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Optimal Investment under Dynamic Risk Constraints

and Partial Information

Wolfgang Putschgl

Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of Sciences

www.ricam.oeaw.ac.at

20th September 2007Joint work with J. Sa (RICAM), Supported by FWF, Project P17947-N12

Workshop and Mid-Term Conference on AdvancedMathematical Methods for Finance

Vienna

http://www.ricam.oeaw.ac.at/http://www.ricam.oeaw.ac.at/
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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Model Setup

Filtered probability space: (,F= (Ft)t[0,T], P)

Finite time horizon: T > 0

Money market: bond with stochastic interest rates r

dS(0)t = S

(0)t rt dt , S

(0)0 = 1 , i.e., S

(0)t = exp

Zt0

rs ds

,

r uniformly bounded and progressively measurable w.r.t. F

Stock market: n stocks with price process St = (S(1)t , . . . , S

(n)t )

, return Rt, and

excess return Rt, where

dSt = Diag(St)(t dt + t dWt) , dRt = t dt + t dWt , dRt = dRt rt dt.

W n-dimensional standard Brownian motion w.r.t. Fand P

drift t Rn Ft-adapted and independent of W

volatility t Rnn progressively measurable w.r.t. FSt ,

t non-singular, and 1

t

uniformly bounded.

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Partial Information

Remark

We consider the case of partial information: we can only observe interest rates and stock prices (Fr,S) but not the drift

The portfolio has to be adapted to Fr,S

we need the conditional density t = E Zt|FSt

we need the filter for the drift t = E t|F

S

t

Assumption

The interest rates r are FS-adapted Fr,S = FS

Z is a martingale w.r.t. Fand P

Lemma

We haveFS = FW = FR the market is complete w.r.t. FS

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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Consumption and Trading Strategy

Definition

Initial capital x0 > 0

Wealth process X satisfies

dXt =

t (t dt + t dWt) + (X

t 1

n t)rt dt

X0 = x0

A strategy is admissible if Xt 0 a.s. for all t [0, T]

t represents the wealth invested in the stocks at time t

t = t/Xt denotes the corresponding fraction of wealth

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Utility Functions

Definition

U: [0,) R {} is a utility function, if U is strictly increasing, strictly concave,

twice continuously differentiable on (0,), and satisfies the Inada conditions:

U() = limx

U(x) = 0 , U(0+) = limx0

U(x) = .

I denotes the inverse function of U.

Assumption

I(y) Kya, |I(y)| Kyb for all y (0,) and a, b, K > 0

Example

Logarithmic utility U(x) = log(x) Power utility U(x) = x/ for < 1, = 0.

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Optimization Problem

Optimization Problem

We optimize under partial information!

Objective: Maximize the expected utility from terminal wealth, i.e.,

maximize E

U(XT)

under (risk) constraints we still have to specify.

The optimization problem consists of two steps:

1. Find the optimal terminal wealth

2. Find the corresponding trading strategy

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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Time-Dependent Convex Constraints

We can write our model under full information with respect to FR as

dRt = t dt + t dVt , t [0, T] .

where the innovation process V = (Vt)t[0,T] is a P-Brownian motion defined by

Vt = Wt + Zt

0

1s (s s) ds = Zt

0

1s dRs Zt

0

1s s ds .

Kt represents the constraints on portfolio proportions at time t t Kt

Kt is a Ft-progressively measurable closed convex set = Kt Rn that contains 0

For each t we define the support function t : Rn R {+} of Kt by

t(y) = supxKt

(xy) , y Rn .

t(y) is Ft-progressively measurable

y t(y) is a lower semicontinuous, proper, convex function on its effective

domain Kt = {y Rn: t(y) < }

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Time-Dependent Convex Constraints

Definition

A trading strategy is called Kt-admissible for initial capital x0 > 0 if Xt 0 a.s. and

t Kt for all t [0, T].

We denote the class of admissible trading strategies for initial capital x0 by AKt(x0).

We introduce the set H of dual processes t : [0, T] Kt which are

F

R

t -progressively measurable processes, satisfying ERT

0`t

2

+ t(t)

dt < .For each dual process H we introduce

a new interest rate process rt = rt + t(t).

a new drift process t = t + t + t(t)1n.

a new market price of risk t = 1t (t rt + t)

a new density process given by dt = t

t dVt

Then:

Solution under constraints = solution under no constraints with new market coefficients!

Problem:

Find optimal !12/34

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Time-Dependent Convex Constraints

Proposition

Supposex0 > 0 andE[U(XT )] < for all AK(x0).

A trading strategy AK(x0) is optimal, if for somey > 0, H

XT = I(y

T) , X

(y) = x0 ,

where T =

T . Further,

and

have to satisfy thecomplementary slacknesscondition

t(t ) + (

t )t = 0 , t [0, T] .

y, solve thedual problem

V(y) = infH

E

U(yT)

,

whereU(y) = supx>0

U(x) xy , y > 0 is theconvex dual functionofU.

IfFR = FV holds, thenan optimal trading strategy exists.

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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Li i d E d L & Li i d E d Sh f ll

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Limited Expected Loss & Limited Expected Shortfall

Suppose we cannot trade in [t, t + t]. Then

Xt = Xt+t X

t = X

t exp

Zt+tt

rs ds Xt + exp

Zt+tt

rs ds

(t )X

t

exp

1

2

Zt+tt

diag(ss ) ds+

Zt+tt

s dWs 1

.

Next, we impose the relative LEL constraint

E

(Xt )|FSt

< t ,

with t = LXt .

Definition

KLELt :=

t Rn E

(Xt )

|FSt

< t

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Li it d E t d L & Li it d E t d Sh tf ll

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Limited Expected Loss & Limited Expected Shortfall

We introduce the relative LES constraint as an extension to the LEL constraint

E(Xt

+ qt)|FS

t < t ,

with t = L1Xt and qt = L2X

t .

LES with L2 = 0 corresponds to LEL with L = L1.

LEL: any loss in [t, t + t] can be hedged with L% of the portfolio value.

LES: any loss greater L2% of the portfolio value in [t, t + t] can be hedged withL1% of the portfolio value.

LEL & LES: For hedging we can use standard European call and put options.

Definition

KLESt :=

t Rn E

(Xt + qt)|FSt

< t

Lemma

KLELt andKLES

t are convex.

For n = 1 we obtain the interval KLESt = [lt, ut ].16/34

b d f LEL d LES

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bounds on for LEL and LES

0

5

0

1

210

0

10

L (in % of Wealth) t (in days)

Boundso

n

0

5

10

0

1

210

0

10

L1L2

Boundso

n

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bounds on for LEL

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bounds on for LEL

0 1 2 3 4 50

2

4

6

8

Upperboun

d

on

t (in days)

L = 2%L = 1.2%L = 0.4%

0 1 2 3 4 58

6

4

2

0

Lowerbound

on

t (in days)

L = 2%L = 1.2%L = 0.4%

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Other constraints

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Other constraints

Value-at-Risk constraint:

Under the original measure Xt is given by

Xt = Xt exp

Zt+tt

rs ds Xt + (

t )X

t

expZt+t

t

`s

1

2diag(s

s )

ds+

Zt+tt

s dWs exp

Zt+tt

rs ds

.

We impose for n = 1 the relative VaR constraint on the loss (Xt ),

P`

(Xt ) > LXt |F

St , t = t < .

VaR is computed under the original measure P.

Under partial information we need the (unknown) value of the drift use e.g. t = t.

For n = 1 we obtain the interval KVaR = [lt, ut ].

If n > 2 then KVaR may not be convex!

Possible to apply a large class of other risk constraints e.g. CVaR.19/34

Strategy

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Strategy

Corollary (Logarithmic utility)

U(x) = log(x), n = 1, no constraints:

ot := t =

1

2t(t rt) .

With constraints:

ct := t =

8>>>>>:

ut ifot >

ut ,

ot ifot

lt,

ut

,

lt ifot <

lt .

Hence, we cut off the strategy obtained under no constraints if it exceeds or falls below

a certain threshold.

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Outline

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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Gaussian Dynamics (GD) for the Drift

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Gaussian Dynamics (GD) for the Drift

Drift: modeled as the solution of the stochastic differential equation (cf. Lakner 98)

dt = ( t) dt + d

Wt ,

0 N(0, 0), n-dimensional,

W is a n-dimensional Brownian motion with respect to (F, P),

We are in the situation of Kalman-filtering with signal , observation R, and

filtert =

EtF

S

t.

Filter: t is the unique FS-measurable solution of

dt =` t(t

t )1

t +

dt + t(tt )1 dRt ,

t = t(tt )1t t t

+ ,

with initial condition (0, 0).

1 satisfies d1t = 1t (t rt1n)

(t )1 dWt.

Proposition

FS = FR = FW = FV an optimal trading strategy exists.

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Bayesian case

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Bayesian case

The Bayesian case is a special case of the Gaussian dynamics for the drift.

Drift: t 0 = ((1)0 , . . . ,

(n)0 ) is an (unobservable) F0-measurable Gaussian

random variable with known mean vector 0 and covariance matrix 0.

Filter: Explicit solution:

t =

1nn + 0

Zt0

(ss )1 ds

10 + 0

Zt0

(ss )1 dRs

,

t =

1nn + 0

Zt0

(ss )1 ds

10 .

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Outline

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Outline

Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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HMM: The Drift

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HMM: The Drift

The drift process of the return, is a continuous time Markov chain given by

t = BYt , B Rnd ,

where Y is a continuous time Markov chain with

state space the standard unit vectors {e1, . . . , ed} in Rd, and

rate matrix Q Rdd, where

Qkl is the jump rate or transition rate from ek to el,

k = Qkk =Pd

l=1,l=k Qkl is the rate of leaving ek,

the waiting time for the next jump is exponentially distributed with parameter k and

Qkl/k is the probability that the chain jumps to el when leaving ek for l= k.

The different states of the drift are the columns of B.

We can write the market price of risk as

t = 1t (t rt1n) =

t Yt , where t :=

1t (B rt1nd) .

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HMM: Filtering

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g

We are in the situation of HMM filtering since Rt = Rt

0BYs ds+ R

t

0s dWs.

We need

the conditional density = (t)t[0,T] = E

Zt|FSt

= 1

1dEt

,

the unnormalized filter E= (Et)t[0,T] = E

Z1T Yt|FSt

,

the normalized filter Y = (Yt)t[0,T] = EYt|FSt =Et

1

d

Et= tEt.

Theorem (Wonham/Elliott)

Et = E[Y0] +

Zt0

QEs ds+

Zt0

Diag(Es)s dWs

Proposition

FS = FR = FW = FV an optimal trading strategy exists.

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Outline

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Model Setup

Problem formulation

Time-Dependent Convex Constraints

Dynamic Risk Constraints

Gaussian Dynamics for the Drift

A hidden Markov Model (HMM) for the Drift

Example

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Example (1/3)

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p ( )

0 0.2 0.4 0.6 0.8 170

80

90

100

110

120

S

Time

0 0.2 0.4 0.6 0.8 1

0.4

0.2

0

0.2

0.4

BY

Time

We consider the HMM for the drift.

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Example ctd (2/3)

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p ( )

0 0.2 0.4 0.6 0.8 1

0.2

0.25

0.3

0.35

Time

0 0.2 0.4 0.6 0.8 1

2

1

0

1

2

ul

Time

For the volatility we consider the Hobson-Rogers model.

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Example ctd (3/3)

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0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

X

Time

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Numerical Results (1/2)

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We consider 20 stocks of the Dow Jones Industrial Index

We use daily prices (adjusted for dividends and splits) for 30 years, 19722001

Parameter estimates are based on five years with starting year 1972, 1973,...,

1996 using a Markov Chain Monte Carlo algorithm.

We apply the strategy in the subsequent year

we perform 500 experiments whose outcomes we average.

We consider LEL- and LES-constraint.

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Numerical Results ctd (2/2)

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U(XT) mean median st.dev. aborted

unconstrained

b&h 0.1188 0.1195 0.2297 0Merton 0.0248 0.0826 0.4815 2

GD -1.2002 -1.0000 0.9580 79

Bayes 0.0143 0.0824 0.5071 2

HMM -0.0346 0.0277 0.9247 13

LEL risk constraint (L=0.5%)

GD 0.0252 0.0294 0.1767 0

Bayes 0.1002 0.0988 0.1595 0

HMM 0.1285 0.1242 0.2004 0

LES risk constraint (L1=0.1%,L2=5%)

GD -0.0395 -0.0350 0.3086 0

Bayes 0.0950 0.0968 0.2752 0

HMM 0.1505 0.1402 0.3434 0

LEL and LES improve the performance of all models.

With LEL and LES we dont go bankrupt anymore.

The HMM strategy with risk constraints outperforms all other strategies.

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Conclusion & Outlook

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Conclusion

We show how to apply dynamic risk constraints usingtime-dependent convex

constraints.

We deriveexplicit trading strategieswith dynamic risk constraints under partial

information.

The numerical results indicate that dynamic risk constraintscan reduce the risk

and improve the performance.

Outlook

Allow for consumption.

More detailed analysis of the multidimensional case.

Explicit strategies for general utility.

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• 8/9/2019 Optimal Investment Under Dynamic Risk Constraints - Slides

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