Stochastic modelling of electricity markets

Jhonny Gonzalez

School of MathematicsThe University of Manchester

Magical books project

23rd August 2012

Electricity marketsA Survey of Electricity MarketsStylised facts of electricity prices

Stochastic models for Energy Spot PricesSpot price modellingGeometric and arithmetic models

Electricity marketsA Survey of Electricity MarketsStylised facts of electricity prices

Stochastic models for Energy Spot PricesSpot price modellingGeometric and arithmetic models

Electricity markets

Clip for this slide� Stock markets. Shares, derivatives, governments bonds.� Commodity markets. Oil, coal, metals, agriculture.� We consider that electricity is a flow commodity.� Energy markets are commodity markets dealing with the trade

and supply of energy. Sometimes refer only to electricity.� We focus particularly on electricity, but other markets such as

gas and temperature markets have many similarities and aresomehow related.

How it used to be

Electricity markets are relatively new.� Prices were set by regulators and reflected the costs implied by

Generation Gas-fired plants, hydroelectricity p., geothermalp., nuclear power p.

Transmission High voltage network.Distribution Low voltage network.

� Thus prices used to change rarely and if so, changes weredeterministic.

Liberalisation, 1990s

� Electricity has gone through a period of liberalisation resultingin the emergence of markets for spot and derivative products.

� Prices are now determined by the fundamental rule of supplyand demand. Bids placed by generators to sell electricity forthe next day are compared to purchase orders.

� Nord Pool (Nordic area), UKPX (UK Power Exchange),Powernext (France), European Power Exchange EEX(Germany), Omel (Spain).

� This generates a problem: price will be moving until aequilibrium point is reached. There is usually an imbalancebetween supply and demand.

� Besides, we do not know supply and demand in advanced.� Liberalisation has made business fairer but also more volatile.

Supply vs demand

Supply vs demand

Figure: Power stack function for the East Center Area Reliability (ECAR)region, in the US. June 1998. Geman and Roncoroni (2006)

But... electricity is differentElectricity is different to other commodities in that it cannot bedirectly stored. The ultimate consumer cannot buy electricity forstorage and smooth out the power output, at least until now.

There are people in Manchester U. and Lancaster U. (and possiblymany other institutions) working on electricity storage and howmarkets would function under this scenario.Lack of storeability has some consequences.

1. Seasonality, if used for heating, consumer needs more inwinter than in summer, cannot store for when they know theywill use more.

2. Spikes, e.g., a nuclear power plant must be closed downunexpectedly, or temperature drops significantly.

3. Markets are regional, usual arbitrage strategies do not workhere. For instance, a difference in price in NordPool and Omeldoes not necessarily imply an arbitrage opportunity.

There are new and challenging problems to solve.

Products

� Products: spot, futures contracts on the spot, options withthe futures contracts as underlying.

How electricity markets work

Electricity markets have market mechanisms to balance supply anddemand. Electricity is traded in an auction system for standardisedcontracts.There are electricity contracts with both physical andfinancial settlement.

1. Physical, actual consumption or production as part ofcontract fulfillment.

2. Financial, contracts are settled in cash.We take by example the NordPool market in the Nordic area.

Electricity contracts with physical delivery

� Since capacity is limited and the supply and demand mustbalance, these markets are supervised by a TransmissionSystem Operator (TSO).

� Restricted to players with proper facilities for production andconsumption.

� The contracts for physical delivery are usually organised intwo different markets, a real-time market and a day-aheadmarket.

Real-time marketReal-time market.

1. The auction specifies both load and time period for generationand consumption.

2. Bids are submitted to the TSO for supply and demand statingprices and volumes.

3. The TSO lists bids in order for each hour according to price.4. This list is then used to balance the power system in the

short-term, as followsUpward regulation:take highest price in thelist. If there is powerdeficit, then increasegeneration or reduceconsumption.

Downward regulation:take lowest price in thelist. If there is powersurplus, then decreasegeneration or increaseconsumption.

(Specific rules for the auction apply in each country.)

Day-ahead market

In the Nordic area this market is the Elspot, and the UKPX in theUK.

Hourly power contracts are traded daily for physical delivery in thenext day’s 24-hour period (12am to 12am).

1. Each morning players submit their bids for purchasing orselling a certain volume of electricity for the different hours ofthe following day.

2. At noon, the day-ahead price is derived for each hour next day.

� Each contract is assigned a specific load for a given futuredelivery: Forward/Futures contract.

Financial settlement

� Main difference is that settlement is made in cash.� Usually traded Over The Counter (OTC).� In the market these are known as futures/forwards, but they

are formally ’swap’ contracts, exchanging a floating spot priceagainst a fixed price.

� These organised markets imply the need to have consistent(stochastic) models describing the price evolution of theproducts.

� Such models should reflect the stylised facts of theelectricity prices observed at the exchanges, and beanalytically tractable to price the relevant derivatives.

Electricity marketsA Survey of Electricity MarketsStylised facts of electricity prices

Stochastic models for Energy Spot PricesSpot price modellingGeometric and arithmetic models

Stylised facts of electricity prices

Clip for this slide1. Spikes: upward jumps shortly followed by a steep downward

trend, caused by the imbalance between supply and demand.2. Seasonality: electricity demand varies with temperature when

power is needed for cooling in areas with warm summertemperatures, or heating in areas with cold winters.

3. Mean reversion towards a seasonally varying mean levelrepresenting marginal cost. It may be constant, periodic orperiodic with trend depending on the particular market.Shared property with other commodities.

Stylised facts of electricity prices

Figure: PJM market prices. Spikes concentrate in summer. Taken fromGeman and Roncoroni (2006)

Related markets. Gas markets

� Natural gas is an important fuel for heating and whengenerating electricity. In 2002 one-third of electricityproduction in the UK came from gas fire plants, in the US14% of gas demand comes from electricity (Benth, 2008).

� Gas prices have many similarities with electricity prices:Spikes (during periods of high demand or shortage ofproduction -low storage). Seasonality (demand depends ontemperature).

� But gas can be stored. So, in certain sense similar to classicalcommodities like oil.

� Gas futures are the most traded products depending on gasprices. Also, spark spread options are popular in this market.Call and put options written on the difference betweenelectricity and gas prices.

Electricity marketsA Survey of Electricity MarketsStylised facts of electricity prices

Stochastic models for Energy Spot PricesSpot price modellingGeometric and arithmetic models

Clip for this slide

We assume of course the electricity spot price to be a stochasticprocess.

Classical models

� For financial assets, GBM

S(t) = S(0)eX (t), X (t) = µt + σW (t)

� For commodity markets (oil, coal, metals), Schwartz model

S(t) = S(0)eX (t), dX (t) = α(µ − X (t))dt + σdW (t)

� Generalisation, use a Lévy process L(t) to capture jumps,leptokurtic behavior, large price variations.

� What about seasonality?

II processes

Brownian Motion

Lévy process

II�

Independent Increments processContinuous in probability

Stationary incrementsIncrements ∼ N(0, t − s)Continuous paths

� Some examples: Time-inhomogeneous compound Poissonprocess, generalised hyperbolic distributions (NIG),Variance-Gamma distributions, CGMY distributions.

� To capture different speeds of mean reversion, jumps andseasonality.

Last week... Ornstein-Uhlenbeck process

The OU process is the unique solution to

dXt = β(µ − Xt)dt + σdWt ,

dXt = (µ2 − βXt)dt + σdWt .

0 10 20 30 40 50 60 70 80 90 10022

24

26

28

30

!1=3

!2=10

"=2, µ=25

1. µ level of mean reversion.2. β speed of mean reversion.3. σ2 volatility.

Spot price modelling with OU processes

DefinitionA RCLL process X (t), s ≤ t ≤ T , is called a (non-Gaussian) OUprocess if it is the unique strong solution of the SDE

dX (t) = (µ(t)− α(t)X (t))dt + σ(t)dI(t), X (s) = x .

µ, α and σ are real-valued continuous functions on [0,T ].

The unique strong solution X (t), s ≤ t is given by

X (t) = x exp�−

� t

sα(v)dv

�+

� t

sµ(u) exp

�−

� t

uα(v)dv

�du

+� t

sσ(u) exp

�−

� t

uα(v)dv

�dI(u).

Electricity marketsA Survey of Electricity MarketsStylised facts of electricity prices

Stochastic models for Energy Spot PricesSpot price modellingGeometric and arithmetic models

Clip for this slideThere are two broad classes of models for the spot energy prices:geometric and arithmetic models.

� Consider n pure jump (semimartingale) II processes Ij(t),j = 1, ..., n, which are independent one each other.

� Assume Wk , k = 1, ..., p, are p independent Brownianmotions.

0 1 2 3 4 5 660

80

100

120

140

160

180

Years

Price

!(t)=a+bt+csin(2"(t!d)/365)

� Seasonal function:average level to whichprices revert back.

� Linear trend: inflation inprice level.

� Seasonal term: variationsover the year.

0 1 2 3 4100

110

120

130

140

0 1 2 3 460

80

100

120

140

160

0 1 2 3 460

80

100

120

140

160

180

200

+more

factorsexplainingvariations

+spykes

Geometric models

ln S(t) = ln Λ(t) +m∑i=1

Xi (t) +n∑j=1

Yi (t),

where, for i = 1, ...,m,

dXi (t) = (µi (t)− αi (t)Xi (t))dt +p∑k=1

σik(t)dWk(t),

and, for j = 1, ..., n,

dYj(t) = (δj(t)− βj(t)Yj(t))dt + ηj(t)dIj(t).

� The deterministic seasonal function Λ : [0,T ] → (0,∞) iscontinuously differentiable. The coefficients µi , αi > 0, δi ,βi > 0, σik and ηj are continuous functions.

� The Xi factors represent short- and long-term fluctuations ofthe spot price. They may be correlated.

Examples1. m = 1, p = 1 and n = 0 is the Schwartz (1997) one-factor

model.dS(t)S(t) =

�Λ�(t)Λ(t) + α(t) ln Λ(t) + 1

2σ2(t)

+ (µ(t)− α(t) ln S(t))�

dt

+σ(t)dW (t).

2. An extension including jumps

dS(t)S(t) =

�Λ�(t)Λ(t) − α(t) (ln S(t)− ln Λ(t)) dt

+σ(t)dW (t) + dI(t).

Same speed of m.r. α(t) for both processes, W (t) for smallvariations, I(t) arrival of info. altering supply/demand. Residualsare leptokurtic.

Examples

3. m = n = p = 1 was used by Benth and Saltyte-Benth (2004)

dX (t) = −α(t)X (t)dt + σ(t)dW (t),dY (t) = −α(t)Y (t)dt + dI(t).

� d ln S(t) = d ln Λ(t) + X (t) + Y (t).� Same speed of m.r. α(t) for both processes� was used for modelling natural gas and oil.� I(t) is an Normal Inverse Gaussian Levy process (comes from

generalised hyperbolic distributions).

Examples4. m = 2, p = 2 and n = 0, Lucia-Schwartz (2002) two-factor

model.

dX1(t) = −α1X1(t)dt + σ1dW1(t),

dX2(t) = µ2dt + σ2

�ρdW1(t) +

�1 + ρ2dW2(t)

�.

� Parameters are constants.� X2 it is a drifted Brownian motion, it does not revert to a

mean.� Correlation is between W1 and second residual is ρ.� Again, X1 for the short-term mean-reverting component, X2

for the long-term equilibrium price level.

5. m = 2, p = 2 and n = 1, Villaplana (2004) is an extension ofthe Lucia-Schwartz (2002) two-factor model.

� Prices are positive with no further assumptions.

S(t) = Λ(t) exp(m∑i=1

Xi (t) +n∑j=1

Yj(t))

� Include many other models.� Sometimes not possible to price swap contracts analytically.

Arithmetic Models

S(t) = Λ(t) +m∑i=1

Xi (t) +n∑j=1

Yj(t),

where Xi (t), Yi (t), i = 1, ...,m, j = 1, ..., n, and Λ(t) are asbefore.

� There is a positive probability that prices are negative.� However, some conditions can be added to get arithmetic

models with probability zero of being negative Benth, Kallsenand Meyer-Brandis (2007).

� Perhaps not quite popular for these extra conditions.� But are more analytically tractable than geometric models.

Examplesm = 0, Benth, Kallsen and Meyer-Brandis (2007)

S(t) = Λ(t) +n∑j=1

Yj(t),

with null mean-reversion terms. We have to reinterpret theseasonal function as a floor seasonal function to which pricesrevert.

0 1 2 3 450

100

150

200

250

300

Benth, F. E., Benth, J. Š. and Koekebakker S..Stochastic modelling of electricity and related markets.World Scientific, London. 2008