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Dynamic Programming for Portfolio Problems
Kenneth L. Judd, Hoover InstitutionYongyang Cai, Hoover Institution
May 27, 2012
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Proportional Transaction Cost and CRRA Utility
I Separability of wealth W and portfolio fractions x .I If u(W ) = W 1−γ/(1− γ), then Vt(Wt , xt) = W 1−γ
t · gt(xt).I If u(W ) = log(W ), then Vt(Wt , xt) = log(Wt) + ψt(xt).I “No-trade” region: Ωt = xt : (δ+t )∗ = (δ−t )∗ = 0, where
(δ+t )∗ ≥ 0 are fractions of wealth for buying stocks, and (δ−t )∗ ≥ 0are fractions of wealth for selling stocks.
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Examples with two stocks
We first look at simple examples:I Two stocks and one bondI Investment begins at t = 0 and is liquidated at t = 6
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i.i.d. returns (uniform distribution on [0.87,1.27])
0.5 0.55 0.6 0.65 0.7 0.75 0.80.5
0.55
0.6
0.65
0.7
0.75
0.8
Stock 1 fraction
Sto
ck 2
fra
ctio
nNo−trade region at stage t=0
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0.5 0.55 0.6 0.65 0.7 0.75 0.80.5
0.55
0.6
0.65
0.7
0.75
0.8
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=3
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0.5 0.55 0.6 0.65 0.7 0.75 0.80.5
0.55
0.6
0.65
0.7
0.75
0.8
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=5
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Correlated returns (discrete)
I Return: R1 = 0.85, 1.08, 1.25; R2 = 0.88, 1.06, 1.2;
I Joint Probability Matrix:
0.08 0.10.12 0.4 0.12
0.1 0.08
0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=0
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0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=3
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0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=5
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Stochastic mean return
I Log-normal return: N(µi − σ2
i /2, σ2i
)with µi = 0.06 or 0.08 and
σi = 0.2, for i = 1, 2
I Transition probability matrix of µ:[
0.75 0.250.25 0.75
]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=0
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=3
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Stock 1 fraction
Sto
ck 2
fra
ctio
n
No−trade region at stage t=5
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Three stocks
I Problem: 3 stocks + 1 bond.I Investment begins at t = 0 and is liquidated at t = 6I Log-normal return: N
(µ− σ2/2, ΛΣΛ
)with σ = (0.16, 0.18, 0.2)>,
µ = (0.07, 0.08, 0.09)>, Λ is the diagonal matrix of σ, and
Σ =
1 0.2 0.10.2 1 0.3140.1 0.314 1
.I degree-7 complete Chebyshev approximationI product Gauss-Hermite quadrature rule with 9 nodes in each
dimension
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0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
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0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
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0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
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Portfolio with Transaction Costs and Consumption
I Problem: 3 stocks + 1 bond, 50 periodsI No-trade regions:
Ω49 ≈ [0.16, 0.33]3,Ω48 ≈ [0.14, 0.24]3, . . . ,
Ωt ≈ [0.19, 0.27]3, for t = 0, . . . , 15,
I Merton’s ratio: x∗ = (ΛΣΛ)−1(µ− r)/γ = (0.25, 0.25, 0.25)>
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Σ12 = Σ13 = 0.2, Σ23 = 0.04
Σ12 = Σ13 = 0.2, Σ23 = 0.04
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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More correlation: Σ12 = Σ13 = 0.4, Σ23 = 0.16
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
0.0
0.1
0.2
0.3
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Portfolios with options
I Problem: 1 stock, 1 put option on the stock, and 1 bondI Investment begins at t = 0 and is liquidated at t = 6 monthsI Proportional transaction costs in stock and option tradesI Return of option is based on pricing of the option.
I We use the binomial lattice method to price the optionI Stock price will be one exogenous state variable
I Separability of wealth W and portfolio fractions x and stock price SI If u(W ) = W 1−γ/(1 − γ), then Vt(Wt , St , xt) = W 1−γ
t · gt(St , xt).I If u(W ) = log(W ), then Vt(Wt ,St , xt) = log(Wt) + ψt(St , xt).
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1 stock and 1 at-the-money put option at t = 0 (liquidate at t = 6months)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1No−Trade Region, Option Price=0.0462351K
Stock fraction
Option fra
ction
I Put option: strike K , expiration time T , payoff max(K − ST , 0)
I stock price S , utility u(W ) = −W−2/2
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Dependence on transaction costs1 at-the-money put option at t = 0
transaction cost ratios: τ1 = 0.01, τ2 = 0.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Stock fraction
Option fra
ction
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1 at-the-money put option at t = 0transaction cost ratios: τ1 = 0.01, τ2 = 0.005
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Stock fraction
Option fra
ction
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1 at-the-money put option at t = 0 (liquidate at t = 6)transaction cost ratios: τ1 = 0.005, τ2 = 0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Stock fraction
Option fra
ction
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1 at-the-money put option at t = 0transaction cost ratios: τ1 = 0.005, τ2 = 0.005
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Stock fraction
Option fra
ction
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Dependence on horizon1 at-the-money put option (τ1 = 0.01, τ2 = 0.01)
expiration time: T = 1 month
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Stock fraction
Option fra
ction
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1 at-the-money put option (τ1 = 0.01, τ2 = 0.01)expiration time: T = 2 months
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Stock fraction
Option fra
ction
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1 at-the-money put option (τ1 = 0.01, τ2 = 0.01)expiration time: T = 3 months
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Stock fraction
Option fra
ction
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1 at-the-money put option (τ1 = 0.01, τ2 = 0.01)expiration time: T = 4 months
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Stock fraction
Option fra
ction
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1 at-the-money put option (τ1 = 0.01, τ2 = 0.01)expiration time: T = 5 months
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Stock fraction
Option fra
ction
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Dependence on strike price (K = 0.8S0)
1 put option at t = 0 (liquidate at t = 6, τ1 = τ2 = 0.01)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02No−Trade Region, Option Price=0.00266625K
Stock fraction
Option fra
ction
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K = 1.2S0
1 put option at t = 0 (liquidate at t = 6, τ1 = τ2 = 0.01)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2No−Trade Region, Option Price=0.154645K
Stock fraction
Option fra
ction
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Social value of optionsValue functions with/without options at t = 0 (liquidate at t = 6)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.483
−0.482
−0.481
−0.48
−0.479
−0.478
−0.477
−0.476
−0.475
−0.474
Value Functions V0(W,S,x,y) at W=1, S=1 and y=0
Stock fraction x
Valu
e
Option unavailableOption available, τ
2=0.02
Option available, τ2=0.01
Option available, τ2=0.005
I (x , y): fractions of money in stock and optionI τ1 = 0.01 and τ2: transaction cost ratios of stock and option
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Summary
I Developed a NDP method with shape-preservingapproximation, stabler
I Developed a NDP method with Hermiteinterpolation, more accurate and more time-saving
I Developed a parallel NDP algorithm, running overhundreds of computers, almost linear speed-up
I Solved arbitrary-number-of-period and many-assetdynamic portfolio problems with transaction costs
I Solved arbitrary-number-of-period dynamicportfolio problems with options and transactioncosts