日経225オプションデータを使ったGARCHオプション価格付けモデルの検証
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1 GARCH GARCH GJREGARCH Duan Duan t GARCHGJREGARCH GARCH GJR EGARCHGARCHGJRt twatanab@bcomp.metro-u.ac.jp
2 Black and ScholesBS Engle ARCHautoregressive conditional heteroskedasticity BollerslevGARCHgeneralized ARCH GARCH GARCH Black ChristieGARCH GlostenJagannathan and Runkle GJRNelsonEGARCHexponential GARCH GARCH t Bollerslev, Engle and NelsonWatanabe t GARCH
3 Duan locally risk-neutral valuation relationship GARCH Duan t GARCHGJREGARCH GARCH GJRBS GARCH GARCHGJREGARCH R t t 1 t t R t = t + t, t t GARCHBollerslev, Engle and Nelson
4 t t z t t = t z t, t > 0, z t i.i.d., E (z t ) = 0, Var (z t ) = 1, t R t t z t t Engle ARCHBollerslevGARCH volatility clustering Engle ARCH Bollerslev GARCH GARCH 2 t = + 2 t 1+ 2 t 1, > 0,, 0. 2 t GARCHt t 1 GARCH GARCH GARCH Glosten, Jagannathan and Runkle GJRNelsonEGARCH GARCHpq p q 2 t = + 2 t + i 2 j t j, > 0,, i =1 j =1 i j ( i = 1, 2,, p; j= 1, 2,, q ). i 0
5 GJR t 1 D t 1 GJR 2 t = + 2 t 1 + ( +D t 1 ) 2 t 1, > 0,,, 0. 2 t t 1 D t 1 = 0 2 t = + 2 t t 1, t 1 D t 1 = 1 2 t = + 2 t 1 + ( + ) 2 t 1, > 0 ARCHGARCHGJR 2 t NelsonEGARCHln ( 2 t ) EGARCH t 1 t 1 z t 1 (= t 1 / t 1 ) EGARCH ln( 2 t )= + ln( 2 t 1 ) + z t 1 + ( z t 1 E ( z t 1 )). z t 1 > 0 ln( 2 t )= + ln( 2 t 1 ) + ( + ) z t 1 E ( z t 1 ), GJRp, q p q 2 t = + 2 t + ( 2 + j t j > 0, i, j ( i = 1, 2,, p; j= 1, 2,, q i j Dt j 2 i t j ),, j 0 ). i =1 j =1 EGARCHp, q p q ln ( 2 = + t ) + i ln ( 2 ) + [ z t j ))], t i j ( z t i =1 j =1 j E ( z t j 1 = 1.
6 z t 1 < 0 ln( 2 t )= + ln( 2 t 1 ) + ( ) z t 1 E ( z t 1 ), EGARCH < 0 GARCHGJREGARCH /2 t 1I t 1 E( 2 t I t 1 )= + ( + ) 2 t 1, E( 2 t I t 1 )= + ( + + /2) 2 t 1, E(ln( 2 t ) I t 1 )= + ln( 2 t 1 ), z t t t 1t R t = (S t S t 1 )/ S t 1 S t S t 1 t t 1 t r R t = r + t, R t = (S t S t 1 )/S t 1 continuous compoundingr t = ln (S t ) ln(s t 1 )
7 z t 1 R t = r 2 t + t, 2 r r R t z t z t t R t = (S t S t 1 )/ S t 1 Engle, Lilien and Robins R t = a + c t 2 + t. GARCH GARCH nonsynchronous trading Campbell, Lo and MackinlaySection R t = r + a + br t 1 + c t 2 + t. a = b = c = 0 z t FamaMandelbrot GARCHz t z t z t Hafner and HerwartzGARCH
8 GARCHz t tbollerslev GEDgeneralized error distributionnelsonbollerslev, Engle and NelsonWatanabe z t t t vegarch E ( z t 1 )z t 2 / t (v 2) /((v 1)/2/(v/2)(.) GARCHz t GARCHn GARCHmz t tgarcht GARCHmGARCH-M GJR GJR-nGJR-mGJR-t EGARCH EGARCH-nEGARCH-mEGARCH-t GARCHGARCH r T + K T C T P T GARCHz t tbollerslev, Engle and NelsonWatanabeWang et al. Verhoeven and McAleer z t t GARCHHafner and HerwartzBauwens and Lubrano GARCHMCMCMarkov chain Monte Carlo GARCH Bauwens and Lubrano
9 C T =(1 + r) E [Max(S T+ K, 0 )]. P T =(1 + r) E [Max(K S T+, 0 )]. S T+ T + GARCH S T+ S (1) T+,,S(n) T+ n E[Max( S T+ K, 0)] E[Max(K, 0)] S T+ 1 n 1 n n Σ Max(S T+ K, 0), i=1 n Σ Max( i=1 (i) (i) S T+ K, 0). GARCHn C T P T {R 1,,R T }GARCHn {z (i), (i) T+1,z T+ }n i=1 GARCHn{R (i), T+1,R(i) T+ }n i=1 T + S (1) T+,,S(n) T+ (i) S T+ (i) = S T Π (1 + R ), i = 1,..., n. T+s s =1 C T P T C T (1 + r) 1 n n Σ Max(S T+ K, 0), i=1 (i) P T (1 + r) 1 n Σ n i=1 Max(K S (i) T+, 0). GJR-nEGARCHn GARCHt
10 R 1,,R T }GARCHt t v vt{z (i), T+1,z (i) T+ }n i=1 v GARCHt {R (i ) T+1,,R (i) T+ }n i=1 GJRt EGARCHt Duan GARCH QP R t I t 1 Q E Q ( R t I t 1 ) = r. Var Q ( R t I t 1 ) = Var P ( R t I t 1 ) a.s.. I t 1 t 1E Q (.) Q Var Q (.)Var P (.)QPDuan Q GARCHm DuanGARCH Hafner and Herwartz GARCH ln (C t C t 1 ) C t C t 1 C t t Duan DuanAppendix
11 R t = + t ξ t ξ t I t 1 i.i.d.n (0,1), λ t = t t r, t 2 = t 1 ( ξ t 1 t 1 ) 2 2 t 1. t GARCHm t = r + a + br t 1 + c 2 t GARCHm Q R 1,,R T }GARCHm {ξ (i), T+1,ξ (i) T+ }n i=1 Q {R (i), T+1,R (i) T+ }n i=1 GJR-mEGARCH-mGJR-mEGARCHm Q t 2 ln( = t 2 t 1 ( + D * t 1 ) ) = + ln( 2 ) + ( ξ t 1 t 1 ) + ξ t 1 t 1 2/. t 1 ( ξ t 1 t 1 ) 2 2 t 1, D t 1 ξ t 1 t 1 < 0 Duan z t GARCHQ z t t G(z t )(.) z t
12 ( z t ) = 1[G ( z t )]. z t t GARCH Q R t = 1 t + t ξ t t ( ), ξ t i.i.d.n (0,1), 2 t = t 1 ( ξ t 1 t 1 ) 2. t E Q 1 ( ξ t t ) I t 1 = r t t. t { T+1,, T+ } t = T + 1,,T + z t t antithetic variatescontrol variates {z (i) } n i=1 = {z(i), T+1,z (i) T+ } n i=1({ξ (i) } n (i) i=1 = {ξ, T+1,ξ (i) T+ } n i=1) { z (i) } n i=1({ ξ (i) } n i=1) nn {z (i) } n i=1 ({ξ (i) } n i=1) {S (i) T }n i=1 { z(i) } n i=1({ ξ (i) } n i=1) {S (i) T }2n i=n+1 BS {z (i) } n i=1 ({ξ (i) } n i=1){ z (i) } n i=1({ ξ (i) } n i=1) GARCHBS (S (1) T+,, S (2n) T+ ) BS
13 T+ = exp 1 T+ r 2 2 (i) + Σ z t, i = 1,..., n, t=t+1 S (i) S (i) T+ = exp 1 T+ 2 2 r + Σ ξ t, i = 1,..., n t=t+1 (i). BS GARCH BST + S (i) GARCH S (i) BS T C GARCH C BS TBS C BS C T = C GARCH C BS C BS. {Max [ S (i) 2n GARCH K, 0 ]} i=1 {Max [ S (i) 2n BS K, 0 ]} i=1 z t t v t {z (i), T+1,z (i) T+ }n i=1 vt v 2 x (i) t w (i) t (i) z t = v (i) 2x t / w (i) t BS z (i) t (i x ) t n = C T = S T (d 1 ) K exp ( r)(d 2 ), P T = S T ( d 1 ) + K exp ( r)( d 2 ), d 1 = ln(s T / K) + (r + 2 / 2), d 2 = ln(s T / K) + (r 2 / 2) (.) BS BSσ T + T C T Var (C T) = Var (C GARCH) + 2 Var (C BS) 2 Cov (C GARCH, C BS). X i = Max (i) S GARCH K, 0 Y i = = Cov (C GARCH, Var (C BS) C BS) Max (i) S BS K, 0 Cov (X i, Y i) =. Var (Y i)
14 GARCH T = S t t t (S t S t 1 )/S t 1 GARCH GARCHGARCH GARCH-nGARCH-mGARCH-t GJR-nGJR-mGJR-t EGARCH-n EGARCH-mEGARCH-t + T = 1,000
15 γ + γ + /2 t t
16 γ t t
17 t GARCH-nGJR-nEGARCH-n GARCH + GJR + + /2EGARCH GJR EGARCH GJR EGARCH t t t z t Bollerslev and Wooldridge n + n γ + + γ /2 γ t n γ t γ
18 a b c t a b c t
19 EGARCH GARCH-mabct H 0 : a = b = c = 0 t abc GJR-m EGARCH-ma = b = c = 0Duan Duan GARCH-mGJR-m EGARCH-mGARCH-n GJR-nEGARCH-n a b c t H 0 a = b = c = 0
20 GARCH GJR EGARCH t GARCH GJR EGARCH
21 t v t t t H 0 : H 1 : z t tgarch-t GJR-t EGARCH-t v z t H 0 : v = t H 1 : v < z t t GARCH r MER : mean error rate RMSER : root mean square error rate C i C i i 1 m Σ m C i C MER = i, i=1 C i 1 m Σ m C i 2 C RMSER = i. i =1 C i MAERmean absolute error ratemaer 1 / m Σ m i =1 ( Ci Ci )/ Ci RMSER =
22 S/K < S/K < S/K < S/K S/K > m m = m = MERRMSER Bakshi, Cao and Chen SK MERRMSER DOTMOTMATM ITMDITMDOTMOTM ATMITMDITMMER RMSERBS MERRMSER GARCH-nGARCH-mGARCH-t MER RMSER GARCH-mGARCH-t GARCH-n GJREGARCH t GARCHmabc GARCH-t GJR-t EGARCH-t t deep-out-of-the-money far-out-of-the-money Bakshi, Cao and Chen deep-out-of-the-money DOTM
23 n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t
24 t z t GARCH-nGJR-nEGARCH-n GARCH-t GJR-t EGARCH-t GARCH t S T = K = r = z t t GARCH-nGJR-nEGARCH-n MERDOTMEGARCH DITMEGARCHBSGARCH GARCHBS z t EMaxS T + K, 0 n t n t n t EMaxK S T +, 0 n t n t n t
25 DOTM OTMATMBSGARCH ATMEGARCHBS ITM DITMBS GARCH GARCH GJREGARCHDOTMOTMTotal ITMDITMGARCHGJREGARCH ITMDITMATMDOTMOTMTotal ATMGJRGARCHEGARCH GARCHGJREGARCH DOTMDITMEGARCH GJR RMSERBS GARCHITMDITMEGARCHDOTMTotalGJR DITMDOTMEGARCHDITM GARCHBSRMSER GARCHGJREGARCHMER DOTMOTM ITMDITMGJREGARCHGARCITMDITM DOTMOTMGARCHGJREGARCHGJR EGARCHEGARCH DOTMDITMGJR GJRGARCHGJR EGARCHBS GJREGARCH GJREGARCH GARCHGJR EGARCH TotalGJREGARCHATMGARCHGJR
26 S T+ GJREGARCH GARCHK K E[Max( S T+ K, 0)] = S T+ f ( S T+ )d S T+, GJREGARCHGARCHS T+ K GJREGARCHGARCH DOTMOTMITMDITM ITMDITMDOTMOTM OTM MERRMSER OTM ITMDITMOTM ITMDITM ITMDITM
27 GARCH MERRMSER GARCH GARCH GARCHGJREGARCH t GARCH DieboldLamoureux and Lastrapes
28
29 n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t
30 n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t n m t
31 GARCH Duan GARCHt t GARCH EGARCH DOTMOTMITMDITMGJREGARCH GARCHITMDITMDOTMOTM GARCHEGARCHGJREGARCH R t = ln (S t ) ln (S t 1 ) R t = + t 1 2 t 2 + t, t = t z t, z t i.i.d.n(0, 1) (26) = r = 0 z t 1/2 2 t t z t t R t =(S t S t 1 )/S t 1
32 EGARCH GJR GARCH GJREGARCHBS GARCHBS GARCH GARCH Bauwens and LubranoMCMCGARCH GARCH Heston and NandiGARCH GARCH SVstochastic volatility SV SV
33 GARCH GARCH Bakshi, G., C. Cao, and Z. Chen, Empirical Performance of Alternative Option Pricing Models, Journal of Finance, 52, 1997, pp Bauwens, L., and M. Lubrano, Bayesian Option Pricing Using Asymmetric GARCH, Journal of Empirical Finance, 9, 2002, pp Black, F., Studies of Stock Market Volatility Changes, Proceedings of the American Statistical Association, Business and Economic Statistical Section, 1976, pp , and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, 1973, pp Bollerslev, T., Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31, 1986, pp , A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rate of Return, Review of Economics and Statistics, 69, 1987, pp , R. F. Engle, and D. B. Nelson, ARCH Models, R. F. Engle and D. McFadden, eds., The Handbook of Econometrics, 4, 1994, pp , Amsterdam: North-Holland., and J. M. Wooldridge, Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances, Econometric Reviews, 11, 1992, pp Campbell, J. Y., A. W. Lo, and A. C. Mackinlay, The Econometrics of Financial Markets, Princeton: Princeton University Press, Christie, A. A., The Stochastic Behavior of Common Stock Variances: Value, Leverage, and Interest Rate Effects, Journal of Financial Economics, 10, 1982, pp Diebold, F. X., Modeling the Persistence of Conditional Variances: A Comment, Econometric Reviews, 5, 1986, pp Duan, J.-C., The GARCH Option Pricing Model, Mathematical Finance, 5, 1995, pp
34 , Conditionally Fat-Tailed Distributions and the Volatility Smile in Options, Working Paper, Department of Finance, Hong-Kong University, Engle, R. F., Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50, 1982, pp , D. M. Lilien, and R. P. Robins, Estimating Time Varying Risk Premia in the Term Structure: The ARCH-M Model, Econometrica, 55, 1987, pp Fama, E. F., The Behavior of Stock Market Prices, Journal of Business, 38, 1965, pp Glosten, L. R., R. Jagannathan, and D. Runkle, On the Relation between the Expected Value and the Volatility of Nominal Excess Returns on Stocks, Journal of Finance, 48, 1993, pp Hafner, C. M., and H. Herwartz, Option Pricing under Linear Autoregressive Dynamics, Heteroskedasticity, and Conditional Leptokurtosis, Journal of Empirical Finance, 8, 2001, pp Heston, S. L., and S. Nandi, A Closed-Form GARCH Option Valuation Model, Review of Financial Studies, 13, 2000, pp Lamoureux, C. G., and W. D. Lastrapes, Persistence in Variance, Structural Change and the GARCH Model, Journal of Business & Economic Statistics, 8, 1990, pp Mandelbrot, B., The Variance of Certain Speculative Prices, Journal of Business, 36, 1963, pp Nelson, D. B., Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59, 1991, pp Verhoeven, P., and M. McAleer, Fat Tails and Asymmetry in Financial Volatility Models, CIRJE Discussion Paper Series, F-211, University of Tokyo (forthcoming in Mathematics and Computers in Simulation), Wang, K.-L., C. Fawson, C. B. Barrett, and J. B. McDonald, A Flexible Parametric GARCH Model with an Application to Exchange Rates, Journal of Applied Econometrics, 16, 2001, pp Watanabe, T., Excess Kurtosis of Conditional Distribution for Daily Stock Returns: The Case of Japan, Applied Economics Letters, 7, 2000, pp
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バリュー・アット・リスクのリスク指標としての妥当性について ― 理論的サーベイによる期待ショートフォールとの比較分析―
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