Lecture Notes in Mathematics 1856
Editors:
J M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris
Subseries:
Fondazione C.I.M.E., Firenze
Adv iser: Pietro Zecca
K. Back T.R. Bielecki C. Hipp
S. Peng W. Schachermayer
Stochastic Methods
in Finance
Lecturesgivenatthe
C.I.M.E E.M.S. Summer School
held in Bressanone/Brixen, Italy,
July 6 12, 2003
Editors: M. Frittelli
W. Runggaldier
123
Editors a nd Authors
Kerry Back
Mays Business School
Department of Finance
310C Wehner Bldg.
College Station, TX 77879-4218, USA
e-mail: [email protected]
Tomasz R. Bielecki
Department of Applied Mathematics
Illinois Inst. of Technology
10 Wes t 32nd Street
Chicago, IL 60616, USA

Financial and Actuarial Mathematics
Vienna University of Technology
Wiedner Hauptstrasse 8/105-1
1040 Vienna, Austria
e-mail: [email protected] ien.ac.at
LibraryofCongressControlNumber:2004114748
Mathematics Subject Classification (2000):
60G99, 60-06, 91-06, 91B06, 91B16, 91B24, 91B28, 91B30, 91B70, 93-06, 93E11, 93E20
ISSN 0075-8434
ISBN 3-540-22953-1 Springer-Verlag Berlin Heidelberg New York
DOI: 10.1007/b100122
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Preface

Risk beneath its different masks.
VI Preface
We begin the tour with expected utility maximization in continuous-time
stochastic markets: this classical problem, which can be traced back to the
seminal works by Merton, received a renewed impulse in the middle of the
1980’s, when the so-called duality approach to the problem was first devel-
oped. Over the past twenty years, the theory constantly improved, until the
general case of semimartingale stochastic models was finally tackled with great
success. This prompted us to dedicate one series of lectures to this traditional
as well as very innovative topic:
“Utility Maximization in Incomplete Markets”, Prof. Walter Schachermayer,
Technical University of Vienna.
This course was mainly focused on the maximization of the expected utility
from terminal wealth in incomplete markets. A part of the course was dedi-
cated to the presentation of the stochastic model of the market, with particular
attention to the formulation of the condition of No Arbitrage. Some results of
convex analysis and duality theory were also introduced and explained, as they
are needed for the formulation of the dual problem with respect to the set
of equivalent martingale measures. Then some recent results of this classical
problem were presented in the general context of semi-martingale financial
models.
The importance of the above-mentioned analysis of the utility maximization
problem is also revealed in the theory of asset pricing in incomplete markets,
where the agent’s preferences have again to be given serious consideration,
since Risk cannot be completely hedged. Different notions of “utility-based”
prices have been introduced in the literature since the middle of the 1990’s.
These concepts determine pricing rules which are often non-linear outside
the set of marketed claims. Depending on the utility function selected, these
pricing kernels share many properties with non-linear valuations: this bordered
on the realm of risk measures and capital requirements. Coherent or convex

The notion of Risk is not limited to finance, but has a traditional and dom-
inating place also in insurance. For some time the two fields have evolved
independently of one another, but recently they are increasingly interacting
and this is reflected also in the financial reality, where insurance companies
are entering the financial market and viceversa. It was therefore natural to
have a series of lectures also on insurance risk and on the techniques to control
it.
“Financial control methods applied in insurance”, Prof. Christian Hipp, Uni-
versity of Karlsruhe.
The methodologies developed in modern mathematical finance have also met
with wide use in the applications to the control and the management of the
specific risk of insurance companies. In particular, the course showed how the
theory of stochastic control and stochastic optimization can be used effectively
and how it can be integrated with the classical insurance and risk theory.
Last but not least we come to the topic of partial and asymmetric information
that doubtlessly is a possible source of Risk, but has considerable importance
in itself since evidently the information is neither complete nor equally shared
among the agents. Frequently debated also by economists, this topic was an-
alyzed in the lectures:
“Partial and asymmetric information”, Prof. Kerry Back, University of St.
Louis.
In the context of economic equilibrium, a survey of incomplete and asymmet-
ric information (or insider trading) models was presented. First, a review of
filtering theory and stochastic control was introduced. In the second part of the
course some work on incomplete information models was analyzed, focusing
on Markov chain models. The last part was concerned with asymmetric in-
formation models, with particular emphasis on the Kyle model and extensions
thereof.
VIII Preface
As editors of these Lecture Notes we would like to thank the many persons

Contents
Incomplete and Asymmetric Information in Asset Pricing
Theory
Kerry Back 1
1 Filtering Theory 1
1.1 Kalman-BucyFilter 3
1.2 Two-StateMarkovChain 4
2 IncompleteInformation 5
2.1 Seminal Work 5
2.2 MarkovChainModelsofProductionEconomies 6
2.3 Markov Chain Models of Pure Exchange Economies . . . . . . . . . . . 7
2.4 HeterogeneousBeliefs 11
3 AsymmetricInformation 12
3.1 AnticipativeInformation 12
3.2 RationalExpectationsModels 13
3.3 KyleModel 16
3.4 Continuous-TimeKyleModel 18
3.5 MultipleInformedTradersinthe KyleModel 20
References 23
Modeling and Valuation of Credit Risk
Tomasz R. Bielecki, Monique Jeanblanc, Marek Rutkowski 27
1 Introduction 27
2 StructuralApproach 29
2.1 BasicAssumptions 29
DefaultableClaims 29
Risk-NeutralValuationFormula 31
DefaultableZero-CouponBond 32
2.2 ClassicStructuralModels 34
Merton’sModel 34
BlackandCoxModel 37

3.4 FurtherDevelopments 88
Default-AdjustedMartingaleMeasure 88
HybridModels 89
Unified Approach 90
3.5 CommentsonIntensity-Based Models 90
4 DependentDefaultsandCredit Migrations 91
4.1 BasketCreditDerivatives 92
The i
th
-to-DefaultContingentClaims 92
CaseofTwoEntities 93
4.2 ConditionallyIndependentDefaults 94
CanonicalConstruction 94
IndependentDefault Times 95
SignedIntensities 96
ValuationofFDC andLDC 96
GeneralValuationFormula 97
DefaultSwapofBasketType 98
Contents XI
4.3 Copula-BasedApproaches 99
DirectApplication 100
IndirectApplication 100
Simplified Version 102
4.4 JarrowandYuModel 103
ConstructionandPropertiesofthe Model 103
BondValuation 105
4.5 ExtensionoftheJarrowandYuModel 106
Kusuoka’sConstruction 107
InterpretationofIntensities 108
BondValuation 108

ObjectiveFunctions 136
Infinitesimal Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Hamilton-Jacobi-BellmanEquations 139
XII Contents
VerificationArgument 141
StepsforSolution 143
4 OptimalInvestmentforInsurers 143
4.1 HJBanditsHandyForm 143
4.2 ExistenceofaSolution 145
4.3 ExponentialClaimSizes 145
4.4 Two orMoreRiskyAssets 147
5 Optimal ReinsuranceandOptimalNewBusiness 148
5.1 OptimalProportionalReinsurance 150
5.2 OptimalUnlimitedXLReinsurance 151
5.3 OptimalXLReinsurance 152
5.4 OptimalNewBusiness 153
6 Asymptotic Behavior for Value Function and Strategies . . . . . . . . . . . . 154
6.1 OptimalInvestment:ExponentialClaims 154
6.2 OptimalInvestment:SmallClaims 154
6.3 OptimalInvestment:LargeClaims 155
6.4 OptimalReinsurance 156
7 A ControlProblemwithConstraint:DividendsandRuin 157
7.1 A SimpleInsuranceModelwithDividendPayments 157
7.2 ModifiedHJBEquation 158
7.3 NumericalExampleandConjectures 159
7.4 EarlierandFurtherWork 161
8 Conclusions 162
References 163
Nonlinear Expectations, Nonlinear Evaluations and Risk
Measures

Upcrossing Inequality of E
g
–Supermartingales and Optional
SamplingInequality 193
3.3 AMonotonicLimitTheorem ofBSDE 199
3.4 g–Martingales and (Nonlinear) g–Supermartingale
DecompositionTheorem 201
4 Finding the Mechanism: Is an F–Expectation a g–Expectation? . . . . . 204
4.1 E
µ
-Dominated F-Expectations 204
4.2 F
t
-ConsistentMartingales 207
4.3 BSDE under F
t
–Consistent Nonlinear Expectations . . . . . . . . . . . 210
4.4 Decomposition Theorem for E-Supermartingales 213
4.5 Representation Theorem
of an F–Expectation by a g–Expectation 216
4.6 How to Test and Find g? 219
4.7 A General Situation: F
t
–Evaluation Representation Theorem . . . 220
5 DynamicRiskMeasures 221
6 NumericalSolutionofBSDEs:Euler’sApproximation 222
7 Appendix 224
7.1 MartingaleRepresentationTheorem 224
7.2 A Monotonic Limit Theorem of Itˆo’sProcesses 226
7.3 Optional Stopping Theorem for E

asymmetric information models [10] have recently been published. In these
notes, I will not attempt to repeat these comprehensive surveys but instead
will give a more selective review.
The first part of this article provides a review of filtering theory, in par-
ticular establishing the notation to be used in the later parts. The second
part reviews some work on incomplete information models, focusing on recent
work using simple Markov chain models to model the behavior of the market
portfolio. The last part reviews asymmetric information models, focusing on
the Kyle model and extensions thereof.
1 Filtering Theory
Let us start with a brief review of filtering theory, as exposited in [33]. Note
first that engineers and economists tend to use the term “signal” differently.
Engineers take the viewpoint of the transmitter, who sends a “signal,” which
is then to be estimated (or “filtered”) from a noisy observation. Economists
tend to take the viewpoint of the receiver, who observes a “signal” and then
uses it to estimate some other variable. To avoid confusion, I will try to avoid
the term, but when I use it (in the last part of the chapter), it will be in the
sense of economists.
K. Back et al.: LNM 1856, M. Frittelli and W. Runggaldier (Eds.), pp. 1–25, 2004.
c
 Springer-Verlag Berlin Heidelberg 2004
2 Kerry Back
We work on a finite time horizon [0,T] and a complete probability space
(Ω,A,P). The problem is to estimate a process X from the observations
of another process Y . In general, one considers estimating the conditional
expectation E[f(X
t
)|F
Y
t

θ
t
denotes for each t a version of E[θ
t
|F
Y
t
]chosen
so that the resulting process (t, ω) →
ˆ
θ
t
(ω) is jointly measurable.
Let W be an n–dimensional Wiener process on its own filtration and define
F
t
to be the σ–field generated by (X
s
,W
s
; s ≤ t) augmented by the P –null
sets in A. We assume for each t that F
t
is independent of the σ–field generated
by (W
v
− W
u
; t ≤ u ≤ v ≤ T ), which simply means that the future changes
in the Wiener process cannot be foretold by X. Henceforth, we will assume

t
),and assume
df
t
= g
t
dt + dM
t
, (2)
for some jointly measurable process g and right-continuous martingale M
such that E

T
0
|g
t
|
2
dt < ∞.IfX is given as the solution of a stochastic
differential equation and f is smooth, the processes g and M can of course be
computed from Itˆo’s formula. We assume further that E[f
2
t
] < ∞ for each t
and E

T
0
f
t

and the other being the random change dW.
The main results of filtering theory, due to Fujisaka, Kallianpur, and Ku-
nita [22], are the following.
1) The innovation process Z is an {F
Y
t
}–Brownian Motion.
Incomplete and Asymmetric Information in Asset Pricing Theory 3
2) For any separable L
2
–bounded {F
Y
t
}–martingale H, there exists a jointly
measurable {F
Y
t
}–adapted 
n
–valued process φ such that E

T
0
φ
t

2
dt < ∞,and
dH
t

t
dt +


fh
t
−
ˆ
f
t
ˆ
h
t
+ˆα
t


dZ
t
, (4)
where

fh
t
denotes E[f
t
h
t
|F
Y

f is updated because f is ex-
pected to change (which is obviously captured by the term ˆg
t
dt) and because
new information from dZ is available to estimate f . The observation process
Y (or equivalently the innovation process Z) is useful for estimating f due to
two factors. One is the possibility of correlation between the martingales W
and M. This is reflected in the term ˆα
t
dZ
t
. The other factor is the correlation
between f and the drift h
t
of Y . This is reflected in the term (

fh
t
−
ˆ
f
t
ˆ
h
t
) dZ
t
.
Note that


dX
t
= aX
t
dt + dB
t
,
dY
t
= cX
t
dt + dW
t
,
4 Kerry Back
where B and W are independent real-valued Brownian motions that are in-
dependent of X
0
. In this case, the distribution of X
t
conditional on F
Y
t
is
normal with deterministic variance Σ
t
.Moreover,
d
ˆ
X

, (7)
where α and −β are the two roots of the quadratic equation 1+2ax−c
2
x
2
=0,
with both α and β positive, λ = c
2
(α+β)andγ =(σ
2
+β)/(α−σ
2
). One can
consult, e.g., [33] or [41] for the derivation of these results from the general
filtering results cited above. In the multivariate case, an equation of the form
(5) also holds, where Σ
t
is the covariance matrix of X
t
conditional on F
Y
t
.In
this circumstance, the covariance matrix evolves deterministically and satisfies
an ordinary differential equation of the Riccati type, but there is in general
no closed-form solution of the differential equation.
1.2 Two-State Markov Chain
A very simple model that lies outside the Gaussian family is a two-state
Markov chain. There is no loss of generality in taking the states to be 0 and
1, and it is convenient to do so. Consider the Markov chain X satisfying

i
. This fits in our earlier framework as
dX
t
= g
t
dt + dM
t
,
where
g
t
=(1− X
t−
)λ
0
− X
t−
λ
1
, and
dM
t
=(1− X
t−
) dM
0
t
− X
t−

t−
.
In terms of our earlier notation, h
t
= h(X
t−
).
Write π
t
for
ˆ
X
t
. This is the conditional probability that X
t
=1.The
general filtering formula (4) implies
1
dπ
t
=

(1 − π
t
)λ
0
− π
t
λ
1

the vector c in the equation
dY
t
= h(X
t−
) dt + dW
t
=

(1 − X
t−
)h(0) + X
t−
h(1)

dt + dW
t
= h(0) dt + cX
t−
dt + dW
t
,
and π
t
(1 − π
t
) is the variance of X
t
conditional on F
Y

which is an {F
S
t
}–Brownian motion. Moreover, we can write
1
Note that (4) implies π is continuous and then from bounded convergence we
have π
t
= E

X
t−
|F
Y
t

,soˆg
t
=(1− π
t
)λ
0
− π
t
λ
1
.
6 Kerry Back
dS
S

documented phenomenon of stochastic volatility. Detemple observes in [17]
that, within a model that is otherwise Gaussian, stochastic volatility can be
generated by assuming non-Gaussian priors. However, more recent work has
focused on Markov chain models.
David in [13] and [14] studies an economy in which the assets are in in-
finitely elastic supply, assuming a two-state Markov chain for which the tran-
sition time from each state is exponentially distributed as in Section 1.2. In
David’s model, there are two assets (i =0, 1), with
dS
i
S
i
= µ
i
(X
t−
) dt + σ
i
dW
i
,
where W
0
and W
1
are independent Brownian motions, X
t
∈{0, 1},and
µ
0

. Then asset 0 is most
productive in state 0 and asset 1 is most productive in state 1. The filtering
equation for the model is (10), with observation process Y =(Y
0
,Y
1
), where
dY
i
t
=
d log S
i
t
σ
i
=

µ
i
(X
t−
)
σ
i
−
σ
i
2



π
t
µ
a
+(1−π
t
)µ
b

dt + σ
1
dZ
1
.
As in [16], [19] and [23], this is equivalent to a complete information model in
which the expected rates of return of the assets are stochastic with particular
dynamics given by the filtering equations, but the volatilities of assets are
constant.
David focuses on the volatility of the market portfolio, assuming a rep-
resentative investor with power utility. The weights of the two assets in the
market portfolio will depend on π
t
(e.g., asset 0 will be weighted more highly
when π
t
is small, because this means a greater belief that the expected return
of asset 0 is µ
a
>µ

market-clearing conditions and hence will be affected fundamentally by the
nature of information.
David and Veronesi (see [44], [45] and [15]) study models of this type
and discuss various issues regarding the volatility and expected return of the
market portfolio. Their models are variations on the following basic model.
Assume there is a single asset, with supply normalized to one, which pays
dividends at rate D. Assume
dD
t
D
t
= α
D
(X
t−
) dt + σ
D
dW
1
, (13)
where X is a two-state Markov chain with switching between states occurring
at exponentially distributed times, as in Section 1.2. Here W
1
is a real-valued
Brownian motion independent of X
0
. Investors observe the dividend rate D
but do not observe the state X
t−
, which determines the growth rate of divi-

σ
H

and µ =

α
D
− σ
2
D
/2
σ
D
,
α
H
σ
H

.
In terms of the innovation process Z =(Z
1
,Z
2
), we have
dD
t
D
t
=

dZ
2
, (16)
and the conditional probability π
t
evolves as
dπ
t
=

(1 − π
t
)λ
0
− π
t
λ
1

dt
+ π
t
(1 − π
t
)

α
D
(1) − α
D

substitution. Specifically, the asset price at time t must be
S
t
= E


∞
t
e
−δ(s−t)
u

(D
s
)
u

(D
t
)
D
s
ds




π
t
,D

This is essentially the same as the early models on incomplete information,
because we have simply specified the expected return
π
t
α
D
(1) + (1 − π
t
)α
D
(0)
as a particular stochastic process.
The case of power utility u(c)=c
γ
/γ is more interesting. Note that for
s ≥ t we have from (13) that
D
γ
s
= D
γ
t
e
γ

s
t

[
α

s
ds




π
t
,D
t

= D
1−γ
t

(1 − π
t
)E


∞
t
e
−δ(s−t)
D
γ
s
ds



t

(1−π
t
)E


∞
t
e
−δ(s−t)
e
γ

s
t

[
α
D
(X
a−
)−σ
2
D
/2
]
da+σ
D
dW

(X
a−
)−σ
2
D
/2
]
da+σ
D
dW
1
a

ds




X
t−
=1

.
Due to the time-homogeneity of the Markovian system (15) and (17), the
conditional expectations in the above are independent of the date t.Denoting
the first expectation by C
0
and the second by C
1
,wehave

+ πC
1
+
(C
1
− C
0
) dD, π
D[(1 −π)C
0
+ πC
1
]
(19)
= something dt + σ
D
dZ
1
+

(C
1
− C
0
)π(1 − π)
(1 − π)C
0
+ πC
1


0
+ πC
1
(20)
introduces stochastic volatility. Thus, stochastic volatility can arise in a model
in which the volatility of dividends is constant.
There are obviously other ways than incomplete information to introduce
a stochastic growth rate of dividends in a Markovian model similar to (15)
and (17). However, this approach leads to a very sensible connection between
investors’ uncertainty about the state of the economy and the volatility of
assets. Note that the factor π
t
(1 − π
t
) in the numerator of (20) is the con-
ditional variance of X
t
—it is largest when π
t
is near 1/2, when investors are
most uncertain about the state of the economy, and smallest when π
t
is near
zero or one, which is when investors are most confident about the state of
the economy. Thus, the volatility of the asset is linked to investors’ confidence
about future economic growth.
Veronesi actually assumes in [44] that the level of dividends (rather than
the logarithm of dividends) follows an Ornstein-Uhlenbeck process as in (13)
and he assumes the representative investor has negative exponential utility
(i.e., he assumes constant absolute risk aversion rather than constant relative

this circumstance.
In [45], Veronesi studies the above model but assuming there are n states
of the world rather than just two. One way to express his model is to let the
state variable X
t
take values in {1, ,n} with dynamics
dX
t
=
n

i=1
(i −X
t−
) dN
i
t
,
where the N
i
are independent Poisson processes with parameters λ
i
.This
means that X jumps to state i at each arrival date of the Poisson process N
i
,
independent of the prior state (in particular, X stays in state i if X
t−
= i and
∆N

Define X
i
t
=1
{X
t
=i}
.ThenE

X
i
t


F
Y
t

, which we will denote by π
i
t
,isthe
probability that X
t
= i conditional on F
Y
t
. The distribution of X
t
conditional


j=i
N
j
t
is a Poisson process with parameter λ
−i
≡

j=i
λ
j
,
because, if X
i
is in state 0, it exits at an arrival time of N
i
, and, if it is
in state 1, it exits at an arrival time of N
−i
. Equation (21) is of the same
form as equation (8), and, therefore, the dynamics of π
i
are given by the
filtering equation (10) for two-state Markov chains. The resulting formula for
the dynamics of the asset price S is a straightforward generalization of (19).
2.4 Heterogeneous Beliefs
Economists often assume that all agents have the same prior beliefs. A ratio-
nale for this assumption is given by Harsanyi in [29]. To some, this rationale
seems less than compelling, motivating the analysis of heterogeneous prior be-

cesses. As an example, consider the economy with dividend process (13) and
observation process (14). We might assume some investors believe the Brow-
nian motions W
1
and W
2
are correlated while others believe they are inde-
pendent, or more generally we may assume investors have different beliefs
regarding the correlation coefficient. Scheinkman and Xiong study a similar
model in [42], though in their model there are two assets. To each asset there
corresponds a process D satisfying (13), though D(t) is interpreted as the
cumulative dividends paid between 0 and t instead of the rate of dividends at
time t. To each asset there also corresponds an observation process of the form
(14). There are two types of investors. One type thinks the observation pro-
cess associated with the first asset has positive instantaneous correlation with
its cumulative dividend process while the other type thinks the two Brownian
motions are independent. The reverse is true for the second asset. Scheinkman
and Xiong intepret this as “overconfidence,” with each investor weighting the
innovation process for one of the assets too highly when updating his beliefs.
They link this form of overconfidence to speculative bubbles, the volume of
trading, and the “excess volatility” puzzle.
3 Asymmetric Information
3.1 Anticipative Information
Recently, a literature has developed using the theory of enlargement of filtra-
tions to study the topic of “insider trading.” See [9], [25], [26], [31], [34], [38]
Incomplete and Asymmetric Information in Asset Pricing Theory 13
and the references therein. One starts with asset prices of the usual form
3
dS
i

∨ σ−(Y )}. By “access to the filtration,” I mean
that the insider is allowed to choose trading strategies that are {G
t
}–adapted.
Some interesting questions are (1) does the model make mathematical
sense—i.e., are the price processes {G
t
}–semimartingales? (2) is there an ar-
bitrage opportunity for the insider? (3) is the market complete for the insider?
(4) how much additional utility can the insider earn from his advance knowl-
edge of Y ? (5) how would the insider value derivatives? . . . . For the answer
to the first question, the essential reference is [32]. In [9], Baudoin describes
the setup I have outlined here as the case of “strong information” and also
introduces a concept of “weak information.”
The study of anticipative information can be useful as a first step to de-
veloping an equilibrium model. Because the insider is assumed to take the
price process (22) as given (unaffected by his portfolio choice) the equilibrium
model would be of the “rational expectations” variety described in the next
section. If one does not solve for an equilibrium, the assumed price dynamics
could be quite arbitrary. Suppose for example that there is a constant riskless
rate r and the advance information Y is the vector of asset prices S
T
.Then
there is an arbitrage opportunity for the insider unless
S
i
t
=e
−r(T −t)
S