A factor model for Forward Price Dynamics Paolo Foschi University - PowerPoint PPT Presentation

A factor model for Forward Price Dynamics Paolo Foschi University of Bologna Padova 2013

Motivation Analyse, study and “predict” electricity forward prices on Italian Market Quotation on forward prices are available from GME, Gestore Mercati Energetici Bloomberg Trading platforms other providers and traders Pros: a lot of data: 5-15 end-of-day prices per day bid-ask interval, last price or both Challenges: Noise Far from delivery forwards are not frequently traded Outliers (inputing errors)

Bid-Ask vs Last quotations Bid-Ask interval Mid price = possible price Reliability/uncertainty proportional to Bid-Ask spread Pro: available much before the first transaction Last price it is a true price transactions happen only near to delivery Example: 82 Jan−Mar 13 78 74 70 2012 2013

Example 82 Jan−Mar 13 78 74 70 2012 2013 time

Example 2 80 75 Apr−Jun 13 70 65 60 2012 2013 time

Problems Red=Ask, Green=Bid, Gray=Last 82 81 Jan−Mar 13 80 79 78 Mar May time

Along time Similar to intraday stock exchange data Our approach: price continuously changes but we observe it at a set of given times Kalman filtering allows to recover the unobserved stochastic process, given “few” observations

Cross section Each day different forwards with different deliveries are tradable: F t , T only aggregate quantities are available (Weekly, Monthly, Quarterly, Calendar): t = 2012−07−20 ● 78 ● ● ● Prices 74 ● ● ● 70 ● 2013 2014 T

Building a model Forwards with long delivery periods can be recovered from smaller ones We have chosen Monthly contracts as a base Each price depends on t : market condition at day t , T : delivery date (i.e. seasonality) T − t : time-to-delivery (i.e. discounting and storage costs) these factors are not fixed but stochastically changes from day to day. F t , T = x t + ϕ t , T + ψ t , T − t , T = T 1 , T 2 , . . . , T n

Building a model F t , T = x t + ϕ t , T + ψ t , T − t , T = T 1 , T 2 , . . . , T n x t scalar Drives parallel shifts T discrete ⇒ ϕ t is a vector Seasonality effect Our choice: periodic, 12 months 11 factors needed: y 1 t , . . . , x 11 t T − t continuous, ψ t is a function of T − t . Back/forwardation, Humped shape, etc... Our choice: t e − λ 1 ( T − t ) + z 2 t e − λ 2 ( T − t ) + z 3 ψ t , T − t = z 1 t e − λ 3 ( T − t ) 3 factors needed: z 1 t , z 2 t , z 3 t . Can reproduce Nelson-Siegel-Svensson factors

Building a model: the observations At each time t , F t , T is a linear combination of x t , y 1 t , . . . , y 11 t , z 1 t , z 2 t , z 3 t : F t , T = x t + α T · y t + β T − t · z t . We do not observe all the monthly contracts F t , T We have only few (and aggregate) observations These observations are linear combinations of unobserved monthly prices ⇒ The observations are l.c. of the variables x t , y t and z t .   x t  + σ t , i ε t , i , y t , i = a t , i · y t ε t , i ∼ iid  z t σ t , i control the reliability of each observation

Building a model: the parameters At each time t , F t , T is a linear combination of x t , y 1 t , . . . , y 11 t , z 1 t , z 2 t , z 3 t : F t , T = x t + α T · y t + β T − t · z t . The dynamics of the parameters control how the model learn from (or adapt to) the observations x t drives the whole curve: Our choice: ARIMA(2,1,2) y t drives seasonality patterns: Quarter component: Mean reverting process (local level) Month component: low-vol martingale z t drives the NSS factors Mean reverting process (local level)

The dynamics (technical) “quarter” factors: dy q t = Λ( µ t − y t ) + C q D q dW q t , d µ t = C q d ˜ W q t “month” factors dy m = C m dW m t t “NSS” factors dz t = Λ( η t − z t ) + C z D z dW z t , d η t = C z d ˜ W z t C q , C m and C z allows to rotate (recombine) the factors in case our choice is not the optimal

b1 Q4a Q3a Q2a Q1a Qa Common −100 0 50 −10 0 5 −5 5 −2 0 2 −5 0 5 −4 0 4 −5 0 5 10 2009 2010 2011 Index 2012 2013

Example: Identifying outliers 10 5 aaa[, 2:4] 0 −5 2009 2010 2011 2012 2013 Index

Example: Identifying outliers (cont.) Q2 12 Q3 12 Q4 12 Q1 13 Q2 13 Q3 13 Q4 13 2012-03-28 80.03 81.75 80.30 73.12 75.05 2012-03-29 75.83 80.20 81.95 80.10 75.80 78.40 2012-03-30 75.65 75.80 80.15 73.65 2012-04-02 79.95 81.80 79.85 73.30 75.60 77.95 2012-04-03 79.95 81.85 73.17 75.60 78.25

Performances one day ahead prediction error 5 4 Abs.Err 3 2 1 0 2009 2010 2011 2012 2013 time

Performances normalized prediction error 5 4 Abs.Err 3 2 1 0 2009 2010 2011 2012 2013 time

Performances normalized prediction error ● 1.5 ● ● 1.0 ● ● Err ● ● ● 0.5 Month Quarter Year Type