Pricing electricity forward contracts under observable information … · Japan Electric Power...
Transcript of Pricing electricity forward contracts under observable information … · Japan Electric Power...
AJRC and RIETI workshop on economic and financial analysis of commodity markets
September 14—15, 2017
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Pricing electricity forward contracts under observable information in JEPX
(Japan Electric Power Exchange)
Faculty of Business SciencesUniversity of Tsukuba, Tokyo, Japan
E-mail: [email protected]://www2.gssm.otsuka.tsukuba.ac.jp/staff/yuji/
Y. Yamada
Acknowledgement
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Supported by Grant-in-Aid for Scientific Research (A) 16H01833 (PI: Yuji Yamada) from Japan Society for the Promotion of Science (JSPS).
(a)Faculty of Business Sciences, University of Tsukuba(b)Faculty of Science and Technology, Tokyo University of Science(c)Faculty of Engineering Division I, Tokyo University of Science(d)Faculty of Science and Engineering, Chuo University
Naoki Makimoto(a), Setsuya Kurahashi(a)
Ryuta Takashima(b), Nobuyuki Yamaguchi(c)
Junya Goto(d)
Collaborators
Japan Electric Power Exchange: JEPX
From short term (half hour) to long term (1 week—1 year)
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Match sell and buy orders for delivering a fixed amount of electricity (kWh) in nine areas, Hokkaido, Tohoku, Tokyo, Chubu, Hokuriku, Kansai, Chugoku, Shikoku, and Kyushu
(http://www.jepx.org/)
Forward contractSpot contract
Japan Electric Power Exchange: JEPX
Spot electricity in JEPX: Every half hour, 48 products a day.Orders are closed at 9am 1 day before the delivery.
(http://www.jepx.org/)
Forward contracts:Unique spot price in the entire 9 areas of Japan
Written on the system price
Illiquid, only traded several times a year!
System price
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Outline
Forward prices under observable information
1. Forward contracts in JEPX, its expression using daily forwardprices, and an application of Esscher transform
2. Derivation of daily forward price via Esscher transformation
Comparison with Girsanov transformation
Modeling of underlying spot price using state space equations
3. Empirical simulation
4. Concluding remarks
(JEPX spot price)
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Financial settlement between forward and system prices for the delivery period from 1 week to 1 year in the future.
Forward contracts in JEPX
: Average system price at day (spot price) , : Forward price with days delivery starting at day (> )
, , ,Float CF
Issue date
Fixed CF
Total payoff: , = × ,
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Expression using daily forward prices, : = , : Daily forward price of delivering at day only
Total payoff: , = ,
, = 1 ,
= × ,No arbitrage!
, , ,Float CF
(Stochastic)
Float CF(Deterministic)
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Spot and daily forward prices, : = , : Daily forward price of delivering at day only
Payoff:
Connection between spot and forward prices
( ): Deterministic function of
- Seasonal and long term trends + Day-of-the-week effects
ln = + = exp +: Spot price process on , , ,
( ): -adopted stochastic process (observed at time )
, Function of
Forward price via Esscher transform (Kijima and Tanaka‘07)
: = ,Change of measure:
Forward price:
,, =
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Esscher transform: = exp + Uncertain random variable
: Esscher paramter
= 1:=
= 1
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Forward price and risk premium
, = 1 ,Forward price with days delivery:
Risk premium Seller > Buyer Seller < Buyer Seller = Buyer
Price
may be implied by matching with the realized forward price.
= 0 (i. e. , = 1) is risk neutral so that , = .
Positive (negative) reflects on the sellers’ (buyers’) risk premium.
, = = 1 = 1
= 0Risk neutral
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Outline
Forward prices under observable information
1. Forward contracts in JEPX, its expression using daily forwardprices, and an application of Esscher transform
2. Derivation of daily forward price via Esscher transformation
Comparison with Girsanov transformation
Modeling of underlying spot price using state space equations
3. Empirical simulation
4. Concluding remarks
(JEPX spot price)
Modeling of underlying spot price
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( ): Seasonal and long term trends + Day-of-the-week effects
ln = + = exp += , = +
: state variable, : white noise, , × , × , : constant matrices
Observed equation: State equation:
( ): -adopted stochastic process (observed at time )
- , e.g., Durbin & Koopman ‘01
e.g., Gibson & Schwartz ‘90, Schwartz & Smith ‘00
- Multi-factor model
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Computation of forward price
, = = ==, Conditionally independent of
, = , ,,= exp + , + , + 1 ,
, : Cumulant generating function
- , : Predicted value of .
= 1
, : Prediction error.
= exp + == + = +
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- Girsanov:
Comparison with Girsanov transformation
= , : Market price of risk
under + under
- Esscher: = , : Esscher parameter
under + under
: -dimensional Brownian motion
Continuous time model: = + + , =
= ( )Equivalence condition:
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= + += = + ( ) + ( )
Girsanov transform
= = ( )( ) =
Esccher transform
Proof: = + + , =Comparison with Girsanov transformation
= ( )Equivalence condition:
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Outline
Forward prices under observable information
1. Forward contracts in JEPX, its expression using daily forwardprices, and an application of Esscher transform
2. Derivation of daily forward price via Esscher transformation
Comparison with Girsanov transformation
Modeling of underlying spot price using state space equations
3. Empirical simulation
4. Concluding remarks
(JEPX spot price)
Step 2) Apply ( ) for { } and compute distribution of prediction errors , .
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Step 1) Estimate trend function and residual using underling spot price data , = 1,… , .
Empirical simulation
, = , ,,
- , = : Predicted value of . , : Prediction error.
Spot price (daily average of 48 system prices released every day)Data period: 2012/4/1 2016/12/31 ( = 1736)
Daily forward price (Esscher):
Step 3) Comparison with realized forward prices.
Step 1) Estimation of seasonal trend using GAM
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: Holiday dummy, : Day dummy (0 or 1)
: Seasonal trend dummy (e.g., 1,…,365 (366))
,0),...,365(364tSeasonal)366(365,...,1tSeasonal
).731(730,...),367(366tSeasonal
Choose spline function for = 1,… , 365 (366).
Generalized Additive Model (GAM; Hastie and Tibshirani‘90)
ln = + × + × + +
Yamada et al. ’14: Apply GAM with repetitive data with
Estimation of trend function by GAMs
ln = + × + × + +
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(Mon—Sat and Holiday) Estimated seasonal trend function
Data period: 2012/4/1 2016/12/31 ( = 1736)
Predicted values for { } using ( ) model with = :
: Current date, : Learning data period, : prediction horizon
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Learning data ( days) Predict
Compute prediction error
Repeat until day
Start ( )
Given learning data,
Step 2) Estimation of prediction errors
, = , × ,, , = , … ,Daily forward price using empirical distribution:
+ 1construct ( ). Predict by , =
and compute , for = ,… , .
Mean Absolute Error (left) and Standard Deviation (right)
Prediction errors of ( ) when = 90 days:
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= 1 + 1 , , = 1 , ,
Term structure of prediction errors
Empirical estimation of daily forward prices
MAE between the daily forward pricewith = 0 and spot price
= ,, : Risk premium rate
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, = , × ,, , = , … ,Estimated value of daily forward price:
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Step 3) Comparison with realized forward prices
, = 1 ,Estimated forward price:
Realized forward prices in JEPX (2012/4/1 2016/12/31):
Year of transaction 2012 2013 2014 2015 20161 week delivery 0 0 8 2 (1)1 month delivery 0 0 2 3 -
888 2 (111)))))))))))))))))))))))) 11 forward prices
Risk premium Seller > Buyer Seller < Buyer Seller = Buyer
Price= 0
Risk neutral
may be implied by matching with the realized forward price.
Risk premium of realized forward price?Estimated sample path of forward price with implied ?
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Step 3) Comparison with realized forward prices
Forward payoff:
= , = 1 1 ,,Payoff normalized by
number of days
Predicted vs. realized values of the underlying vs. forward payoffs?
Risk premium Seller > Buyer Seller < Buyer Seller = Buyer
Price= 0
Risk neutralPredicted
value of the underlying
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Predicted and realized values vs. forward payoff
“Realized Predicted of the underlying”& Normalized forward payoff Implied by AR and RW predictions
- Realized payoff is significantly negative.AR RW Payoff
0.427 0.498 0.918**
MAE 0.770 1.18 1.10 - Implied by AR prediction is always positive.
Case with maximum payoff size
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AR prediction RW prediction
- In the case of RW prediction, the forward price fluctuates with the daily spot price and is affected by “spikes.”
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Case with minimum payoff size
AR prediction RW prediction
- In the case of AR prediction, the forward price is smoothand seems to reflect on the long term average of spot prices.
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Concluding remarks
In the case of RW prediction, the forward price fluctuates with the daily spot price and is affected by “spikes.”
Generalized additive model (GAM)
In the case of AR prediction, the forward price is smoothand seems to reflect on the long term average of spot prices.
Forward prices under observable information (JEPX spot price)
Empirical simulation
Modeling of spot price dynamics and forward prices
- Trend functionState space equation formula- Stochastic partEsscher transformation- Forward price