energy models for pricing and risk management

Energy models for pricing and RiskManagement

Master IMEF– PwC

Venezia, 26 Gennaio 2010

CONFIDENTIAL DOCUMENT

PricewaterhouseCoopers

26 gennaio 2010IMEF MeetingConfidential Document

Agenda

Stylized facts on Energy

Energy Markets

Energy Models : spot price modelling

Risk Management in the Energy context


26 gennaio 2010IMEF MeetingConfidential Document Slide 3

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Price dynamics

• Energy prices have a seasonal pattern and the seasonalitychanges on different time scales. We can observe periodic behaviorof both high frequency (intraday) and low frequency (annual).

• Prices are determined by the match between supply and demandand show the property of mean reversion, which represents themarginal costs of production: after a sudden jump, prices tend torevert to a mean value. The mean reversion becomes very strongwhen jumps happen, since the rate of reversion to normal prices isvery high.

• The mean reversion can be periodic or periodic with a trend.

• Persistence of volatility: persistence of volatility means that theconditional volatility changes slowly over time, remaining high or lowfor long periods.

• Time series of price show spikes, sudden jumps in the distributionof returns: returns distribution is not normal. Spikes have: periodicrecurrence; sequentiality (sequence of positive jumps followed bysequence of negative jumps); mean reversion through shocks (jumpreversion).

• According to the characteristics listed above, we can describe thedynamics of the price as the sum of the following parts:

Stylized facts on energy

Main Features

•Energy cannot be stored: unlike other commodities, a cash and carrystrategy is not feasible, and there isn’t a simple relation between spotand forward prices.

• Energy, as well as gas, is a commodity based on a stream to bereceived in a given period of time; so it’s important also the hourlydynamics of energy because it affects the valuation of derivativecontracts.

• These phenomena are reflected in:

• Very low elasticity of demand

• Limited storage possibilities

• Rigidity on the supply side

Et = at + st + Wt + Jt

Trend Seasonality Noise Spikes



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Spikes

• Spikes are foundamental feature of electricity. They are very sharpmovements upwards shortly followed by drops of the same amplitude.

• From an economic standpoint, spikes are explained by the followingfacts:

1 An equilibrium between supply and demand needs to be secured atany time.

2 Demand is a fairly inelastic function of price. Residential customersmust be serviced at all times. Interruption rights may be exercised bythe utility in the direction of some industrial customers but not all ofthem. (High tech companies, for instance, are adversely affected bypower blackout).

3 Supply may abruptly change in the case of a plant outage or a failurein the transmission network. The non-existence of the buffering effect ofinventory as in the case of oil and gas explains the price spikesregularity observed in power markets worldwide.

Stylized facts on energy

Seasonality

• Electricity demand is strongly influenced by economic activities andweather. For example, in some countries where the summer iswarmer, the energy demand is greater and therefore the price isaffected by the increased demand. For this reason there are manykind of seasonality which can be intra-day, weekly, monthly, etc.

• We find seasonality not only in the spot prices but also in forwardprices.

• When modeling energy, before estimate the parameters of thedeterministic component of the price, you must clean the series ofthose numbers, as the seasonal component and a jump, that canbias the estimation of parameters.

• Generally the seasonal component can be decomposed into twoparts: trend part and periodic part. There are different methods ofmodeling according to the frequency with which the seasonalcomponent is presented in the series.



Energy Project

•Structure of electricity markets:

Country Date Name

England and Wales 1990-19992001

Electricity PoolUK power Exchange (UKPX)

Norway 19931996

Nord Pool ScandinaviaNord Pool

Spain 1998 OMEL

Netherlands 1999 Amsterdam Power Exchange (APX)

Germany 20002001

Leipzig Power Exchange (LPX)European Power Exchange (EEX)

Poland 2000 Polish Power Exchange (PPX)

France 2001 Powernext

Italy 2004 Gestore Mercato Elettrico (GME)

Spot power markets



Energy Project

Spot power markets

•Figures depicts the shapes of the supply functions in two region of the US; one coveredby the Energy Regulatory Council of Texas (ERCOT) and the other by the East CenterArea Reliability Coordination Agreement (ECAR).


26 gennaio 2010IMEF MeetingConfidential DocumentPricewaterhouseCoopers Slide 7

Electricity Market

The Electricity Supply-Chain (1 of 2)



Electricity Market

The Electricity Supply-Chain (2 of 2)

- Wholesaler: it’s the person or entity that buys and sells electricity withoutexercising activities' of production, transmission and distribution in EuropeanUnion countries.

- Eligible customers: it’s the person or entity that has the ability to enter intosupply contracts with any producer, distributor or wholesaler, both in Italy andabroad.

- Captive customers: it’s the final user who does not fall into the category ofeligible customers, and so it is not able to draw up exclusive supply contractswith any producer but it is in binding over choosing among local energysuppliers.


26 gennaio 2010IMEF MeetingConfidential DocumentPricewaterhouseCoopers

The Electricity Market Structure

Slide 9

Electricity Market



Electricity Market

The Electricity Market Actors (1 of 3)

GME

Activities:• Organizing and managing the electricity market according to criteria oftransparency, neutrality and objectivity in order to promote competitionbetween producers also ensures the economic management of anadequate supply of reserve capacity• Counterparty on MGP (day ahead market) and MA (adjustment market)• Organize and maintain the premises trading of green certificates (showingthe generation of energy from renewable sources) and energy efficiencycertificates (so-called "white certificates" attesting the execution of policiesto reduce energy consumption).



Electricity Market


Terna

Activity (previously exercised by GRTN):• Dispatching Services• Balancing Service• Service to resolve congestion• Service for the reserve• Transmission and network development• Counterparty on MSD(dispacement services market)



Electricity Market


Acquirente Unico

Activities:• Role of ensuring the supply of electricity to the captive market• Has the task to purchase electricity at the most favorable prices on themarket and resell it to distributors

Modes of supply:• entering into contracts, including multi-year, for an amount of energy notexceeding a quarter of total demand in the captive market



Electricity Market

A decree of the Minister of Industry (May 9, 2001) approved the Disciplineof the electricity market designed by the operator of the electricity market.The market structure is composed as follows:• Day-ahead energy market;• Adjustment Market (Market reserve);• Market to solve congestion;• Balancing market.

The Electricity Market


26 gennaio 2010IMEF MeetingConfidential DocumentPricewaterhouseCoopers

Timing of the Electricity Market

Slide 14

Electricity Market



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Valuation approaches

•There are two fundamental approaches regarding valuation and riskmanagement of derivatives having commodity price as underlying.

• The first approach concerns the modeling of spot price and othervariables such as the convenience yield. The main problem of “spot”models is that forward prices, endogenously derived from spot prices,are not consistent with the observed forward market prices.

• The second line of research relates to the modeling of the wholeforward curve using only a few stochastic factors and taking the initialterm structure as input data. This approach is based on the Heath-Jarrow-Morton (1992) structure, applied to interest rate markets. Suchmodels are "market consistent" and are also called "market models".

•The third approach concerns the so-called hybrid models.

• As in the case of spot price modeling, all variables that characterizethe price, such as the seasonal component, must be identified andconsidered even for forward prices modeling.

Main Energy models

Spot Price Modelling

• For the evaluation of energy derivatives, there isn’t a standard

approach.

• Spot price modeling: the evaluation is based on the spot pricedynamic, that represents the characteristics listed on the side. It maybe a one factor model (spot price) or a two factors model (spot priceand convenience yield).

• Spot price models can be divided in:

Structural or equilibrium models: the price of electricity is obtainedfrom the intersection of demand and supply or from the price-loadrelation.

Synthetic or reduced form models: the price is obtained through acalibration process on market data. Prices reproduce thecharacteristics of historical prices and include the market price ofrisk.



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Forward Price modelling

• Forward price models try to address the shortcomings of spot pricemodels, based on use of unobservable parameters (such as theconvenience yield).

• The main advantage using forward price models is that quantitiesunder analysis are all determined by observable market data.Furthermore such models are intuitive from a trading point of view.For example, they allow to perform a "what if" analysis, changingforward curves on the basis of underlying historical movements.

.

Main Energy models

Hybrid model

•Hybrid models combine empirical approaches with stochasticmodels.

•Hybrid models provide a natural representation of energy dynamicsin terms of mechanisms of generation.

•They use the forward curve and empirical data: in this sense aresimilar to structural models.

•They give the possibility to extend the information that is availablefrom historical data on energy prices.



Energy Project

•A financial forward contract on an underlying S guarantees to purchase a given amountof S in a future time T for a fixed price K, called "forward price".

•An forward on energy ensure a continuous energy flow over a period of time [T1, T2] fora fixed price K(T1,T2).

•A forward on energy can be compared to a portfolio of financial forward with delivery in allmoments of time between T1 and T2 at the same price.

•Energy forward are also called “swap”, because at every time the energy flow is swappedfor a fixed price K(T1,T2).

•Because of the difference between financial forward and energy forward, also theirmathematical payoff description at maturity comes to be different:

- financial forward : S(T) - K

- energy forward : S(T1,T2) – K(T1,T2), where S is the average of S on [T1, T2]

Financial Forward Vs. Electric Forward



Energy Project

•In the case of a non-dividend paying stock and assuming no-arbitrage, the spot price S(t)is related to the T forward price f(t) by the relationship:

F(t) = S(t) er(T-t)

•The proof will simply come from application of the no-arbitrage assumption to a wellchosen portfolio (cash & carry strategy).

• At the inception the forward contract value is equal to zero and the delivery price K isequal to the forward price F(t). During contract life the value can be positive or negativeaccording to the following formula

f(t) = (F(t) -K) e-r(T-t)

Spot–forward relation for a non-dividend paying stock

t T

Buy the stock S -S(t) Delivery

Borrow to finance the purchase S(t) -S(t) er(T-t)

Sell a forward contract, written S, for maturity T - F(t)



Energy Project

• When going to a storable commodity the spot–forward relationship becomes

F(t) = S(t) e(r-y)(T-t)

where y means the convenience yield coming from holding the physical commodity.Expressed as a rate, the convenience yield tells us that the benefit for the holder of thecommodity will be equal to S(t)·y·dt over the interval [t; t + dt].

• If we decompose y into two components, the pure benefit from the physical commodityy1 and the holding cost c in the following way

y = y1 – c

The spot-forward relation becomes

F(t) = S(t) e(r+c-y1)(T-t)

Can we use the same formulas also for describe energy forward price?

Spot–forward relation for a storable commodity



Energy Project

• The convenience yield, with its economic representation, cannot be extended toelectricity. Electricity cannot be stored, hence there is no possibility of “carrying” it over theinterval (t,T) and the timing option does not exist (except for hydroelectricity, wichrepresents a small fraction of the electricity produced worldwide).

• The mathematical representation of spot-forward price relation for a storable commodityis not appropriate too. According to this relation, when t goes to T, the forward price shouldconverge smoothly to the spot price, due to the exponential in the formula. This certainlydoes not happen during price spikes.

• In the case of electricity, we should think the spot-forward price relation in terms of:

Forward Price = Spot Price + Risk Premium p(t,T)

where p(t,T) varies over the time t with maturity T and may have a different sign fordifferent values of T-t

Cash & Carry relation in electricity markets



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Main characteristics

Commodities can be classified by price behaviour / modellingrequirements

• Storable vs Non-Storable

• Continuous vs Seasonal Production

• Continuous vs Seasonal Demand

• Local vs Regional vs Global

• Elasticity of Supply or Demand to Price

Applying the classical notion of volatility – the standard deviation ofreturns- we obtain that measured on a daily scale

•treasury bills and notes have a volatility of less than 0.5%

•stock indices have a moderate volatility of about 1–1.5%

•commodities like crude oil or natural gas have volatilities of 1.5–4%

•very volatile stocks have volatilities not exceeding 4%

Electricity exhibits extreme volatility – up to 50.%!

.

Summary on models

Main types of models

This facts should be reflected in the main types of models

1 Reduced-form pure spot/ price Fundamental Equilibrium - explicitmatching of supply and demand

2. Forward Price Models

3. Hybrid / Structural - particularly for electricity, there are a numberof models which fall between these categories.

Modelling choices often include:

• Are many state variables are needed? Observable andunobservable?

• What data is available? How much to include?

• Explicit modelling of supply and demand? Discrete or continuoustime?



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Ornstein Uhlenbeck model

The model can be shown in its form of stocastic differential equation

or in the equivalent integral formulation

One can derive the following expressions for mean and variance of the process

Note that the process has a limited variance and tends in the long term to the constant value a. Parameters s and l respectivelyindicate the volatility and the speed of mean reversion of the process.

Parameters are calibrated on a historical basis, using the maximum likelihood method, i.e. maximizing the density distribution of theobserved sample.

Spot price modelling

)())(()( tdWdttXtdX

)())()1()( )()()( sdWeetXeTXT

t

sTtTtT

)(2

2

)()(

1)(Var

1)()(E2 tTQ

t

tTtTQt

eTX

etXeTX



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Ornstein Uhlenbeck model + Jumps

The assumption of normality, underlying the Ornstein Uhlembeck model, is generally violated when analyzing gas and electricityprices, characterized by features of discontinuity such spikes. One possible solution to overcome this problem is to introducerandom jumps in the dynamic of the process. Thus the differential equation assumes the following formulation:

where the jump process J(t) is defined as following

A common choice is to use a Poisson process N (t) with intensity ly to count the number of jumps that occur in the interval [0, T].Indicating with x the number of jumps for which we want to calculate the probability and with ly the expected value of the processon a time unit, the probability distribution of the Poisson process is the following

The amplitude of jumps is modeled by lognormal random variables Yi iid, with parameters ay, sy.


)()())(()( tdJtdWdttXtdX X

,...2,1,0!

)(),(

xx

tetxf

xY

t

YP

Y

)(

1

)(tN

iiYtJ



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Ornstein Uhlenbeck model + Jumps

X (T) is the sum of the solution of standard OU process and a part dependent on the process of jumps

and one can derive the following expressions for the mean and variance of the process


XYY

tTQt

YtT

YQt

Y

eTX

tXeTX

X

X

YYY

X

/E

1)(Var

)()(E)(2

2

)(

)(

222

Jumps

T

t

sTT

t

sTtTtT sdJesdWeetXeTX XXXX )()())()1()( )(

OU

)()()(



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Ornstein Uhlenbeck model + seasonal trend + Jumps

One can also think to model in some way the seasonality of prices. Geman and Roncoroni (2006) proposed a mixed linear-sinusoidal trend which is implemented in the graph below for the U.S. market PJM


PJM log prices 1996-2002

2.4

2.9

3.4

3.9

4.4

4.9

5.4

5.9

6.4

28-ott-95 11-mar-97 24-lug-98 06-dic-99 19-apr-01 01-set-02

log prices PJM percentile 90% trend



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Ornstein Uhlenbeck model + seasonal trend + Jumps

The trend is defined by the following formula

represents a fixed cost of electricity, b represents a linear growth rate of prices and the periodic part has annual and semi-annualseasonality

Introducing now the trend in the model OU + jump, we get

the model can be further enriched by introducing a seasonality also in volatility and intensity of the process with jumps and forcingeven the process of the jumps to be mean reverting. For these devices you can refer to Geman and Roncoroni (2006).


)4cos()2cos((t) ttt

)()())((t)((t)')( tdJtdWdttXtdX X



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Models calibration

Here is a simple application of the OU model on PUN market, using the logarithm of spot prices observed in 2008 to estimatemodel parameters.

OU model

The simplicity of the model allows an analytical solution for the maximization of the log-likelihood.

l = 4.4555

s = 3.0218

θ = 193.4498

OU model + Jumps

A recursive procedure applied to the historical series of data allows to separate jumps from the OU process.

parameters lY, ay , sy are calibrated on jumps, while lX, s , θ are obtained from filtered data

lX = 4.4555 lY = 7

s = 3.0218 ay = 0.47

θ = 193.4498 sy = 0.0396




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Models calibration

OU model + seasonal trend + jumps

Step 1: Filtering of jumps and estimation of jump process parameters;

Step 2: Estimation of linear trend by least squares regression, performed on all the filtered data

Step 3: Estimation of seasonal trend using nonlinear least squares regression on annual data filtered (in the case of data overseveral years, averages on the same day for multiple years should be considered)

Step 4: Estimation of parameters sigma and theta, maximizing likelihood

a = 4.36285 b = 0.164293

g = 0.120489 d = 0.0157106

e = 1.75182 z = 6.33266

s = 3.0218 θ = 193.4498

lY = 7 sy = 0.0396

ay = 0.47




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Models application

Once the parameters have been estimated using econometric techniques, our model is ready to be used to simulate the futureevolution of electricity spot prices.




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VaR definition

•Value at Risk (VaR) measures the maximum negative change interms of mark to market (fair value), which can affect a particulardeal or a given portfolio with an established level of statisticalconfidence (97.5% - 99%) in a predetermined time interval (from 1 to10 days, close to the maximum liquidation time of the position,depending on the calculation purpose).

•VaR is a suitable measure for measuring risk of a portfolio ofstandard products (OTC or exchange traded), for which there issome market liquidity allowing to close positions at any time withouthaving to pay (excessive) costs.

•VaR summarizes in one number the risk exposure of the portfoliodefined by a certain level of accuracy (defined “a priori”): it is usefulto define the maximum exposure limits to which we want to expose.

Value based risk measures

Calculation methodologies

• Value at Risk is a benchmark risk measure when you managederivative portfolios in a perspective of full immunization of the valuefrom market fluctuations. This situation is common for a financialbroker or a firm whose business is subjectedto significant marketrisks.(Example: the price risk of jet fuel for an airline company)

• Methods of calculating Value at Risk can be divided into threeclasses:

• Analytical VaR

• Montecarlo VaR

• Historical VaR

• Other risk measures based on possible variations in the value ofthe portfolio are Expected Shortfall (ES) and Conditional VaR(CVaR).



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When to use VaR

• When the fair price of a product is not available due to lack of reference liquid markets or other reasons (incomplete market, …),any "value based" risk valuation is unnecessary and wrong.

•When the closure of a financial position in a reasonable time is not possible, because of the lack of a real counterparty, the risk isnot properly described by a variation of value but by a reduction of finacial/economic flows, until the natural maturity of theunderlying deal.

• The so-called "Flow-based risk measures" as PAR (Profit at Risk), EAR (Earnings at Risk) or CFAR (cashflows at risk) are moreappropriate in this situation.

• Structured products (e.g. exotic derivatives) and real assets (e.g. power plant, oil rig, the power transmission network, refinery)typically fall into this category.

Flow based risk measures



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The calculation of VaR / PAR involves the construction of "n" marketscenarios for risk factors, which may occur for the portfolio over apredetermined time horizon.

The scenarios used for the calculation may be based on historical orforward data (depending on values and volatilities of market variables).The picture on the right side shows an example of possible scenarios ofthe underlying.

The statistical distribution of market scenarios is the basis for PARcalculation.

In the figure on the right we present an example of statistical distributionof Profit & Loss function.

In particular, PAR is given by the 5% (1 – α) percentile of that distribution. As can be seen looking at the red tail in the figure, PARtakes value of -200 € (i.e. in 5% of cases there will be a loss of at least200 €)

As described in the formula on the right side, PaR measures (Pr) themaximum negative variation (DP) which the portfolio will have in 95% ofcases (a)

VaR/PaR calculation

Example of possible scenarios

Time

€

Frequency

αVaR/PaR)Pr(Δ P

PaR Formula



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Definition of risk restrictions

•High commodity price volatility can generate a loss of value in a commodity book. It affects P&L of the firm, dipendent on the stop-loss policy used. The advantage of VaR/Par is to provide a risk measure in order to define stop-loss restrictions, regardless ofhedging derivatives presence or absence.

•Price risk is affected by commodity trading activities which generate a risk position, hence risk restrictions must be established.The risk management office generally determines these limits.

•When managing portfolios of trading/optimization of risk coming from derivatives contracts, an “ad hoc” (VaR/PaR) risk restrictionhave to be used, because of their high rotation. This method allows to have the risk of the open positions between set levels, evenwhen there are perimeter changes (e.g. strong variation in unit sales, ...).

Restrictions calculation

energy models for pricing and risk management

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