スポット価格予測に基づくJEPX先渡価格付けモデルの構築

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1 RIETI Discussion Paper Series 17-J-072

2 RIETI Discussion Paper Series 17-J 年 12 月スポット価格予測に基づく JEPX 先渡価格付けモデルの構築 * 山田雄二 ( 筑波大学ビジネスサイエンス系 ) 要旨電力システム改革に基づく小売電力完全自由化を背景に, 日本卸電力取引所 (JEPX) における卸電力の取引所取引が, 現在, 注目を浴びている. 特にスポット電力市場は,2016 年度実績で約定量が総需要の 3% 程度に達し,2005 年の開設以来, 継続的な取引が行われている. 一方, 先渡市場については, 開設から 10 年以上が経過した現在においても, 取引は年に十数件程度とデータがまばらにしか観測されず, 実績先渡価格のみから先渡価格のモデルを構築することは困難である. そこで本研究では, 現時点で連続的なデータが観測される JEPX スポット価格をベースに, 価格予測の考え方を用いて先渡価格モデルを構築することを考える. 本論文では,JEPX スポット価格の対数系列をトレンドと残差に分解し, 金融工学分野におけるデリバティブ価格付け手法の一つである測度変換を適用することで先渡価格を導出する. 具体的な手順は以下の通りである. まず,JEPX スポット価格系列を確定的なカレンダーパラメータのトレンド関数とそれ以外の残差項を用いて表現し, 残差項を状態空間モデルとして記述する. つぎに, 残差に対して時系列モデルを当てはめ, 残差の条件付期待値から将来価格を予測するスポット価格予測モデルを構築する. さらに, スポット価格予測モデルに対して, 測度変換手法の一つであるエッシャー変換を適用し, 先渡価格を定式化する. 最後に実証分析を実施し, スポット価格の実績データに対して提案手法を適用した際の先渡受け渡し期間におけるスポット価格の予測精度, および実績先渡ペイオフから推定されるリスク回避度について考察する. キーワード : 日本卸電力取引所 (JEPX), 先渡取引, 価格予測モデル, エッシャー変換, リスクプレミアム JEL classification: C14, C15, C22, C32, C44, C53, G17, L94 RIETI ディスカッションペーパーは専門論文の形式でまとめられた研究成果を公開し活発な議論を喚起することを目的としています論文に述べられている見解は執筆者個人の責任で発表するものであり所属する組織及び ( 独 ) 経済産業研究所としての見解を示すものではありません * 本稿は ( 独 ) 経済産業研究所におけるプロジェクト商品市場の経済ファイナンス分析の成果の一部である 1

3 1 (Japan Electric Power Exchange; JEPX) [11] JEPX [8, 10, 12, 13, 18] (TOCOM) JEPX JEPX 1 30 (0.5 ) 48 JEPX % JEPX Black-Scholes [1] JEPX JEPX JEPX,,, 2

4 JEPX ( ) 2 JEPX 2.1 JEPX JEPX [11].. ( ) JEPX ( ) (JEPX ). 2.2 (CF) 24 t (t = 0, 1, 2,...) (24 1kWh ) S t T 0 m ( ) F (m) t,t 0 CF [11] T 0 +m 1 T =T 0 [ ] T 0 +m 1 S T F (m) t,t 0 = T =T 0 S t S T m F (m) t,t 0 (2.1) JEPX 1 ( )

5 ( ) ( ), JEPX JEPX ( ) ( ) JEPX 2.3 F (m) t,t 0 m 1 (7 ) 1 1 m m = 1 m F (m) t,t 0 S t m = 1 F t,t := F (1) t,t (2.2) F t,t m F (m) t,t 0 T 0 m t m m T 0 +m 1 T =T 0 [S T F t,t ] = T 0 +m 1 T =T 0 T 0 +m 1 S T F t,t (2.3) T =T 0 F (m) t,t 0 (2.1) (2.1) (2.3) T 0 +m 1 m F (m) t,t 0 = F t,t F (m) t,t 0 = 1 T 0 +m 1 F t,t (2.4) m T =T 0 T =T 0 m m (2.3) S T F t,t, T = T 0,..., T 0 + m 1 4

6 S T S t ( ) Black-Scholes [1] 2 Step 1: S t, Step 2: ( ) ( ) Black-Scholes ( ) Esscher JEPX f(t) η t ln S t = f(t) + η t S t = exp [f(t) + η t ] (3.1) (3.1) f t f(t) (Ω, F, P) {η t } {F t } t F t - P S T ( ) ( ) 1. 5

7 P P [5] Esscher JEPX Esscher Esscher 2. [5] Esscher Z t [ e λη T ] Z t := E E [e λη T ] F t, 0 t T (3.2) Esscher F t,t, F t,t = Ẽ [S T F t ] = 1 Z t E [S T Z T F t ], 0 t T (3.3) T λ Esscher Ẽ P 3 P (A) := E [I A Z T ], A F T (3.4) {η t } 4 P η T ψ ηt (u) = E [exp (uη T )] = exp ( u 2 σ 2 η T /2 ) ( σ ηt η T ) P η T ψ ηt (u) = Ẽ [exp (uη T )] = E [exp (uη T ) Z T ] = E [exp ((u + λ) η T )] /E [exp (λη T )] = ψ ηt (u + λ) /ψ ηt (λ) = exp ( uλσ 2 η T + u 2 σ 2 η T /2 ) (3.5) η T P λσ 2 η T S T = e f(t )+η T η T 0 ( F 0,T ) λ = 0 F 0,T λ=0 = E [S T ] λ > 0 F 0,T > E [S T ], λ < 0 F 0,T < E [S T ] Esscher λ > 0 λ < 0 ( ) 2 [5] Esscher. 3 ( [5, 16] ) (3.6) {η t }. 6

8 3.2 {η t } x t+1 = Ax t + Bw t (3.6) η t = C x t A R n n, B R n l, C R n ( ) x t R n w t N (0, Σ) l (Σ R l l ) (3.1), (3.6) A 5 AR (p) : η t = a 1 η t a p η t p + c + ε t (3.7) ARMA(p, q) : η t = a 1 η t a p η t p + b 1 ε t b q ε t q + c + ε t (3.8) {η t } AR (3.7), ARMA (3.8) (3.6) [4] (3.1) {η t } 6 AR (p) {η t }. {η t } (3.6) T η T, S T (3.6) η T T t η T = C A T t s Bw t+s + C A T t x t (3.9) s=1 t [0, T ] η T g t,t g t,t := E [η T F t ] = C A T t x t (3.10) η T g t,t η T T t η T g t,t = C A T t s Bw t+s (3.11) s=1 t + 1 F t S T E [S T F t ] [ ] E [S T F t ] = E e f(t )+η T F t ] = e f(t ) E [e (η T g t,t )+g t,t Ft = e f(t )+g t,t E [ e η T g t,t ] (3.12) (3.12) S T E [S T F t ] η T g t,t 5 a i, b j, c {ε t }. 6 {η t }. 7

9 3.3 (3.3) Esscher g t,t F t - η T g t,t F t (3.2) Z t Z t = [ e λη Ẽ T ] E [e λη T ] F t Ẽ [ e ] λ{(η T g t,t )+g t,t } F t = E [ e λ{(η T g t,t )+g t,t } ] = eλg t,t Ẽ [ e λ(η T g t,t ) F t ] E [ e λ(η T g t,t ) ] E [e λg t,t ] = e λg t,t E [e λg t,t ] (3.13) F t,t ] F t,t = Ẽ [S T F t = 1 E [S T Z T F t ] Z t = E [ ] e λg [ t,t e λg E e f(t )+η e λη ] T T t,t E [e λη T ] F t = e f(t ) E [ ] [ ] e λg t,t e (λ+1){(η T g t,t )+g t,t } e λg E t,t E [ e ] λ{(η T g t,t )+g t,t } F t = e f(t )+g E [ e ] (λ+1)(η T g t,t ) t,t E [ e ] (3.14) λ(η T g t,t ) (3.14) η T g t,t σ ηt g t,t ( B.1 ) 5 η T g t,t 4 (3.1) f(t) 4.1 GAM. [19] 8

10 . S t ln S t, t = 1,..., N (Generalized Additive Model; GAM [7]) 7. ln S t = f(t) + η t h (Seasonal t ) + β 1 Mon t + β 2 T ue t + + β 6 Sat t + β 7 Hoilday t + β 8 P eriod t + η t (4.1) h GAM β i, i = 1,..., 8 η t E [η t ] = 0 Seasonal t : (= 1,..., 365 (or 366)). Mon t, T ue t,..., Sat t :. Mon t =1 ( ) or 0 ( ) Holiday t : ( 1 0) P eriod t : (= 1,..., N). GAM (4.1) f(t) ln S t GAM (4.1) f(t) η t 1 h ( ) ( ). h GAM (4.1) Y = X b + e ( Y X, e ) Y X Y = X Y X (ˆb = e b + e e (4.2) ( X X ) 1 X Y ) h GAM (4.1) 1. Y ( ) X S (i) 1 (a), (b), (c) Seasonal (i) t (a) 2 29 : (i = 1, 2, 3) Seasonal (1) t = 365,..., 0, Seasonal (2) t = 1,..., 366, Seasonal (3) t = 366,..., R3.0.2 ( mgcv gam() GAM gam() [17]. 9

11 (b) (a) : (c) : Seasonal (1) t = 364,..., 0, Seasonal (2) t = 1,..., 365, Seasonal (3) t = 367,..., 731 Seasonal (1) t = 364,..., 0, Seasonal (2) t = 1,..., 365, Seasonal (3) t = 366,..., GAM (4.1) Y S (1) X Y, S (2) X (4.3) Y S (3) X 3. Seasonal (2) t = 1,..., 365 (366) GAM 8. (2 29 ) 1. (4.1) f GAM (4.1) h β i, i = 1,..., 8 t = 1,..., N f 4.2 GAM (4.1) {η t } 3.2 {η t } AR (p) t = 1,..., N {η t } k L AR a i, i = 1,..., q τ η k+τ {η t } AR (p) η k+τ B.2 (B.4) 1. (t = 1,..., N) JEPX GAM (4.1) h, β i, i = 1,..., 8 S t η t 2. k L k τ k = L,..., N τ 8 t t 10

12 (a) AR (3.7) 9 (B.4) ĝ k,k+τ := E [η k+τ F k ] (b) ĝ k,k+τ k τ ( k + τ) η k+τ ĝ k,k+τ 3. λ, t, T = t + τ (3.14) F t,t+τ ˆF t,t+τ (t = L,..., N τ) ˆF t,t+τ = e f(t+τ)+ˆη t,τ N τ e (λ+1)(η k+τ ĝ k,k+τ ) k=l N τ e λ(η k+τ ĝ k,k+τ ) k=l, t = L,..., N τ (4.4) ( ) k η k η k+τ (Random walk; RW) AR RW ĝ k,k+τ η k (4.4) λ = 0 ˆF t,t+τ λ=0 t + τ S t+τ ( ) JEPX JEPX Fig. 5.1 Mon t,..., Sat t β 1,..., β 6 β 7 β 8 c β 8 : , c: % 9 MATLAB ARfit [9] ( AR ARfit q Schwarz s Bayesian Criterion. 10 JEPX JEPX ( N = 1, 698 (t = 1,... N). 11

13 º h(seasonal period) Seasonal period Fig. 5.1: β i (, ) Fig. 5.2: GAM Fig. 5.2 (4.1) h ( ) 95% h 1,..., 365 (366) 1 GAM R % 5.2 η t (3.7) τ η t+τ ĝ t,t+τ 4.2 η k+τ ĝ k,k+τ τ k = L, L+1,..., N τ ( L ) (Mean absolute error; MAE) (Standard deviation; SD) N τ 1 N τ MAE = η k+τ ĝ k,k+τ, SD = 1 { } 2 (ηk+τ ĝ k,k+τ ) η k+τ ĝ k,k+τ (5.1) N τ L + 1 N τ L k=l η k+τ ĝ k,k+τ η k+τ ĝ k,k+τ AR L = 90 ( ) RW (5.1) ĝ k,k+τ η k MAE, SD Fig. 5.3 AR RW ( 0 ) MAE SD MAE SD τ RW AR MAE, SD k=l 12

14 0.18 MAE of residuals for AR and RW predictions 0.2 SD of residuals for AR and RW predictions Mean absolute error (MAE) Standard deviation (SD) AR RW Prediction horizon 0.06 AR RW Prediction horizon Fig. 5.3: MAE ( ) ( ) 90 RW MAE SD AR AR RW MAE, SD 2.2 MAEs of AR and RW predictions 2 Mean absolute error (MAE) AR RW Prediction horizon Fig. 5.4: MAE MAE, SD η k+τ ĝ k,k+τ, k = L, L + 1,..., N τ (4.4) ˆF t,t+τ t + τ E [S t+τ F t ] ( ˆF t,t+τ λ=0 ) S t+τ 13

15 MAE SD Fig. 5.4 MAE τ MAE 11 MAE = 1 N τ L + 1 N τ k=l S k+τ ˆF λ=0 k,τ (5.2) AR RW MAE MAE MAE AR MAE JEPX t 0 t 1 t 0 + τ ( ) 1 t 1 m 1 1 ˆF t,τ+t0 +i t 1 t m t=t 0 τ t 0 24 m = 7 ( ) (5.3) λ = 0 Fig. 5.5 ((5.3) λ = 0 ) ( ) 13. AR RW 14 3 (AR RW ), i= , , (5.3) 11 MAE SD HP JEPX

16 3 Prediction errors of AR and RW predictions with forwad payoff 10 Risk aversion coefficients implied by AR and RW predictions Prediction error (Real-Pred) or Forward payoff : 01-Feb : 08-Feb : 15-Mar : 05-Apr : 12-Apr : 10-May : 24-May : 05-Jul : 18-Jul : 25-Jul : 24-Dec-2016 AR pred. RW pred. Forward payoff Forward contract number Risk aversion coefficient () : 01-Feb : 08-Feb : 15-Mar : 05-Apr : 12-Apr-2014 AR pred. RW pred. 6: 10-May : 24-May : 05-Jul : 18-Jul : 25-Jul : 24-Dec Forward contract number Fig. 5.5: AR RW ( ) ( ) 5% AR RW 5% t , , AR Fig. 5.5 λ AR RW AR 8 10 Fig. 5.6 Fig. 5.7 Fig , 2,..., AR ( ) RW ( ) λ = 0 ( ) x 2 2 Fig. 5.5 λ AR RW AR RW RW ( ) AR 15

17 Forward price of 7 days delivery from 05-Jul-2014 (AR, lambda = ) Predicted Esscher Daily spot Foward Realized Forward price of 7 days delivery from 05-Jul-2014 (RW, lambda = ) Predicted Esscher Daily spot Foward Realized Price 17 Price Time in days before start of delivery Time in days before start of delivery Fig. 5.6: AR ( ) RW ( ) Forward price of 7 days delivery from 24-Dec-2016 (AR, lambda = ) Predicted Esscher Daily spot Foward Realized Forward price of 7 days delivery from 24-Dec-2016 (RW, lambda = ) Predicted Esscher Daily spot Foward Realized Price Price Time in days before start of delivery Time in days before start of delivery Fig. 5.7: AR ( ) RW ( ) 16

18 ( ) RW AR 6 JEPX JEPX Esscher JEPX (AR ) ( ) ( ) [1] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, pp ,

19 [2] M.J. Brennan and E.S. Schwartz, Evaluating Natural Resource investements, The Journal of Business, 58(2), pp , [3] R. Carmona and M. Ludkovski, Spot convenience yield models for the energy markets, Contemporary Mathematics, vol. 351, pp , [4] J. Durbin and S.J. Koopman, Time Series Analysis by State Space Methods: Second Edition, Oxford Statistical Science Series (Vol. 24), Oxford University Press, [5],, ( ),, [6] R. Gibson and E. Schwartz, Stochastic Convenience Yield and the Pricing of Oil Contigent Claims, The Journal of Finance, 45(3), pp , [7] T. Hastie and R. Tibshirani, Generalized Additive Models, Chapman & Hall, [8] H. Miyauchi and T. Misawa, Regression Analysis of Electric Power Market Price of JEPX, Journal of Power and Energy Engineering, 2, , [9] A. Neumaier and T. Schneider, Algorithm 808: ARfit. A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models, ACM Trans. on Mathematical Software, 27(4), 58 65, [10],, JAFEE, , [11] Ver. 2.00, ( [12],, JEPX, (B)128 1, 57 66, [13],, JEPX, (B)133 8, , [14] E.S. Schwartz, The stohastic Behavior of Commodity Prices: Implications of Valuation and Hedging, The Journal of Finance, 52(3), pp , [15] E.S. Schwartz and J.E. Smith, Short-Term Variations and Long-Term Dynamics in Commodity Prices, Management Science, 46(7), pp , [16] S.E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, [17],, R GAM, 34(1), , [18], JEPX,, Y06006, [19],,, JEPX -, JAFEE 14, pp. 8-39,

20 A Girsanov t W t = [W 1,t,..., W n,t ] R n P n Brown {F t } {W t } A.1 ( ) Brennan and Schwartz [2] Brown Ornstein-Uhlenbeck ( [14]) 1 Gison and Schwartz [6] 1, 2 Schwartz and Smith [15] 2 Gison and Schwartz [6] 3 ( [3] ) (3.6) 2 Schwartz and Smith [6] ( SS ) (3.6) SS 2 dχ t = κχ t dt + σ χ dw 1,t dξ t = µ ξ dt + σ ξ dw 2,t (A.1) dt t (A.1) χ (t+1) t = (1 κ t ) χ t t + σ χ t w 1,t ξ (t+1) t = ξ t t + µ ξ t + σ ξ t w 2,t (A.2) t t = 0, 1, 2,..., x t+1 χ (t+1) t, x t χ t t, w t w 1,t, A = 1 κ t 0 ξ (t+1) t ξ t t w 2,t 0 1, B = σ χ t 0 0 σ ξ t (A.3) (A.2) (3.6) (A.2) 2 (3.1) f(t) 19

21 A.2 Girsanov Girsanov 15 ( Girsanov ) Esscher Girsanov Z t P 16 Z t := E E e T 0 θ tdw t ] 0 θ tdw t [ e T F t, 0 t T (A.4) P (A) = E [I A Z T ], A F T Girsanov W t P Brown d W t = dw t θ t dt (A.5) P n Brown W t R n η t dx t = (Ax t + D) dt + BdW t (A.6) η t = C x t A R n n, B R n n, C R n, D R n, (A.6) ( ) x t R n, η t, W t R n, (A.1) (A.2) (A.6) (3.6). 1 x T η T = C x T T T η T = C x T = C e AT x 0 + C e AT e At Ddt + C e AT e At BdW t T = C e AT x 0 + C e A(T t) Ddt T 0 0 C e A(T t) BdW t (A.7) (A.7) Esscher Z t, (3.2) T Z t := E eλ 0 C e A(T t) BdW t [ E e λ ] T 0 C e A(T t) BdW t F t, 0 t T (A.8) (A.8) Girsanov (A.4) Z t θ t λc e A(T t) B (A.9) (A.4) θ t (A.9) Girsanov Esscher 15 [5]. 16 θ t R n F t- n θ t R t. 20

22 B AR B.1 (3.14) η T g t,t ϕ ηt g t,t η T g t,t σ ηt g t,t ϕ ηt g t,t ϕ ηt g t,t (x) = σ2 η T g t,t 2 x 2 (3.14) F t,t = e f(t )+g t,t exp { ϕ ηt g t,t (λ + 1) } exp { ϕ ηt g t,t (λ) } = exp { f(t ) + g t,t + ϕ ηt g t,t (λ + 1) ϕ ηt g t,t (λ) } { ( = exp f(t ) + g t,t + ση 2 T g t,t λ + 1 )} 2 (B.1) B.2 AR (3.7) AR (p) p = 1 (3.7) η t+τ η t+τ = a τ 1η t + ( a τ a 1 + I ) c + ( a τ 1 1 ϵ t a 1 ϵ t+τ 1 + ϵ t+τ ) E [η t+k F t ] = 0, k = 1,..., τ E [η t+τ F t ] E [η t+τ F t ] = a τ 1η t + ( a τ a ) c (B.2) p 2 p = 1 η t η t 1. η t p+1 }{{} e t a 1 a 2 a p = } {{ 0 } A 1 η t 1 η t 2. η t p }{{} e t 1 c c + +. c }{{} c ϵ t 0. 0 }{{} ε t e t = A 1 e t 1 + c + ε t (B.3) (B.2) a 1 A 1, η t e t, c c E [e t+τ F t ] [ E [η t+τ F t ] = ] E [e t+τ F t ] (B.4) E [η t+τ F t ] (3.6) 21

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w