Risk simulation with optimally stratified importance...
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Risk simulation withoptimally stratified importance sampling
W. Hörmann and I. BasogluBogazici University Istanbul
Industrial Engineering Department
June 12, 2014
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Simulation of Portfolio Risk Some Risk Measures
Some Risk Measures
Tail loss probability for given threshold level, τ :Pr(Loss ≥ τ) = E [1 {Loss ≥ τ}]
VaR(α): Value-at-risk, 1− α quantile of the loss distribution.
Conditional excess: E [Loss|Loss ≥ τ ]
Conditional value-at-risk: CVaR(α): E [Loss|Loss ≥ VaR(α)]
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Simulation of Portfolio Risk Some Risk Measures
Some Risk Measures
Tail loss probability for given threshold level, τ :Pr(Loss ≥ τ) = E [1 {Loss ≥ τ}]
VaR(α): Value-at-risk, 1− α quantile of the loss distribution.
Conditional excess: E [Loss|Loss ≥ τ ]
Conditional value-at-risk: CVaR(α): E [Loss|Loss ≥ VaR(α)]
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Simulation of Portfolio Risk Some Risk Measures
Some Risk Measures
Tail loss probability for given threshold level, τ :Pr(Loss ≥ τ) = E [1 {Loss ≥ τ}]
VaR(α): Value-at-risk, 1− α quantile of the loss distribution.
Conditional excess: E [Loss|Loss ≥ τ ]
Conditional value-at-risk: CVaR(α): E [Loss|Loss ≥ VaR(α)]
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Simulation of Portfolio Risk Some Risk Measures
Some Risk Measures
Tail loss probability for given threshold level, τ :Pr(Loss ≥ τ) = E [1 {Loss ≥ τ}]
VaR(α): Value-at-risk, 1− α quantile of the loss distribution.
Conditional excess: E [Loss|Loss ≥ τ ]
Conditional value-at-risk: CVaR(α): E [Loss|Loss ≥ VaR(α)]
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The Stock Portfolio Model
t-copula for modeling log-returns
For modeling dependence among log-returns the importance ofcopulae is stressed (Frey and McNeil, 2001).t-copula models have a good fit to the joint distribution of logreturns. (for example Mashal et al. 2003; Kole et al. 2007).
Return (T) =D∑
d=1
wdecd G−1d (Fν(Td ))
cd =
√σ2
d252
1vard
T = (T1, . . . ,TD)′ =LZ√Y /ν
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The Stock Portfolio Model Our Objectives
Our Objectives
For a linear asset portfolio of moderate size (2 to 10 assets):Efficient estimation of a single tail loss probability, Pr(Loss ≥ τ)
Efficient estimation of a single conditional excessE [Loss|Loss ≥ τ ].
Efficient estimation of multiple tail loss probabilities or multipleconditional excesses in a single simulation.(Important to calculate VaR and CVaR.)
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The Stock Portfolio Model Our Objectives
Our Objectives
For a linear asset portfolio of moderate size (2 to 10 assets):Efficient estimation of a single tail loss probability, Pr(Loss ≥ τ)
Efficient estimation of a single conditional excessE [Loss|Loss ≥ τ ].
Efficient estimation of multiple tail loss probabilities or multipleconditional excesses in a single simulation.(Important to calculate VaR and CVaR.)
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The Stock Portfolio Model Our Objectives
Our Objectives
For a linear asset portfolio of moderate size (2 to 10 assets):Efficient estimation of a single tail loss probability, Pr(Loss ≥ τ)
Efficient estimation of a single conditional excessE [Loss|Loss ≥ τ ].
Efficient estimation of multiple tail loss probabilities or multipleconditional excesses in a single simulation.(Important to calculate VaR and CVaR.)
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
General Principles of Monte Carlo Simulation:
Estimation of an expectation: x = Ef [q (X )]
X ∈ RD and X has density f (.),q : RD → R is the "simulation function",Ef[q2 (X )
]<∞.
x = Ef [q (X )] =
∫x∈RD
q (x)f (x) dx
The naive estimator: xNV = N−1∑N
n=1q (X n)
Generate iid sample X 1, . . . ,X N from density f (.).
Central Limit Theorem:xNV − xσ/N
→ N (0,1).
Error bound: xNV ± Φ−1 (α/2)σ/√
N.
To get more precise results: Variance Reduction MethodsLook for new simulation function q(.) with the same expectation andsmaller variance
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
![Page 27: Risk simulation with optimally stratified importance samplingstatmath.wu.ac.at/research/talks/resources/slideshoermann.pdf · Simulation of Portfolio Risk Some Risk Measures Some](https://reader034.fdocument.org/reader034/viewer/2022051815/603cbb7ba400cf74f42e767c/html5/thumbnails/27.jpg)
The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling (IS)
IS is a frequently used method for rare event situations
Ef [q (X )] =
∫x∈RD
q (x) f (x) dx
=
∫x∈RD
q (x)f (x)
fIS (x)fIS (x) dx
=
∫x∈RD
q (x) ρ (x) fIS (x) dx
= EfIS [q (X ) ρ (X )] .
Generate iid sample X 1, . . . ,X n from density fIS(.),
Evaluate xIS = N−1∑n
i=1q(
X i)ρ(
X i)
.
Regularity conditions are necessary to prove that the estimate isunbiased
6/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling cont.
V [xIS] is minimal for f ∗IS (x) =|q (x)| f (x)∫
x∈RD |q (x) | f (x) dxThat density is unknown for relevant applications.
In practice an IS density is typically taken from a parametric family(often the same as f (.)). The parameters are selected such thatthe IS density imitates |q (x) f (x)|.The cross entropy method is a general approach to select theparameters of the IS density.
Often the variance reduction reached with IS decreases fast withthe dimension of the problem.
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling cont.
V [xIS] is minimal for f ∗IS (x) =|q (x)| f (x)∫
x∈RD |q (x) | f (x) dxThat density is unknown for relevant applications.
In practice an IS density is typically taken from a parametric family(often the same as f (.)). The parameters are selected such thatthe IS density imitates |q (x) f (x)|.The cross entropy method is a general approach to select theparameters of the IS density.
Often the variance reduction reached with IS decreases fast withthe dimension of the problem.
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling cont.
V [xIS] is minimal for f ∗IS (x) =|q (x)| f (x)∫
x∈RD |q (x) | f (x) dxThat density is unknown for relevant applications.
In practice an IS density is typically taken from a parametric family(often the same as f (.)). The parameters are selected such thatthe IS density imitates |q (x) f (x)|.The cross entropy method is a general approach to select theparameters of the IS density.
Often the variance reduction reached with IS decreases fast withthe dimension of the problem.
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The Stock Portfolio Model Monte Carlo Simulation
Importance Sampling cont.
V [xIS] is minimal for f ∗IS (x) =|q (x)| f (x)∫
x∈RD |q (x) | f (x) dxThat density is unknown for relevant applications.
In practice an IS density is typically taken from a parametric family(often the same as f (.)). The parameters are selected such thatthe IS density imitates |q (x) f (x)|.The cross entropy method is a general approach to select theparameters of the IS density.
Often the variance reduction reached with IS decreases fast withthe dimension of the problem.
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The Stock Portfolio Model Monte Carlo Simulation
IS Algo for Tailloss Probabilities
Sak, WH and Leydold (2010) use IS for the iid normal input Z and thechi-square random variate Y .
Problem: Even for heuristic approach necessary to find a gooddirection for Z . It depends on the threshold τ . Thus a numericoptimization is required.Disadvantage of IS: Selection of the IS density and its parametersdifficult.
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The Stock Portfolio Model Monte Carlo Simulation
IS Algo for Tailloss Probabilities
Sak, WH and Leydold (2010) use IS for the iid normal input Z and thechi-square random variate Y .
Problem: Even for heuristic approach necessary to find a gooddirection for Z . It depends on the threshold τ . Thus a numericoptimization is required.Disadvantage of IS: Selection of the IS density and its parametersdifficult.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling
Let ξi , i = 1, . . . , I be a partition of RD into I strata,Assume pi = Pr {X ∈ ξi} are known for i = 1, . . . , I.Let Xi be the random vector that follows the conditional distributionof X given X ∈ ξi .
x = Ef [q (X )] =I∑
i=1
piEf [q (X ) |X ∈ ξi ] =I∑
i=1
piEf [q (Xi)]
The stratified estimator: xSTRS =∑I
i=1piN−1
i
∑Ni
n=1q (X n
i )
Ni replications in stratum i , N =∑I
i=1Ni
Generate iid sample X 1i , . . . ,X
Nii in each stratum.
How should we select Ni , the samplesize for each stratum?standard stratification uses proportional allocationpossible generalisation to QMC
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
10/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Sampling with optimal allocation
Define allocation fractions πi = Ni/N
xSTRS =I∑
i=1
piN−1i
Ni∑n=1
q (X ni ) = N−1
I∑i=1
piπ−1i
πi N∑n=1
q (X ni )
Let σ2i = V [q (X ) |X ∈ ξi ] = V [q (Xi)], then:
V [xSTRS] = N−1∑I
i=1p2
i σ2i π−1i ≥ N−1
(∑I
i=1piσi
)2
Optimal allocation fractions: π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
Use a pilot sample allocated proportional to pi ,Estimate conditional standard deviations σi ,Calculate optimal allocation fractions π∗ = (π1, . . . , πI)
′,Use optimal allocation in the main run.
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
11/ 33
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
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The Stock Portfolio Model Monte Carlo Simulation
Adaptive Optimal Allocation Algorithm
The Adaptive Optimal Allocation (AOA) algorithm terminates in Kiterations.The total sample size N is divided between iterations with anon-decreasing order (e.g., K = 3, 0.1N,0.4N,0.5N).In the first iteration, the sample is allocated proportional to stratumprobabilities pi .
In each iteration, the conditional standard deviations, σi ,i = 1, . . . , I, are estimated using all previous drawings.The estimates σi determines the allocation fractions for the nextiteration.
Étoré and Jourdain (2010) show that the stratified estimator ofAOA is unbiased and asymptotically normal.
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The Stock Portfolio Model Monte Carlo Simulation
IS vs Optimal Allocation Stratification (OAS)
OAS can be interpreted as IS using the product of the originaldensity and a step function as weight function.Advantage of OAS is the simple formula for the optimal allocationfractions.High dimensional stratification not possible in practice. Thus (likefor IS) one (or two) main directions are used for most applications.Disadvantage of OAS: Many strata (or an adaptive stratastructure) are necessary for rare event simulations.Is it possible and sensble to combine IS and OAS to increase thevariance reduction?
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The Stock Portfolio Model Monte Carlo Simulation
IS vs Optimal Allocation Stratification (OAS)
OAS can be interpreted as IS using the product of the originaldensity and a step function as weight function.Advantage of OAS is the simple formula for the optimal allocationfractions.High dimensional stratification not possible in practice. Thus (likefor IS) one (or two) main directions are used for most applications.Disadvantage of OAS: Many strata (or an adaptive stratastructure) are necessary for rare event simulations.Is it possible and sensble to combine IS and OAS to increase thevariance reduction?
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The Stock Portfolio Model Monte Carlo Simulation
IS vs Optimal Allocation Stratification (OAS)
OAS can be interpreted as IS using the product of the originaldensity and a step function as weight function.Advantage of OAS is the simple formula for the optimal allocationfractions.High dimensional stratification not possible in practice. Thus (likefor IS) one (or two) main directions are used for most applications.Disadvantage of OAS: Many strata (or an adaptive stratastructure) are necessary for rare event simulations.Is it possible and sensble to combine IS and OAS to increase thevariance reduction?
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The Stock Portfolio Model Monte Carlo Simulation
IS vs Optimal Allocation Stratification (OAS)
OAS can be interpreted as IS using the product of the originaldensity and a step function as weight function.Advantage of OAS is the simple formula for the optimal allocationfractions.High dimensional stratification not possible in practice. Thus (likefor IS) one (or two) main directions are used for most applications.Disadvantage of OAS: Many strata (or an adaptive stratastructure) are necessary for rare event simulations.Is it possible and sensble to combine IS and OAS to increase thevariance reduction?
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The Stock Portfolio Model Monte Carlo Simulation
IS vs Optimal Allocation Stratification (OAS)
OAS can be interpreted as IS using the product of the originaldensity and a step function as weight function.Advantage of OAS is the simple formula for the optimal allocationfractions.High dimensional stratification not possible in practice. Thus (likefor IS) one (or two) main directions are used for most applications.Disadvantage of OAS: Many strata (or an adaptive stratastructure) are necessary for rare event simulations.Is it possible and sensble to combine IS and OAS to increase thevariance reduction?
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The Stock Portfolio Model Monte Carlo Simulation
Stratified Importance Sampling: Single Estimate Case
Stratified Importance Sampling (SIS): Applies OAS to an ISalgorithm.Numerical results for IS of (Sak et al. 2010) and SIS for t-copulamodel with generalized hyperbolic marginals (parametersestimated from NYSE data)
P(R < t) ≈ 0.05 P(R < t) ≈ 0.001IS SIS IS SIS
d VR TM ER VR TM ER VR TM ER VR TM ER2 6.1 0.23 6.5 296.5 0.30 243.5 183.5 0.21 201.7 4741.2 0.28 4046.95 8.4 0.79 8.4 110.3 0.90 96.8 278.5 0.79 247.3 3495.7 0.91 2684.9
10 5.3 1.59 5.0 11.4 1.75 9.8 66.6 1.64 60.5 198.3 1.91 154.1
Variance reduction factors: VR (x) = V [xNV ]/V [x ]Efficiency ratios: ER (x) = VR (x) TM [xNV ]/TM [x ]TM exeuction time, N ≈ 100,000 for all simulations.
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The Stock Portfolio Model Monte Carlo Simulation
Why works combination of IS and OAS so well?
IS and OAS use the same direction and are very similar methods.
We demonstrate their synergy effects for a one-dimensionalexample
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The Stock Portfolio Model Monte Carlo Simulation
simple example IS
y = Eφ [q (Z )], q (x) = {ex − 3.6}+ and Z ∼ N(0,1).
The original density φ (x),
the shifted IS density fIS (x) and
−2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Input Domain
Den
sity
Val
ues
φ(x) fIS(x)
fIS* (x) the optimal IS density f ∗IS (x).
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The Stock Portfolio Model Monte Carlo Simulation
simple example with OAS
IS density corresponding to OAS with 100 and 500 strata.
−2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(b)
Input Domain
Den
sity
Val
ues
φ(x) fIS* (x)
fOA(x)
−2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(c)
Input Domain
Den
sity
Val
ues
φ(x) fIS* (x)
fOA(x)
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The Stock Portfolio Model Monte Carlo Simulation
simple example with SIS
IS density corresponding to SIS with 10 and 25 strata.
−2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(b)
Input Domain
Den
sity
Val
ues
fIS(x) fIS* (x)
fOA(x)
−2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(c)
Input Domain
Den
sity
Val
ues
fIS(x) fIS* (x)
fOA(x)
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The Stock Portfolio Model Monte Carlo Simulation
simple example: Variance Reduction Factors
Table: Comparison of Naive, IS, OAS 1000, and SIS 100.exact solution 0.2815896024 .
Estimate Variance VRFNaive 0.27970 2.17E-05 1
IS 0.28192 3.77E-07 58OAS 1000 0.28155 2.73E-09 7950SIS 100 0.28159 8.53E-11 2.5e5
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The Stock Portfolio Model Monte Carlo Simulation
Advantages of combining IS and OAS
We have observed the following advantages when combining IS andOAS for risk simulations:
IS helps that there are a smaller number of strata with return 0.Thus a smaller number of stratification intervals still leads tosubstantial variance reduction.IS helps stratification to obtain better estimates for the variancesin the strata and thus better allocation fractions.
NEW IDEAStratification can help to estimate many tailloss probabilities fordifferent thresholds τj in a single simulation.That is important when estimating VaR and CVaR.
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The Stock Portfolio Model Monte Carlo Simulation
Advantages of combining IS and OAS
We have observed the following advantages when combining IS andOAS for risk simulations:
IS helps that there are a smaller number of strata with return 0.Thus a smaller number of stratification intervals still leads tosubstantial variance reduction.IS helps stratification to obtain better estimates for the variancesin the strata and thus better allocation fractions.
NEW IDEAStratification can help to estimate many tailloss probabilities fordifferent thresholds τj in a single simulation.That is important when estimating VaR and CVaR.
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The Stock Portfolio Model Monte Carlo Simulation
Advantages of combining IS and OAS
We have observed the following advantages when combining IS andOAS for risk simulations:
IS helps that there are a smaller number of strata with return 0.Thus a smaller number of stratification intervals still leads tosubstantial variance reduction.IS helps stratification to obtain better estimates for the variancesin the strata and thus better allocation fractions.
NEW IDEAStratification can help to estimate many tailloss probabilities fordifferent thresholds τj in a single simulation.That is important when estimating VaR and CVaR.
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The Stock Portfolio Model Monte Carlo Simulation
Advantages of combining IS and OAS
We have observed the following advantages when combining IS andOAS for risk simulations:
IS helps that there are a smaller number of strata with return 0.Thus a smaller number of stratification intervals still leads tosubstantial variance reduction.IS helps stratification to obtain better estimates for the variancesin the strata and thus better allocation fractions.
NEW IDEAStratification can help to estimate many tailloss probabilities fordifferent thresholds τj in a single simulation.That is important when estimating VaR and CVaR.
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Multiple Estimates Case
Suppose we are interested in probability estimates for distinctthreshold values, τj , j = 1, . . . , J.We estimate the j-th tail loss probability and its relative error withthe following formula:
xj =I∑
i=1
pi xij , RE[xj]
=Φ−1 (0.975)
xj√
N
√√√√ I∑i=1
p2i σ
2ij
πi
General questionsHow should we define the "overall error" of our J estimationproblems?How can we minimize that "overall error"?
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Multiple Estimates Case
Suppose we are interested in probability estimates for distinctthreshold values, τj , j = 1, . . . , J.We estimate the j-th tail loss probability and its relative error withthe following formula:
xj =I∑
i=1
pi xij , RE[xj]
=Φ−1 (0.975)
xj√
N
√√√√ I∑i=1
p2i σ
2ij
πi
General questionsHow should we define the "overall error" of our J estimationproblems?How can we minimize that "overall error"?
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Multiple Estimates Case
Suppose we are interested in probability estimates for distinctthreshold values, τj , j = 1, . . . , J.We estimate the j-th tail loss probability and its relative error withthe following formula:
xj =I∑
i=1
pi xij , RE[xj]
=Φ−1 (0.975)
xj√
N
√√√√ I∑i=1
p2i σ
2ij
πi
General questionsHow should we define the "overall error" of our J estimationproblems?How can we minimize that "overall error"?
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Multiple Estimates Case
Suppose we are interested in probability estimates for distinctthreshold values, τj , j = 1, . . . , J.We estimate the j-th tail loss probability and its relative error withthe following formula:
xj =I∑
i=1
pi xij , RE[xj]
=Φ−1 (0.975)
xj√
N
√√√√ I∑i=1
p2i σ
2ij
πi
General questionsHow should we define the "overall error" of our J estimationproblems?How can we minimize that "overall error"?
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Multiple Estimates Case
Suppose we are interested in probability estimates for distinctthreshold values, τj , j = 1, . . . , J.We estimate the j-th tail loss probability and its relative error withthe following formula:
xj =I∑
i=1
pi xij , RE[xj]
=Φ−1 (0.975)
xj√
N
√√√√ I∑i=1
p2i σ
2ij
πi
General questionsHow should we define the "overall error" of our J estimationproblems?How can we minimize that "overall error"?
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Definition of Overall Error
xij = Prob(Loss > τj |X ∈ ξi)
xj = Prob(Loss > τj) =∑I
i=1pi xij
sjki = Cov(xij , xik ), j , k = 1, . . . , J
For the vector x the variance-covariance matrix Σ depends on theallocation fractions:
Σjk (π) = N−1I∑
i=1
π−1i p2
i sjki .
We define an overall error function:
ω(π) = g(Σ(π)) .
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Relevant Overall Error functions
ωMSE (π) =∑J
j=1Σjj(π), the mean squared error of all estimates,
ωMSR(π) =∑J
j=1x−2
j Σjj (π), the mean squared relative error of
all estimates,ωMAXE (π) = max{j : Σjj (π)}, the maximum of the squared errorsof all estimates,ωMAXR(π) = max{j : x−2
j Σjj (π)}, the maximum of the squaredrelative errors of all estimates,
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Relevant Overall Error functions
ωMSE (π) =∑J
j=1Σjj(π), the mean squared error of all estimates,
ωMSR(π) =∑J
j=1x−2
j Σjj (π), the mean squared relative error of
all estimates,ωMAXE (π) = max{j : Σjj (π)}, the maximum of the squared errorsof all estimates,ωMAXR(π) = max{j : x−2
j Σjj (π)}, the maximum of the squaredrelative errors of all estimates,
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Relevant Overall Error functions
ωMSE (π) =∑J
j=1Σjj(π), the mean squared error of all estimates,
ωMSR(π) =∑J
j=1x−2
j Σjj (π), the mean squared relative error of
all estimates,ωMAXE (π) = max{j : Σjj (π)}, the maximum of the squared errorsof all estimates,ωMAXR(π) = max{j : x−2
j Σjj (π)}, the maximum of the squaredrelative errors of all estimates,
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Relevant Overall Error functions
ωMSE (π) =∑J
j=1Σjj(π), the mean squared error of all estimates,
ωMSR(π) =∑J
j=1x−2
j Σjj (π), the mean squared relative error of
all estimates,ωMAXE (π) = max{j : Σjj (π)}, the maximum of the squared errorsof all estimates,ωMAXR(π) = max{j : x−2
j Σjj (π)}, the maximum of the squaredrelative errors of all estimates,
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
mean squared relative error
Using the variances defined above we get:
ωMSR(π) =J∑
j=1
x−2j Σjj (π) = N−1
I∑i=1
π−1i p2
i
J∑j=1
x−2j sjj
i ,
We know from AOS that for a single estimate:
V[xj]
= N−1∑I
i=1π−1
i p2i σ
2i ≥ N−1
(∑I
i=1piσi
)2
.
It attains its lower bound for π∗i =piσi∑I
k=1 pkσk, i = 1, . . . , I.
We can see that ωMSR(π) has the same structure as V[xj].
Replacing σ2i by
J∑j=1
x−2j sjj
i we thus can minimize ωMSR(π).
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
mean squared relative error, cont.
ωMSR(π) ≥ N−1
I∑i=1
pi
J∑j=1
x−2j sjj
i
1/2
2
ωMSR(π) attains its lower bound selecting
π∗i = pi
J∑j=1
x−2j sjj
i
1/2/I∑
l=1
pl
J∑j=1
x−2j sjj
i
1/2
, i = 1, . . . , I.
This idea and the closed form solution can be generalized to all ω(π)
that are linear functions of the sjki .
(Theorem requires non-negativity condition. Assuming positivecorrelations is no problem for applications to simulation.)
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Maximal relative error: Optimization Model
New objective: Minimize the maximum relative error using thedecision variables π = (π1, . . . , πI)
′.We denote aij = x−2
j p2i σ
2ij and add constraints which guarantee
that the πi are positive and sum to one.
min max
{j :
I∑i=1
aij
πi
}
s.t.I∑
i=1
πi = 1,
πi > 0, i = 1, . . . , I.
⇒
min ω
s.t. ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
I∑i=1
πi = 1,
πi > 0, i = 1, . . . , I.
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Maximal relative error: Optimization Model
New objective: Minimize the maximum relative error using thedecision variables π = (π1, . . . , πI)
′.We denote aij = x−2
j p2i σ
2ij and add constraints which guarantee
that the πi are positive and sum to one.
min max
{j :
I∑i=1
aij
πi
}
s.t.I∑
i=1
πi = 1,
πi > 0, i = 1, . . . , I.
⇒
min ω
s.t. ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
I∑i=1
πi = 1,
πi > 0, i = 1, . . . , I.
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Maximal relative error: Optimization Model
New objective: Minimize the maximum relative error using thedecision variables π = (π1, . . . , πI)
′.We denote aij = x−2
j p2i σ
2ij and add constraints which guarantee
that the πi are positive and sum to one.
min max
{j :
I∑i=1
aij
πi
}
s.t.I∑
i=1
πi = 1,
πi > 0, i = 1, . . . , I.
⇒
min ω
s.t. ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
I∑i=1
πi = 1,
πi > 0, i = 1, . . . , I.
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Stratification for Multiple Tail Loss Probabilities Optimization Model for Multiple Estimates
Maximal relative error: Optimization Model
New objective: Minimize the maximum relative error using thedecision variables π = (π1, . . . , πI)
′.We denote aij = x−2
j p2i σ
2ij and add constraints which guarantee
that the πi are positive and sum to one.
min max
{j :
I∑i=1
aij
πi
}
s.t.I∑
i=1
πi = 1,
πi > 0, i = 1, . . . , I.
⇒
min ω
s.t. ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
I∑i=1
πi = 1,
πi > 0, i = 1, . . . , I.
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Stratification for Multiple Tail Loss Probabilities The Allocation Heuristic
The Allocation Heuristic
For the previous optimization model, we have developed aheuristic which searches for a suboptimal solution in the convexhull of the respective optimal solutions πj , j = 1, . . . , J.
min ω
s.t . ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
πi −J∑
j=1
λjπji = 0, i = 1, . . . , I,
J∑j=1
λj = 1,
λj ≥ 0, j = 1, . . . , J.
The heuristic method terminates with an average 2 percentsub-optimality for our simulation instances.
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Stratification for Multiple Tail Loss Probabilities The Allocation Heuristic
The Allocation Heuristic
For the previous optimization model, we have developed aheuristic which searches for a suboptimal solution in the convexhull of the respective optimal solutions πj , j = 1, . . . , J.
min ω
s.t . ω −I∑
i=1
aij
πi≥ 0, j = 1, . . . , J,
πi −J∑
j=1
λjπji = 0, i = 1, . . . , I,
J∑j=1
λj = 1,
λj ≥ 0, j = 1, . . . , J.
The heuristic method terminates with an average 2 percentsub-optimality for our simulation instances.
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Numerical Results
Numerical Results
We consider D = 5 stocks under the t-copula model withGeneralized Hyperbolic marginals and J = 10 equidistantthreshold values.For IS we use a mixture of two densities.For SIS, we simply used a single IS density selected for thethreshold τ∗ = 0.75τmax + 0.25τmin.The parameters of the distributions were estimated from NYSEdata.
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Numerical Results
Numerical Results
We consider D = 5 stocks under the t-copula model withGeneralized Hyperbolic marginals and J = 10 equidistantthreshold values.For IS we use a mixture of two densities.For SIS, we simply used a single IS density selected for thethreshold τ∗ = 0.75τmax + 0.25τmin.The parameters of the distributions were estimated from NYSEdata.
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Numerical Results
Numerical Results
We consider D = 5 stocks under the t-copula model withGeneralized Hyperbolic marginals and J = 10 equidistantthreshold values.For IS we use a mixture of two densities.For SIS, we simply used a single IS density selected for thethreshold τ∗ = 0.75τmax + 0.25τmin.The parameters of the distributions were estimated from NYSEdata.
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Numerical Results
Numerical Results
We consider D = 5 stocks under the t-copula model withGeneralized Hyperbolic marginals and J = 10 equidistantthreshold values.For IS we use a mixture of two densities.For SIS, we simply used a single IS density selected for thethreshold τ∗ = 0.75τmax + 0.25τmin.The parameters of the distributions were estimated from NYSEdata.
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Numerical Results
Numerical Results cont.
Following table shows the percentage relative errors and thevariance reduction factors obtained under NV, IS and SIS. Theexecution times are 0.75, 0.95 and 1.25 seconds respectively.
NV IS SISτ Pr (r(T) ≤ τ) RE RE VR RE VR0.895 0.0010 ±19.89% ±2.65% 56.3 ±0.77% 667.20.909 0.0014 ±16.67% ±2.37% 49.5 ±0.70% 567.10.917 0.0020 ±14.13% ±2.12% 44.4 ±0.69% 419.40.924 0.0028 ±12.09% ±1.89% 40.9 ±0.68% 316.10.932 0.0040 ±9.89% ±1.77% 31.2 ±0.66% 224.50.939 0.0060 ±7.96% ±1.72% 21.4 ±0.69% 133.10.947 0.0094 ±6.46% ±1.82% 12.6 ±0.66% 95.80.954 0.0154 ±4.91% ±2.14% 5.3 ±0.65% 57.10.962 0.0267 ±3.75% ±2.26% 2.8 ±0.64% 34.30.973 0.0500 ±2.69% ±9.23% 0.1 ±0.69% 15.2
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Numerical Results
Estimating Conditional Excess
Simulating the conditional excess: E [Loss|Loss ≥ τ ] requires aratio estimate; this makes variance reduction more difficult.Literature: "Use the same IS density as for tail-loss probabilities."The variance of the ratio estimate:V [x1/x2] ≈ x2
1 x−42 Σ22 (π)− 2x1x−3
2 Σ12 (π) + x−22 Σ11 (π) = ω (π) .
To reach optimal allocation for stratification we can use thetheorem above to minimize the variance of a ratio estimate.
For Conditional Excess we obtained stratification and stratified ISalgorithms with optimal allocation for:
a single thresholdfor several thresholds minimizing the mean squared relative error.
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Conclusions & Future Work
Conclusions
For practically relevant examples SIS (combination of IS andstratification) increases the efficiency of tail loss probabilityestimates under the t-copula model.Compared to the methods in the literature, the variance of theestimates are substantially reduced without a significant increasein the execution time.The SIS method allows the efficient estimation of tail lossprobabilities for multiple threshold values in a single simulation.
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Conclusions & Future Work
Conclusions
For practically relevant examples SIS (combination of IS andstratification) increases the efficiency of tail loss probabilityestimates under the t-copula model.Compared to the methods in the literature, the variance of theestimates are substantially reduced without a significant increasein the execution time.The SIS method allows the efficient estimation of tail lossprobabilities for multiple threshold values in a single simulation.
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Conclusions & Future Work
Conclusions
For practically relevant examples SIS (combination of IS andstratification) increases the efficiency of tail loss probabilityestimates under the t-copula model.Compared to the methods in the literature, the variance of theestimates are substantially reduced without a significant increasein the execution time.The SIS method allows the efficient estimation of tail lossprobabilities for multiple threshold values in a single simulation.
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Conclusions & Future Work
Conclusions
The mean square (relative) error of all estimates can be minimizedusing the variance estimates of a pilot run and a simple closedform formula for the allocation fractions.To minimize the maximal squared (relative) error of all estimateswe have developed a fast and simple heuristic to find close tooptimal allocation fractions.The idea to obtain optimal variance reduction for several estimatesin a single simulation using optimal allocation stratification (OAS)can be used for all simulation problems for which OAS leads goodvariance reduction.
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Conclusions & Future Work
Conclusions
The mean square (relative) error of all estimates can be minimizedusing the variance estimates of a pilot run and a simple closedform formula for the allocation fractions.To minimize the maximal squared (relative) error of all estimateswe have developed a fast and simple heuristic to find close tooptimal allocation fractions.The idea to obtain optimal variance reduction for several estimatesin a single simulation using optimal allocation stratification (OAS)can be used for all simulation problems for which OAS leads goodvariance reduction.
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Conclusions & Future Work
Conclusions
The mean square (relative) error of all estimates can be minimizedusing the variance estimates of a pilot run and a simple closedform formula for the allocation fractions.To minimize the maximal squared (relative) error of all estimateswe have developed a fast and simple heuristic to find close tooptimal allocation fractions.The idea to obtain optimal variance reduction for several estimatesin a single simulation using optimal allocation stratification (OAS)can be used for all simulation problems for which OAS leads goodvariance reduction.
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Conclusions & Future Work
Questions?
THANK YOU !
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Conclusions & Future Work
References
Basoglu, I., W. Hörmann, and H. Sak. 2013.“Optimally Stratified Importance Sampling for Portfolio Risk withMultiple Loss Thresholds”.Optimization 62 (11): 1451–1471.
Basoglu, I. and W. Hörmann, 2014.“Efficient stratified sampling implementations in multiresponsesimulation”.submitted to Winter Simulation Conference 2014.
Etoré, P., and B. Jourdain. 2010.“Adaptive Optimal Allocation in Stratified Sampling Methods”.Methodology and Computing in Applied Probability 12 (3):335–360.
Sak, H., and W. Hörmann. 2012.“Fast Simulations in Credit Risk”.Quantitative Finance 12 (10): 1557–1569.
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