New Trends in Energy Derivatives

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					New Trends in Energy

   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
  – 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
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
• 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 dP P at any
                                                T   T
     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.      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

                  dt  v(t ) dW1                  price process

     dv  t      v  t   dt   v(t ) dW2   volatility process

             E  dW1 dW2    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 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 ,VOM T , 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
                 H            H  eh

 • Power Price
                     P  e   g h
   Hybrid Model (Mahoney, 2004)

               dg   g  g  g  dt   g dWg            - fuel

            d     t    dt    dW               - temperature

        dh   h ( Lh  M h   h)dt   h dwh  jh dqh    - Heat Rate

E  dWg dW    gdt       E  dWg dWh    gh dt      E  dW dWh    h dt

               qh - Poisson process with intensity h
                             h   h   h 
                             jh ~ N   h , h 
                                                           - jump magnitude
      At t0 the value of the power plant at a future time T is
         computed as a conditional expectation
                                                                                          
                                                         T  gT  hT            T  gT
                   V (t0 , g , h, )  Et0 , g ,h, e                    H 0e

      Using characteristic function
                                                                           A  g  B h C   D , 
f  , ; t0 , g , h,    Et0 , g ,h , e
                                                   i gT i hT
      the value of the plant can be represented as
             

            
                 dgT dhT (e T  gT hT  H 0 e T  gT )  Pr( gT , hT | g , h, ) 
                               

                              
                                   dgT dhT (e T  gT hT  H 0 e T  gT ) 
                                              
                                   (2 )2
                                                
                                              
                                                   e  i gT ihT e A( ) gT  B ( ) hT C ( ) T  D ( , ) 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
What are the alternatives?

• Structural models
• Correlation independent bounds;
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
• 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

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