Energy Derivatives New Developments and Challenges

Alexander EydelandMorgan Stanley

New players in energy derivatives markets• Traditional users of energy derivatives:

energy producers, marketers and end-users.

• Main objective: to hedge energy exposure• New players: “investors” - banks,

institutional investors, hedge funds

Increased interest in commodity-linked products: the investors point of view• spectacular returns in the last few years• diversification

– historically commodity returns are weakly correlated with equity or fixed income products and can be used as a separate asset class (Gorton and Rouwenhorst, 2004)

– protection against inflation caused by economic growth

– commodities are correlated with non-economic drivers: weather, environmental issues, supply constraints, etc.

Increased interest in commodity-linked products: the issuer point of view

• Frequently the products can be split into several components that can be used as a long-term hedge of existing commodity market risks - a useful feature particularly when the markets are illiquid

Example 1. Product: Energy commodity-linked bond. Issuer: Credit Suisse, 1998. Time maturity: 10 years.

• At redemption, holder receives par, In addition, holder receivessemi-annual coupon. Each coupon is calculated according to the formula:

Coupon = .73 x Percent_GainNYMEX_WTI

• In this formula the coupon is calculated using percentage gain of the NYMEX WTI contract during the coupon period, provided that this gain is positive. For example, if during the coupon period the NYMEX WTI contract moves up from $50 to $55 per barrel, coupon payment on $1000 par bond will be .73*(.1)*1000 or $73. Next coupon would be determined using a new base price of $55.

Example 2. Product: Energy commodity-linked note.

Issuer: Canadian Imperial Bank, 1998. Time to maturity: 5 years.

• The note payment is determined by the price of the underlying basket consisting of a weighted combination of the light sweet crude and natural gas. If the basket price drops, the note holder receives

par at maturity. If it increases, holder receives

Par x (1 + 1.1 x Percent_GainBasket)

• If the basket price moves from $10 to $11 the holder of $1000 par bond receives $1000*(1+1.1*.1) = $1110

Example 3. Product: Commodity-linked note. Issuer: Sao Paolo IMI, 1998. Time to maturity: 5 years.

• Underlying index: a basket of petroleum, copper, gold, aluminum and zinc. Holder is guaranteed 17% gain even if the price of thebasket drops. At maturity, holder is paid

Par x (1 + 1.2 x Percent_GainBasket)

• If the basket price has moved from $100 to $120 holder receives $1240

• There are similar products involving energy commodities; mostly, oil products or natural gas (rarely, electricity)

New Developments: Increased volume and complexity

Examples:

• The issuer guarantees that the sum of all coupons is greater than a specified percentage value. If the value is not reached over the life of the product, the deficit is compensated at maturity.

• Coupon dependent on the history of commodity prices during the coupon period, and not just on one price at the coupon payment day. For example, it may depend on the sum of all previous coupon payments.

• Various deal interruption condition can be included such as note call provisions, or stipulations that the product will be terminated when commodity prices reach specified levels.

Frequent feature: use of baskets of instruments as an underlying index

Baskets:• may consist entirely of energy commodities

• may include, in addition to energy commodities, other commodities, such as metals

• may be combination of commodities with practically any other group of indexes

• we often witness products dependent on combination of crude, natural gas, metals, SP500, Treasury yields, LIBOR, etc.

• Creation of these baskets has become progressively easier in recent years with popularization of the Goldman-Sachs Commodity Index (GSCI) and its various sub-indexes

Hybrid Products

• Depend on several market/non-market drivers

• We interested in hybrid products which are exposed to at least one commodity

• Pricing requires analysis of correlation structure (in addition to volatility)

Hybrid Products: Examples

• Price/Price – spark spread options, crack spread options

• Price/Volume – load following deals

• Price/Temperature products

• Basket products – Rainbow options, Himalayan options

• Interest rates/FX/Equity contingent commodity products – swaps, swaptions

• Credit/Commodity products – cds linked to commodity price

Spark Spread Options

• Tolling deals– call on power with strike price dependent on the cost of fuels,

emission and variable costs = option on spread between power prices and prices of fuels and emission

– basket of correlated commodity products (three or four products in the basket)

– objectives:• power operator will guarantee stable cash flows stream (option

premium) typically from an institution with higher credit rating• power plant operator may also use these options to hedge against

adverse power and fuel market movements • marketers use these options to financially replicate power plant

operation without taking on operational and other risks associated with running the plant

Tolling Deals: Examples

• Unit Contingent Toll with Callback on High Gas– Standard Toll: Buyer has the right to call for power. When the

right is exercised the buyer pays the cost:Number MWh x Price of 1MMBtu of NG x Heat Rate + costs

– Callback: Seller has the right not to deliver power during not more than 10% of all hours of the year (if a specified unit is forced out)

• Tolling Deal with Limited Number of Start-ups during the year - complex path-dependent option

• Tolling deals with fuel substitution option

Challenges: Correlation Structure• Correlation has a complex term structure: seasonality,

dependence on time to maturity• “Correlation smile”: in Black-Scholes-type models used

to price complex spread options correlation parameters may depend on underlying prices

• Example: Correlation vs Power_price/NG_price

Price/Volume Products

• Swing options• Load following contracts

– receiving fixed payments – paying costs of serving the load: Price x Load

• Challenges:– Potentially strong non-linearity (if the correlation is high)– Complex correlation structure– Inability to hedge all risks, particularly, risks associated with load

fluctuations and load shape dynamics– Need new approaches to valuation

Basket Products

• Options on basket price – basket components may include crude, NG, equity indices,

bonds, etc.

• Rainbow or Best-of basket products– pays the best annual return of the basket components

• Himalayan option– every year pays the return of the best performing basket

component and then this component is removed from the basket

• Challenges:– Finding distribution of basket prices– How to construct the volatility structure of the basket from the

volatility structures of the individual components?

Commodity-contingent interest rate/equity products• Commodity-contingent interest rate swap

– floating leg - LIBOR– “fixed” leg - fixed rate multiplied by the number of days

(expressed as a fraction of the payment period) during which crude or other commodity prices are above a certain level

• Commodity-contingent interest rate swaption (typically, Bermudan style)

• Bermudan-style commodity-contingent guaranteed minimum coupon knock-out option– Pays coupon dependent on the commodity price levels at the

payment time– Disappears after the total coupon reaches a specified level– If at the end of the deal the total value of paid coupons is less

than the specified value the last coupon pays the difference

Modeling challenges• Test: terminal distributions of returns at any

time T is normal - justification for the use of geometric Brownian motion (GBM) as a modeling process

• SP500: distribution of returns is close to normal

T TdP P

Modeling Challenges

Power, NG and crude prices: normality must be rejected; distribution has fat tails

Modeling Challenges

Crude: Fat tails of the distribution

Modeling Challenges

Distribution Parameters (A. Werner, Risk Management in the Electricity Market, 2003)

3.330.00423%DAX

76.822.079238%NP 6.p.m.

26.341.468182%Nord Pool

KurtosisSkewnessAnnual. Volatility

Empirical characteristics of energy commodities

• Implied volatility surface: – implied volatility increases with time– volatility depends on strike

In addition, for spread options we must consider• Correlation surface:

– Correlation depends on time to expiration– Correlation depends on time between contracts– Correlation depends on “strike” (heat rate, in case of

spark spread options)

WTI futures implied volatility curve

Correlation between returns of Jan ’04 NYMEX WTI futures contract and Feb’04 - Jun’05 WTI contracts

Volatility has “smiles”, “smirks”, and “frowns”

Stochastic Volatility (Heston, 1993)

Volatility is a random variable

price process

volatility process

1( )t

t

dP dt v t dWP

μ= +

( ) ( )( ) 2( )dv t v t dt v t dWκ θ σ= − +

( )1 2E dW dW dtρ=

Stochastic volatility process generates more realistic price distributions

Tails of CDF for terminal distributions generated by stochastic volatility process and by GBM

New Developments

• Levy Stable Processes (for review see Boyarchenkoand Levendorskii, 2002 )

• Levy Processes with Stochastic Volatility: CGMY model (Carr, Geman, Madan, Yor, 2003)

• Regime-switching models

Historic Power Prices vs. GBM paths

Hybrid Power Price ModelPower is a function of principal drivers

1. Demand

2. Fuel Prices3. Outages

Hybrid Power Price Model (Eydeland, Wolyniec, 2001)

Model uses fundamental and market data• sgen - function determined by technical characteristics of

all power plants (efficiency, operational constraints, etc.)

• D - demand• U - fuel(s) used• Ω - outages

( )1 2 3( ; , , , , , )genT T T T T T TP s D T U E VOM Cα α λ α= Ω

Hybrid Model generates realistic paths

Actual prices vs. Modeled prices

Risks• Quantifiable risks

– market/price risk– credit/default risk– modeling/valuation risk– financing/financial risk– operations risk– volumetric risk– business continuity risk– environmental risk

• Source: Committee of Chief Risk Officers (CCRO), 2002

• Non-quantifiable risks– strategic risk– operational risk– staffing/organization risk– regulatory risk– political risk– technological risk– legal risk

Managing market price risk

Two methods of risk reduction

• Diversification• Hedging

Hedging in the perfect world

V - option valueF - price of a tradable instrument (for

example, futures price)f - option pricing formula (for example,

Black-Scholes)

( )V f F=

Hedging in the perfect worldDelta Hedging:

Combine long position in one option and short position in Δ futures

VF

∂Δ =

∂

Hedging in the perfect worldChange in option value vs. change in hedge value

Hedging = risk reductionReduction of the uncertainty of the future cash flows

Energy deals and assets = spread options

• Power plants, tolling contracts = spark spread options

• Gas storage = calendar spread options• Transmission lines, pipelines, transmission

right contracts = geographical spread options

Power Plant• In the simplest case (immediate response

to price changes, one fuel -- natural gas), the plant cash flow at time T:

– C -- capacity– HR -- heat rate– P T and GT are power and nat. gas prices at time T– VOM -- variable costs

0max( ,0)

T

t tt

CF C P HR G VOM=

= ⋅ − ⋅ −∑

Hedging spread options in the perfect world

• What is needed for valuation and hedging?

– Joint distribution– Evolution processes for power and fuel prices– Cashflow function– Sufficient amount of tradable hedge instruments

(futures, forwards, options)

Hedging in energy markets: real world

• Mismatch in asset/hedge maturities: long maturity of assets vs. short maturity of hedges

• Mismatch in granularity: fine (daily, hourly) granularity of assets vs. coarse (monthly, quarterly) granularity of hedges

• Mismatch in underlying commodity, “dirty” hedges. – For example, fuel contracts in one location are used to

hedge exposure in other locations

Hedging in energy markets: real world

• Liquidity constraints:– Price may depend on the volume– Execution time may depend on the volume– Wider bid/ask spreads– Higher hedging costs– distributions are hard to calibrate because of

biases due to liquidity constraints• implication: different hedging strategies may

produce different option values

Hedging in energy markets: short and medium term• Hedge instruments are available although the

set of hedges is not complete and mismatches persist

• Consequently, hedges are “dirty” resulting in residual cashflow variance

• Liquidity, in general, is not a problem for medium size deals

• Possible to follow the “perfect world” hedging methodology

Hedging in energy markets: short and medium term• Problems in defining the right evolution

process– empirical power price data shows mean

reversion, spikes, high kurtosis, regime switching; processes difficult to calibrate due to lack of data and its non-stationarity

Hedging in energy markets: short and medium termHow is the joint distribution defined?

– Typically, by supplying correlation coefficient implicitly assuming that the joint distribution belongs to the elliptic family of distributions

– The correlation coefficient is computed using historical data

• Problem: correlation coefficient between power and natural gas (or oil) is not a constant; it has structure. Taking average will overestimate the option and miscalculate hedges

Hedging in energy markets: short and medium term• Other issues:

– Standard approach will have difficulties incorporating structural changes of the stack or demand

– Unlike cashflows of financial products, the cashflows of energy assets are determined by complex operating strategies: dispatch strategy for power plants or injection/withdrawal strategy for gas storage

Alternative Method: Hybrid Model for Power Prices

Model uses fundamental and market data• sgen - function determined by technical characteristics of

all power plants (efficiency, operational constraints, etc.)

• D - demand• U - fuel(s) used• Ω - outages

( )1 2 3( ; , , , , , )genT T T T T T TP s D T U E VOM Cα α λ α= Ω

Hybrid method

• The benefits of the method:– captures distribution properties (high kurtosis,

spikes, volatility and correlation structure)– correlation is not an input– matches market data– allows incorporation of future structural

changes

Hedge efficiency

Back testing of hedging

Managing other risks

• Credit risk - credit derivatives• Operational risk - insurance• Demographic, economic growth risks -

contractual clauses• All this increases the cost of risk

management; these costs should be taken into consideration at the valuation stage

References• Boyarchenko, Svetlana and Sergei Levendorskii, Non-Gaussian Merton-Black-

Scholes Theory, World Scientific, 2002• Eydeland, Alexander and Krzysztof Wolyniec, Energy and Power Risk

Management: New Developments in Modeling, Pricing and Hedging, Wiley, 2002

• Carr, Peter and Helyette Geman, Dilip Madan, Marc Yor, Stochastic Volatility for Levy Processes, Mathematical Finance, Vol. 13, No. 3 (2003)

• Rouwenhorst, K. Geert and Gorton, Gary B., "Facts and Fantasies about Commodity Futures" (February 28, 2005). Yale ICF Working Paper

• Heston, Steven, A Closed-Form Solution for Options with Stochastic Volatility,Review of Financial Studies, Vol. 6, No. 2 (1993)

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