optimal investment under dynamic risk constraints - slides
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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
Trading strategy t: n-dimensional, FS-adapted, measurable
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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