Context selection for volume forecast

Transcription

1 Prediction of time series and non stationary time series, 10 february 2012 Context selection for volume forecast The influence of news on traded volume Nathanaël Mayo

2 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

3 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

4 Definition News and events News definition Macroeconomic number publication and special market conditions Context list, from 03.Jan.2007 to 29.Jun Total= 1274 observations Goals Study the strong heterogeneity of volume profiles when event occur Select prevalent contexts amongst rare events with important cooccurence Notations for news e k j : indicatrix variable of news k on day j e j = (e 1 j,..., ek j ) : set of news We only use the indicatrix variable of news occurrences We do not use time of publication or published information

5 Definition Exemple of volume curves

6 Definition Contextualization of location model Location model Usual setting to forecast iid data X j Location parameter θ : X j = θ + ɛ j Using external information in forecast Conditionnal location model X j = θ(e j ) + ɛ j θ(e) is the best theoretical forecast when e j = e occurs θ(e) is the average of X when e j = e occurs, and could be estimated as such The full contextualization is irrelevant in practice Most general case : e j = (e 1 j,..., e K j ) Every combination of news can be a context Huge parameter set (2 K dim(θ)) Most combinations (e 1,..., e K ) are never or rarely observed We need constrain e θ

7 Combining events Constraining θ Mixing : All occuring news have an impact Events have mixed impacts when they occur simultaneously θ(e j ) = mean k Nj (θ(ej k )) (average over contexts occuring today) Impacts decrease with number of events N j = k ek j Estimation of θ(e k ) is based on the full sample of rawdata (all occurrences of e k ) Equivalent to linear regression model of X j on 1 N j e k j Multicolinearity due to cooccurence effect Ranking : Some news mask other There is one prevailing event amongst occuring contexts θ(e j ) = θ(ej k ) (k is the prevailing context occuring today) Prevailance can be defined using rankings : e k < e k if rank(e k ) < rank(e k ) Leads to successive orthogonalization of contexts O(e k ) = 1 e k = 1 and e k = 0 for better news (e k < e k ) Multicolinearity is partly adressed with orthogonalization Estimation of θ(e k ) is based on a robust subsample

8 Notations Notations Indexing variables j = 1,..., J days (most often implicit) t [0, T ] time of the day (possibly binned : t = 1,..., T ) Volume V t,j : volume process (cumulated traded volume up to time t on day j) V T,j : daily volume v t,j = V t,j V T,j : relative volume process (intraday repartition, grows from 0 to 1) X T,j = V t,j : instantaneous volume x T,j = v t,j : instantaneous relative volume Price Market price P t,j, volatility σ t,j ω t,j : Volume Weighted Average Price (VWAP) Forecast : Z : realized random variable Z f : forecast value

9 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

10 Principle 1 Estimate extreme volume profiles Non-negative Matrix Factorization is fitted to positive data x Extract generating vertices of a cone. Matrix low-rank decomposition x F.B = p F p B p Positive factors F = (F 1,..., F P ) and positive weights B = (B 1,..., B P ) L 1 distance to obtain sparse representations Arbitrary extract 10 factors Normalize B p to 1 extreme volume profiles 2 Match factors expression with news Matching between factor F p and event e k m p,k := 1 E[F p ] E[F p e k = 1] For readability reason, we plot min(m p,k, 2.0)

11 NMF of volume B 1 : Close effect B 8 : Derivative expiry (12 : 00) B 10 : US Employment Situation (14 : 45) Factors usually located where variance is higher (NMF depends on scale)

12 Matching of news

13 Identifiability issues Smart news orthogonalization Full orthogonalization is irrelevant Most news occur simultaneously Not enough observations for news occuring alone Partial orthogonalization procedure Detect cooccurent news clusters with MCA These groups raise acute identifiability issues Orthogonalize news relatively to their cluster only Conclusions Important volume features are caused by events Cooccurences biaises the analysis Some effects appear or disappear when filtering events We use rankings and orthogonalization

14 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

15 State of the art Volume forecast, what for? VWAP contracts (Volume Weighted Average Price) The instantaneous volume available for sell or buy is limited, and trading a large quantity at once is costly Solution : dispatch the order through some time period I = [0, T ] VWAP ω = 1 V T T 0 Pt dvt is the usual benchmark P geometric diffusion (e.g GBM) V doubly stochastic point process (e.g Cox) Theoretical objective : VWAP replication (from [1, McCulloch&al,2009] ) [ 1 ] 2 min IE [ω w] 2 = min IE P t (dv t dν ν ν t) 0 Depends only on relative volume v t = Vt V T, adapted in G t = σ(f t, V T ) Projection (G t-adapted process F t-adapted process) risk= IE 1 0 (vt νt)2 d P t ν = IE v t when P V Improving forecast of v t reduces the risk

16 State of the art Volume forecast : state of the art Static forecast Use the historical average of x (i.i.d location model with x and V T independent) Estimation issues : estimator choice, bin size, taking logarithm... Improve forecast using auxiliary information Corr(x t, σ t) impact the optimal profile : ν = IE(Xt σ2 t ) (see [5, Konishi,2008]) IE(σ t 2) Autocorrelations of volumes (Dynamic forecast model) IE((V t,..., V T ) V 1,..., V s) Multistock. Uses strong correlations across stocks volumes to improve forecast ([2, Bialkowsky&al,2008]) Estimation issues in high dimension : filtering covariance matrices [3, Mayo,2010] Events affect usual procedures Creates dependance between x and V T (e.g expiries) Strongly affects autocorrelation. Explains part of stock cross-dependance (most news are shared by all stocks)

17 How to evaluate performance? Performance evaluation (1 :usual indicators) Volume forecast quality L 2 = (x t xt f ) 2 dt L 1 = x t xt f dt L KS = sup X t Xt f Operationnal criteria C (VWAP cash slippage) VWAP replication error for an execution spreading over the entire day IE ω j ω f j Absolute value allways on the worst side (buy or sell) Evaluated on 1 min bins C = t (x j,t x f j,t) ω j,t Intrinsic risk R (from [1, McCulloch and al,2009]) Upper bound on IE(ω t ω f t ) 2 under strong assumptions : Independance between P and V No microstructure Best forecast of x is the historical mean R = Var(X t) σ 2 t p 2 t dt For constant volatility, R L 2 distance on volumes

18 How to evaluate performance? Performance evaluation (2 :Multiscale slippage) Problems with uniscale C Full sample error= j t (x j,t x f j,t) ω j,t Assumes executions spread over the entire day Involves deviation at timescale > day, and biais at timescale < day Corr(L 2, C) is low Improving volume forecast does not necessarily improves C Usual volume predictors have poor aggregative properties (The median has usually lowest L 2 error, but highest L KS error, while C is aggregated over the day) Solution : multiscale C Involve risk at timescale < day Take absolute values more often Computation : Define C dt as C on a subinterval dt of the day Compute (C k ) for randomly generated subintervals (dt k ) k...k Define multiscale C as a weighted average k w kc k Weights use discretized control variables Z from historical contract values Likely weights regarding time open, time end, duration, use open and use close

19 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

20 Algorithm Algorithm (1 : learning) 1/ Initial rankings Maximize some distance between x e k = 1 and x e k = 0 Distance between occurrence sample and non occurrence sample Successive minima of φ(x e k = 1, x e k = 0), 2-sample distance function Orthogonalization (remove ranked news from the sample) min φ(x O(e k ) = 1; x e k = 0) Parameters φ : try various 2-sample functions Nb.occurence : remove contexts with too few (orthogonalized) occurrences Threshold : ignore context if φ(x O(e k ) = 1; x e k = 0) threshold bin size : we use 15 min aggregated volume 2 sample function Explanation Formula φ std (x, y) std ratio t std(x t ) t std(y t ) φ mpv (x, y) Single MANOVA pvalue(one-way [H0] : both sample have same mean multivariate analysis of variance) φ pv (x, y) Quantile 10% ANOVA s pv on bin t

21 Algorithm Algorithm (2 : validation) 2/ Backtest Use jacknife (leave-one-out anticipative backtest) Tries various estimator (mean, median) NB : only estimators are jacknifed, not the full ranking procedure 3/ Inhibition Chaotic contexts := good initial rankings, but poor backtest C Initial rankings use only volumes while C uses price information φ detect only some perturbation of the distribution (time varying event) Inhibition : since θ(e k ) is a poor predictor, use a noncontextual predictor instead Thanks to orthogonalization This does not affect other contexts No need to backtest inhibition (would lead to the same results) Inhibition acts as filtering of extreme news, since occurrences are removed of the following contexts

22 Algorithm Algorithm (3 : Model selection) 4/ Model competition and selection Proceed steps 1 3 for various models Model parameter = (φ, estimator, threshold, nb.occurence) Select the model having lowest operationnal criteria C on the full sample C is computed on 1min bin, using constant rate interpolation 5/ Usual days forecast Usual Days := no ranked context occur Various non anticipative models are backtested using sliding windows Usual day context contains enough observations for classical backtest The best model is selected using lowest C Parameters of usual day forecast models Estimator(mean, median, EWMA, threemean) Historical sample length Removes ranked context occurrences or not

23 Results Exemple of volume curves

24 Results A few volume curves (2)

25 Results Global performance

26 Results Selected parameters Figure: Parameters in contexts (top) and in Usual days (bottom) Mostly selects median estimator On short historic length (<50 days) Allmost never removes occurences of ranked news

27 Results Main contexts : rankings

28 Results Performance in contexts

29 Outlook 1 Events Definition Combining events Notations 2 Descriptive approach with factor models 3 Traded volumes forecast State of the art How to evaluate performance? 4 Ranking procedure and forecast Algorithm Results 5 Multivariate perspective

30 Intrinsic risk and illiquidity Intrinsic risk R underevaluates risk by 20% Illiquidity cost Penalize bins where traded volume > available volume on market Minimalistic market impact t (x f t x t) + spread t

31 Clustering NMF factors across stocks

32 Portfolio volume error (15min) L 2 forecast errors are correlated across stocks News and events account for 30% of the correlation over the CAC40

33 Portfolio execution risk (1min) C are correlated across stocks Under normal market condition (contradicts [5, Konishi,2008]) Events have no influence on correlations? Non syncronization biais non treated

34 Conclusion Events Events strongly affect the repartition of traded volume Rankings is a possible solution to identifiability issues (cooccurrences) Theoretical issues Why are volume errors, operationnal risk and intrinsic risk so different? Include microstructure effects Extensions Include events in usual approaches (autocorrelations, multistock) More events smart news orthogonalization in high dimension Real time news Impact on other variables (volatility, price)

35 J.Mc Culloch and V.Kazakov, Optimal VWAP Trading Strategy and Relative Volume, Draft January, J.Bialkowsky, S.Darolles and G.Le Fol : Improving VWAP strategies : A dynamic volume approach, Journal of Banking and Finance, 32, (2008), N.Mayo : Les modèles à variables cachées et leurs applications en Finance : risque systématique, détection d arbitrage et prévision des volumes., Chapter 5, (2010). Available at samm.univ-paris1 J.Mc Culloch : Relative Volume as a Doubly Stochastic Binomial Point Process, (2004). Available at ssrn. H. Konishi : Optimal slice of a VWAP trade, Journal of Banking and Finance, 32, (2008), D.D.Lee and H.S.Seung : Algorithms for Non-negative Matrix Factorization,NIPS (2000). Available at citeseer.