nonlinear models for volatility prediction in the financial markets

OutlineIntroduction

Prediction ModelsResults

Modeling GUIConclusions

Nonlinear Modelsfor Volatility Predictionin the Financial Markets

Matteo Ainardi

Advisors: Prof. Gianpiero Cabodi, Prof. Derong Liu

University of Illinois at ChicagoMaster of Science in Electrical and Computer Engineering

Matteo Ainardi Nonlinear Models for Volatility Prediction in the Financial Markets

OutlineIntroduction

Outline

1 IntroductionVolatility

2 Prediction ModelsVolatility Models

3 ResultsVolatility PredictionInvestment Strategy

4 Modeling GUI

5 Conclusions

OutlineIntroduction

Volatility

Thesis goal

Development and evaluation of models for volatility prediction.

Qualitative definition

Volatility: degree of price variation over time.

Why Volatility?

Investors assess expected returns of an asset against its risk.

Financial institutions want to ensure that the value of theirassets does not fall below some minimum level that wouldexpose them to insolvency.

OutlineIntroduction

Volatility

Thesis goal

Why Volatility?

OutlineIntroduction

Volatility

Thesis goal

Why Volatility?

OutlineIntroduction

Volatility

Thesis goal

Why Volatility?

OutlineIntroduction

Volatility

Volatility Features

not directly observable

not constant over time

Quantitative Definition

Time varying volatility measure from the price time series:

σ2t = (rt − r)2

rt= pt−pt−1

pt−1: return on day t,

r : mean return over the last 200 days period.

OutlineIntroduction

Volatility

Volatility Features

σ2t = (rt − r)2

rt= pt−pt−1

OutlineIntroduction

Volatility

Volatility Features

σ2t = (rt − r)2

rt= pt−pt−1

OutlineIntroduction

Volatility

Volatility Features

σ2t = (rt − r)2

rt= pt−pt−1

OutlineIntroduction

Volatility Models

Prediction Models

Let’s assume that the system to forecast can be described by aregression equation of the form

yt+1 = f0(ϕt) + dt

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

t ∈ N: time [days],

yt : volatility at day t,

ϕt : regressor,

dt : noise affecting the system.

OutlineIntroduction

Volatility Models

Prediction Models

yt+1 = f0(ϕt) + dt

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

ϕt : regressor,

OutlineIntroduction

Volatility Models

Prediction Models

yt+1 = f0(ϕt) + dt

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

ϕt : regressor,

OutlineIntroduction

Volatility Models

Prediction Models

yt+1 = f0(ϕt) + dt

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

ϕt : regressor,

OutlineIntroduction

Volatility Models

Prediction Models

yt+1 = f0(ϕt) + dt

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

ϕt : regressor,

OutlineIntroduction

Volatility Models

Prediction Models

A prediction model f can be defined as an approximation of f0,providing a prediction yt+1 of yt+1:

yt+1 = f (ϕt),

where ϕt represents an estimate of the true regressor ϕt .

OutlineIntroduction

Volatility Models

Prediction Models - Statistical/Parametric Approach

The traditional approach followed to build a model implies a choiceof a specific structure for the functional form f0 and statisticalassumptions on the noise dt affecting the system.

if possible, physical/economical laws are used to obtain aparametric representation of the system f (ϕ, θ)

as a parametric combination of basis functions (polynomial,sigmoid, ...)

The parameters θ are then estimated from data by optimizingLeast Squares or Maximum Likelihood functions.

OutlineIntroduction

Volatility Models

GARCH Models

Linear Regression / Moving Average Models

Neural Network Models

Nonlinear Set Membership Models

OutlineIntroduction

Volatility Models

GARCH Models

Generalized Autoregressive Conditional HeteroskedasticityThis methodology is the most widely adopted and led its creator,Prof. Robert Engle, to the Nobel Prize in Economy in 2003.

The volatility prediction is based on

long run constantvolatility

most recent return

previous volatilityprediction

yt+1 = ω + αr2t + βyt

The GARCH regressor is ϕt = [rt , yt ]

OutlineIntroduction

Volatility Models

GARCH Models

Generalized Autoregressive Conditional HeteroskedasticityThis methodology is the most widely adopted and led its creator,Prof. Robert Engle, to the Nobel Prize in Economy in 2003.The volatility prediction is based on

most recent return

OutlineIntroduction

Volatility Models

GARCH Models

most recent return

OutlineIntroduction

Volatility Models

GARCH Models

most recent return

OutlineIntroduction

Volatility Models

GARCH Models

most recent return

OutlineIntroduction

Volatility Models

Volatility Models - Linear Models

Linear Regression Models

yt+1 =N−1∑i=0

αi r2t−i +

M−1∑j=0

βj yt−j

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

αi and βi are the parameters that must be estimated.

Moving Average Models

yt+1 =1

N−1∑i=0

r2t−i , ϕt = [rt , . . . , rt−N+1]

Particular case of linear regression: αi = 1N , βj = 0 for each i , j .

OutlineIntroduction

Volatility Models

Volatility Models - Linear Models

Linear Regression Models

yt+1 =N−1∑i=0

αi r2t−i +

M−1∑j=0

βj yt−j

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

αi and βi are the parameters that must be estimated.

Moving Average Models

yt+1 =1

N−1∑i=0

r2t−i , ϕt = [rt , . . . , rt−N+1]

Particular case of linear regression: αi = 1N , βj = 0 for each i , j .

OutlineIntroduction

Volatility Models

Volatility Models - Neural Networks

Neural Network Models

yt+1 =r∑

αiσ(βiϕt − λi ) + ζ

r : number of neurons in the network

σ(x) = 21+e−2x : sigmoidal function (neuron transfer function)

ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

αi , βi , λi : parameters estimated through a network trainingalgorithm

ζ: noise affecting the system

OutlineIntroduction

Volatility Models

Observations

These models require the selection of a specific functional form off0 and statistical assumptions on the noise (usually supposed to begaussian).

Search of the appropriate functional form and parameters:complex and computationally heavy

Wrong f0 choice ⇒ degradation in model accuracy

OutlineIntroduction

Volatility Models

Observations

OutlineIntroduction

Volatility Models

Observations

OutlineIntroduction

Volatility Models

Volatility Models - Nonlinear Set Membership

NSM Methodology

Nonlinear Set Membership models do not require the choice of aparametric form of f0, but more relaxed assumptions on theanalyzed system:

bound on its derivatives

bound on the noise affecting the system

Advantages:

avoid the complexity/accuracy problems posed by the choiceof the model function and of the parameters.

NSM Local Approach: check if a given model can be improvedby identifying a NSM model to forecast its residual process.

OutlineIntroduction

Volatility Models

NSM Methodology

Advantages:

OutlineIntroduction

Volatility Models

NSM Methodology

Advantages:

OutlineIntroduction

Volatility Models

NSM Models

yt+1 =1

2(f (ϕt) + f (ϕt)),

f (ϕT ) = mink=1,...,T−1

(yk+1 + εk + γ ‖ϕT − ϕk‖)

f (ϕT ) = maxk=1,...,T−1

(yk+1 − εk − γ ‖ϕT − ϕk‖)

where γ (gradient) and ε (noise) bounds are derived from pastmeasured data through a validation procedure.NSM Regressor: ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

OutlineIntroduction

Volatility Models

NSM Models

yt+1 =1

2(f (ϕt) + f (ϕt)),

f (ϕT ) = mink=1,...,T−1

(yk+1 + εk + γ ‖ϕT − ϕk‖)

f (ϕT ) = maxk=1,...,T−1

(yk+1 − εk − γ ‖ϕT − ϕk‖)

OutlineIntroduction

Volatility Models

NSM Models

yt+1 =1

2(f (ϕt) + f (ϕt)),

f (ϕT ) = mink=1,...,T−1

(yk+1 + εk + γ ‖ϕT − ϕk‖)

f (ϕT ) = maxk=1,...,T−1

(yk+1 − εk − γ ‖ϕT − ϕk‖)

OutlineIntroduction

Volatility Models

NSM Models

yt+1 =1

2(f (ϕt) + f (ϕt)),

f (ϕT ) = mink=1,...,T−1

(yk+1 + εk + γ ‖ϕT − ϕk‖)

f (ϕT ) = maxk=1,...,T−1

(yk+1 − εk − γ ‖ϕT − ϕk‖)

where γ (gradient) and ε (noise) bounds are derived from pastmeasured data through a validation procedure.

NSM Regressor: ϕt = [rt , . . . , rt−N+1, yt , . . . , yt−M+1]

OutlineIntroduction

Volatility Models

NSM Models

yt+1 =1

2(f (ϕt) + f (ϕt)),

f (ϕT ) = mink=1,...,T−1

(yk+1 + εk + γ ‖ϕT − ϕk‖)

f (ϕT ) = maxk=1,...,T−1

(yk+1 − εk − γ ‖ϕT − ϕk‖)

OutlineIntroduction

Volatility Models

OutlineIntroduction

Volatility Models

Example

Availablemeasurements:

t ϕt yt+1

1 [3, 3] 3

2 [1, 3] 4

3 [3, 2] 5

Let’s assume

γ = 0.5, εr = 0.2

ϕ4 = [2, 1].

How to determine thevalue of y5?

OutlineIntroduction

Volatility Models

Example

1) compute εt

t yt εt

2 3 0.6

3 4 0.8

4 5 1.0

2) compute ‖ϕ4 − ϕt‖

t ϕt ‖ϕ4 − ϕt‖1 [3, 3] 2.24

2 [1, 3] 2.24

3 [3, 2] 1.41

f (ϕ4) = mink=1,...,3

(yk+1 + εk + γ ‖ϕ4 − ϕk‖)

f (ϕ4) = maxk=1,...,3

(yk+1 − εk − γ ‖ϕ4 − ϕk‖)

OutlineIntroduction

Volatility Models

Example

1) compute εt

t yt εt

2 3 0.6

3 4 0.8

4 5 1.0

t ϕt ‖ϕ4 − ϕt‖1 [3, 3] 2.24

2 [1, 3] 2.24

3 [3, 2] 1.41

f (ϕ4) = mink=1,...,3

(yk+1 + εk + γ ‖ϕ4 − ϕk‖)

f (ϕ4) = maxk=1,...,3

(yk+1 − εk − γ ‖ϕ4 − ϕk‖)

OutlineIntroduction

Volatility Models

Example

1) compute εt

t yt εt

2 3 0.6

3 4 0.8

4 5 1.0

t ϕt ‖ϕ4 − ϕt‖1 [3, 3] 2.24

2 [1, 3] 2.24

3 [3, 2] 1.41

f (ϕ4) = mink=1,...,3

(yk+1 + εk + γ ‖ϕ4 − ϕk‖)

f (ϕ4) = maxk=1,...,3

(yk+1 − εk − γ ‖ϕ4 − ϕk‖)

OutlineIntroduction

Volatility Models

Example

f (ϕ4) = min (4.72, 5.92, 6.71) = 4.72

f (ϕ4) = max (1.28, 2.08, 3.30) = 3.30

The NSM prediction for y5 value will be

2(f (ϕ4) + f (ϕ4)) = 4.01.

OutlineIntroduction

Volatility Models

Example

f (ϕ4) = min (4.72, 5.92, 6.71) = 4.72

f (ϕ4) = max (1.28, 2.08, 3.30) = 3.30

The NSM prediction for y5 value will be

2(f (ϕ4) + f (ϕ4)) = 4.01.

OutlineIntroduction

Volatility Models

OutlineIntroduction

Volatility PredictionInvestment Strategy

Models Results

Models are evaluated on real financial time series:

both in terms of predictive performance and by simulating aninvestment strategy.

IBM Dataset

The IBM dataset is composed of 2248 daily closure prices, from2000 to 2009.

Identification dataset: samples 1 to 1500.

Validation dataset: samples 1501 to 2248.

OutlineIntroduction

Results Evaluation

Trivial Models

Persistent Modelyt+1 = yt ,

Constant Modelyt+1 = const,

Exact Modelyt+1 = yt+1

These models are used as terms of comparisons.

OutlineIntroduction

IBM Volatility

Performance Evaluation

Model RMSE R2MZ

Persistent 10.66× 10−4 2.84%

GARCH 7.82× 10−4 12.20%

MAV 7.74× 10−4 13.59%

NN 7.68× 10−4 14.03%

NSM 7.86× 10−4 12.77%

RMSE (Root Mean Squared Error)

RMSE =

√∑nt=1(yt − yt)2

OutlineIntroduction

IBM Volatility

Model RMSE R2MZ

Persistent 10.66× 10−4 2.84%

GARCH 7.82× 10−4 12.20%

MAV 7.74× 10−4 13.59%

NN 7.68× 10−4 14.03%

NSM 7.86× 10−4 12.77%

RMSE (Root Mean Squared Error)

RMSE =

√∑nt=1(yt − yt)2

OutlineIntroduction

IBM Volatility

Model RMSE R2MZ

Persistent 10.66× 10−4 2.84%

GARCH 7.82× 10−4 12.20%

MAV 7.74× 10−4 13.59%

NN 7.68× 10−4 14.03%

NSM 7.86× 10−4 12.77%

R2MZ (Mincer Zarnowitz’s Regression R2)It is widely adopted in econometrics to assessthe practical utility of the volatility model.

OutlineIntroduction

Investment Strategy

Scenario

An investor can choose among a risk free asset which gives aconstant return and a risky asset.The composition of the investor’s portfolio can be changed everyday.

Investment Strategy

The investor accepts a certain level of volatility for its portfolio ifthere is the opportunity of an higher profit.

↘ predicted volatility ⇒↗ risky asset weight

↗ predicted volatility ⇒↘ risky asset weight

OutlineIntroduction

Investment Strategy

Scenario

An investor can choose among a risk free asset which gives aconstant return and a risky asset.The composition of the investor’s portfolio can be changed everyday.

Investment Strategy

The investor accepts a certain level of volatility for its portfolio ifthere is the opportunity of an higher profit.

↘ predicted volatility ⇒↗ risky asset weight

↗ predicted volatility ⇒↘ risky asset weight

OutlineIntroduction

Investment Strategy Results

Strategy Profit (×10−3) ∆(×10−3) Risk (×10−3) ProfitRisk

Constant 65.7 — 142 0.47

Exact Vol 163 97.3 141 1.16

Persistent 54.4 −11.3 348 0.16

GARCH 110 44.3 158 0.70

MAV 126 60.3 179 0.70

NN 115 49.3 162 0.71

NSM 152 86.3 216 0.70

Profit: mean portfolio return

∆: improvement with respect to the constant strategy

Risk: portfolio return standard deviation

Profit refers to the annualized mean portfolio return.For example the NSM strategy showsabout a +15% annual portfolio variation.

OutlineIntroduction

Constant 65.7 — 142 0.47

Exact Vol 163 97.3 141 1.16

Persistent 54.4 −11.3 348 0.16

GARCH 110 44.3 158 0.70

MAV 126 60.3 179 0.70

NN 115 49.3 162 0.71

NSM 152 86.3 216 0.70

OutlineIntroduction

Constant 65.7 — 142 0.47

Exact Vol 163 97.3 141 1.16

Persistent 54.4 −11.3 348 0.16

GARCH 110 44.3 158 0.70

MAV 126 60.3 179 0.70

NN 115 49.3 162 0.71

NSM 152 86.3 216 0.70

OutlineIntroduction

Constant 65.7 — 142 0.47

Exact Vol 163 97.3 141 1.16

Persistent 54.4 −11.3 348 0.16

GARCH 110 44.3 158 0.70

MAV 126 60.3 179 0.70

NN 115 49.3 162 0.71

NSM 152 86.3 216 0.70

OutlineIntroduction

Data Load GUI

OutlineIntroduction

Model Identification GUI

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Automatic Model Identification GUI

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Display Results GUI

OutlineIntroduction

Conclusions

GARCH, MAV, NSM and NN models show significantimprovements with respect to the persistent model and toconstant volatility.

NSM optimality analysis, based on local approach, indicatesthat GARCH models performance can not be significantlyimproved.

NN models often produce unstable results in presence ofvolatility dynamics very different from the ones on which theyare trained.

NSM models are able to produce robust results even inpresence of extreme volatility variations.

OutlineIntroduction

Conclusions

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Conclusions

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Conclusions

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Conclusions

All the experiments have been implemented and executed inMatlab, the development of the GUI resulted into about 2500lines of Matlab code.

More than 30 prediction models have been identified andevaluated on the different datasets, relying on differentvolatility measures proposed in financial literature.

Future developments: try different regressor forms, apply themodels to a larger number of markets and timeframes.

OutlineIntroduction

Conclusions

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Conclusions

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Conclusions

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Thank You

nonlinear models for volatility prediction in the financial markets

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