Lecture 1 Erik Lindstrm FMS161/MASM18 Financial Statistics Erik - - PowerPoint PPT Presentation

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Lecture 1

Erik Lindström

FMS161/MASM18 Financial Statistics

Erik Lindström Lecture 1

Intro Stylized facts

People and homepage

◮ Erik Lindstrom: erikl@maths.lth.se, 222 45 78, MH:129

(Lecturer)

◮ Carl Åkerlindh: carl.akerlindh@matstat.lu.se, 222 04 85 ,

MH:125a (Computer exercises)

◮ Maria Lövgren: marial@maths.lth.se, 222 45 77, MH:225

(Course secretary)

◮ http:

//www.maths.lth.se/matstat/kurser/fms161mas229/

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Purpose:

» The course should provide tools for analyzing data, and use these tools in combination with economic theory. The main applications are valuation and risk management. » The course used to be the inference part in a package of courses consisting of ’TEK180 Financial Valuation and Risk Management’, ’FMS170 Valuation of Derivative Assets’ and this course.

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Recommended courses:

» Mathematical statistics: FMS045/MASC04 Stationary processes (G2), FMSF05/MASC01 Probability theory, intermediate course (G2), FMS051/MASM17 Time series analysis (A), FMS155/MASM15 Extreme Value Theory (A), FMS170/MASM19: Valuation of Derivative Assets (A) and FMS161/MAS18: Financial Statistics (A). » Economics: TEK135 Microeconomics (G2), TEK180 Financial Valuation and Risk Management (A), TEK103 Financial economics, advanced course (A), TEK110 Empirical Financial economics (A) » Other: Programming courses (G2), Optimization (G2), Advanced analysis and prob. theory courses...

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Inference problems?

◮ Forecast future prices, interest rates, volatilities (P and Q) ◮ Filtering of data (e.g. estimating hidden states such as

stochastic volatility)

◮ Distribution of prediction errors; can we improve the

model?

◮ What about extreme events? ◮ How do we estimate parameters in general models? ◮ Cross covariance and auto covariance. ◮ Non-linear, Non-Gaussian, Non-stationary...

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Example I – Daily interest data

Q1−92 Q2−92 Q3−92 Q4−92 Q1−93 10 100 500 STIBOR and REPO Yields 1992 REPO STIBOR 1W STIBOR 1M STIBOR 3M STIBOR 6M

◮ Models and distributions. ◮ Forecasts - 0.5 % or 500 %? ◮ Events ◮ Covariation with of market factors?

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Example II – Forward prices on Nordpool

What can we expect??

◮ Traders are interested in predicting price movements on

the futures on Nordpool on yearly contracts.

◮ Or predicting the movements on short horizons (days or

weeks).

◮ Expected movement and/or prob. of declining prizes. ◮ What about fundamental factors?

Hydrological situation is the energy stored as snow, ground water or in reservoirs. Time to maturity. Perfect or imperfect markets.

◮ Other factors – suggestions?

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Electricity spot price and Hydrological situation

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Ex – Forwards on Nordpool, contd.

There is a strong dependence between the hydrological situation and the price.

◮ How do we model this dependence, e.g. what model

should we use?

◮ How do we estimate the chosen model? ◮ How do we know if the model is working (or is good

enough)?

◮ One supermodel or several models? ◮ Adaptive models?

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Contents

The course treats estimation, identification and validation in non-linear dynamical stochastic models for financial applications based on data and prior knowledge. There are rarely any absolutely correct answers in this course, but there are often answers that are absolutely wrong

• r to put it differently

All models are wrong– but some are useful!

Think for yourself, and question the course material!

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Contents, 2

◮ Discrete and continuous time. ◮ Parameter estimation, model identification and model

validation.

◮ Prediction. ◮ Modelling of variance, ARCH, GARCH, ..., and other

approaches.

◮ Calibration ◮ Risk ◮ Stochastic calculus and SDEs. ◮ State space models, and filters

(Kalman filters and versions thereof), particle filters

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Course goals -Knowledge and Understanding

For a passing grade the student must:

◮ handle variance models such as the GARCH family,

stochastic volatility, and models use for high-frequency data,

◮ use basic tool from stochastic calculus: Ito’s formula,

Girsanov transformation, martingales, Markov processes, filtering,

◮ use tools for filtering of latent processes, such as Kalman

filters and particle filters,

◮ statistically validate models from some of the above model

families.

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Course goals -Skills and Abilities

For a passing grade the student must:

◮ be able to find suitable stochastic models for financial data, ◮ work with stochastic calculus for pricing of financial

contracts and for transforming models so that data becomes suitable for stochastic modelling,

◮ understand when and how filtering methods should be

applied,

◮ validate a chosen model in relative and absolute terms, ◮ solve all parts of a modelling problem using economic and

statistical theory (from this course and from other courses) where the solution includes model specification, inference, and model choice,

◮ present the solution in a written technical report, as well as

• rally,

◮ utilise scientific articles within the field and related fields.

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Literature

Literature:

◮ Lindström, E., Madsen, H., Nielsen, J. N. Statistics for

Finance, Lund, 2014.

◮ Handouts

Course program.

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Properties of financial data

◮ No Autocorrelation in returns ◮ Unconditional heavy tails ◮ Gain/Loss asym. ◮ Aggregational Gaussianity ◮ Volatility clustering ◮ Conditional heavy tails ◮ Significant autocorrelation for abs. returns - long range

dependence?

◮ Leverage effects ◮ Volume/Volatility correlation ◮ Asym. in time scales

Evaluate claims on OMXS30 data.

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Autocorrelation in returns

5 10 15 20 25 30 −0.2 0.2 0.4 0.6 0.8 1 1.2 lag Autocorrelation, returns

No or little autocorrelation.

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Unconditional distribution

−0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability Normal Probability Plot

Normplot of the unconditional returns.

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Gain/Loss asym.

Jan95 Jan00 Jan05 Jan10 Jan15 10

2

10

3

log(Index) OMX S30

Losses are larger than gains (data is log(Index)).

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• Aggr. Gaussianity

−0.05 0.05 0.1 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data r Normal Probability Plot −0.05 0.05 0.1 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data r2 Normal Probability Plot −0.1 0.1 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data r4 Normal Probability Plot −0.1 0.1 0.2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data r8 Normal Probability Plot −0.2 −0.1 0.1 0.2 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data r16 Normal Probability Plot −0.1 0.1 0.2 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 Data r32 Normal Probability Plot

Returns are increasingly Gaussian. Interpretation?

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• Vol. Clustering

Nov91 Nov94 Nov97 Nov00 Nov03 Nov06 −0.1 −0.05 0.05 0.1

Volatility clusters. Average cluster size?

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Conditional distribution

Nov91 Nov96 Nov01 Nov06 −6 −4 −2 2 4 6 −6 −4 −2 2 4 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability Normal Probability Plot

Normplot of the conditional returns (GARCH(1,1) filter).

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Dependence in returns

50 100 150 200 250 300 −0.2 0.2 0.4 0.6 0.8 1 1.2 lag Autocorrelation, abs returns

Significant autocorrelation. Long range dependence?

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Leverage effects

◮ Most assets are negatively correlated with any measure of

volatility.

◮ One popular explanation is corporate debt. ◮ Makes sense if you are risk averse.

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Volume/Volatility correlation

◮ Trading volume is correlated with the volatility. ◮ Sometimes modelled with ’business time’ in option

valuation community.

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• Asym. in time scales

◮ Coarse-grained measurements can predict fine scaled

volatility

◮ While fine scaled volatility have difficulties predicting

coarse scale volatility

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Extra material

Feel free to download the paper (you need a Lund University IP

• address to access the paper.)

Cont, R. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, Vol. 1, No. 2 (March 2001) 223-236.

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