Making money in electricity markets

Transcription

1 Making money in electricity markets Risk-minimising hedging: from classic machinery to supervised learning Martin Tégner Department of Engineering Science & Oxford-Man Institute of Quantitative Finance January 18, 2018

2 Introduction Imagine you want to be certain about the price to pay for next month s electricity

3 Introduction Imagine you want to be certain about the price to pay for next month s electricity You sign a fixed price agreement with your energy supplier! Source:

4 Introduction Imagine you want to be certain about the price to pay for next month s electricity You sign a fixed price agreement with your energy supplier! Your energy supplier s perspective: The contract seller faces two risks: price and quantity uncertainty The focus of this work is to manage these risks, by finding a hedging strategy which replicates the fixed price agreement

5 Model based hedging

6 Model based hedging Fixed price agreement has pay-off at expiry time T F L T S T L T Fixed price F contracted at initiation Spot price S T and consumption L T unknown before T

7 Model based hedging Fixed price agreement has pay-off at expiry time T F L T S T L T Fixed price F contracted at initiation Spot price S T and consumption L T unknown before T Hedge with long position V in forward contract with settlement price F ; pay-off S T V FV. = Total pay-off (contract + hedge position) π T = (S T F )(V L T )

8 Model based hedging Fixed price agreement has pay-off at expiry time T F L T S T L T Fixed price F contracted at initiation Spot price S T and consumption L T unknown before T Hedge with long position V in forward contract with settlement price F ; pay-off S T V FV. = Total pay-off (contract + hedge position) π T = (S T F )(V L T ) How determine forward position V for hedge to be efficient?

9 Model based hedging Consider convex risk measures to determine V Expected positive loss: E[max( π T, 0)] = E[max( (S T F )(V L T ), 0)] Expected quadratic loss: E[( π T ) 2 ] = E[(S T F ) 2 (V L T ) 2 ]

10 Model based hedging Consider convex risk measures to determine V Expected positive loss: E[max( π T, 0)] = E[max( (S T F )(V L T ), 0)] Expected quadratic loss: E[( π T ) 2 ] = E[(S T F ) 2 (V L T ) 2 ] Need dynamical model for spot-price consumption process (S, L)

11 Model based hedging Decomposed spot-price and consumption model Spot and consumption exhibit periodic patters (24h, weekly, yearly etc.) Π(t) = α 0 + i α i sin(2πt/τ i + φ i ).

12 Model based hedging Decomposed spot-price and consumption model Spot and consumption exhibit periodic patters (24h, weekly, yearly etc.) Π(t) = α 0 + i α i sin(2πt/τ i + φ i ). Residual, de-seasonalized spot S and consumption L d S = κ S (θ S S)dt + σ S dw 1 d L = κ L (θ L L)dt + σ L dw 2 such that S = S + Π S and L = L + Π L

13 Model based hedging Model set-up (Gaussianity) gives neat expression for risk f (V ) = E[max( (S T F )(V L T ), 0)] = minimise f (V ) for optimal model-hedge Similarly, calculate expected quadratic loss g(v ) = E[(S T F ) 2 (V L T ) 2 ] and minimize g(v )

14 Model based hedging Model set-up (Gaussianity) gives neat expression for risk f (V ) = E[max( (S T F )(V L T ), 0)] = minimise f (V ) for optimal model-hedge Similarly, calculate expected quadratic loss and minimize g(v ) g(v ) = E[(S T F ) 2 (V L T ) 2 ] Special case: Quadratic hedging with ρ W = 0: = V = E[L T ] In fact, this average load strategy is usually employed by industry

15 Empirical study

16 Empirical study Test empirical performance of model-hedge vs. average-load hedge on data from Danish energy market

17 Empirical study Test empirical performance of model-hedge vs. average-load hedge on data from Danish energy market Contracts have monthly expires: e.g. January-13 contracts a fixed price for consumption during each hour of the month = adjust and minimise loss functions accordingly

18 Empirical study Test empirical performance of model-hedge vs. average-load hedge on data from Danish energy market Contracts have monthly expires: e.g. January-13 contracts a fixed price for consumption during each hour of the month = adjust and minimise loss functions accordingly The seasonal model is fitted to data: (1) amplitudes and phases of periodic functions estimated with a regression approach, (2) parameters of bivariate OU process fitted to residual time-series with maximum likelihood

19 Empirical study

20 Empirical study

21 Empirical study Next: calculate optimal model-hedge for 24 monthly contracts from For each contract, employ six months of spot-consumption data preceding the expiry month to fit model parameters Compare: realised hourly pay-off for each contract, for opmal model-hedge and average-load hedge

22 Empirical study January-13 contract Average pay-off: 3,208 EUR (left) -2,192 EUR (right) Standard deviation: 10,224 EUR (left) 4,808 EUR (right) Probability of loss: 22.6% (left) 63.2% (right)

23 Empirical study January-13 contract Left: accumulated profit-&-loss over time Right: density of hourly pay-off, average load (black), model hedge (red)

24 Empirical study Results for 24 contracts January-13 results are typical: higher average pay-off for model hedge, lower standard deviation for average-load hedge

25 Empirical study Accumulated monthly profit-and-loss in million euros of the strategies for all 24 contracts

26 Empirical study Results for 24 contracts Looking at entire 24-month period, model hedge yields highest hourly pay-off 66% of the time

27 Empirical study Results for 24 contracts Looking at entire 24-month period, model hedge yields highest hourly pay-off 66% of the time To make the two strategies equivalent, add 0.72 EUR/MWh ( 1.5% of fixed price) to FPA for the average load strategy to have the same reward-to-variability ratio

28 Postlude: A supervised learning task?

29 A supervised learning task Classical modelling approach: estimation of Ornstein-Uhlenbeck process GP + Matérn 1/2 Deterministic periodic model = GP + periodic covariance Price spikes = GP + noise Decomposed model = joint estimation

30 A supervised learning task Classical modelling approach: estimation of Ornstein-Uhlenbeck process GP + Matérn 1/2 Deterministic periodic model = GP + periodic covariance Price spikes = GP + noise Decomposed model = joint estimation Unsupervised approach = supervised learning? Fixed loss functions = learning of optimal form?

31 Thank you for your attention! Tegnér, M., Poulsen, R., Ramsdal-Ernstsen, R. and Skajaa, A. (2017), Risk-minimization in electricity markets: Fixed price, unknown consumption, Energy Economics 68,