This document discusses optimal weighted distributions and their applications to financial time series. It addresses the problems of minimizing distances between weighted sample distributions and target distributions for 1) a single population, 2) two populations, and 3) N populations. Explicit formulas are derived for the optimal weights in each case. An iterative algorithm is presented to compute the target distribution as a fixed point. Applications to Google stock data are shown. Possible developments include extending the analysis to more populations, relaxing assumptions, and considering alternative distances.

1. Optimal Weighted Distributions
and
Applications to Financial Time Series
Alessandro Zanatta
Relatore: Prof. Marco Maggis
Corelatore: Prof. Giacomo Aletti
Universit`a Degli Studi di Milano
Corso di Laurea in Matematica
Anno Accademico 2013/2014
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2. Introduction
Given a probability space (Ω, F, P), a vector of observations {Y i
}n
i=1 with fixed
n ∈ N and a process {Xt}t∈[0,T] with unknown distribution at maturity T we
define
Weighted Sample Distribution ⇒ F(u) =
n
i=1
wi 1{Y i ≤u}
where w = (w1, . . . , wn) ∈ ∆n
with
∆n
= w ∈ Rn
: wi ≥ 0 for any i = 1, ..., n and
n
i=1
wi = 1
the n-simplex in Rn
and
Target Distribution ⇒ FX (u) = P(XT (ω) ≤ u)
for all ω ∈ Ω and for all u ∈ R.
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3. Optimization Problem
Problem 1
Given a vector of observations {Y i
}n
i=1, n ∈ N, we want to minimize the following
distance
d(F, FX ) = EP
R
F(u) − FX (u)
2
du
with respect the weights w1, . . . , wn ∈ ∆n
, where F is a weighted sample
distribution and FX is a target distribution.
Using the optimal weights w∗
1, . . . , w∗
n obtained solving the minimum problem we
can compute the
Optimal Weighted Distribution ⇒ F∗(u) =
n
i=1
w∗
i 1{Y i ≤u}
for all u ∈ R.
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4. Optimization Problem
Let us fix the following objects
Ak =
R
FY k (u) (1 − FY k (u)) + (FX (u) − FY k (u))
2
du
Bk,j =
R
P Y k
≤ u, Y j
≤ u − FY k (u)FY j (u)+
+ FX (u)2
− FX (u)FY k (u) − FX (u)FY j (u) + FY k (u)FY j (u)du
Remark
Bk,j is symmetric
if j = k ⇒ Bk,k = Ak
if {Y i
}n
i=1 is an i.i.d. sample for XT then
Ak =
R
FX (u) (1 − FX (u)) du and Bk,j = 0
for all k, j = 1, . . . , n
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5. Optimization Problem
The k-optimal weight that solves Problem 1 has the following form
wk = −
1
2Ak
n
j=k,j=1
wj Bk,j +
1
Ak
n
i=1
1
Ai
+
n
i=1
n
j=i,j=1
wj Bi,j
Ai
2Ak
n
i=1
1
Ai
,
for all k = 1, . . . , n.
Remark
the k-optimal weight has an implicit form
we are not doing any assumptions about the vector of observations {Y i
}n
i=1
If we assume the observations Y i
, i = 1, . . . , n, independent and identically
distributed (i.i.d.) the k-optimal weight that solves Problem 1 have the following
form
wk =
1
n
, ∀k = 1, . . . , n
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6. Populations
a population is a complete set of items that share at least one property in
common that is the subject of a statistical analysis
a statistical sample is a subset drawn from the population to represent the
population in a statistical analysis
a subset of a population is called a subpopulation if they share one or more
additional properties
populations consisting of subpopulations can be modeled by mixture models,
which combine the distributions within subpopulations into an overall population
distribution
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7. The Two Populations Problem
Consider two populations Ω1 and Ω2, we assume
nk the number of observation in population Ωk , k = 1, 2, with n1 + n2 = n
the observation in the same populations an i.i.d. sample
Problem 2
Given two populations Ω1 = {W i
}n1
i=1 and Ω2 = {Zj
}n2
j=1, n1, n2 ∈ N, we want to
minimize the following distance
d(F, FX ) = EP



R


n1
i=1
w11{W i ≤u} +
n2
j=1
w21{Zj ≤u} − FX (u)


2
du



with respect the weights n1w1, n2w2 ∈ ∆2
, where FX is a target distribution.
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8. The Two Populations Problem
Let us fix the following objects
ai =
R
(Fi (u) − FX (u))
2
du vi =
R
Fi (u) (1 − Fi (u)) du
bi,j =
R
(Fi (u) − FX (u)) (Fj (u) − FX (u)) du
ci,j =
R
CovP


n1
i=1
1{W i ≤u};
n2
j=1
1{Zj ≤u}

 du
where Fi is the distribution of the population Ωi , i = 1, 2.
Remark
ai is the square of the norm ||Fi − FX ||L2
bi,j and ci,j are both symmetric
ai and bi,j depend on target distribution FX
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9. The Two Populations Problem
The optimal weights that solve Problem 2 have the following form
n1w∗
1 =
a2 + v2
n2
− b1,2 −
c1,2
n1n2
(a1 + v1
n1
− b1,2 −
c1,2
n1n2
) + (a2 + v2
n2
− b1,2 −
c1,2
n1n2
)
n2w∗
2 =
a1 + v1
n1
− b1,2 −
c1,2
n1n2
(a1 + v1
n1
− b1,2 −
c1,2
n1n2
) + (a2 + v2
n2
− b1,2 −
c1,2
n1n2
)
Remark
the two optimal weights have an explicit form
the weight n1w∗
1 depends on a2 and vice versa
the two optimal weights depend on the target distribution FX
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10. The N Populations Problem
Consider N populations Ω1, . . . , ΩN , we assume
nk the number of observation in population Ωk , k = 1, . . . , N, with
n1 + · · · + nN = n
the observation in the same populations an i.i.d. sample
Problem 3
Given N populations Ω1 = {Y n1
}, . . . , ΩN = {Y nN
}, n1, . . . , nN ∈ N, we want to
minimize the following distance
d(F, FX ) = EP



R


N
i=1
ni
j=1
wi 1{Y ni ≤u} − FX (u)


2
du



with respect the weights n1w1, . . . , nN wN ∈ ∆N
, where FX is a target distribution.
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11. The N Populations Problem
The i-optimal weights that solves Problem 3 has the following form
ni w∗
i = −
1
2(ai + vi
ni
)
N
k=i,k=1
nk wk (bi,k +
ci,k
ni nk
)+
+
1
ni (ai + vi
ni
)
N
j=1
1
n2
j (aj +
vj
nj
)
+
N
j=1
N
k=j,k=1
nk wk (bj,k +
cj,k
nj nk
)
aj +
vj
nj
2ni (ai + vi
ni
)
N
j=1
1
n2
j (aj +
vj
nj
)
for all i = 1, . . . , N.
Remark
the i-optimal weight has an implicit form
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12. Financial Time Series
a time series is a collection of numerical observations described through random
variables, and indexed according to the order
when we think of a time series, we usually think of a collection of values
{Xt}n
t=1 in which the index t indicates the time at which the datum Xt is
observed
the study of time series has diverse applications ranging from biology to finance
a key feature that distinguishes financial time series from other type of time
series consists in the presence of an element of uncertainty and randomness
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13. The Algorithm
Given two populations Ω1 and Ω2 with distribution functions F1 and F2 consider
the optimal weights deduced from Problem 2 with the extra assumption
ci,j = 0
w1 =
a2 + v2
n2
− b1,2
(a1 + v1
n1
− b1,2) + (a2 + v2
n2
− b1,2)
w2 =
a1 + v1
n1
− b1,2
(a1 + v1
n1
− b1,2) + (a2 + v2
n2
− b1,2)
the optimal weights as function of the Target distribution FX
an initial guess F0
X ∈ X for the Target distribution FX , where X is the convex
hull formed by distributions F1 and F2
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14. The Algorithm
Given two populations Ω1 and Ω2 with distribution functions F1 and F2 consider
the optimal weights deduced from Problem 2 with the extra assumption
ci,j = 0
w1 =
a2 + v2
n2
− b1,2
(a1 + v1
n1
− b1,2) + (a2 + v2
n2
− b1,2)
w2 =
a1 + v1
n1
− b1,2
(a1 + v1
n1
− b1,2) + (a2 + v2
n2
− b1,2)
the optimal weights as function of the Target distribution FX
an initial guess F0
X ∈ X for the Target distribution FX , where X is the convex
hull formed by distributions F1 and F2
Iterative Algorithm
Fi+1
X = w1 Fi
X F1 + w1 Fi
X F2, for i = 0, 1, 2, . . .
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15. Properties
Given the operator T : X → X such that
T[·] = w1 (·) F1 + w2 (·) F2
a fixed point for the operator T[·] is a distribution G ∈ X such that T[G] = G
G ∈ X ⇒ G = αF1 + (1 − α)F2 for α ∈ [0, 1]
than we want to find α ∈ [0, 1] such that
αF1 + (1 − α)F2 = T[αF1 + (1 − α)F2].
Lemma
Given the previous considerations we have that
α =
v2/n2
v1/n1 + v2/n2
and 1 − α =
v1/n1
v1/n1 + v2/n2
where
v1 =
R
F1(u) (1 − F1(u)) du and v2 =
R
F2(u) (1 − F2(u)) du
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16. Numerical Applications
Google time series
black: populations Ω1
green: stress populations Ω2
log return time series
black: populations Ω1
green: stress populations Ω2
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17. Numerical Applications
red: Target distribution
black: populations Ω1
green: stress populations Ω2
optimal weights
w∗
1 = 0.84686
w∗
2 = 0.15314
α = 0.84686
1 − α = 0.15314
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18. Numerical Applications
Figura: Different initial guess and corresponding Target distributions
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19. Possible developments
numerical extension to the case of N populations
minimal hypothesis
change the distance
19 / 20

20. Possible developments
numerical extension to the case of N populations
minimal hypothesis
change the distance
Thanks for your attention
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