Rafał Weron. Hugo Steinhaus Center Wrocław University of Technology

Transcription

1 Rafał Weron Hugo Steinhaus Center Wrocław University of Technology

2 Options trading at Nord Pool commenced on October 29, 1999 with two types of contracts European-style Electric/Power Options (EEO/EPO) Written on the exchange traded standardized forward contracts and Asian-style Electric/Power Options (AEO/APO) Exercised and settled, in retrospect, against the average spot price during the settlement period 2

3 Asian-style Options settled against the average spot price during the settlement period NOT during the trading period Settlement periods: 4 week blocks corresponding to futures contracts Probably the only exchange traded Asian-style option How to price it? 3

4 Derivatives, risk premia and the market price of risk Power markets in a nutshell Jump-diffusion models Case study: Nord Pool and Asian-style options 4

5 Standard approach: Construct a portfolio that will replicate the payout This generally involves storing the underlying What if it cannot be stored? Electricity itself cannot be stored Storage of energy sources is limited and costly Futures and forwards can be stored Derivatives written on these contracts can be priced with the no-arbitrage principle Can the information contained in forward prices be used to price derivatives on the spot price? 5

6 For commodities, the relationship spot price P t vs. forward price F t,t is often explained in terms of the convenience yield y t the premium to a holder of a physical commodity as opposed to a forward contract written on it: Kaldor, 1939; Working, 1949 Does the notion of the convenience yield make sense in the context of electricity? Or generally: a non-storable commodity? 6

7 The reward for holding a risky investment rather than a risk-free one, i.e. The difference between the spot price forecast and the forward price: Botterud et al., 2002; Diko et al., 2006; Hirshleifer, 1989; Pindyck, 2001; Weron, 2006 The expectation E t is taken today with respect to the real-world or risky probability measure and concerns the price at a future date T Note, that F t,t is the expectation of P T with respect to the risk-neutral or risk-adjusted measure 7

8 Mixed evidence on the sign and variability RP can be both positive and negative Can vary throughout the year and/or the day Can differ from market to market Inconsistent definitions Some authors use the notion of the forward premium (or forward risk premium): FP = RP Bessembinder & Lemmon, 2002; Longstaff & Wang, 2004; Villaplana, 2003 Others use the term risk premium, but define it like the forward premium Benth et al., 2006; Bessembinder, 1992; Eydeland & Wolyniec, 2003; Geman,

9 To price derivatives we need to take into account the RP observable in the market Forward prices already include the RP No need to care about it for forward curve derivatives Spot price models have to calibrated not only to spot prices but also to forward (derivative) prices Naturally, E t (P T ) is then model dependent In math-finance language: change the measure The market price of risk (λ) is the difference between the drift in the original risky probability measure and the drift in the risk-neutral measure in the SDE for P t 9

10 Derive spot dynamics Price derivatives on spot price Market price of risk (Risk premium) Model forward curve Price derivatives on forward curve 10

11 Model spot dynamics Price derivatives on spot price Market price of risk (Risk premium) Derive forward curve Price derivatives on forward curve 11

12 Derivatives, risk premia and the market price of risk Power markets in a nutshell Jump-diffusion models Case study: Nord Pool and Asian-style options 12

13 Time to delivery Bilateral Contracts Bilateral Contracts Futures and Forward Markets Futures, Forward and Option Markets Day-Ahead Market Day-Ahead (Spot) Market Financial Physical delivery Intraday Market Balancing (Real-time) Market Market OTC Power Exchange (Power Pool) System Operator 13

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16 Seasonality Daily, weekly, annual Weather dependency Demand, also supply Non- or limited storability Transmission constraints Spikes in prices and loads (consumption) Extreme volatility, up to 50% for daily returns Inverse leverage effect Prices and volatility are positively correlated Both are negatively related to the inventory level Samuelson effect Volatility of forward prices decreases with maturity 16

17 Derivatives, risk premia and the market price of risk Power markets in a nutshell Jump-diffusion models Case study: Nord Pool and Asian-style options 17

18 Typically the spot electricity price is assumed to follow some kind of a jump-diffusion (JD) process: Generalized drift Generalized volatility Pure jump process with given intensity and severity a compound Poisson process, e.g. dq(x,t) = XdN t Clewlow & Strickland, 2000; Eydeland & Geman, 2000; Kaminski,

19 Originally developed for spot interest rate modeling Speed (rate) of mean reversion Conditional distribution of X at time t is normal Long term mean 19

20 ABM GBM Vasicek α/β 20

21 Geometric JD (Merton, 1976) N t is a Poisson (point, counting) process Used by Kaminski (1997) for electricity derivatives Mean reverting JD (MRJD) J is the jump size (Gaussian, lognormal, Pareto,...) Simulation straightforward Estimation problematic 23

22 The likelihood function L includes an infinite sum over all possible numbers of jump occurrences in a given time interval (t,t+δt) Assume that only one jump can take place in (t,t+δt), approx. JD by a mixture of normals (Ball & Torous, 1983) Truncate it (Huisman & Mahieu, 2001) Use the characteristic function ML and partial ML (PML) estimation based on Fourier inversion of the conditional characteristic function (CCF) Quasi ML (QML) based on conditional moments computed from the derivatives of the CCF at 0 See Cont & Tankov (2003), Singleton (2001) for reviews 24

23 ML-type methods tend to converge on the smallest and most frequent jumps We would rather want to capture the lower frequency, large jumps The obtained estimates (especially of the jump component) tend to be unstable and unreasonable Solution: hybrid or stepwise approach Filter the jumps estimate jump intensity and size Next, the mean-reverting jump-free diffusion is calibrated from the filtered series, e.g. using ML 25

24 After a jump the price is forced back to its normal level by mean reversion mean reversion coupled with downward jumps Deng 1999; Escribano et al., 2002; Geman & Roncoroni, 2006 linear combination of mean reversions with different rates Benth et al., 2007 Alternatively, a positive jump may be always followed by a negative jump of (approx.) the same size especially on the daily scale Weron et al., 2004; Weron,

25 Derivatives, risk premia and the market price of risk Power markets in a nutshell Jump-diffusion models Case study: Nord Pool and Asian-style options 27

26 1/97-4/00 28

27 Spot Price [NOK/MWh] Spot prices Linear trend Annual sinusoidal cycle Days [30/12/ /03/2000] 29

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29 We model the spot price as: p t = s t + S t + exp(j t dq t + X t ) s t is the weekly seasonal component (average week), S t = A sin(2 (t + B)/365) + Ct is the annual seasonal component (with linear trend), J t dq t is the jump component: q t is a Poisson r.v. with intensity J t is a random jump size, e.g. a lognormal r.v. log J t ~N(, 2 ) truncated at the maximum price attainable in the market X t is a Vasicek type process: 31

30 Vasicek price log(spot Price) Days [30/12/ /03/2000]

31 1-F(x) Spike size Normal fit Lognormal fit Power law fit c*x x 33

32 Spot Price [NOK/MWh] Trading period until Settlement period Asian option payout (A-K) Spot prices Annual cycle and linear trend Days 34

33 Change the measure to obtain risk-neutral dynamics: Now the process has a new level of mean reversion Find by calibrating the model to prices of derivatives The choice of λ uniquely determines the equivalent martingale measure for pricing derivatives Therefore calibrating it to futures prices should do the same job as calibrating it to option prices 35

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37 Over a longer time horizon cannot be approximated by a linear function For most of the time λ is Negative, i.e. the futures price is an upward biased predictor of the future spot price ( contango ) There is a higher incentive for hedging on the demand side Non-storability of electricity vs. (limited and costly) storage capabilities of fuel - especially water, but also coal, oil, gas Consistent with studies of Botterud et al., 2002; Geman and Vasicek, 2001; Longstaff and Wang,

38 For most of the time λ is increasing In agreement with the equilibrium model of Bessembinder & Lemmon (2002) RP>0 when expected demand and demand risk is low It decreases when expected demand or demand variance is high, because of positive skewness in the spot electricity price distribution As maturity approaches the uncertainty about the demand at delivery declines, hence RP (and λ) increases But what happened in 2001? 40

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40 Based on: R. Weron (2008) Market price of risk implied by Asian-style electricity options and futures, Energy Economics 30, Reviews: A. Eydeland, K. Wolyniec (2003) Energy and Power Risk Management, Wiley, Chichester. H. Geman (2005) Commodities and Commodity Derivatives, Wiley, Chichester. R. Weron (2006) Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach, Wiley, Chichester. Selected references: F.E. Benth, A. Cartea, R. Kiesel (2006) Pricing Forward Contracts in Power Markets by the Certainty Equivalence Principle: Explaining the Sign of the Market Risk Premium, SSRN Working Paper. H. Bessembinder (1992) Systematic risk, hedging pressure, and risk premiums in futures markets, Rev. Fin. Stud. 5, H. Bessembinder, M. Lemmon (2002) Equilibrium pricing and optimal hedging in electricity forward markets, J. Finance 57, A. Botterud, A.K. Bhattacharyya, M. Ilic (2002) Futures and spot prices an analysis of the Scandinavian electricity market, Proceedings of North American Power Symposium, Tempe, Arizona. P. Diko, S. Lawford, V. Limpens (2006) Risk premia in electricity forward prices, Stud. Nonlin. Dyn. Econom. 10 (3) (Article 7). H. Geman, O. Vasicek (2001) Forwards and Futures on Non Storable Commodities: The Case of Electricity, Risk (August). F.A. Longstaff, A.W. Wang (2004) Electricity forward prices: a high-frequency empirical analysis, J. Finance 59, R. Weron, I. Simonsen, P. Wilman (2004) Modeling highly volatile and seasonal markets: evidence from the Nord Pool electricity market. In: H. Takayasu (ed.), The Application of Econophysics. Springer, Tokyo, pp