Modeling the Spot Price of Electricity in Deregulated Energy Markets

Transcription

1 in Deregulated Energy Markets Andrea Roncoroni ESSEC Business School September 22, 2005 Financial Modelling Workshop, University of Ulm

2 Outline Empirical Analysis of Electricity Spot Prices Physical vs. financial assets Trajectorial vs. statistical properties of market prices Endogenous factors A Jump-Diffusion Model Desirable features Modules: trend, noise, and spikes The model Model Calibration Step 1: Fitting structural elements Step 2: Parameter estimation Empirical results Financial Modelling Workshop, University of Ulm 1

3 Empirical Analysis of Electricity Spot Prices I. Introduction Market context Deregulation of energy market price fluctuation new typology of risk hedging needed Issue Determine and quantify these relations Method Empirical analysis modelling (qualifying risk) calibration (quantifying risk) test (performance analysis) Financial Modelling Workshop, University of Ulm 2

4 II. A Special Underlying Arbitrage pricing = derivative price is the minimal capital required to set up a self-financing hedging portfolio of tradeable assets Hypothesis = the underlying is transferable in time (at a cost = interest rate) and space (at some transaction cost). Electricity - Limited space transferability (capacity constraints, line losses, physical market segmentation). - Almost impossible time transferability (power cannot be easily stored). singular price dynamics new typology of nondiversifiable risk. Financial Modelling Workshop, University of Ulm 3

5 III. Trajectorial Properties of Market Prices 2500 ECAR Market Price Path 500 PJM Market Price Path Jan 6,1997 Jan 6,1998 Jan 6,1999 Dec 31,1999 Date Jan 6,1997 Jan 6,1998 Jan 6,1999 Dec 31,1999 Date 100 COB Market Price Path Average Trend Functions COB PJM ECAR Jan 6,1997 Jan 6,1998 Jan 6,1999 Dec 31,1999 Date 0 Time HL Financial Modelling Workshop, University of Ulm 4

6 III. Trajectorial Properties of Market Prices /5/92 4/5/93 4/5/94 4/5/95 4/5/96 4/5/97 4/5/98 4/5/99 4/5/00 1) Drift = Periodical trend + Mean reversion 2) Volatility = Local perturbations 3) Spikes = Periodical occurrence + Jump reversion Warning: spike = sequence of upward jumps followed by downward jumps 6= two jumps with opposite sign. Financial Modelling Workshop, University of Ulm 5

7 IV. Statistical Properties of Market Prices Daily price return of 1 MWh. Empirical distribution: January, December, ECAR PJM COB NP APX Mean Std.Dev Skew Kurt.Ex Financial Modelling Workshop, University of Ulm 6

8 Market: Inelastic demand function V. Driving Factors Public service: Producers floating Utilities fixed Consumers Financial Modelling Workshop, University of Ulm 7

9 V. Driving Factors Economy: Real economy growth, generation asset prices Weather: Hot waves, cold winters Physical: Grid balancing marginal prices; transmission constraints line congestion; production system: -thermal(pjm) non storable capacity shortfalls due to demand excess - hydro (WSCC) stable shortfalls due to outages Financial Modelling Workshop, University of Ulm 8

10 A Jump-Diffusion Model I. Desirable Features: A Model Should be... Representative and Flexible (fit trajectorial and statistical properties across different markets + embed all risk factors) Tractable (Markovian, reduced-form) Valuation and hedging Easily implementable Scenario simulation Statistically stable Robustness of hedging prescriptions Simple Acceptance by market operators Tested with respect to these criteria Financial Modelling Workshop, University of Ulm 9

11 II. Trend and Local Volatility Variable Trend E (t) :=log(spot Price (t)) µ (t) =α + βt + γ cos [ε +2πt]+δ cos [ζ +4πt] Mean reversion de (t) =Dµ (t) dt + θ 1 [µ (t) E (t )] dt +... Local shock... + σdw (t)+... Financial Modelling Workshop, University of Ulm 10

12 III. Spikes Jump component... + h (E (t )) dj (t) B Jump sign h (t )=1,ifE (t ) <µ(t)+ ; 1 otherwise. B Jump size J i i.i.d. p (x; θ 3,ψ) e θ 3f(x), 0 x ψ. B Jump occurrence N (t) counting process with freq. ι (t) =θ 2 s (t). B Jump frequency shape s (t). B Cumulative jump size J (t) = P N(t) i=1 J i. Financial Modelling Workshop, University of Ulm 11

13 IV. The Model Dynamics de (t) =µ 0 (t) dt + θ 1 [µ (t) E (t )] dt + σdw (t)+h (E (t )) dj Input parameters Structural elements (market specific) µ (t) Mean trend s (t) p (x; θ 3 ) Jump frequency shape Jump direction switch Jump size distribution Parameters (time series specific) σ θ 1 θ 2 θ 3 Local volatility Mean reversion force Jump frequency magnitude Jump size parameter Financial Modelling Workshop, University of Ulm 12

14 Model Calibration I. Fitting Structural Elements Trend µ (t) =α + βt + γ cos [ε +2πt]+δcos [ζ +4πt] ³ 2 Jump frequency shape s (t) = 1+ sin[π(t τ)/k] τ = max. freq. date (summer peak τ =0.5) - k = period (annual periodicity k =1) Jump direction switch: = α-quantile of detrended prices Jump size distribution: p (x; θ 3 )=truncated exponential Financial Modelling Workshop, University of Ulm 13

15 II. Parameter Estimation Idea Continuous time process X Discretization Discrete time approxim. X n Transition Exact Likelihood (approx.proc.) L X n Continuous time process X Girsanov Exact Likelihood (cont.observ.) L Piecewise constant obs. Approx. Likelihood (discr. observ.) L n X Data disentangling E (t) E c (t)+ E d (t) Financial Modelling Workshop, University of Ulm 14

16 II. Parameter Estimation Local volatility σ modified local covariance estimator σ = q Pn 1 i=0 ( Ec (t i ) θ 1 (µ (t i ) E (t i )) ) 2 Parameters θ 1,θ 2,θ 3 new estimator L θ 0,E (θ) =P n 1 i=0 (µ(t i ) E i )θ 1 σ 2 [ E c i µ0 (t i ) t] t 2 P n 1 i=0 ³ 2 (µ(ti ) E i )θ 1 σ (θ 2 1) P n 1 i=0 s (t i) t +lgθ 2 N (t) + P h i µ n 1 i=0 (θ 3 1) Ed i h(e i ) + N (t)lg 1 e θ 3 ψ θ 3 (1 e ψ ) Financial Modelling Workshop, University of Ulm 15

17 III. Empirical Results: Price Paths Trajectorial properties (ECAR, PJM, COB) 2500 ECAR Simulated Price Path 600 PJM Simulated Price Path 175 ECAR Simulated Price Path ECAR Market Price Path 600 PJM Market Price Path 175 ECAR Market Price Path Financial Modelling Workshop, University of Ulm 16

18 III. Empirical Results: Price Paths Trajectorial properties (ECAR market under varying resolution) 2500 ECAR Simulated Price Path 500 ECAR Simulated Price Path 100 ECAR Simulated Price Path ECAR Market Price Path 500 ECAR Market Price Path 100 ECAR Market Price Path Financial Modelling Workshop, University of Ulm 17

19 IV. Empirical Results: Statistics ECAR PJM COB EMP SIMUL Average Std. Dev Skewness Kurtosis EMP SIMUL EMP SIMUL Financial Modelling Workshop, University of Ulm 18

20 V. Comparative Analysis of Alternative Model Specifications Reduction: upward jumps only Extension: price dependent jump frequency (random intensity) ECAR Up-jump (det. freq.) Sgn-jump (det. freq.) Sgn-jump (random freq.) Average Std. Dev Skewness Kurtosis Financial Modelling Workshop, University of Ulm 19

21 VI. Conclusion: our class of models... (1) Matches both trajectorial and statistical properties of price dynamics (2) Fits all markets (3) Embeds risk factors (noise, spikes) (4) Reproduces forecastable trends (drift, periodical jumps) (5) Is Markovian (6) Can be easily estimated and simulated Financial Modelling Workshop, University of Ulm 20

22 References 1. Geman, H., Roncoroni, A., Understanding the Fine Structure of Electricity Prices. Forthcoming in The Journal of Business (downloadable from 2. Roncoroni, A., Commodity Forward Prices: Dynamic Modelling and Calibration. Research Report, Gaz-De-France, Paris. Financial Modelling Workshop, University of Ulm 21

23 About the Author Andrea Roncoroni is Assistant Professor of Finance at ESSEC Business School (France). He is regular lecturer at University Paris Dauphine, Bocconi University, and the Italian Stock Exchange. He holds PhD s in Applied Mathematics and Finance. His research interests cover quantitative modeling and risk management in energy markets. He has consulted for private companies (Gaz de France, Fideuram Capital Asset Management) and public institutions (International Energy Agency, Italian Authority for Electricity and Gas). He is author of the book Implementing Models in Quantitative Finance: Methods and Cases (with G. Fusai), Springer-Verlag, forthcoming roncoroni@essec.fr Financial Modelling Workshop, University of Ulm 22