New Trends in Energy Derivatives : New Trends in Energy Derivatives
Alexander Eydeland
Morgan Stanley
Increased interest in commodity-linked products: the investors point of view : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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/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 : 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
Modeling Challenges : Modeling Challenges Power, NG and crude prices: normality must be rejected; distribution has fat tails
Modeling Challenges : Modeling Challenges Crude: Fat tails of the distribution
Modeling Challenges : Modeling Challenges Distribution Parameters (A. Werner, Risk Management in the Electricity Market, 2003)
Stochastic Volatility (Heston, 1993) : Stochastic Volatility (Heston, 1993) Volatility is a random variable
price process
volatility process
Stochastic volatility process generates more realistic price distributions : Stochastic volatility process generates more realistic price distributions Tails of CDF for terminal distributions generated by stochastic volatility process and by GBM
New Developments : 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 : Historic Power Prices vs. GBM paths
Hybrid Power Price Model Power is a function of principal drivers : 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) : 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
Hybrid Model generates realistic paths Actual prices vs. Modeled prices : Hybrid Model generates realistic paths Actual prices vs. Modeled prices
Hybrid Model: Analytical Approximation (Mahoney, 2004) : Hybrid Model: Analytical Approximation (Mahoney, 2004) Fuel Price
(t) - seasonal factor
Market Heat Rate
Power Price
Hybrid Model (Mahoney, 2004) : Hybrid Model (Mahoney, 2004)
Slide27 : 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
Correlation Risk : 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 : Difficulties in managing correlation risk
correlation is not traded
historical data is poor
data is nonstationary, markets are evolving
What are the alternatives? : What are the alternatives?
Structural models
Correlation independent bounds; super/sub-replication
Managing other risks : 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 : 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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