# Real Option Theory and Electricity Forwards

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```					Real Option Theory and Electricity Forwards

John van der Hoek, Mahmoud Hamada and Tony Hall

University of Adelaide, and University of Technology of Sydney

Quantitative Methods in Finance Conference

December 14th, 2005
Motivation and General Context

   National Electricity Market liberalized ( deregulated )

   Wholesale electricity prices are determined by intersecting demand and supply every 5 min.

   Very high volatility ( \$40/MWh to \$10,000/MWh ) over few hours

   Retailer buys at a floating price and sells at a fixed price: huge market risk

   Need for hedge : Electricity forwards

   Question: What is the fair price of electricity forwards?

2
Pricing Methodologies

       In the literature: different methods for pricing electricity forwards
-    No-arbitrage pricing
-    Equilibrium pricing

In this paper:

       Applying Real option theory to electricity, because it has similar characteristics to real asset
-   Not storable

       In Real option pricing literature, there are two main approaches:
-    Replicating the non-tradeable asset by a portfolio of tradeable assets
-    Applying some new principals with roots in the actuarial sciences (Musiela, Henderson,

 This paper applies a new approach by Elliott and van der Hoek (2003), a generalization of the
two approaches

3
New approach for valuing Real Options

 Let X(t) denote the investor’s wealth at time t and
V 0 x  max E u X 1
where the maximum is taken over all portfolios in tradeable assets with X 0  x and they have
uncertain value X 1 in one period’s time. We also let
V G x  max E u X 1  G
where G is the value of the contingent claim at time 1 and the maximum is taken over all
portfolios in tradeable assets with X 0  x.
 We define  b G , the bid price of G, as the solution of
VG x   V0 x
In the presence of optimal investment in tradeable assets at a given level of wealth x we are
indifferent between  now and G in one year’s time.
u x
 If the utility function u is such that Arrow-Pratt risk aversion parameter u x is independent
of the states of the nature, then

     1 E G  1 u W var G
R          2 u W
u W
   Similar to the actuarial variance principal with loading factor of   u W

4
Electricity Forward Pricing

   Definition of Electricity forward contract
Electricity Forward Contract Difference Payments

80

70

60

50
Spot price values

40

30

20

10

0
24/01/2005 0:00   25/01/2005 0:00    26/01/2005 0:00   27/01/2005 0:00   28/01/2005 0:00   29/01/2005 0:00   30/01/2005 0:00   31/01/2005 0:00
Time in Half-hours

Spot Price     Strike Price

   Difference payments from a forward contract spanning n weeks is
n                      Mw                                       n
e    rw
Si         F L                         e      rw
MwL Pw                     F
w1                      i1                                   w1
where M w is the number of half-hours contracted in week w, S i is the pool spot price for the
half-hour i, P w is the average price for week w and F : the strike price of the contract (\$/MWh)

5
 Suppose that the contract spans 1 week. If S t were known, then the fair forward price F could
be chosen such that
Mw
F    1               St
Mw
t1

   The usual approach in no-arbitrage models consists of putting
Mw
F    1             EQ St
Mw
t1

6
Electricity Forward Pricing

   Electricity prices characteristics: seasonality, cycles, autocorrelation, by time partitions
   Modelling is done by Peak and offpeak

NSW Historical Off Peak            Weekly                     Extracting cycles
Pool Spot Prices (S)               Averaging (P)   Ln (P)
(x) = Ln(P) - s

Fitting Normal                                        Estimating AR and
Residuals (Noise) (ε)       Removing them from (x)
Distribution

After fitting a linear model to x, we obtain an AR 3 model
ln P t  s t  x t
where £s t ¤ is a deterministic seasonal component, and £x t ¤ is modelled by an AR 3 process
x t  + 1 x t 1  + 2 x t 2  + 3 x t 3  ( t

7
Pricing Electricity Forwards

   Electricity is a real asset so:   P t   1   and G  #P t where 0  #  1 is a constant.

  log # can be thought of as a storage cost (of fuel) per megawatt hour per week. With
exponential utility functions:

P t 1  1 E t 1 #P t  var t 1 #P t
R              2

   The risk aversion parameter  t    1can be inferred from the above as follows
E t 1 #P t  RP t 1
t 1       #2
2
var t 1 P t

where R  e rt , r is the risk-free rate of interest and t  1/52 (1 week).

8
A Recursion Formula for Electricity Forward Prices

   The settlement amount for a forward contract spanning n weeks is
n     Mw                    n
 e rw  S i    F L       e rw M w L¡P w   F¢
w1    i1                 w1

 where M w is the number of half-hours contracted in week w, S i is the pool spot price for the
half-hour i, P w is the average price for week w and F is the strike price.
 Let V n be the present value of this contract (asking price), then V n is the ask price of
t                                                              t
L t1 P t1 F  V t11  Z t1
n

   So

V n  1 E t Z t1  t var t Z t1
t
R             2
where
E t Z t1  L t1 E t P t1                      F  E t V t11
n

and
2
var t Z t1  L t1 var t P t1  var t V n 1  2 L t1 cov t P t1 , V n 1
t1                           t1
and

  RP
t
t
#2
#E t ¡P t1 ¢
var t ¡P t1 ¢
2

9
Mathematical Preliminary

We assume that the process of weekly average prices, P t tK0 is such that
P t  e s t x t
where £s t ¤ is a deterministic seasonal component, and £x t ¤ is modelled by an AR 3 process
xt  +1xt 1  +2xt 2  +3xt 3  (tt
   Lemma
+            +       +3
E t P t1  M t, P t 1 P t 12 P t                   2
where
M t,    exp     s t1   +1 st      +2 st        1        +3 st        2    1 ( t1
2      2
2
   and
2 +1      2 +2 2 +3
var t P t1  N t, P t                 Pt 1 Pt 2
where
2(2
N t,      1        e              M t, 2
   For each time t  0 and for any real numbers %,  0, we have
%                                     ( 2 %            % + 1        % + 2   % +
cov t P t1 , P t1  M t, %   1        e                 Pt             Pt    1      Pt 2 3

10
Week-Ahead Forward Contract (Case n  1

The present value of the one week contract is

V 1  1 E t Z  t var t Z
t
R            2
where
Z  L t1 P t1 F
   This gives a starter value for V n in the recursion formula.
t

   Therefore, the week-ahead forward price is given by
+     +      +
F t 1  11 a 1 P t  b 1 M t P t 1 P t 21 P t 32
t       t
ct
where

2
a1
t          L t1
#2
2
b1
t       L t1
R
L t1
#R

c1 
t
L t1
R

11
General case

 By induction, we can prove that the forward price corresponding to the n-period contract is
given by

 a n k P  P t 1 P t 2
n   n    n
F t n  1n        t     t
k   k    k
ct
k

where
c n1  1 L t1  12 L t2 . . .  1 L tn1
t
R         R                R n1

and a n k  n ,
t     k
n
k   and  n are deterministic coefficient, computed in a recursive fashion.
k

12
Further development

   Calibration of the forward prices to market data

   Markov chains probabilities approximation for efficient calculation of a n k  n ,
t     k
n
k   and  n
k

13

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