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

▶

Mar 30, 2023 143 likes •344 views

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

SLIDE 1

A factor model for Forward Price Dynamics

Paolo Foschi

University of Bologna

Padova 2013

SLIDE 2

Motivation

Analyse, study and “predict” electricity forward prices

n Italian Market

Quotation on forward prices are available from

GME, Gestore Mercati Energetici Bloomberg Trading platforms

ther 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)

SLIDE 3

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:

2012 2013 70 74 78 82 Jan−Mar 13

SLIDE 4

Example

2012 2013 70 74 78 82 time Jan−Mar 13

SLIDE 5

Example 2

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

SLIDE 6

Problems

Red=Ask, Green=Bid, Gray=Last

Mar May 78 79 80 81 82 time Jan−Mar 13

SLIDE 7

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

SLIDE 8

Cross section

Each day different forwards with different deliveries are tradable: Ft,T

nly aggregate quantities are available (Weekly, Monthly,

Quarterly, Calendar):

2013 2014 70 74 78 T Prices

t = 2012−07−20

SLIDE 9

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. Ft,T = xt + ϕt,T + ψt,T−t, T = T1, T2, . . . , Tn

SLIDE 10

Building a model

Ft,T = xt + ϕt,T + ψt,T−t, T = T1, T2, . . . , Tn xt scalar

Drives parallel shifts

T discrete ⇒ ϕt is a vector

Seasonality effect Our choice: periodic, 12 months 11 factors needed: y1

t , . . . , x11 t

T − t continuous, ψt is a function of T − t.

Back/forwardation, Humped shape, etc... Our choice: ψt,T−t = z1

t e−λ1(T−t) + z2 t e−λ2(T−t) + z3 t e−λ3(T−t)

3 factors needed: z1

t , z2 t , z3 t .

Can reproduce Nelson-Siegel-Svensson factors

SLIDE 11

Building a model: the observations

At each time t, Ft,T is a linear combination of xt, y1

t , . . . , y11 t , z1 t , z2 t , z3 t :

Ft,T = xt + αT · yt + βT−t · zt. We do not observe all the monthly contracts Ft,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 xt, yt and zt. yt,i = at,i ·   xt yt zt   + σt,iεt,i, εt,i ∼ iid σt,i control the reliability of each observation

SLIDE 12

Building a model: the parameters

At each time t, Ft,T is a linear combination of xt, y1

t , . . . , y11 t , z1 t , z2 t , z3 t :

Ft,T = xt + αT · yt + βT−t · zt. The dynamics of the parameters control how the model learn from (or adapt to) the observations xt drives the whole curve: Our choice: ARIMA(2,1,2) yt drives seasonality patterns:

Quarter component: Mean reverting process (local level) Month component: low-vol martingale

zt drives the NSS factors

Mean reverting process (local level)

SLIDE 13

The dynamics (technical)

“quarter” factors: dyq

t = Λ(µt − yt) + CqDqdW q t ,

dµt = Cqd ˜ W q

t

“month” factors dym

t

= CmdW m

t

“NSS” factors dzt = Λ(ηt − zt) + CzDzdW z

t ,

dηt = Czd ˜ W z

t

Cq, Cm and Cz allows to rotate (recombine) the factors in case our choice is not the optimal

SLIDE 14

−5 5 10

Common

−4 4

Qa

−5 5

Q1a

−2 2

Q2a

−5 5

Q3a

−10 5

Q4a

2009 2010 2011 2012 2013

Index

−100 50

b1

SLIDE 15

Example: Identifying outliers

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

SLIDE 16

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

SLIDE 17

Performances

2009 2010 2011 2012 2013 1 2 3 4 5 time Abs.Err

ne day ahead prediction error

SLIDE 18

Performances

2009 2010 2011 2012 2013 1 2 3 4 5 time Abs.Err

normalized prediction error

SLIDE 19

Performances

Month