2 Recovery Theorem Spears [2013]Audrino et al. [2015]Backwell [2015] Spears [2013] Ross [2015] Audrino et al. [2015] Recovery Theorem Tikhonov (Tikhon
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1 Recovery Theorem Forward Looking Recovery Theorem Ross [2015] forward looking Audrino et al. [2015] Tikhonov Tikhonov 1. Tikhonov 2. Tikhonov forward looking *1 Recovery Theorem Ross [2015] forward looking Recovery Theorem 2 1 Recovery Theorem (Carr and Yu [2012]Dubynskiy and Goldstein [2013]Walden [2014]Park [2015]Qin and Linetsky [2015]) (Martin and Ross [2013]), kiriutakuya@gmail.com, hibiki@ae.keio.jp *1 Breeden and Litzenberger [1978]Melick and Thomas [1997]Bliss and Panigirtzoglou [2002]Ludwig [2015] Bliss and Panigirtzoglou [2004] Alonso et al. [2005] Zdorovenin and Pézier [2011] [2014] 1
2 2 Recovery Theorem Spears [2013]Audrino et al. [2015]Backwell [2015] Spears [2013] Ross [2015] Audrino et al. [2015] Recovery Theorem Tikhonov (Tikhonov ) 13 S&P500 Backwell [2015] Recovery Theorem Audrino et al. [2015] Tikhonov Tikhonov 3 1. Tikhonov 2. Tikhonov 3. 2 Ross [2015] Recovery Theorem 3 Recovery Theorem Recovery Theorem (Ross [2015]) Ross [2015] Recovery Theorem Recovery Theorem 1 θ i (i = 1,..., n) r i θ i θ j *2 p i,j, q i,j, f i,j n n P, Q, F *3 P *4 Q F P *5 Q P Q P *2 p i,j θ j 1 θ i *3 Q F *4 i, j (M k ) i,j > 0 k N M. *5 3 2
3 q i,j = p i,j n k=1 p i,k (i, j = 1,..., n). (1) F Q P Ross [2015] P F U(c i ) + δ n f i,j U(c j ) (i = 1,..., n) (2) j=1 c i θ i U(c i ) δ(> 0) U(c) (U (c) > 0) f i,j p i,j f i,j = 1 δ U (c i ) U (c j ) p i,j (i, j = 1,..., n) (3) p i,j f i,j ϕ i,j ϕ i,j := p i,j = δ U (c j ) f i,j U (c i ) (i, j = 1,..., n) (4) P = P λ v Recovery Theorem v i v i δ = λ (5) U (c i ) = v 1 i (i = 1,..., n) (6) Recovery Theorem P F P λ v F f i,j = 1 v j p i,j (i, j = 1,..., n) (7) λ v i Ross [2015] Recovery Theorem P F = Q 3 2 P P Spears [2013] 1 3 Step Step 3 2 Recovery Theorem Step 2 P Step 1 Step 2 Step 2 P 3
4 1 3.1 Step 1: S s j,τ (τ = 1,..., m) i 0 θ j τ n m S S m θ i0 (i 0 = (n + 1)/2, n ) Step 1 S Breeden and Litzenberger [1978] 2 Bliss and Panigirtzoglou [2002]()Melick and Thomas [1997]()Ludwig [2015]() S Step 2 Step Step 2: S P Step 2 *6 n m S n n P 4.5 n m S 1 s 1 P i 0 p i0 j p i0,j s 1 = p i0 (8) P 1 s τ, s τ+1, P s τ+1 = s τ P (τ = 1,..., m 1) (9) S (m 1) n AS 1 (m 1) n B (9) AP = B (10) P p i,j 0 (i, j = 1,..., n) (8) min P AP B 2 2 (11) subject to s 1 = p i0 (12) p i,j 0 (i, j = 1,..., n) (13) *6 Backwell [2015] 4
5 2 Audrino et al. [2015] S&P A (ill-posed problem) Audrino et al. [2015] Tikhonov Tikhonov (11) min P AP B ζ P 2 2 (14) 2 ζ (14) n I O min P [ ] [ ζi A B 2 P O] Tikhonov P *7 P p i,j ζ P = O P Recovery Theorem ζ F P ζ F P 2 Info 1. s 1 p i0 ( (8)) Info 2. p i,j p i+k,j+k (i, j = 1,..., n; k Z, 1 i + k n, 1 j + k n) P 2 2 (15) min P min P AP B ζ P P 2 2 (16) [ ] [ ] ζi A B 2 P ζ (17) P 2 *7 5
6 P = = p 1,1 p 1,2 p 1,i0 1 p 1,i0 p 1,i0 +1 p 1,n 1 p 1,n p i0 1,1 p i0 1,2 p i0 1,i 0 1 p i0 1,i 0 p i0 1,i 0 +1 p i0 1,n 1 p i0 1,n p i0,1 p i0,2 p i0,i 0 1 p i0,i 0 p i0,i 0 +1 p i0,n 1 p i0,n p i0 +1,1 p i0 +1,2 p i0 +1,i 0 1 p i0 +1,i 0 p i0 +1,i 0 +1 p i0 +1,n 1 p i0 +1,n p n,1 p n,2 p n,i0 1 p n,i0 p n,i0+1 p n,n 1 p n,n i0 k=1 s k,1 s i0+1,1 s n 1,1 s n, k=1 s k,1 s 3,1 s i0,1 s i0 +1,1 s i0 +2,1 s n,1 0 s 1,1 s 2,1 s i0 1,1 s i0,1 s i0 +1,1 s n 1,1 s n,1 n 0 s 1,1 s i0 2,1 s i0 1,1 s i0,1 s n 2,1 k=n 1 s k, n s 1,1 s 2,1 s i0 1,1 k=i 0 s k,1 (18) (19) P 2 *8 P (15) (17) P 2 ζ F 2 P ( n k=1 s k,1) ζ (2 )ζ F Q F Q Recovery Theorem 4 2 F H Φ H *9 2 P H S H Step 1 S H S H S N e i,j i.i.d. 0, σ 2 s N i,j = s H i,j(1 + e i,j ) (i, j = 1,..., n) (20) S N Step 1 Tikhonov S N P N (Step 2)P N Recovery Theorem F N (Step *8 (16) Tikhonov Tikhonov P (19) Tikhonov P = O *
7 Φ Φ 2 3) F N F H % 2% 31 r 1 = 30%, r 16 = 0%, r 31 = 30% m S S m m < n m (m = n) * (m < n) * 11 F (i 0 ) f i0 f N i 0 f H i 0 Kullback-Leibler(KL) D KL (f N i 0 f H i 0 ) KL 2 ( ) n f N D KL (f N i 0 f H i 0 ) := fi N i0,j 0,j ln (21) j=1 2 D KL (f N i 0 f H i 0 ) = f N i 0 f H i 0 (10 20 ) KL * 12 f H i 0,j 4.2 Φ H, F H *10 Audrino et al. [2015] m < n S (m = n) *11 m > n m = n *12 F KL D KL (f N i 0 f H i 0 ) D KL (f N i 0 f H i 0 ) 7
8 [a] [b] 3 : KL Φ H CRRA U(c) = c 1 γ R /(1 γ R ) γ R = 3, δ = F H S&P500 () θ j r j 1% r j + 1% 29% (29% ) θ 1 (θ 31 ) F H Φ H, F H A H Step ζ = 10 14, ,..., , 10 2 σ = 0%, 1%, 2%, 5% Tikhonov KL 3 RND(Risk Neutral Distribution) q H i 0 KL D KL (q H i 0 f H i 0 ) KL Recovery Theorem 1 S N σ = 0% ζ = 0 KL 0 KL ζ = KL ζ = 0 ζ RND ζ Tikhonov ζ = KL
9 [a] U(c) : CARA [b] γ R : 10 [c] δ : : KL [d] F H : 225 ζ Tikhonov KL S N (σ = 1%, 2%, 5%) (ζ = 0) (σ = 0%) Tikhonov ζ ζ Tikhonov Tikhonov 4.4 4a CARA γ A = 3 δ KL 4b γ R = 10 4c δ = d S&P Step 2 δ KL γ R 9
10 [a] σ = 0% [b] σ = 1% 5 m < n : KL (1) Tikhonov (2) Tikhonov (m < n) S H 31 S 31 CBOE S&P S * 13 n = 31 S H m = 6, 12, 31 5a σ = 0% m 6 12 ζ m = 31 5b σ = 1% * KL *13 n m *14 σ = 2% 5% 10
11 6 () 7 h(ζ) KL () Step 2 (16) 2 1 ( AP B 2 2) y fit 2 ζ ( P P 2 2) P N P y reg ζ y fit y reg 6 ζ y fit y reg y fit y reg σ = 1% ζ = 1 y fit ζ = 10 6 y reg y fit y reg ζ y fit y reg h(ζ) ζ h(ζ) := y fit(ζ) y fit (0) y fit ( ) y fit (0) + y reg(ζ) y reg ( ) y reg (0) y reg ( ) (22) y fit (ζ) y reg (ζ) 6 ζ h(ζ) y(0) y( ) P = P y reg ( ) 0 y fit (ζ) y reg (ζ) h(0) = h( ) = 1 ζ h(ζ) ζ ζ h(ζ) KL 7 σ = 1%, 2%, 5% h(ζ) ζ KL ζ ζ ζ y fit y reg ζ 5 Recovery Theorem forward looking 11
12 Audrino et al. [2015] Tikhonov Tikhonov forward looking F. Alonso, R. Blanco, and G. Rubio. Testing the forecasting performance of ibex 35 option-implied risk-neutral densities. Working Paper, Banco de Espana, F. Audrino, R. Huitema, and M. Ludwig. An empirical analysis of the ross recovery theorem. Working Paper, Available at SSRN , A. Backwell. State prices and implementation of the recovery theorem. Journal of Risk and Financial Management, 8(1):2 16, R.R. Bliss and N. Panigirtzoglou. Testing the stability of implied probability density functions. Journal of Banking & Finance, 26(2): , R.R. Bliss and N. Panigirtzoglou. Option-implied risk aversion estimates. The Journal of Finance, 59 (1): , D. Breeden and R. Litzenberger. Prices of state-contingent claims implicit in option prices. Journal of Business, 51: , P. Carr and J. Yu. Risk, return, and ross recovery. Journal of Derivatives, 20(1):38, S. Dubynskiy and R.S. Goldstein. Recovering drifts and preference parameters from financial derivatives. Working Paper, Available at SSRN , M. Ludwig. Robust estimation of shape-constrained state price density surfaces. The Journal of Derivatives, 22(3):56 72, I. Martin and S. Ross. The long bond. Working Paper, Stanford University, 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):91 115, H. Park. Ross recovery with recurrent and transient processes. Working Paper, arxiv preprint arxiv: , L. Qin and V. Linetsky. Positive eigenfunctions of markovian pricing operators: Hansen-scheinkman factorization and ross recovery. Working Paper, arxiv preprint arxiv: , S. Ross. The recovery theorem. The Journal of Finance, 70(2): , T. Spears. On estimating the risk-neutral and real-world probability measures. PhD thesis, Oxford University, J. Walden. Recovery with unbounded diffusion processes. Working Paper, Available at SSRN , V.V. Zdorovenin and J. Pézier. Does the information content of option prices add value for asset allocation? ICMA Centre Discussion Paper No. DP , 2011.,.. Transactions of the Operations Research Society of Japan, 57: ,
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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