4Part I: Modeling... in which we see what basic properties of stock prices/indices we want to capture
5Contents Returns and their (static) properties Pricing models Time series properties of returns
6Why returns? Prices are generally found to be non-stationary Makes life difficult (or simpler...)Traditional statistics prefers stationary dataReturns are found to be stationary
7Which returns? Two type of returns can be defined Discrete compounding Continuous compounding
8Discrete compoundingIf you make 10% on half of your money and 5% on the other half, you have in total 7.5%Discrete compounding is additive over portfolio formation
9Continuous compounding If you made 3% during the first half year and 2% during the second part of the year, you made (exactly) 5% in totalContinuous compounding is additive over time
10Empirical properties of returns MeanSt.dev.Annualized volatilitySkewnessKurtosisMinMaxIBM-0.0%2.46%39.03%-23.51-138%12.4%(corr)0.0%1.64%26.02%-0.2815.56-26.1%S&P0.95%15.01%-1.439.86-22.9%8.7%Data period: July December 2004; daily frequency
12Pricing modelsFinance considers the final value of an asset to be ‘known’as a random variable , that isIn such a setting, finding the price of an asset is equivalent to finding its expected return:
13Pricing models 2As a result, pricing models model expected returns ...... in terms of known quantities or a few ‘almost known’ quantities
14Capital Asset Pricing Model One of the best known pricing modelsThe theorem/model states
15Black-Scholes Also Black-Scholes is a pricing model (Exact) contemporaneous relation between asset prices/returns
16Time series properties of returns Traditionally model fitting exercise without much financemostly univariate time series and, thus, less scope for tor the ‘traditional’ cross-sectional pricing modelslately more finance theory is integratedFocuses on the dynamics/dependence in returns
17Random walk hypothesis Standard paradigm in thePrices follow a random walkReturns are i.i.d.Normality often imposed as wellCompare Black-Scholes assumptions
19Linear time series analysis Box-Jenkins analysis generally identifies a white noiseThis has been taken long as support for the random walk hypothesisRecent developmentsSome autocorrelation effects in ‘momentum’Some (linear) predictabilityLargely academic discussion
21Risk predictabilityThere is strong evidence for autocorrelation in squared returnsalso holds for other powers‘volatility clustering’While direction of change is difficult to predict, (absolute) size of change isrisk is predictable
22The ARCH model First model to capture this effect No mean effects for simplicityARCH in mean
23ARCH properties Uncorrelated returns Correlated squared returns martingale difference returnsCorrelated squared returnswith limited set of possible patternsSymmetric distribution if innovations are symmetricFat tailed distribution, even if innovations are not
24The GARCH modelGeneralized ARCHBeware of time indices ...
25GARCH model Parsimonious way to describe various correlation patterns for squared returnsHigher-order extension trivialMath-stat analysis not that trivialSee inference section later
26Stochastic volatility models Use latent volatility process
27Stochastic volatility models Also SV models lead to volatility clusteringLeverageNegative innovation correlation means that volatility increases and price decreases go togetherNegative return/volatility correlation(One) structural story: default risk
28Continuous time modeling Mathematical finance uses continuous time, mainly for ‘simplicity’Compare asymptotic statistics as approximation theoryEmpirical finance (at least originally) focused on discrete time models
29ConsistencyThe volatility clustering and other empirical evidence is consistent with appropriate continuous time modelsA simple continuous time stochastic volatility model
30Approximation theoryThere is a large literature that deals with the approximation of continuous time stochastic volatility models with discrete time modelsImportant applicationsInferenceSimulationPricing
31Other asset classes So far we only discussed stock(indices) Stock derivatives can be studied using a derivative pricing modelsFinancial econometrics also deals with many other asset classesTerm structure (including credit risk)CommoditiesMutual fundsEnergy markets...
32Term structure modeling Model a complete curve at a single point in timeThere exist modelsin discrete/continuous timedescriptive/pricingfor standard interest rates/derivatives...
34Contents Parametric inference for ARCH-type models Rank based inference
35Analogy principleThe classical approach to estimation is based on the analogy principleif you want to estimate an expectation, take an averageif you want to estimate a probability, take a frequency...
36Moment estimation (GMM) Consider an ARCH-type modelWe suppose that can be calculated on the basis of observations if is knownMoment condition
37Moment estimation - 2 The estimator now is taken to solve In case of “underidentification”: use instrumentsIn case of “overidentification”: minimize distance-to-zero
38Likelihood estimation In case the density of the innovations is known, say it is , one can write down the density/likelihood of observed returnsEstimator: maximize this
39Doing the math ... Maximizing the log-likelihood boils down to solving with
40Efficiency consideration Which of the above estimators is “better”?Analysis using Hájek-Le Cam theory of asymptotic statisticsApproximate complicated statistical experiment with very simple onesSomething which works well in the approximating experiment, will also do well in the original one
41Quasi MLEIn order for maximum likelihood to work, one needs the density of the innovationsIf this is not know, one can guess a density (e.g., the normal)This is known asML under non-standard conditions (Huber)Quasi maximum likelihoodPseudo maximum likelihood
42Will it work?For ARCH-type models, postulating the Gaussian density can be shown to lead to consistent estimatesThere is a large theory on when this works or notWe say “for ARCH-type models the Gaussian distribution has the QMLE property”
43The QMLE pitfall One often sees people referring to Gaussian MLE Then, they remark that we know financial innovations are fat-tailed ...... and they switch to t-distributionsThe t-distribution does not possess the QMLE property (but, see later)
44How to deal with SV-models? The SV models look the sameBut now, is a latent process and hence not observedLikelihood estimation still works “in principle”, but unobserved variances have to be integrated out
45Inference for continuous time models Continuous time inference can, in theory, be based oncontinuous record observationsdiscretely sampled observationsEssentially all known approaches are based on approximating discrete time models
46... in which we discuss the main ideas of rank based inference
47The statistical model Consider a model where ‘somewhere’ there exist i.i.d. random errorsThe observations areThe parameter of interest is someWe denote the density of the errors by
48Formal modelWe have an outcome space , with the number of observations and the dimension ofTake standard Borel sigma-fieldsModel for sample size :Asymptotics refer to
49Example: Linear regression Linear regression model (with observations )Innovation density and cdf
50Example ARCH(1) Consider the standard ARCH(1) model Innovation density and cdf
51Maintained hypothesis For given and sample size , theinnovations can be calculated from theobservationsFor cross-sectional models one may even often writeLatent variable (e.g., SV) models ...
52Innovation ranks The ranks are the ranks of the innovations We also write for the ranksof the innovations based ona value for the parameter of interestRanks of observations are generally not very useful
53Basic properties The distribution does not depend on nor on permutation ofThis is (fortunately) not true forat least ‘essentially’
54InvarianceSuppose we generate the innovations as transformation with i.i.d. standard uniformNow, the ranks are even invariant with respect to
55ReconstructionFor large sample size we haveand, thus,
56Rank based statisticsThe idea is to apply whatever procedure you have that uses innovations on the innovations reconstructed from the ranksThis makes the procedure robust to distributional changesEfficiency loss due to ‘ ’?
57Rank based autocorrelations Time-series properties can be studied using rank based autocorrelationsThese can be interpreted as ‘standard’ autocorrelationsrank basedfor given reference density and distribution free
58RobustnessAn important property of rank based statistics is the distributional invarianceAs a result: a rank based estimator is consistent for any reference densityAll densities satisfy the QMLE property when using rank based inference
59Limiting distribution The limiting distribution of depends on both the chosen reference density and the actual underlying densityThe optimal choice for the reference density is the actual densityHow ‘efficient’ is this estimator?Semiparametrically efficient
60RemarkAll procedures are distribution free with respect to the innovation densityThey are, clearly, not distribution free with respect to the parameter of interest
62Why ranks?So far, we have been considering ‘completely’ unrestricted sets of innovation densitiesFor this class of densities ranks are ‘maximal invariant’This is crucial for proving semiparametric efficiency
63Alternatives Alternative specifications may impose zero-median innovationssymmetric innovationszero-mean innovationsThis is generally a bad idea ...
64Zero-median innovations The maximal invariant now becomes the ranks and signs of the innovationsThe ideas remain the same, but for a more precise reconstructionSplit sample of innovations in positive and negative part and treat those separately
65But ranks are still ... Yes, the ranks are still invariant ... and the previous results go throughBut the efficiency bound has now changed and rank based procedures are no longer semiparametrically efficient... but sign-and-rank based procedures are
66Symmetric innovations In the symmetric case, the signed-ranks become maximal invariantsigns of the innovationsranks of the absolute valuesThe reconstruction now becomes still more precise (and efficient)