FMS161/MASM18 Financial Statistics Lecture 1, Introduction and - - PowerPoint PPT Presentation

FMS161/MASM18 Financial Statistics Lecture 1, Introduction and stylized facts

Erik Lindström

People and homepage

◮ Erik Lindström: erikl@maths.lth.se, 222 45 78,

MH:221 (Lecturer)

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

222 04 85 , MH:223 (Computer exercises)

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

MH:225 (Course secretary)

◮ http:

//www.maths.lth.se/matstat/kurser/fmsn60masm18/

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 is intended to provide necessary

statistical tools supporting courses like 'TEK180 Financial Valuation and Risk Management' or 'FMSN25/MASM24 Valuation of Derivative Assets'.

Inference problems?

◮ Forecast prices, interest rates, volatilities (under

the P and Q measures)

◮ Filtering of data (e.g. estimating hidden states

such as stochastic volatility or credit default intensity)

◮ 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. Often results in Non-linear, Non-Gaussian, Non-stationary models...

Inference problems?

◮ Forecast prices, interest rates, volatilities (under

the P and Q measures)

◮ Filtering of data (e.g. estimating hidden states

such as stochastic volatility or credit default intensity)

◮ 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. ◮ Often results in Non-linear, Non-Gaussian,

Non-stationary models...

Example I -- Daily interest data - big crisis in Sweden during the early 1990s

See Section 2.4 in the book for more information.

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

Forecasts - 0.5 % or 500 %? Covariation with of market factors? - Can this happen again? Models and distributions.

Example I -- Daily interest data - big crisis in Sweden during the early 1990s

See Section 2.4 in the book for more information.

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

◮ Forecasts - 0.5 % or 500 %? ◮ Covariation with of market factors? - Can this

happen again?

◮ Models and distributions.

Electricity spot price and Hydrological situation

Example II -- Forward prices on Nordpool

◮ 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?

• 1. Hydrological situation is the energy stored as

snow, ground water or in reservoirs

• 2. Time to maturity.
• 3. Perfect or imperfect markets.

◮ Other factors -- suggestions?

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? Is the relation linear?

◮ How do we fit the chosen model? ◮ How do we know if the model is good enough? ◮ One supermodel or several models? ◮ Adaptive models?

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. This was expressed by George Box as

All models are wrong - but some are useful!

Think for yourself, and question the course material!

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. This was expressed by George Box as

All models are wrong - but some are useful!

Think for yourself, and question the course material!

Contents, 2

Discrete and continuous time.

◮ Parameter estimation (LS, ML, GMM, EF), model

identification and model validation.

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

• ther approaches.

◮ Stochastic calculus and SDEs. ◮ State space models and filters

Kalman filters (and versions thereof) and particle filters

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.

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,

◮ understand when and how filtering methods

should be applied,

◮ validate a chosen model, ◮ 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,

◮ utilise scientific articles within the field and

related fields.

◮ present the solution in a written technical

report, as well as orally,

Literature

◮ Lindström, E., Madsen, H.,

Nielsen, J. N. (2015) Statistics for Finance, Chapman & Hall, CRC press.

◮ Handouts (typically articles on

the course home page) Course program.

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.

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.

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.

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)). This may contradict the EMH, see Nystrup et al. (2016).

• 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?

• Vol. Clustering

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

Volatility clusters. Average cluster size?

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).

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

• r other reason? Hint: Nystrup et al., (2015, 2016)

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.

Volume/Volatility correlation

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

• ption valuation community - cf. Time Shifted

Levy processes models, Def 7.12, such as NIG-CIR model.

• Asym. in time scales

◮ Coarse-grained measurements can predict fine

scaled volatility

◮ While fine scaled volatility have difficulties

predicting coarse scale volatility

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. http://dx.doi.org/10.1080/713665670

◮ Nystrup, P., Madsen, H., & Lindström, E. (2015).

Stylised facts of financial time series and hidden Markov models in continuous time. Quantitative Finance, 15(9), 1531-1541. http: //dx.doi.org/10.1080/14697688.2015.1004801

◮ Nystrup, P., Madsen, H., & Lindström, E.

(2016).Long Memory of Financial Time Series and Hidden Markov Models with Time-Varying

• Parameters. Journal of Forecasting,

http://dx.doi.org/10.1002/for.2447