on the construction of the hpfc for electricity prices - conference … · 2017-06-12 · page...

On the construction of the HPFC for electricity prices

Conference on Computational Management Science 2017/Bergamo

Rudiger Kiesel 1 Florentina Paraschiv2 Audun Sætherø 1 | 1Chair of Energy Trading and Financial Services, University Duisburg-Essen, Essen, Germany.

2NTNU Business School, Norwegian University of Science and Technology, 7491 Trondheim, Norway | Universitt Duisburg-Essen

Construction of the HPFC |

Table of contents

What is the HPFC

Construction methods

Empirical Results

Conclusion

References

Audun Sætherø | Bergamo, Italy | June 01, 2017

Construction of the HPFC | What is the HPFC

Introduction



HPFC

DefinitionIs the current price of electricity with delivery at a certain point (on anhourly scale) in the future.

I Is an OTC product used by producers of electricity to sellspecialized electricity contracts

I Products traded on the exchanges are typically insufficient formost producers/consumers, making the HPFC crucial!

I Should reflect the believes about how electricity prices evolve,and be arbitrage free to traded Futures



I The goal of this talk is to explain how the HPFC is constructedI We will present three different methods, two from the Literature

and one novel approachI We will make a comparative analyzes of these models explaining

the strengths and weaknesses, and how these methods can beimproved.


Construction of the HPFC | Construction methods

I Consists of two parts:I Seasonality curve:

I Represents historical spot prices, weather etc.I Adjustment function:

I Makes sure the curve is arbitrage free to traded Futures products



Construction Methods



I Seasonality Curve (Blochlinger, 2008, PHD-Thesis) [1]:I Dummy Variables:I

f2yd = a0 +6∑

i=1

biDdi +12∑

i=1

ciMdi +3∑

i=1

diCDDdi +3∑

i=1

eiHDDdi

I Assigns different days into blocks and assigns a mean price tothose blocks of days

I Problem: A non-smooth change between months



0 100 200 300

3545

Seasonality curve − Dummy VariablesDays (1 Year)

Price

(Eur

o/M

Wh)

0 100 200 300

3438

4246

Seasonality curve − Dummy Variables (Without daily seasonality)Days (1 Year)

Price

(Eur

o/M

Wh)



Adjustment Curve

I Has as a primary focus to make sure that the PFC fits to theobserved Futures products

I Our methods:I Fleten et al. (2003) [2]I Benth et al. (2007) [3]I Novel approach based on a combined fitting of the seasonal

directly to the Futures pricesI Our three main questions:

I Do they take care of the smoothing?I What happens when the price of a Futures product change?I What happens when a Futures product is cascaded into several

smaller products?



0 100 200 300

3545

Seasonality curve − Dummy VariablesDays (1 Year)

Price

(Eur

o/M

Wh)

0 100 200 300

3040

5060

Days

Price

s Mean of seasonality curveFutures prices

0 100 200 300

−11

−8−5

Days

Adjus

tmen

t Fun

ction



Fletens approach

The approach suggested by Fleten et al. (2003)[2] reads as follows:I For a seasonality curve st

I

minft

[T∑

t=1

(ft − st)2 + λ

T−1∑t=2

(ft−1 − 2ft + ft+1)2

](1)

I given:∑

A ft = FuturesPrice(A)I Problem:

I Suppresses the weekly/daily seasonality (Blochlinger, 2008,PHD-Thesis) [1]

I λ seems arbitrary



Daily Profile

0 100 200 300

3545

days

price

0 100 200 300

3236

40

days

price



Benth Approach

Benth et al. (2007) [3] suggests a method as follow:I FC(t) = st + εt

I

εt =

a1t4 + b1t3 + c1t2 + d1t + e1 t ∈ [t0, t1)a2t4 + b2t3 + c2t2 + d2t + e2 t ∈ [t1, t2)

...ant4 + bnt3 + cnt2 + dnt + en t ∈ [tn−1, tn]

I

x = {a1, · · · ,b1, · · · , c1, · · · }I

minx

∫ tn

t0[ε′′(t ; x)]2dt



Introducing a new Futures Product

0 100 200 300

−20

−15

−10

−50

Days (1 Year)

Euro

/Mwh

Original CurveImplied CurveImplied Futures



Spline Curve

I

FC(t) = C+6∑

i=1

ai sin(2πi(t + S(m))

12 ·M(m)) + bi cos(

2πi(t + S(m))

12 ·M(m)) +

aQ(m)4 sin(

8π(t + S(m))

12 ·M(m)) + bQ(m)

4 cos(8π(t + S(m))

12 ·M(m))+

aQ(m)12 sin(

24π(t + S(m))

12 ·M(m)) + bQ(m)

12 cos(24π(t + S(m))

12 ·M(m))

IT (m) + S(m)

M(m)= 0,1,2,3,4,5,6,7,8,9,10,11,12



minimizex

‖Ax − y‖2

subject to Cx = d

where x are the 26 coefficients,C and d assures the resulting curve isarbitrage free with respect to the Futures products

x = (a1, . . . ,b1, . . . ,a14, ...a

44,b

14, ...,b

44,a

112, ...a

412,b

112, ...,b

412)

I Requires a higher than normal number of parametersI May lead to overfitting!

I Needs to pre-specify the number of Futures products



Derivative of Adjustment Function

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●

●

●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0 100 200 300

0.000

0.020

Fleten Method

deriv

ative

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

0 100 200 300

−0.10

0.05

Fred Method

deriv

ative

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●

●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●

0 100 200 300

0.00

0.02

0.04

Novel Method

deriv

ative

Figure: The derivative of the adjustment functions with respect to the MarchFuture, with 3-12 monthly products as input.


Construction of the HPFC | Empirical Results

Test Data Benth Novel Fleten 1 Fleten 2MAPE IS 32% 29% 45% 32%MAPE OoS 67% 42% 57% 41%

AD IS 4.79 4.57 6.05 4.85AD OoS 8.76 6.79 7.15 6.32AD IS 5.95 5.83 7.15 5.98AD OoS 9.92 8.09 8.55 7.67SD IS 65.71 61.69 91.46 65.89SD OoS 181.69 116.86 139.76 109.25

Table: Comparing the different models to the realized spot prices, Test 1-4 isfor a daily scale, while Test 5-8 is on an hourly scale. AD=AbsoluteDifference, SD=Square Difference, IS=In Sample, OoS=Out of sample


Construction of the HPFC | Conclusion

Conclusion

I Important to know what your construction method does, andwhat you actually want it to do

I One should be careful about what the different parts of theseasonality curve does

I One should be careful about adding new parameters when theFutures are cascading (or when other new information appears)

I If your adjustment function has a smoothing term, you have to besure what you want to smooth (and what not to smooth)


Construction of the HPFC | References

References,Blochlinger, Lea (2008). Power prices–A regime-switching spot/forward pricemodel with Kim filter estimation. PHD-Thesis

Fleten, Stein-Erik and Lemming, Jacob (2003). Constructing forward price curvesin electricity markets, Journal of Energy Economics, 5, 409–424

Benth, Fred Espen and Koekkebakker, Steen and Ollmar, Fridthjof(2007).Extracting and applying smooth forward curves from average-basedcommodity contracts with seasonal variation, The Journal of Derivatives, 15,52–66

Sætherø, Audun Sviland and Paraschiv, Florentina and Kiesel, Ruediger, On theConstruction of the Hourly Price Forward Curve for Electricity Prices (May 17,2017). Available at SSRN.

Thank you for your attention...


on the construction of the hpfc for electricity prices - conference … · 2017-06-12 · page...

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