FMSN60/MASM18 Financial Statistics Lecture 1, Introduction and stylized facts. Magnus Wiktorsson

FMSN60/MASM18 Financial Statistics Lecture 1, Introduction and stylized facts Magnus Wiktorsson

People and homepage Magnus Wiktorsson: magnusw@maths.lth.se, 222 86 25, MH:130 (Lecturer) Samuel Wiqvist: samuel.wiqvist@matstat.lu.se, 222 79 83, MH:326 (Computer exercises) Carl Åkerlindh: carl.akerlindh@matstat.lu.se, 222 04 85, MH:223 (Computer exercises) Susann Nordqvist: susann.nordqvist@matstat.lu.se, 222 85 50, MH:221 (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 EXTQ35 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?

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. STIBOR and REPO Yields 1992 500 REPO STIBOR 1W 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. STIBOR and REPO Yields 1992 500 REPO STIBOR 1W 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 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.

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 other 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 S&P 500 data.

Autocorrelation in returns 10-5 10 Covariance log returns 8 6 4 2 0-2 0 10 20 30 40 50 60 70 80 90 100 Lag No or little autocorrelation.

Unconditional distribution Normplot unconditional log returns 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 0.001-0.2-0.15-0.1-0.05 0 0.05 0.1 Data Normplot of the unconditional returns.

Gain/Loss asym. 3000 S & P 500 2500 2000 1500 1000 500 0 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 Losses are larger than gains (data is Index). This may contradict the EMH, see Nystrup et al. (2016).

Aggr. Gaussianity Normplot log returns Daily Normplot log returns Monthly Probability 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001 Probability 0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001-0.2-0.15-0.1-0.05 0 0.05 0.1 Data -0.3-0.25-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 Data 0.999 0.997 0.99 0.98 0.95 0.90 Normplot log returns Quarterly 0.99 0.98 0.95 0.90 Normplot log returns Yearly 0.75 0.75 Probability 0.50 0.25 Probability 0.50 0.25 0.10 0.10 0.05 0.05 0.02 0.01 0.02 0.01 0.003 0.001-0.3-0.2-0.1 0 0.1 0.2 0.3-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 Data Data Returns are increasingly Gaussian. Interpretation?

Vol. Clustering 0.15 S & P 500 log returns 0.1 0.05 0-0.05-0.1-0.15-0.2-0.25 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 Volatility clusters. Average cluster size?

Dependence in absolute returns 6-5 10 Covariance absolute log returns 5 4 3 2 1 0-1 0 10 20 30 40 50 60 70 80 90 100 Lag Significant autocorrelation. Long range dependence or other reason? Hint: Nystrup et al., (2015, 2016)

Conditional distribution 8 Conditional log returns Normplot conditional log returns 6 0.999 0.997 4 0.99 0.98 2 0.95 0.90 0 0.75-2 Probability 0.50-4 0.25 0.10-6 0.05-8 -10 0.02 0.01 0.003 0.001-12 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020-10 -8-6 -4-2 0 2 4 6 Data Normplot of the conditional returns (GARCH(1,1) filter).

No correlation in conditional absolute returns 0.45 Covariance cond absolute log returns 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-0.05 0 10 20 30 40 50 60 70 80 90 100 Lag

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 option 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 or STIL login - 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://ludwig.lub.lu.se/login?url=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://ludwig.lub.lu.se/login?url=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://ludwig.lub.lu.se/login?url=http: //dx.doi.org/10.1002/for.2447