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 dPT PT 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
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)
Annual.
Volatility
Skewness
Kurtosis
Nord Pool
182%
1.468
26.34
NP 6.p.m.
238%
2.079
76.82
DAX
23%
0.004
3.33
Stochastic Volatility (Heston, 1993)
Volatility is a random variable
dPt
  dt  v(t ) dW1
Pt
dv  t      v  t   dt   v(t ) dW2
E  dW1 dW2    dt
price process
volatility process
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 Model
Power is a function of principal drivers
1. Demand
2. Fuel Prices
3. Outages
Hybrid Power Price Model
(Eydeland, Wolyniec, 2001)
PT  1s gen (DT ; T ,UT , T 2  , ET ,VOMT ,3CT )
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
Hybrid Model generates realistic paths
Actual prices vs. Modeled prices
Hybrid Model: Analytical Approximation
(Mahoney, 2004)
• Fuel Price
G  e g
 (t) - seasonal factor
• Market Heat Rate
P
H
G
H  eh
• Power Price
P  e   g h
Hybrid Model (Mahoney, 2004)
dg   g  g  g  dt   g dWg
d     t    dt    dW
dh   h ( Lh  M h   h)dt   h dwh  jh dqh
E  dWg dW    g  dt
E  dWg dWh    gh dt
- fuel
- temperature
- Heat Rate
E  dW dWh    h dt
qh - Poisson process with intensity h
h   h   h 
jh ~ N   h ,  h2 
- jump magnitude
At t0 the value of the power plant at a future time T is
computed as a conditional expectation

V (t0 , g , h, )  Et0 , g ,h, e
T  gT  hT
 H 0e
Using characteristic function
f  , ; t0 , g , h,   Et0 , g ,h , e
i gT i hT
e
T  gT


A  g  B hC   D , 
the value of the plant can be represented as
 

T  gT  hT
T  gT 
dg
dh
(
e

H
e
) Pr( gT , hT | g , h, ) 
0
 T T
 
 

T  gT  hT
T  gT 
dg
dh
(
e

H
e
)
0
 T T
 
 (21 )2
 

 i gT i hT A ( ) gT  B ( ) hT C ( ) T  D ( , )
e
e
d d

 
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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