Primbs 1 Receding Horizon Control Receding Horizon Control for Constrained for Constrained Portfolio Optimization Portfolio Optimization James A. Primbs Management Science and Engineering Stanford University (with Chang Hwan Sung) Stanford-Tsukuba/WCQF Workshop Stanford University March 8, 2007
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Primbs1 Receding Horizon Control for Constrained Portfolio Optimization James A. Primbs Management Science and Engineering Stanford University (with Chang.
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Primbs 1
Receding Horizon Control for Receding Horizon Control for Constrained Portfolio OptimizationConstrained Portfolio Optimization
A Motivational Problem from Finance: Index Tracking
Let ni ...1)())(1()1( kSkwkS iiiii represent the stochastic dynamics of the prices of n stocks, where i is the expected return per period, i is the volatility per period, and wi is an iid noise term with mean zero and standard deviation 1.
n
iii kSkI
1
)()(
An index is a weighted average of the n stocks:
The index tracking problem is to trade l<n of the stocks and a risk free bond in order to track the index as “closely as possible”.
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A Motivational Problem from Finance: Index Tracking
)()()()()1()1(1
kukwrkWrkW i
l
iiifif
If we let ui(k) denote the dollar amount invested in Si(k) for l<n, and we let rf denote the risk free rate of interest, then the total wealth W(k) of this strategy follows the dynamics:
0
22 ))()((mink
k
ukIkWE
One possible measure of how closely we track the index is given by an infinite horizon discounted quadratic cost:
where <1 is a discount factor.
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A Motivational Problem from Finance: Index Tracking
Limits on short selling: 0)( kui
Limits on wealth invested: )()( kWku ii
Value-at-Risk: 1))()(Pr( kIkW
etc...
Finally, it is quite common to require that constraints be satisfied, such as
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Index Tracking
subject to:
ni ...1
0
22 ))()((mink
k
ukIkWE
)())(1()1( kSkwkS iiiii
)()()()()1()1(1
kukwrkWrkW i
l
iiifif
n
iii kSkI
1
)()(
nl
10)()()(11
n
iii
l
iii kSckubkaWP
Primbs 7
Linear systems with State and Control Multiplicative Noise:
0 )(
)(
)(
)(min
k
T
u ku
kxM
ku
kxE
subject to:
q
iiii kwkuDkxCkBukAxkx
1
)()()()()()1(
dckubkxaP TT 1))()((
00
0
R
QM 0Qwhere
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The unconstrained version of the problem is known as the Stochastic Linear Quadratic (SLQ) problem.
It’s solution has been well studied...
Willems and Willems, ‘76Yao et. al., ‘01El Ghaoui, ’95Ait Rami and Zhou, ’00McLane, ’71Wonham, ’67, ‘68Kleinman, ’69
The constrained version has received less attention...
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Receding Horizon Control (also known as Model Predictive Control) has been quite successful for constrained deterministic systems.
Can receding horizon control be a successful tool for (constrained) stochastic control as well?
Dynamic resource allocation: (Castanon and Wohletz, ’02)Portfolio Optimization: (Herzog et al, ’06, Herzog ‘06)Dynamic Hedging: (Meindl and Primbs, ’04)Supply Chains: (Seferlis and Giannelos, ’04)Constrained Linear Systems: (van Hessem and Bosgra, ’01,’02,’03,’04)Stoch. Programming Approach: (Felt, ’03)Operations Management: (Chand, Hsu, and Sethi, ’02)
By exploiting and imposing problem structure, constrained stochastic receding horizon control can be implemented in a computationally tractable manner for a number of problems.
It appears to be a promising approach to solving many constrained portfolio optimization problems.
Future Research
There are many interesting theoretical questions involving properties of this approach, especially concerning stability and performance.
We are pursuing applications to other portfolio optimization problems, dynamic hedging, and coupled with filtering methods.