FMS161/MASM18 Financial Statistics Lecture 1, Introduction and stylized facts Erik Lindström
People and homepage MH:221 (Lecturer) 222 04 85 , MH:223 (Computer exercises) MH:225 (Course secretary) //www.maths.lth.se/matstat/kurser/fmsn60masm18/ ◮ Erik Lindström: erikl@maths.lth.se , 222 45 78, ◮ Carl Åkerlindh: carl.akerlindh@matstat.lu.se , ◮ Maria Lövgren: marial@maths.lth.se , 222 45 77, ◮ http:
Purpose: data, and use these tools in combination with economic theory. The main applications are valuation and risk management. statistical tools supporting courses like 'TEK180 Financial Valuation and Risk Management' or 'FMSN25/MASM24 Valuation of Derivative Assets'. ◮ The course should provide tools for analyzing ◮ The course is intended to provide necessary
Inference problems? such as stochastic volatility or credit default intensity) the model? How do we estimate parameters in general models? Cross covariance and auto covariance. Often results in Non-linear, Non-Gaussian, Non-stationary models... ◮ Forecast prices, interest rates, volatilities (under the P and Q measures) ◮ Filtering of data (e.g. estimating hidden states ◮ Distribution of prediction errors; can we improve ◮ What about extreme events?
Inference problems? such as stochastic volatility or credit default intensity) the model? models? Non-stationary models... ◮ Forecast prices, interest rates, volatilities (under the P and Q measures) ◮ Filtering of data (e.g. estimating hidden states ◮ Distribution of prediction errors; can we improve ◮ What about extreme events? ◮ How do we estimate parameters in general ◮ Cross covariance and auto covariance. ◮ Often results in Non-linear, Non-Gaussian,
Example I -- Daily interest data - big crisis in Sweden during the early 1990s happen again? Covariation with of market factors? - Can this Forecasts - 0.5 % or 500 %? Models and distributions. See Section 2.4 in the book for more information. STIBOR and REPO Yields 1992 REPO STIBOR 1W 500 STIBOR 1M STIBOR 3M STIBOR 6M 100 10 Q1−92 Q2−92 Q3−92 Q4−92 Q1−93
Example I -- Daily interest data - big crisis in Sweden during the early 1990s See Section 2.4 in the book for more information. happen again? STIBOR and REPO Yields 1992 REPO STIBOR 1W 500 STIBOR 1M STIBOR 3M STIBOR 6M 100 10 Q1−92 Q2−92 Q3−92 Q4−92 Q1−93 ◮ Forecasts - 0.5 % or 500 %? ◮ Covariation with of market factors? - Can this ◮ Models and distributions.
Electricity spot price and Hydrological situation
Example II -- Forward prices on Nordpool movements on the futures on Nordpool on yearly contracts. (days or weeks). prizes. 1. Hydrological situation is the energy stored as snow, ground water or in reservoirs 2. Time to maturity. 3. Perfect or imperfect markets. ◮ Traders are interested in predicting price ◮ Or predicting the movements on short horizons ◮ Expected movement and/or prob. of declining ◮ What about fundamental factors? ◮ Other factors -- suggestions?
Ex -- Forwards on Nordpool, contd. There is a strong dependence between the hydrological situation and the price. model should we use? Is the relation linear? ◮ How do we model this dependence, e.g. what ◮ 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. identification and model validation. other approaches. Kalman filters (and versions thereof) and particle filters ◮ Parameter estimation (LS, ML, GMM, EF), model ◮ Modelling of variance, ARCH, GARCH, ..., and ◮ Stochastic calculus and SDEs. ◮ State space models and filters
Course goals -Knowledge and Understanding For a passing grade the student must: family, stochastic volatility, and models use for high-frequency data, formula, Girsanov transformation, martingales, Markov processes, filtering, Kalman filters and particle filters, above model families. ◮ handle variance models such as the GARCH ◮ use basic tool from stochastic calculus: Ito's ◮ use tools for filtering of latent processes, such as ◮ statistically validate models from some of the
Course goals -Skills and Abilities For a passing grade the student must: related fields. model choice, includes model specification, inference, and and from other courses) where the solution economic and statistical theory (from this course report, as well as orally, should be applied, financial contracts and for transforming models, financial data, ◮ be able to find suitable stochastic models for ◮ work with stochastic calculus for pricing of ◮ understand when and how filtering methods ◮ validate a chosen model, ◮ solve all parts of a modelling problem using ◮ utilise scientific articles within the field and ◮ present the solution in a written technical
Literature Nielsen, J. N. (2015) Statistics for Finance , Chapman & Hall, CRC press. the course home page) Course program. ◮ Lindström, E., Madsen, H., ◮ Handouts (typically articles on
long range dependence? Properties of financial data Evaluate claims on OMXS30 data. ◮ No Autocorrelation in returns ◮ Unconditional heavy tails ◮ Gain/Loss asym. ◮ Aggregational Gaussianity ◮ Volatility clustering ◮ Conditional heavy tails ◮ Significant autocorrelation for abs. returns - ◮ Leverage effects ◮ Volume/Volatility correlation ◮ Asym. in time scales
Autocorrelation in returns No or little autocorrelation. 1.2 1 0.8 Autocorrelation, returns 0.6 0.4 0.2 0 −0.2 0 5 10 15 20 25 30 lag
Unconditional distribution Normplot of the unconditional returns. Normal Probability Plot 0.999 0.997 0.99 0.98 0.95 0.90 0.75 Probability 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Data
Gain/Loss asym. may contradict the EMH, see Nystrup et al. (2016). OMX S30 3 10 log(Index) 2 10 Jan95 Jan00 Jan05 Jan10 Jan15 Losses are larger than gains (data is log ( Index ) ). This
Aggr. Gaussianity Returns are increasingly Gaussian. Interpretation? Normal Probability Plot Normal Probability Plot Normal Probability Plot 0.999 0.999 0.999 0.997 0.997 0.997 0.99 0.99 0.99 0.98 0.98 0.98 0.95 0.95 0.95 0.90 0.90 0.90 0.75 0.75 0.75 0.50 r2 0.50 r4 0.50 r 0.25 0.25 0.25 0.10 0.10 0.10 0.05 0.05 0.05 0.02 0.02 0.01 0.02 0.01 0.01 0.003 0.003 0.001 0.003 0.001 0.001 −0.05 0 0.05 0.1 −0.05 0 0.05 0.1 −0.1 0 0.1 Data Data Data Normal Probability Plot Normal Probability Plot Normal Probability Plot 0.999 0.999 0.997 0.997 0.997 0.99 0.99 0.99 0.98 0.98 0.98 0.95 0.95 0.95 0.90 0.90 0.90 0.75 0.75 0.75 r16 r32 r8 0.50 0.50 0.50 0.25 0.25 0.25 0.10 0.10 0.10 0.05 0.05 0.05 0.02 0.02 0.01 0.02 0.01 0.01 0.003 0.003 0.001 0.003 0.001 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 −0.1 0 0.1 0.2 Data Data Data
Volatility clusters. Average cluster size? Vol. Clustering 0.1 0.05 0 −0.05 −0.1 Nov91 Nov94 Nov97 Nov00 Nov03 Nov06
Conditional distribution Normplot of the conditional returns (GARCH(1,1) filter). Normal Probability Plot 6 0.999 0.997 4 0.99 0.98 0.95 0.90 2 0.75 Probability 0 0.50 0.25 −2 0.10 0.05 0.02 −4 0.01 0.003 0.001 −6 Nov91 Nov96 Nov01 Nov06 −6 −4 −2 0 2 4 Data
Dependence in returns Significant autocorrelation. Long range dependence or other reason? Hint: Nystrup et al., (2015, 2016) 1.2 1 0.8 Autocorrelation, abs returns 0.6 0.4 0.2 0 −0.2 0 50 100 150 200 250 300 lag
Leverage effects measure of volatility. ◮ Most assets are negatively correlated with any ◮ One popular explanation is corporate debt. ◮ Makes sense if you are risk averse.
Volume/Volatility correlation option valuation community - cf. Time Shifted Levy processes models, Def 7.12, such as NIG-CIR model. ◮ Trading volume is correlated with the volatility. ◮ Sometimes modelled with 'business time' in
Asym. in time scales scaled volatility predicting coarse scale volatility ◮ Coarse-grained measurements can predict fine ◮ While fine scaled volatility have difficulties
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