Optimal Investment Under Dynamic Risk Constraints - Slides

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  • 8/9/2019 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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