutions for the quadrtatic hedging strategies. In section 4.1 we introduce the class of time-homogeneous affine processes and in sections 4.2 and 4.3 we apply the theory of LRM and MVH on two affine stochastic volatility models. We focus our attention on hedging European-style options and we denote by H the payoff of such options. Under the assumption that the risk-free rate r is non-stochastic, there is no

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loss of generality assuming that r equals zero and S represents the discounted asset price.

4.1 Introduction to affine stochastic volatility models

A key role in this chapter is played by affine Markov processes and their characterization in terms of the generalized Riccati equations. They are studied in depth in Duffie et al. (2003) and Filipovi´c (2005). We focus here on the subclass needed for sections 4.2 and 4.3.

Definition 4.1. Let X = (υ, z) an R+× Rd-valued semimartingale. We call X

time-inhomogeneous affine process if its local characteristics (b, c, F ) are affine functions of υ(t−)

b(t) = β0(t) + υ(t−)β1(t), (4.1)

c(t) = γ0(t) + υ(t−)γ1(t), (4.2)

F (t, A) = φ0(t, A) + υ(t−)φ1(t, A) ∀A ∈ Bd+1, (4.3)

where (βi(t), γi(t), φi(t)), i = 0, 1 are strongly admissible L´evy-Khintchine triplets on

Rd+1 in the sense of [Definition 3.1 Kallsen (2006)]. In case that (βi(t), γi(t), φi(t)),

i = 0, 1 do not depend on t, we call X time-homogeneous affine process. In the whole chapter we mostly focus on the time-homogeneous case.

Theorem 4.2. Let (βj, γj, φj), j = 0, 1 be admissible L´evy-Khintchine triplets and

de-note by ψj the corresponding L´evy exponents in the sense of (1.1). Suppose in addition

that

Z

|x|≥1

|xk|φ₁(dx) < ∞ for k ≤ 1.

Let X be a time-homogeneous affine process as in Definition 4.1. Then the conditional exponential moment E[eutrX(s+t)|F_s_{] for u ∈ iR}d+1 _{is given by}

E[eu tr_X(s+t) |F (s)] = exp Ψ0(t, u) + Ψ1(t, u)X1(s) + d+1 X i=2 Xi(s)ui (4.4)

where Ψ0 and Ψ1 solve the following system of generalized Riccati equations:

∂1Ψ1(t, u) = ψ1(Ψ1(t, u), u2, · · · , ud+1), Ψ1(0, u) = u1

∂1Ψ0(t, u) = ψ0(Ψ1(t, u), u2, · · · , ud+1), Ψ0(0, u) = 0.

Proof. This is proved in Theorem 3.2 of Kallsen (2006).

We remark that the formula (4.4) holds for u ∈ Cd+1if either side of (4.4) is well defined for some t ≤ T and u ∈ Rd+1.

Quadratic hedging strategies in affine models

MSc Stochastics and Financial Mathematics

Master Thesis