New Trends in Energy Derivatives

Alexander Eydeland

Morgan Stanley

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

– 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

Examples: Commodity-linked bonds

• At redemption, holder is paid par if the GSCI has fallen. If the the GSCI price has risen, holder receives par (1 + a percentage gain in the GSCI)

• At redemption, holder receives

85% of par + par * (2 * percentage rise in gold price)

For example, if gold grows from $400 to $440 then the holder of a $1000 par bond gets $1000(.85) + $1000 * 2(.1) = $1050

Examples: Commodity-linked bonds

• At redemption, holder receives par. In addition, holder receives semi-annual coupon. Those payments are .82 (percentage gain in the NYMEX WTI). Say the NYMEX WTI goes from $50/bbl to $55/bbl, coupon payment on a $1000 par bond would be .82 (.1) (1000) or $82. Next coupon payment would be determined off a

new base price of $55.

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

Stochastic Volatility (Heston, 1993)

Volatility is a random variable

price process

volatility process

1( )t

t

dPdt v t dW

P

2( )dv t v t dt v t dW

1 2E dW dW dt

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 Boyarchenko and 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 C

Hybrid Model generates realistic paths

Actual prices vs. Modeled prices

Hybrid Model: Analytical Approximation (Mahoney, 2004)

• Fuel Price

(t) - seasonal factor

• Market Heat Rate

• Power Price

gG e

hPH H e

G

g hP e

Hybrid Model (Mahoney, 2004)

g g g h gh h hE dW dW dt E dW dW dt E dW dW dt

- temperatured t dt dW

( ) - Heat Rateh h h h h h hdh L M h dt dw j dq

- fuelg g g gdg g dt dW

- Poisson process with intensity h hq

h h h

2~ , - jump magnitudeh h hj N

At t0 the value of the power plant at a future time T is computed as a conditional expectation

Using characteristic function

the value of the plant can be represented as

00

,, , ,, ; , , , T T A g B h C Di g i h

t g hf t g h E e e

0( )T T T T Tg h gT Tdg dh e H e

2

( ) ( ) ( ) ( , )1(2 )

T T T T Ti g i h A g B h C De e d d

00 , , , 0( , , , ) T T T T Tg h gt g hV t g h E e H e

0( ) Pr( , | , , )T T T T Tg h gT T T Tdg dh e H e g h g h

Correlation Risk

• Correlation structure is complex

• Term structure: dependence on time to expiration, time interval between two contracts; seasonality

• Sensitivity to correlation is high

• How to manage correlation risk?

Difficulties in managing correlation risk

• correlation is not traded

• historical data is poor

• data is nonstationary, markets are evolving

What are the alternatives?

• Structural models

• Correlation independent bounds; super/sub-replication

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)

•

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

• Mahoney, Daniel, A New Spot Model for Power Prices, Preprint, 2004

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