1 Jensen et al.[6] GRT S&P500 GRT RT GRT Kiriu and Hibiki[8] Jensen et al.[6] GRT 3 GRT Generalized Recovery Theorem (Jensen et al.[6])

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1 Generalized Recovery Theorem Ross[11] Recovery Theorem(RT) RT forward looking Kiriu and Hibiki[8] Generalized Recovery Theorem(GRT) Jensen et al.[6] GRT RT Kiriu and Hibiki[8] 1 backward looking forward looking (Breeden and Litzenberger[4]) Ross[11] (time-separable utility) Recovery Theorem(RT) (time-homogeneity) RT forward looking Kiriu and Hibiki[8] Audrino et al.[1] Tikhonov RT Generalized Recovery Theorem( GRT) Jensen et al.[6]. GRT RT masatakeitoh@gmail.com UFJ (MTEC) kiriu@mtec-institute.co.jp hibiki@ae.keio.ac.jp 1

2 1 Jensen et al.[6] GRT S&P500 GRT RT GRT Kiriu and Hibiki[8] Jensen et al.[6] GRT 3 GRT Generalized Recovery Theorem (Jensen et al.[6]) Jensen et al.[6] GRT GRT s(= 1,..., s 0,..., S) s 0 r s r s0 = 0% s 0 τ(= 1,..., T ) s π τ,s, q τ,s, p τ,s, m τ,s T S Π, Q, P, M Π q τ,s π τ,s q τ,s π τ,s q τ,s = π τ,s S k=1 π τ,k (τ = 1,..., T ; s = 1,, S) (2.1) π τ,s, m τ,s, p τ,s (2.2) π τ,s = m τ,s p τ,s (τ = 1,..., T ; s = 1,, S) (2.2) p τ,s π τ,s m τ,s GRT δ (0, 1] s u s > 0 m τ,s (2.3) h s s 0 1 m τ,s = δ τ ( us u s0 (2.2) (2.3) ) = δ τ h s (τ = 1,..., T ; s = 1,..., S) (2.3) π τ,s = δ τ p τ,s h s (τ = 1,..., T ; s = 1,..., S) (2.4) 1 GRT RT 2

3 S S H =diag(h 1, h 2,..., h S ), τ T T D =diag(δ, δ 2,..., δ T ) (2.4) (2.5) Π = DP H (2.5) P e P e = e (2.5) H 1 e ΠH 1 e = DP e = De (2.6) π 1,1 π 1,S..... π T,1 π T,S 1. s s S δ =. δ T (2.7) (2.7) T δ, 1,..., h 1 s 0 1, h 1 s 0 +1,..., h 1 S S S T (2.7) δ, 1,..., h 1 s 0 1, h 1 s 0 +1,..., h 1 S (2.7) δ τ δ τ δ = δ 0 1 (2.8) (2.7) δ τ a τ + b τ δ (τ = 1,..., T ) (2.8) a τ = (τ 1)δ τ 0 (τ = 1,..., T ) (2.9) b τ = τδ τ 1 0 (τ = 1,..., T ) (2.10) b 1 π 1,1 π 1,s0 1 π 1,s0 +1 π 1,S b T π T,1 π T,s0 1 π T,s0 +1 π T,S δ 1. s 0 1 s S B, h δ, a π a 1 π 1,s0 =. a T π T,s0 (2.11) Bh δ = a π (2.12) (2.12) δ, 1,..., h 1 s 0 1, h 1 s 0 +1,..., h 1 S min h δ Bh δ a π 2 2 (2.13) subject to 0 < δ 1 (2.14) s > 0 (s = 1,..., s 0 1, s 0 + 1,..., S) (2.15) 3

4 (2.4) p τ,s (2.13)-(2.15) p τ,s p τ,s = 1 δ τ h 1 s π τ,s (τ = 1,..., T ; s = 1,..., S) (2.16) 3 GRT P 3.1 Π 3.2 GRT S T 3.1 Π Breeden and Litzenberger[4] τ k π(τ, k) c(τ, k) π(τ, k) = 2 c(τ, k) k 2 (τ = 1,..., T ) (3.1) s k s π τ,s S&P500 S = 21, T = 21 Π B ( (2.13)-(2.15)) 4 B 4 (2.13)-(2.15) 3.3 ζ δ, h 1 s (s = 1,..., s 0 1, s 0 +1,..., S) 2 (Melick and Thomas[10]), (Bliss and Panigirtzoglou[2], [7]), (Ludwig[9]) 3 δ 0 = (δ0 12 = 0.99) δ

5 h δ ( ) 2 min Bh δ a π ζ δ δ + h δ δ S s=1,s s 0 ( h 1 s s h 1 s ) 2 (3.2) subject to 0 < δ 1 (3.3) s > 0 (s = 1,..., s 0 1, s 0 + 1,..., S) (3.4) (3.2) (2.4) δ, h 1 s (s = 1,..., s 0 1, s 0 + 1,..., S) p τ,s p τ,s = π τ,s δ τ h s (τ = 1,..., T ; s = 1,..., s 0 1, s 0 + 1,..., S) (3.6) Kiriu and Hibiki[8] RT Kiriu and Hibiki[8] GRT RT GRT 2 Kiriu and Hibiki[8] RT 3.4 ( δ) ( h s ) min δ ( T δ τ τ=1 ) 2 S π τ,s (3.7) δ 6 δ 0 s=1 5 δ s min h δ Bh δ a π ζ 1 ( δ δ) 2 + ζ 2 S ( h 1 s s=1,s s 0 1) 2 h s (3.5) 2 2 (3.5) 6 5

6 3.4.2 (1) 1 Kiriu and Hibiki[8] GRT s = 1(s = 1,..., s 0 1, s 0 + 1,..., S) (2) 2 Shackleton et al.[12] Q(x) P (x) C(x) P (x) = C(Q(x)) (3.8) C(x) h s = C (Q(s))(s = 1,..., s 0 1, s 0 + 1,..., S) Jackwerth[5] Bliss and Panigirtzoglou[3] 2 forward looking 1 (3) GRT 3 GRT 1 CRRA CRRA γ R s = (1 + r s ) γ R(s = 1,..., s 0 1, s 0 + 1,..., S) (2.11) δ, γ R 2 ( (3.2)-(3.4)) P H P N 1 P H 6

7 1: 3.4 GRT P H δ H, h H s (s = 1,..., S) P H, δ H, h H s (2.4) ΠH 3. Π H Π N π N τ,s = π H τ,s(1 + σe τ,s ) (τ = 1,, T ; s = 1,..., S) (4.1) σ e τ,s 4. GRT Π N () P E, δ E, h E s (s = 1,..., S) 5. P H P E s r s 0% 3% r 1 = 45%, r s0 = r 16 = 0%, r 31 = 45% T = S = Kullback-Leibler(KL) P H P E KL D KL (P E P H ) D KL (P E P H ) (4.2) 8 ( ) T S p D KL (P E P H ) = p E E τ,s τ,s ln (4.2) τ=1 s=1 100 KL D KL (Q E P H ) KL D KL (P E P H ) 7 Π T T S < T Audrino et al.[1] S < T Π T = S 8 D KL (P E P H ) = 0 0 P E,Q E,P H (10 20 ) KL p H τ,s 7

8 (1) 0.99 δ H = (0.99) 1 12 = γ R = 3 CRRA h H s = (1 + r s ) γ R 9 (2) P H S&P ,60,, % (43.5% ) 1( 31) P H P H KL 1: (4.2) Q E P E KL Q E Π N (2.1) δ H, h H s (s = 1,..., S) ζ KL 0 KL KL 4.3 σ = 1% Π N τ = 3(90 ) 2 2(a) GRT 9 Bliss and Panigirtzoglou[3] CRRA S&P γ R = γ R = 10 8

9 2(b) ζ = 10 6 (3.2) 2 (a) 2: σ = 1% (b) 4.4 σ = 1%, 5% ζ(= 10 12, ,..., 10 2 ) KL 3 KL ζ = 0 ζ KL KL ζ = ζ KL KL ζ = Π H ζ KL 0 3(a) σ = 1% KL ζ KL ζ = 10 6 KL (b) σ = 5% σ = 1% KL KL GRT KL KL KL ζ σ = 1% ζ δ H = 0.99 δ H =

10 (a) σ = 1% (b) σ = 5% 3: KL 5 CRRA γ R = 3 γ R = 10 GRT σ = 1% KL 0.060(ζ = ) KL σ = 5% KL 6 CRRA γ A = 3 CARA σ = 1% KL KL σ = 5% KL 7 P H S&P GRT σ = 1% KL ζ σ = 1% 8(a) (3.7) δ 12 = 0.96, 0.97, 0.98, 0.99, 1.00 KL δ KL CRRA γ R = 10

11 (a) σ = 1% (b) σ = 5% (a) σ = 1% (b) σ = 5% 図 4: ロバスト性の検証 δ = 0.95 図 5: ロバスト性の検証 γr = 10 (a) σ = 1% (a) σ = 1% (b) σ = 5% 図 6: ロバスト性の検証 CARA 型効用 (b) σ = 5% 図 7: ロバスト性の検証日経 225 6, 3, 0, 3, 6, 9 と変化させた場合について分析する γr = 0 の場合が提案法リスク中立 γr = 3 の場合が提案法完全先験情報に対応する分析結果を図 8(b) に示すまず先験情報として設定するリスク回避度が真のリスク回避度 (γr = 3) よりも小さい場合 (γr = 3, 0) の最小の KL 情報量を比較すると γr = 3 の場合に比べて γr = 0 の推定精度が高くリスク回避度が大きい場合 (γr = 6, 9) の最小の KL 情報量を比較すると γr = 9 の場合に比べて γr = 6 の推定精度が高いこのことから先験情報として設定したリスク回避度が真のリスク回避度に近いほど精度の良い解が得られる傾向にあることがわかったしかし最小の KL 情報量に対応する正則化パラメータ ζ は先験情報として設定した γr によって異なっており明確な特徴は見られなかったこの点について詳細に分析することは ζ の設定方法を考える上でも重要な問題であるが今後の課題とする (a) 主観的割引係数 (b) 限界効用 γr の変更図 8: 先験情報に関する感度分析 5 結論と今後の課題本研究では GRT を用いてオプション価格から forward looking な実分布を推定する方法について議論した GRT を用いて実分布を推定するには非適切問題を解く必要があるそこで本研究では実分布に関する先験情報を与えて解を安定化する推定方法を提案したこの方法は状態価格の均一性の仮定と先験情報の 2 つの観点から Kiriu and Hibiki[8] の RT を用いた実分布の推定方法を一般化した推定方法であるまた仮想データを用いた数値分析によってリスク中立分布を先験情報として与えた場合の提案法の特徴および有効性に関する検証を行った数値分析の結果から提案法を用いた場合には先験情報として与えた分布や従来の推定方法と比べて推定精度の高い実分布の推定値が得られる可能性があることがわかったただし今回の分析に用いたデータセットは限られており今後多くの先験情報の設定や仮想データの設定のもとでロ 11

12 ζ (3.5) 10 [1] F. Audrino, R. Huitema, and M. Ludwig. An empirical analysis of the ross recovery theorem. Working Paper, Available at SSRN: [2] R. R. Bliss and N. Panigirtzoglou. Testing the stability of implied probability density functions. Journal of Banking & Finance, 26(2), , [3] R. R. Bliss and N. Panigirtzoglou. Option-Implied Risk Aversion Estimates. Journal of Finance, 59(1), , [4] D. Breeden and R. Litzenberger. Prices of state-contingent claims implicit in option prices. Journal of Business, 51, , [5] J. C. Jackwerth. Recovering risk aversion from option prices and realized returns. Review of Financial Studies, 13(2), , [6] C. S. Jensen, D. Lando, and L. H. Pedersen. Generalized Recovery. Working Paper, Available at SSRN: [7],.. Transactions of the Operations Research Society of Japan, 57, , [8] T. Kiriu and N. Hibiki. Estimating Forward Looking Distribution with the Ross Recovery Theorem. Working Paper, Available at 2/Kiriu&Hibiki_ pdf, [9] M. Ludwig. Robust Estimation of Shape-Constrained State Price Density Surfaces. Journal of Derivatives, 22(3), 56-72, [10] W. R. Melick, and C. P. Thomas. Recovering an asset s implied PDF from option prices: an application to crude oil during the Gulf crisis. Journal of Financial and Quantitative Analysis, 32(1), , [11] S. Ross. The Recovery Theorem. Journal of Finance, 70(2), , [12] M. B. Shackleton, S. J. Taylor and P. Yu. A multi-horizon comparison of density forecasts for the S&P 500 using index returns and option prices. Journal of Banking & Finance, 34(11), , Kiriu and Hibiki[8] RT 12

2 Recovery Theorem Spears [2013]Audrino et al. [2015]Backwell [2015] Spears [2013] Ross [2015] Audrino et al. [2015] Recovery Theorem Tikhonov (Tikhon

2 Recovery Theorem Spears [2013]Audrino et al. [2015]Backwell [2015] Spears [2013] Ross [2015] Audrino et al. [2015] Recovery Theorem Tikhonov (Tikhon Recovery Theorem Forward Looking Recovery Theorem Ross [2015] forward looking Audrino et al. [2015] Tikhonov Tikhonov 1. Tikhonov 2. Tikhonov 3. 3 1 forward looking *1 Recovery Theorem Ross [2015] forward