Ecmi presentation

The problem Univariate Analysis Multivariable Analysis Conclusion

How mathematicians predict the future?

Mattia Zanella

Group 5
Costanza Catalano, Angela Ciliberti, Goncalo S. Matos, Allan S. Nielsen,
Olga Polikarpova, Mattia Zanella

Instructor: Dr inz. Agnieszka Wyłomańska (Hugo Steinhaus Center)
˙

December 22, 2011

How mathematicians predict the future? ECMI
European Consortium for Mathematics in Industry


Introduction and deﬁnitions

Introduction

SPOT RATE
INFLATION RATE
NOMINAL RATE
REAL RATE



Introduction and deﬁnitions

Datas



Detecting Trends



Continous Case

Ornstein-Uhlenbeck Process
Definition
Let (Ω, F, P) a probability space and F = (Ft )t≥0 a filtration
satisfying the usual hypotheses. A stochastic process Xt is an
Ornstein-Uhlenbeck process if it satisfies the following stochastic
differential equation

dXt = λ (µ − Xt ) dt + σdWt
X0 = x 0

where λ ≥ 0, µ and σ ≥ 0 are parameters, (Wt )t≥0 is a Wiener
process and X0 is deterministic.
If (St )t≥0 is the process implied/real/nominal inflation we will in
our model consider St = exp Xt ∀t ≥ 0.


Continous Case

Model Calibration
Maximum Likelihood Estimation

Let (Xt0 , ..., Xtn ) n + 1 − observations, the Likelihood Function of
Xti |Xti −1 is
n
fi Xti ; λ, µ, σ|Xti −1
i=1



Continous Case

Model Calibration

Let (Xt0 , ..., Xtn ) n + 1 − observations, the Likelihood Function of
Xti |Xti −1 is
n
fi Xti ; λ, µ, σ|Xti −1
i=1

The Log-Likelihood function is deﬁned as
n
L(X , λ, µ, σ) = log f (Xti ; λ, µ, σ|Xti −1 ).
i=1



Continous Case

Model Calibration

Now we have to ﬁnd

arg max L(X , λ, µ, σ)
λ∈R,µ∈R,σ∈R+

putting conditions of the ﬁrst and second order.



Continous Case

Model Calibration
Results

λ=9.9241 µ = 2.8656 σ = 2.2687



Continous Case

Model Calibration
Results

λ=9.9241 µ = 2.8656 σ = 2.2687

λ=5.8952 µ = 4.4358 σ = 3.1919



Continous Case

Model Calibration
Results

λ=9.9241 µ = 2.8656 σ = 2.2687

λ=5.8952 µ = 4.4358 σ = 3.1919

λ=4.5916 µ = 1.5487 σ = 2.3572



Continous Case

Numerical Approximations

Consider a general SDE

dXt = a(Xt )dt + b (Xt ) dWt , t ∈ [0, T ]

and a partition of the time interval [0, T ] into n equal subintervals
of width δ = Tn
0 = t0 < t1 < ... < tn = T



Continous Case

Methods

Euler-Maruyama scheme:

Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi



Continous Case

Methods

Euler-Maruyama scheme:

Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi

Millstein scheme:
1
Yi+1 = Yi +a (Yi ) δ+b (Yi ) ∆Wi + b (Yi ) b (Yi ) (∆Wi )2 − δ
2



Continous Case

Numerical Results
Implied Inﬂation



Continous Case

Numerical Results
Nominal Inﬂation



Continous Case

Numerical Results
Real Inﬂation



Continous Case

Empirical Distributions
Implied Inﬂation



Continous Case

Empirical Distributions
Nominal Inﬂation



Continous Case

Empirical Distribution
Real Inﬂation



Discrete Case

Autoregressive Model
AR(p)

Deﬁnition
The AR(p) model is deﬁned as
p
Xt = c + ϕi Xt−i + εt
i=1

where ϕ1 , ..., ϕp are the parameters of the model, c a constant and
εt is normally distributed.



Discrete Case

Autocorrelation

Definition
We define autocorrelation coefficient of a random variable X
observed at times t and s
E [(Xt − µt ) (Xs − µs )]
R(s, t) = .
σs σt

If R = 1: perfect correlation
If R = −1: anti-correlation
If R = 0: non correlated



Discrete Case

Autocorrelation Plots



Discrete Case

First Order Autoregressive Model
AR(1)

Our model takes the form

Xt+1 = c + ϕXt + εt .



Discrete Case

AR(1)

Our model takes the form

Xt+1 = c + ϕXt + εt .

Or equivalently

Xt+1 = µ + ϕ (Xt − µ) + N 0, σ 2 .



Discrete Case

Numerical Results Real Spot Rate



Discrete Case

PDF Evolution
Real Spot Rate



Discrete Case

Conﬁdence Bands
Entire Data Set



Discrete Case

Conﬁdence Bands
Partial Data Set



Multiple Regression

Multiple Regression

     
y1 1 x11 x12   ε1
 y2   1 x21 x22  β0  ε2 
. = . . .  β1  +  .
     
 .
. .
. .
. . 
. .
.

β2
    
yn 1 xn1 xn2 εn



Multiple Regression

Multiple Regression

     
y1 1 x11 x12   ε1
 y2   1 x21 x22  β0  ε2 
. = . . .  β1  +  .
     
 .
. .
. .
. . 
. .
.

β2
    
yn 1 xn1 xn2 εn

Or in equivalently
y = Xβ + ε



Multiple Regression

Regressors



Multiple Regression

Assumption on the Model
y = Xβ + ε

E (εi ) = 0
Var (εi ) = σ 2 ∀i = 1, . . . , n
Cov(εi , εj ) = 0 ∀i = j



Multiple Regression

Least Square Estimation
β Coeﬃcients

If X X is invertible the LSE of β is

ˆ −1
β= XX Xy



Multiple Regression

β Coeﬃcients


ˆ −1
β= XX Xy

{β0 , β1 , β2 } = {−0.0068, −0.9869, 0.9935}



Multiple Regression

β Coeﬃcients


ˆ −1
β= XX Xy

{β0 , β1 , β2 } = {−0.0068, −0.9869, 0.9935}

Real = β0 + β1 Nominal + β2 Implied + ε



Multiple Regression

Numerical Results



Multiple Regression

About the Noise



Multiple Regression

Conﬁdence Bands



Conclusion

Ornstein-Uhlenbeck
AR(1)
Regression



Conclusion

Ornstein-Uhlenbeck
AR(1)
Regression
Validation of the classical Fisher hypothesis

rr = rn − π e .



The end

Thank you for attention


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