Scenario Generation Algortihm using R

Vu Pham
Scenario Generation Algorithm using R
(without optimization)
26.12.2016 Scenario Generation Algorithm using R 1Ашу Пракаш | Семенов Михаил Евгеньевич
Ashu Prakash
Indian Institute of Technology Kanpur, India
ashupk@iitk.ac.in
Dr Mikhail Semenov
Tomsk Polytechnic University, Russia
sme@tpu.ru

Vu Pham
I’m a third-year undergraduate student majoring in 4 year bachelorship
programme of Mathematics and Scientific Computing in the Department of
Mathematics and Statistics at the Indian Institute of Technology Kanpur,
India
My research area includes parts of Applied Mathematics, Finance,
Microeconomics and Business Analytics. I’m also interested in Computer
Programming and Multivariable Calculus.
I worked as an intern under Prof Mikhail Semenov, Tomsk Polytechnic
University, Russia. My project was to generate scenarios for stock markets
using different techniques (mainly without using optimisation)
A little about myself

Vu Pham
I worked on an algorithm for moment-matching scenario generation. I tried
to produce scenarios and probability weights that match different moments.
In this approach, the statistical properties of the joint distribution are
specified in terms of moments, basically including the covariance matrix.
Optimisation is not employed in this scenario generation process and thus
this method is computationally more advantageous than previous
approaches. The algorithm is used for generating scenarios in a mean-CVaR
portfolio optimisation model.
This method is proposed in the paper "An algorithm for moment-matching
scenario generation with application to financial portfolio optimisation" by
Ponomareva, Roman, Date, (Published 2015)
Scenario Generation

Vu Pham
I also worked on the paper “Generating Scenario Trees for Multistage
Decision Problems” by Høyland and Wallace, (Published 2001)
This paper deals with Tree Scenario Generation for multistage decision
problems with approximating the uncertainties by a limited number of
discrete outcomes.
The motivation for developing the method presented in this paper is the
implementation of a stochastic multistage asset allocation model.
There is a different paper which is actually a comment on this paper, by
Klaassen. "Comment on Generating Scenario Trees for Multistage Decision
Problems" by Pieter Klaassen, (Published 2002). This deals with removal of
arbitrage opportunities.
Scenario Generation

Vu Pham
I have used the programming language R for all the statistical coding for the
generation of proposed algorithms.
My code followed the paper "An algorithm for moment-matching scenario
generation with application to financial portfolio optimisation" by
Ponomareva, Roman, Date, (Published 2015).
I tried to generate probabilities associated with the moment-matching
scenario generation. Mean, Variance, Skewness, Kurtosis are matched in the
proposed algorithm.
The input of the algorithm follows a covariance matrix, mean vector,
skewness and kurtosis. The aim was to generate a vector of positive
probabilities.
Scenario Generation Algorithm

Vu Pham
Let us assume we have an N – variate random vector r = (r1, r2, r3, … , rN)T
We know the distribution of this random vector in terms of following:
µ : target mean vector
∑ : target covariance matrix (positive definite)
kj : marginal third central moment (skewness) of rj, j = 1, 2, 3, … , N
ƺj : marginal fourth central moment (kurtosis) of rj, j = 1, 2, 3, … , N
We denote the average marginal moments as
Mean skewness : k’ =
1
𝑁
σ 𝑗=1
𝑁
kj
Mean kurtosis : ƺ’ =
1
𝑁
σ 𝑗=1
𝑁
ƺj
An even number of scenarios, 2Ns, proportional to the vector’s dimension,
are generated such that they match the first and second moments. These
scenarios are symmetrically distributed around the expected value such that
the variance–covariance matrix is matched

Vu Pham
Three additional scenarios are generated in order to match the average
marginal skewness and the average marginal kurtosis of each individual
component of the random vector r.
Parameters for Scenario Generation
We have the inputs µ, ∑, kj , ƺj . Our programme must choose an arbitrary
positive integer s, an arbitrary nonzero deterministic vector Z, such that
∑ - ZZT > 0, and a scalar ρ ϵ 0,1
We randomly generate a ρ using a normal distribution.
Also, Z is generated using this ρ . We take Zj = ρ σjj (ρ = 0.45, in my case)
Here ∑jj = vector corresponding to the diagonal elements of ∑

Vu Pham
The next part is to find L. In the paper, L is defined as a positive definite
matrix such that LLT = ∑ - ZZT . We may use Cholesky decomposition method
on ∑ - ZZT to find L.
Now we generate a random natural number ‘s’ after which we can find the
probability weights. Let pi’s be real scalars such that pi ∈ 0,1 . There are
many ways for generating pi’s.
Note that number of scenarios are independent on the dimension of the
random vector r as ‘s’ is a randomly generated positive integer.
pi’s can be generated by sampling from a uniform distribution or gamma
distribution method. The detailed explanation is provided in the next slide.

Vu Pham
One method to generate pi is such that it satisfies
pi =
𝑠
𝑁𝛾
+ (
1
2𝑁𝑠
−
𝑠
𝑁𝛾
) U , where U ∈ 0,1 is a uniformly generated random
variable where 𝛾 is defined as
𝛾 = 2𝑠2
𝑁ƺ’ − 3
4
σ 𝑗=1
𝑁
𝑍4
𝑗
𝑁k'
σ 𝑗=1
𝑁
𝑍3
𝑗
2
σ𝑙 σ 𝑘 𝐿4
𝑙𝑘
Another method of generating pi is from gamma distribution. We generate U
in between (
1
𝑒
, 1 ) by gamma distribution, and taking pi = loge(U). In this
method, we must normalise the weights.
p𝑖′ =
𝑝𝑖
𝑁𝑠(2 σ 𝑝𝑖 + max(𝑝𝑖))

Vu Pham
Now, after finding pi’s we must check that the following two conditions are
satisfied.
1.) σ𝑖=1
𝑠
𝑝𝑖 <
1
2𝑁
2.) σ𝑖=1
𝑠 1
𝑝𝑖
< 𝛾 if 𝛾 𝑖𝑠 𝑢𝑠𝑒𝑑
We define a new element in the vector p, ps+1 = 1 – 2N σ𝑖=1
𝑠
𝑝𝑖
Now we will define 𝜑1 𝑎𝑛𝑑 𝜑2 which will be used to find probabilities.
𝜑1 =
𝑁𝑘′ ps
+
1
σ 𝑗=1
𝑁
𝑍3
𝑗
and 𝜑2 = ps+1
𝑁ƺ’ −
1
2𝑠
2 σ 𝑙 σ 𝑘 𝐿4
𝑙𝑘
(σ𝑖=1
𝑠 1
𝑝𝑖
)
σ 𝑗=1
𝑁
𝑍4
𝑗
After determining the above values, we’ll find 𝛼 and 𝛽.

Vu Pham
We define 𝛼 and 𝛽 as follows :
𝛼 = +
1
2
𝜑1 +
1
2
4𝜑2 − 3𝜑1
2
𝛽 = −
1
2
𝜑1 +
1
2
4𝜑2 − 3𝜑1
2
In our algorithm, we must have 4𝜑2 − 3𝜑1
2 ≥ 0
We define w1, w2 and w0 .
w1 =
1
𝛼(𝛼+𝛽)
, w2 =
1
𝛽(𝛼+𝛽)
and w0 = 1 -
1
𝛼𝛽
Observe that w1 + w2 + w0 = 1

Vu Pham
After finding these values, our aim is to find the probability weights P.
We define, S = 2Ns + 3. Given p1, p2, … , ps+1 and w0 , w1, w2 from previous
steps. Vector P is formed as follows :
𝑃 = 𝑝1, 𝑝2, … , 𝑝 𝑠, 𝑝1, 𝑝2, … , 𝑝 𝑠, … , 𝑝 𝑠 + 1 𝑤0, 𝑝 𝑠 + 1 𝑤1, 𝑝 𝑠 + 1 𝑤2
All the elements of P must be non-negative. Also, σ 𝑃 = 1
Note that there is no optimisation involved in this whole algorithm.
Since there is no algorithm involved in this method, it’s the fastest way for
the moment-matching scenario generation.
My code in R is described in coming slides.

Vu Pham
My code in R
Initialising inputs for the algorithm.
Implementation of Algorithm in R

Vu Pham
My code in R
We randomly generate ‘s’ (an integer ranging between 1 and 10000)
L is generated using Cholesky decomposition method.
Note: L must be positive definite

Vu Pham
My code in R (Loop is used to find the vector P with only positive elements)

Vu Pham
My code in R
𝛼, 𝛽, 𝑤1, 𝑤2 𝑎𝑛𝑑 𝑤0 𝑎𝑟𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
Vector P with S = 2Ns + 3
elements is formed
For loop is created so that only
elements with positive values are in the
output result.
This code also checks
4𝜑2 − 3𝜑1
2 ≥ 0

Vu Pham
My code in R (Finalising the code)
The main loop is closed.
In the output vector P, sum of all the elements must be 1.
sum(P) = 1 ??? Yes
Finally we can get the value of ‘s’ at which the loop terminated.

Vu Pham
I also did some works with Prof Semenov. I tried to understand one of his
research paper on Options Portfolio.
I’m still working with an algorithm in that paper and like to work further in
the future.
I read some more research papers on Stock Market Scenario Generation.
I thank Prof Mikhail Semenov for mentoring me in this one month project. I
learned a lot during this time period.
I also thank National Research Tomsk Polytechnic University, Russia for
giving this research internship opportunity.
Internship at TPU

Vu Pham
IIT Kanpur and TPU has an agreement on academic research collaboration
which was signed on 06.07.2015
IITK and TPU have many areas of common interest in education and
research.
This collaborative agreement has programmes like
a) Faculty/Scientist/Staff exchanges
b) Student exchanges (at both UG and PG level)
TPU is the only university in Russia with which IITK has a collaboration.
Our agreement is valid up to 06.07.2020
IITK & TPU Collaboration

Vu Pham
Ponomareva, Roman, Date, “An algorithm for moment-matching scenario
generation with application to financial portfolio optimisation”. (European
Journal of Operational Research, 2015)
Høyland and Wallace, “Generating Scenario Trees for Multistage Decision
Problems”. (Norwegian University of Science and Technology, 2001)
Pieter Klaassen, Comment on “Generating Scenario Trees for Multistage
Decision Problems”. (The Netherlands, Vrije Universiteit, 2002)
Hoffman, “Combinatorial optimization: Current successes and directions for
the future”. (Journal of Computational and Applied Mathematics 124 (2000)
341-360)
Mico Loretan, “Generating market risk scenarios using principal components
analysis: methodological and practical considerations”. (Federal Reserve
Board, March 1997)
Wurtz, Chalabi, Chen, Ellis, “Portfolio Optimization with R/Rmetrics”.
(Zurich, Rmetrics Association & Finance Online, May 2009)
References

Vu Pham
СПАСИБО
Thank You !

Scenario Generation Algortihm using R

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