part i: portfolio selection in one period part ii ... · part iii: advanced topics in portfolio...

Part I: Portfolio Selection in One PeriodPart II: Portfolio Selection in Continuous TimePart III: Advanced Topics in Portfolio Theory

Portfolio Theory

Gorazd Brumen

Morgan Stanley

September 11-12, 2009

Gorazd Brumen Portfolio Theory


Topics I will cover

1 Part I: Portfolio Selection in One Period

2 Part II: Portfolio Selection in Continuous Time

3 Part III: Advanced Topics in Portfolio Theory



Requirements, prior knowledge

Basic linear algebra, optimization techniques.

Basic probability theory.

Stochastic integration, SDE.

Basic microeconomics.



Part I

Part I: Portfolio Selection inOne Period



Historical perspective on portfolio selection

Even though the concept of diversification is firmly groundedin today’s economic thinking, this was not always the case.

Before Markowitz’s seminal contribution investors did notconsider portfolio diversification but rather stock picking: Ofall stocks in a market pick the one which brings you highestcombination of dividends (and capital gains).

Portfolio theory answers the question which risks are pricedand in what extent.



Framework and Notations

One period model. Vectors will be underlined, such as x , matricesare boldface, e.g. Γ. Begining of period at time 0, end of period at1. Return on an asset i = 1, . . . ,N in terms of its price Si in thisperiod is

Ri =Si(1) − Si(0)

Si(0)

(Ri is a random variable) Let the number of units of asset i be ni .The value of the portfolio X (t) at time t holding ni units is

X (t) = n′S(t). The return on such a portfolio is RX = X (1)−X (0)X (0) .

If xi(0) = xi = niSi (0)X (0) then x ′1 = 1 and

RX = x ′R.

(proof left as an exercise.)Gorazd Brumen Portfolio Theory


Analysis of mean and variance

We focus on the first two moments of R. Let µ = (µi) = E(Ri)and Γ = (σij) = (cov(Ri ,Rj )). Then

µX = E(RX ) = x ′µ

var(RX ) =∑

i ,j

xixjσij = xΓx



Portfolio with risk-less asset

We introduce the riskless asset by S0(1) = S0(0)(1 + r) andadditionally

x0 = 1 −N∑

i=1

xi

Portfolio returns in this case are

RX = x0r +N∑

i=1

xiRi = r +N∑

i=1

xi(Ri − r)

RX − r = x ′(µ − r1),

i.e. portfolio excess return is a linear combination of excess returnsof individual stocks.



Simple relations

Definition

Assets i = 1, . . . ,N are redundant if there exists N scalarsλ1, . . . , λN such that

∑Ni=1 λiRi = k for some constant k. The

portfolio λ is risk-free.

Proposition

The assets i = 1, . . . ,N are not redundant if and only if Γ ispositive definite.

(exercise.)



Efficiency criteria and optimization program

Definition

Portfolio (x∗,X ∗) is efficient if for every other portfolio y we havethat if σY < σX∗ then µY < µX∗ and σY = σX∗ impliesµY ≤ µX∗ .

Portfolio optimization problem:

maxx

E(RX ) s.t. x ′Γx = k x ′1 = 1.

The Lagrangian of this problem is

L(x ,θ

2, λ) = x ′µ −

θ

2x ′Γx − λx ′1

First order condition gives us

µ − θΓx∗ − λ1 = 0 (1)



Optimization program (contd.)

Equivalently

µi = λ + θ

N∑

j=1

x∗

j σij .

FOC are neccessary and sufficient, since the second derivative isstrictly concave (Γ is positive definite).



Connection to utility theory

Criterium of portfolio efficiency is consistent with the economicagents with the following utility

u(x) = E(RX ) −θ

2var(RX ),

= x ′µ −θ

2x ′Γx .

where the Lagrange parameter θ now represents some degree ofrisk aversion, i.e. the higher θ is, the more averse the agent is wrt.(variance) risk.



Competitive economic equilibirum

A set of agents i = 1, . . . , I .

A set of assets Sj , j = 1, . . . ,N in net supply y .

Definition (Competitive equilibrium)

Portfolio x∗ and price system S is a competitive equilibrium if

x∗

i is the solution to the optimization problem

maxx i

ui (x) s.t. x ′

iS = Wi

Markets clear:

I∑

i=1

x∗

i = y



Two funds separation (Black)

Consider any two efficient portfolios x and y . Then

Theorem

Any convex comination of x and y, i.e. ux + (1 − u)y isefficient.

Any efficient portfolio is a combination of x and y (notnecessarily convex).

The efficient frontier is a parabola in the expectedreturn-variance space (µ, σ2) and a hyperbola in the expectedreturn-standard deviation space (µ, σ).

Due to the first bullet point above, any efficient portfolio can bedescribed as a convex combination of just 2 portfolios. Proof ofthe first two bullet points left as an exercise.



Proof of last bullet of two fund separation

We have shown in (1) that x∗ = θΓ−1(µ − λ1). From 1′x∗ = 1 we

get that λ =1′Γ

−1µ−θ

1′Γ1. Therefore x∗ = k1 + θk2 for appropriate

k1 and k2. The efficiency set is given by

ES = x∗ : x∗ = k1 + θk2, θ > 0

from where it follows that portfolio return µ′x∗ is linear and the

variance x∗′

Γx∗ is quadratic in θ. The efficiency frontier is aparabola in this space.



Efficiency set with riskless asset

Theorem

Asset 0 is efficient.

Any combination uRX + (1 − u)r of asset 0 and a portfolio Xlies on the straight line between 0 and X in the (µ, σ) space.

The straight line between asset 0 and asset M is the efficientfrontier called the Capital Market Line.

(Tobin’s two fund separation) Any efficient portfolio is acombination of only 2 portfolios (e.g. 0 and M).



Efficiency set with riskless asset (contd.)

Theorem (contd.)

Any efficient portfolio satisfies

x∗ = θΓ−1(µ − r1)

Tangent (market) portfolio (m,M) is

m = θMΓ−1(µ − r1)

θM =1

1′Γ−1(µ − r1)

(Proof left as an exercise.)



Capital Market Equilibirium

Since the market portfolio is efficient there exists scalars λ and θsuch that

µi = λ + θcov(RM ,Ri ),

It follows that for any portfolio we have

E(RX ) =

N∑

i=1

xiµi

=

N∑

i=1

xi(λ + θcov(RM ,Ri))

= λ + θcov(RM ,RX )



CAPM equilibrium

In particular for market portfolio it holds that

µM = λ + θσ2M

from where it follows that θ = µM−λσ2

M

and therefore

µi = λ + θcov(RM ,Ri) = λ + (µM − λ)βi

where βi = cov(RM ,Ri )σ2

M

. If we set Ri = r the risk-less asset we get

E(Ri) = r + βi (µM − r).



CAPM as a Pricing Model

Question: If a security delivers V (1) at time 1, what is the priceV (0) of this security at time 0? Assuming that the risk-free assetexists then

E(V (1))

V (0)= E(1 + R) = 1 + r + θcov

(

V (1)

V (0),RM

)

where θ = µM−r

σ2M

Solving for V (0) gives us

V (0) =E(V (1)) − θcov(V (1),RM)

1 + r.



Further topics and Relevant literature

Relevant literature:

Book by Huang/Litzenberger.

Mossin (Econometrica paper), Sharpe, Cass-Stiglitz.

Further topics:

Arbitrage pricing theory (Ross).

Behavioral portfolio theory. (Kahneman and Tversky)

Factor models. (partially given in the exercises)



Part II

Part II: Portfolio Selection inContinuous Time



Continuous time financial market

Financial market:

Risk-free security: dBt = Btrt dt, process rt is progressivelymeasurable (adapted with cadlag paths) and

∫ T

0 ru du < ∞.

d stocks with dynamics:

dS t + Dt dt = IS (µtdt + σt dW t)

where dS are stocks’ capital gains and D dividends.

IS = diag(S1,S2, . . . ,Sd).

(µ,σ) is a progressively measurable process such that∫ T

0 µtdt < ∞ and

∫ T

0 σtσ′

t dt < ∞.

If σ is invertible, the financial market is complete.

Market price of risk (Sharpe ratio): θt = σ−1t (µ − r1).



Investors

Progressively measurable consumption process c > 0, such that∫ T

0 ct dt < ∞ and U(c) < ∞ such that

U(c) = E

[∫ T

0u(ct , t)dt

]

where u(·, t) : R+ → R is strictly increasing, strictly concave andtwice continuously differentiable and satisfies the Inada condition:u′(0, t) = ∞, u′(∞, t) = 0 for every t ∈ [0,T ].



Examples of utility functions considered

u(c , t) = ρtu(c) where ρt is subjective discount factor, e.g.ρt = exp(−

∫ t

0 βv dv).

u(c) = c1−R

1−R, R ≥ 0 is an example of CRRA utilities.

u(c) = 11−R

(c + δ)1−R , R ≥ 0, δ > 0 an example of HARAutilities.



Investor’s wealth dynamics

Progressively measurable portfolio process π generates investor’swealth dynamics

dXt = π′

t((IS)−1(dS t + Dt dt)) + (Xt − π′

t1)rt dt − ct dt

= π′

t [(µt− rt1)dt + σt dW t ] + (Xtrt − ct)dt,

where X0 = x given. The first term is the return on stockportfolio, the second the return on bonds and the third theconsumption part.

Definition

(c , π) is admissible (belongs to A(x) iff Xt ≥ 0 for allt ∈ [0,T ].

(c , π) is optimal (belongs to A∗(x) iff there does not exist(c , π) such that U(c) > U(c).



Static approach

Let

ηt = exp

(

−

∫ t

0θ′v dW v −

1

2

∫ t

0θ′vθv dv

)

.

Due to Novikov condition η is a martingale. Therefore we canchange the measure dQ = ηT dP. This implies the following:

It can be proven that W t = W t +∫ t

0 θv dv is a Brownianmotion.

Sv = Ev [Stξv ,t +∫ t

vξv ,sDs ds] where ξt = btηt and ξv ,t = ξt

ξv.

Arrow-Debreu prices are then ξT dP , i.e. a security paying 1ω

is ξT (ω)dP(ω).



Equivalence of approaches

Let

B(x) =

c : E

(∫ T

0ξtct dt

)

≤ x

.

Theorem

We have the following implications:

(a) If (c , π) ∈ A(x) then c ∈ B(x)

(b) If c ∈ B(x) then there exists π such that (c , π) ∈ A(x).



Proof for (a)

Let us have X ≥ 0, c ≥ 0, then ξtXt +∫ t

0 ξvcv dv ≥ 0 for everyt ∈ [0,T ]. LHS is a positive local martingale, which implies that it

is a supermartingale. Therefore E

[

ξTXT +∫ T

0 ξvcv dv]

≤ x and

therefore E

[

∫ T

0 ξvcv dv]

≤ x which implies that c ∈ B(x).



Proof of (b)

Let

Et

[∫ T

t

ξvcv dv

]

= Et

[∫ T

0ξvcv dv

]

−

∫ t

0ξvcv dv

= E0

[∫ T

0ξvcv dv

]

+

∫ t

0φv dWv −

∫ t

0ξvcv dv

Choose φt = ξt [π′

tσt − Xtθ′

v ]. Then by the equation () frombefore we have that

ξtXt +

∫ t

0ξvcv =

∫ t

0φv dWv + x

= Et

[∫ T

t

ξvcv dv

]

− E0

[∫ T

0ξvcv dv

]

+ x

from where it follows that ξtXt ≥ Et

[

∫ T

tξvcv dv

]

≥ 0, i.e.

Xt ≥ 0 for all t ∈ [0,T ]. This proves that (c , π) ∈ A(x).Gorazd Brumen Portfolio Theory


Static portfolio optimization

Constructing the Lagrangian:

L = E

[∫ T

0u(ct , t)dt

]

− y

E

[∫ T

0ξvcv dv

]

− x

Process c is optimal if there does not exist a process ∆t such thatL(c + ε∆) > L(c). The necessary condition is therefore∂L∂ε |ε=0 = 0 for every ∆. We have

∂L

∂ε|ε=0 = E

[∫ T

0u′(ct , t)∆t dt

]

− yE

[∫ T

0ξv∆v dv

]

= E

[∫ T

0(u′(ct , t) − yξt)∆t dt

]

from where it follows that u′(ct , t) = yξt , i.e. marginal utilityequals marginal costs. y is fixed by the condition

E

[

∫ T

0 ξvcv dv]

= x , y ≥ 0.



Summary of the portfolio optimization

Theorem

Optimal portfolio optimization gives us

c∗t = I (y∗ξt , t) where I = (u′)−1

y∗ : x = E

[∫ T

0ξv I (yξv , v)dv

]

π∗

t = Xt(σ′

t)−1θt + ξ−1

t (σ′

t)−1φ∗

t

φ∗

t = Et [F∗] − E[F ∗]

F ∗ =

∫ T

0ξvc∗v dv

X ∗

t = Et

[∫ T

t

ξt,vc∗v dv

]



Malliavin calculus (Stochastic calculus of variations)

Motivation: If F ∈ L2 then there exists by the martingalerepresentation theorem a progressively measurable process φ suchthat

F = E[F ] +

∫ T

0φv dWv

How to extract φ? The question is not important only in portfoliotheory but also in derivatives pricing for hedging purposes.



Hedging of derivative securities

Fundamental theorem of asset pricing states that the price of aderivative security with payoff ϕ(ST ) at time T is given by

EQ[ϕ(ST )].

The replicating portfolio is given by the process u such that

ϕ(ST ) = EQ[ϕ(ST )] +

∫ T

0ut dSt.

Malliavin calculus gives us an answer to what is u.



Malliavin calculus definitions

Definition

Let S be the space of smooth Brownian functionals, i.e.

S = f (Wt1 , . . . ,Wtn) : f ∈ C∞

p (Rdn)

and where C∞

p is the space of functions on Rdn which are infinitelydifferentiable and of polynomial growth.Then the Malliavin derivative DF = DtF : t ∈ [0,T ] is ad-dimensional stochastic process defined by

Di ,tF =

n∑

j=1

∂f

∂xij· 1[0,tj ](t)

for every i = 1, . . . , d (i corresponds to rows).



Simple properties of Malliavin calculus

Malliavin derivative is the generalization of the Frechetderivative for stochastic processes.

Malliavin derivatives are not adopted (anticipating processes).

DtF = 0 if F ∈ F s and s < t.

Theory can be extended to appropriate spaces for stochasticprocesses called D2,1.

If ST = S0 exp((µ − 1/2σ2)T + σWT ) thenDtST = STσ1[0,T ](t).



Properties of Malliavin calculus

Chain rule: Let g : Rm → R with bounded derivatives andF1, . . . ,Fm ∈ D2,1. Then

Dtg(F1, . . . ,Fm) =

m∑

i=1

∂g

∂FiDtFi

Dt(Ev [F ]) = Ev (DtF ) for v ≥ t

Clark-Ocone formula: Let F ∈ D2,1. Then

F = E(F ) +

∫ T

0φv dWv

where

φv = Ev (DvF ).



Malliavin calculus rules

More rules:

If F1 =∫ T

0 φ1t dt then

DtF1 =

∫ T

0Dtφ1v dv

=

∫ T

t

Dtφ1v dv

If F2 =∫ T

0 φ2t dWt then

DtF2 =

∫ T

t

Dtφ2v dWv + φ2t



Malliavin Calculus of Stochastic Processes

Let

dSt = µ(t,St)dt + σ(St , t)dWt

then

ST = St +

∫ T

t

µ(Sv , v)dv +

∫ T

t

σ(Sv , v)dWv .

Applying the rules from before we get that

DtST =

∫ T

t

∂µ

∂S(Sv , v)DtSv dv +

∫ T

t

∂σ

∂S(Sv , v)DtSv dWv + σ(St , t)

from where it follows that

dDtSv =∂µ

∂S(Sv , v)DtSv dv +

∂σ

∂S(Sv , v)DtSv dWv

with initial condition DtSt = σ(St , t).Gorazd Brumen Portfolio Theory


Optimal portfolios

Theorem

We have

π∗

t = ξ−1t Et

[∫ T

t

c∗vRv

ξv dv

]

(σ′

t)−1θt

−ξ−1t (σ′

t)−1Et

[∫ T

t

c∗v (1 −1

Rv)ξvHt,v dv

]

where

Rt = −u′′(c∗t , t)

u′(c∗t , t)c∗t relative risk aversion

Ht,v =

∫ v

t

Dtrs ds +

∫ v

t

Dtθ′

v(dW v + θv dv)

Proof left as an exercise.Gorazd Brumen Portfolio Theory


Optimal portfolio (corollary)

In the case of deterministic opportunity set (meaningθt = σ

−1(µ − r1) and r are constant) we have that Ht,v = 0 andwe get

π∗

t = Xt

Et [∫ T

tc∗vRv

ξv dv ]

Et [∫ T

tξvc∗v dv ]

(σ′

t)−1θt

In case when u(c , t) = ρ log c and ρ deterministic we get R = 1and π∗

t = Xt(σ′

t)−1θt showing that the logarithmic utility function

exhibits myopic behavior.



Asset pricing

We follow Lucas (1978) model in continuous time. Let stocks havedividends that follow

dD jt = D j

t(γjt dt + λj

t dW t)

where j = 1, . . . , d . We also assume that the aggregateconsumption C =

∑dj=1 D j follows

dCt = Ct [µCt dt + σC

t dW t ]

where

µCt =

d∑

j=1

D jt

Ctγjt

σCt =

d∑

j=1

D jt

Ct

λjt



Asset pricing (II)

Bonds are in zero-net supply (no exogenous supply of bonds).

Stocks are in unit supply.

Single (representative) investor with endowment (1, 0) at time0.

Definition

Equilibrium is the set of S0, µ, σ, r and (c , π) such that

(c , π) ∈ A∗(x0) given S0, µ, σ, r .

Market clearing conditions: c = C = D ′1, π = S andX − π′1 = 0.



Asset pricing (III)

Theorem

Rational expectations equilibrium exists and the following holds:

ξt = m0,t =u′(ct , t)

u′(c0, 0)

rt = −∂u′(ct ,t)

∂t

u′(c0, 0)+ Rtµ

Ct −

1

2RtPt(σ

Ct )(σC

t )′

Pt = −u′′′(ct , t)

u′′(ct , t)ct Prudence coefficient

θt = Rt(σCt )′

πt = S t

Xt = S ′

t1

(Proof is left as an exercise.)Gorazd Brumen Portfolio Theory


Remarks and Corollary

rt = −Et [ dξt/ dt]ξt

is the expected growth rate of SPD.

θt = −σξ

ξtgrowth rate volatility of SPD.

We have the following:

St = Et

[∫ T

t

ξt,vDv dv

]

= EQt

[∫ T

t

bt,vDv dv

]

= Et

[∫ T

t

mt,vDv dv

]



Consumption CAPM (Breeden (1979))

From before we get that

θt = Rt(σCt )′ = σ

−1t (µ

t− rt1)

from where it follows that

µt− rt1 = Rtσtσ

C ′

t

Applying this to the market portfolio we get

µm − rt = Rtσm′

t σC ′

t

from where it follows that Rt = µmt −rt

σm′

t σC ′

t

. We get

µt− rt1 = βC

t (µmt − rt) βC

t =σtσ

C ′

t

σm′

t σC ′

t

.



Equity premium puzzle

From the CCAPM (d = 1) we have that

µmt − rt = Rtσ

mt σC

t

or equivalently

µmt − rtσm

t

= RtσCt

Usual values for θmt ≈ 0.37, σC

t ≈ 0.036 and Rt ≈ 10.27. Mehraand Prescott (1985) obtained that in this case µm

t − rt ≈ 0.4% forlevels of risk aversion R = 2, 3, 4 whereas in reality this is appx.6 − 8%.



Risk-free rate puzzle (Weil (1989))

In case u(c , t) = ρu(c) where ρt = exp(−∫ t

0 βv dv) we have frombefore

rt = βt + RtµCt −

1

2RtPtσ

Ct σC ′

t .

Empirically r ≈ 6 − 7% contrary to model prediction of 0.8%although this is questionable with the new data.



Volatility of stocks

We have that

S jtu

′(ct , t) = Et

[∫ T

t

u′(cv , v)D jv dv

]

for j = 1, . . . , d . Using Ito formula on both sides of the equationgives (equating the volatility terms)

u′′ctσCt S j

t + u′S jtσ

jt = Et

[∫ T

t

u′′(cv , v)DtcvD jv dv +

∫ T

t

u′(cv , v)DtD

Further we have that

Dtcv = cv (σC ′

t + HCt,v )

where

HCt,v =

∫ v

t

Dt(µCu −

1

2σC

u σC ′

u )du +

∫ v

t

(DtσCu )dWu

DtDjv = D j

v (λj ′

t + HD,jt,v )



Stock volatility (II)

After rearranging (exercise) we get that

σjt = λj

t + hedging terms.

Empirically, the σmt ≈ 0.2 while λj

m ≈ 0.036. This is the volatilitypuzzle (Schiller; Grossman and Schiller).



Further topics

Multiple agents equilibrium does not resolve the puzzles.

Habit formation and connection to Forward-Backward SDE(Constantinides (1990), Detemple and Zapatero (1991)).

Incomplete and asymmetric information in the continuoustime portfolio theory.

Mathematical aspects: Forward-Backward stochasticdifferential equations.



Part III

Part III: Advanced Topics inPortfolio Theory



Risk Measures

In one-period portfolio optimization, variance is taken as ameasure of risk and there is no reason to do so.

In practice, the popular measure of risk is VaR(Value-at-Risk):

VaRα(X ) = − infx : P(X ≥ x) ≤ 1 − α,

i.e. it is a quantile, e.g. VaR99%(X ) = 100M says that theprobability of a 100M loss over a certain time horizon is lessthan 1%. This risk measures was mandated in the Basel IIdocument for bank risk management.

Heath, Artzner, Delbaen and Eber postulated axioms that anyrisk measure should fulfil.



Axioms of coherence

Fix some probability space (Ω,F ,P) and a time horizon ∆.Denote by L0(Ω,F ,P) the set of rvs which are almost surely finiteand a convex cone M ⊂ L0 which we interpret as portfolio lossesover time period ∆: If L1, L2 are in M then alsoL1 + L2, λL1 ∈ M for λ > 0. Risk measures are real valuedfunctions ρ : M → R. ρ(L) is the amount of capital that should beadded to the position to become acceptable.A function ρ : L → R is coherent if it satisfies the following set(HADE) of axioms:

1 Monotonicity: If Z1,Z2 ∈ L and Z1 ≤ Z2 then ρ(Z2) ≤ ρ(Z1).2 Sub-additivity: If Z1,Z2 ∈ L then ρ(Z1 +Z2) ≤ ρ(Z1)+ ρ(Z2).3 Positive homogeneity: If α ≥ 0 and Z ∈ L then

ρ(αZ ) = αρ(Z ).4 Translation invariance: If a ∈ R and Z ∈ L then

ρ(Z + a) = ρ(Z ) + a.



Rationale behind Subadditivity Axiom

Risk can be reduced by diversification. Non-subadditive riskmeasures can lead to very risky portfolios.

Breaking a firm into subsidiaries would reduce regulatorycapital.

Decentralization of risk-management system: Trading desksL1 and L2. Risk manager wants to ensure thatρ(L1 + L2) < M. It is enough to ensure that ρ(L1) < M1 andρ(L2) < M2 with M1 + M2 = M.

Positive homogeneity insures there is no diversification ofmultiplying a portfolio.



VaR is not Coherent

VaR is not in general a coherent risk measure - it does notrespect the sub-additivity, which implies that VaR mightdiscourage diversification. An example: Let X1,X2, . . . ,Xn berevenues from different business lines, which can be equitytrading desk, interest rate trading desk, etc. Let us assumethat the capital requirements for operating a business line Xi

are exactly VaRα(Xi). Then the capital requirement fromoperating all business lines is greater than operating each oneseparately. This is in direct contradiction to the diversificationprinciple.

VaR is coherent for the class of elliptically distributed losses(e.g. normally distributed).



Risk measures as generalized scenarios

Denote by P the set of probability measures on the underlyingspace (Ω,F). Let MP = L : EQ(L) < ∞ for all Q ∈ P andρP : MP → R such that

ρP(L) = supEQ(L) : Q ∈ P.

Theorem

(a) For any set P of probability measures (Ω,F) the risk measureρP is coherent on MP (Exercise.)

(b) Suppose that Ω = ω1, . . . , ωd is finite and letM = L : Ω → R. Then for any coherent risk measure ρ onM there is a set P of probability measures on Ω such thatρ = ρP.



Examples of Coherent Risk Measures

Expected shortfall, defined as

ESα(X ) = mine

E(X − e|X ≥ VaRe(X ))

is coherent.



Mean-VaR portfolio optimization

Instead of taking variance as a risk measure one could consider thefollowing optimization problem:

maxπ

E(X ) − θVaRα(X ).

This problem was considered in, for example Basak (2001).

Further reading (a lot):

Literature on convex risk measures where the subadditivityaxiom is replaced by the convexity axiom (Foellmer, Schied).

Dynamic risk measures (Delbaen, Cheridito, El-Karoui,Ravanelli).



Statistical Arbitrage

Understanding that arbitrage in a financial market is impossible toachieve, statistical arbitrage tries to be as close to it. Let usconsider two different stocks

S1t = ρ1Mt + ε1

t

S2t = ρ2Mt + ε2

t

where Mt is a market factor and εit is a market residual for this

stock. Constructing a portfolio

ρ2S1t − ρ1S

2t = ρ2ε

1t + ρ1ε

2t

we only have the residual risk. Notice that arbitrage does not existhere.



Trading strategy in this example

Assuming that the process Ut = ρ2ε1t − ρ1ε

2t follows an

Ornstein-Uhlenbeck process

dUt = −ρUt dt + σ dWt

a trading process can for example optimize the following: Select abuy-time τ1 and a sell-time τ2, such that τ1 < τ2 and

maxτ1,τ2

E(−e−rτ1Uτ1 + e−rτ2Uτ2).

This was solved and there exists boundaries A and B such that

τ1 = inft : Ut ≤ −A

τ2 = inft : t > τ1,Ut ≥ B

where A,B solve integral equations. The strategy is then a simplebuy-and-hold strategy.


part i: portfolio selection in one period part ii ... · part iii: advanced topics in portfolio...

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