Sock Volailiy Models and he Pricing of Warrans 36005 005 0 Hong & Li 005 Absrac This paper used a lo of popular volailiy models o sudy he dynamic behavior of underlying sock and hen used Hong & Lee (005) nonparameric specificaion es o compare he model specificaion errors of differen models. Based on he resul of volailiy esimaiesimaion, we used he Mane-Carlo simulaions o price Baogang warran, Chinese firs Warran afer he reform of sock marke, and compare he pricing resuls of differen models. The pricing resuls showed ha Baogang Warran is seriously overpriced. There is a srong manipulaion in he marke. We sudied some possible insiuional reasons for such manipulaion and proposed some policy suggesions in he end. Key Words: Warrans Pricing, Jump, Disribuion, Specificaion Tes
Black & Scholes (973) (Duan, 995) volailiy cluseringleverage effec Engle (98) ARCH Bollerslev (986) ARCHGARCH Engle, Lilien & Robbins (987) ARCH-MNelson(99), Zakoian (994) EGARCHTGARCH() IPO (Ball & Torous (983), Vlaar & Palm (993), Das (00)) (Jump) 3 (GED) (00) (994), (998), (998), (999), (000) Galai & Schneller (978), Schulz & Trauman (989,994) Corunhy & Galai (99) GARCH Hong & Li (005) Hong & Li005
GARCH GARCH EWMA TGARCH GARCH-M 3 GARCH TGARCH GARCH-M 4 GARCH- TGARCH- GARCH-M- µ ( r, θ ) σ ( r, θ ) (a) µ σ GARCH µ EWMA µ TGARCH µ EGARCH µ (b) GARCH h ln h h h h h = α + αξ + β h = λξ + ( λ) h h h = α + ( α + ϕd ) ξ + β h = α + α ξ / h + βξ / h + β ln h GARCH-M GARCH TGARCH GARCH µ δ + h µ + Jdq J ~ N( ϑ, γ ) (c) (Jump) dq ~... i i d B() q µ + Jdq J ~ N( ϑ, γ ) dq ~ i.. i d. B( q ) µ + δ h + Jdq J ~ N( ϑ, γ ) h h h h = α + αξ + β h = α + αξ + β h h h = α + ( α + ϕd ) ξ + β h h h = α + αξ + β h dq ~... i i d B() q (d)
GARCH µ TGARCH µ h h = α + αξ + β h h h = α + ( α + ϕd ) ξ + β h GARCH-M µ δ + h h h = α + αξ + β h r = µ ( r, θ) + ξ θ ξ σ( r, θ) z (a), (b), (c) z ~... iid N(0,) (d) z ~ ( v) = v+ z Γ ( )( + ) v () = v v Γ vπ v+ Hong & Li (005) Hong & Li (005) { r } P 0 ( x, y, s) r s y x θ 0 θ { p( x, ys,, θ ) = P( x, ys, )} 0 0 { r } n τ τ = Z ( θ ) r τ ( τ ) Z ( θ) = px (, τ x,( τ ), θ) dx τ =,, n τ θ 0 θ { p( x, ys,, 0) P0( x, ys, )} θ = { Z ( 0)} n τ = Zτ θ τ= iid.. U(0,) Hong and Li (005) U (0,) { Zτ, Zτ j } g j ( z, z) gˆ j ( z, z) g j ( z, z) gˆ j ( z, z) n ˆ ˆ j = h τ h τ j τ = j+ gˆ ( z, z ) ( n j) K ( z, Z ) K ( z, Z )
x y h k( )/ k( u) du, x [0, h] h ( x/ h) x y Kh( x, y) = h k( ), x [ h, h] h ( x) / h x y h k( )/ k( u) du, x [ h,] h k() i [,] k( u) du =, uk( u) du = 0 ukudu ( ) ( u ) < 5 ku ( ) = ( u) X I 6 n i ( u ) i= I Z ˆ Z ( ˆ ) τ = τ θ ˆ /6 θ θ 0 h= SˆZ n S ˆZ { Z } n τ τ = gˆ j ( z, z) M ˆ ( j) M ˆ ( j) Q ˆ( j) Mˆ j g z z dzdz ( ) = [ ˆ j (, ) ] 0 0 Qˆ ( j) = [( n j) hmˆ ( j) A ]/ V 0 / h 0 b 0 h = + 0 b 0 = [ [ ( + ) ( ) ] ] A ( h ) k ( u) du k ( u) dudb V k u v k v dv du Hong & Li (005) ˆ ( ) d Q j N (0,) { Z ( 0)} n τ = Zτ θ τ= iid.. U (0,) ˆ( p Q j) 3 60009 00 005 6 7 08 99
0.5 600 0. 500 0.05 400 0 300 00-0.05 00-0. 00-003-6 003-004-6 004-005-6 0-0. -0.05 0 0.05 0. 0.5 3 MLE BHHH GUASS6.0, GARCH GARCH GARCH-M TGARCH GARCH EWMA EGARCH GARCH % GARCH GARCH GARCH 3 GARCH Hong & Li (005) 3 3 Q ˆ( j ) j= j=5 j=0 j= j=5 j=0 69.33 65.69 6.09 -GARCH-Jump 0.94 0.5 6.56 -GARCH 54.98 53.3 46.78 -TGARCH-Jump.08 0.9 6.98 -EWMA 39.45 36.77 35.68 -GARCH-M-Jump. 0.67 7.08 -TGARCH 54.9 5.94 46.76 -GARCH- 8.63 7.8 5.74 -EGARCH 57.40 58.5 48.6 -TGARCH- 0.36 8.83 7.04 -GARCH-M 58.84 55.74 49.89 -GARCH-M- 9.94 9.77 8.0
Q ˆ( j ) GARCH Q ˆ( j ) GARCH EWMA Q ˆ( j ) GARCH EWMA GARCH Q ˆ( j ) GARCH TGARCH GARCH-M Q ˆ( j ) GARCH- TGARCH- GARCH-M- 4 4 Galai & Schneller (978), Schulz & Trauman (989,994) Corunhy & Galai (99) max(0, S X) S 0 =4.6 0.3 X=4.5 r f =0.05 (005 8 ) T=5 4 50000 4 3 T 4 0.5950 7 _GARCH_jump 0.475
_GARCH 0.6683 8 _GARCH_M_jump 0.4650 3 _GARCH_EWMA 0.563 9 _TGARCH_jump 0.449 4 _GARCH_M 0.6883 0 _GARCH_ 0.633 5 _EGARCH 0.6889 _GARCH_M_ 0.683 6 _TGARCH 0.6675 _TGARCH_ 0.655 3 ( 4 ) 0.5950GARCH 0.63-0.68 EWMA 0.563 0.44-0.47 0.6 4.5% 4.5% 0.%, 5 5 r f =.3% r f =.7% r f =.3% r f =.7% 0.5965 0.603 _GARCH_jump 0.467 0.4706
_GARCH 0.6638 0.6737 _GARCH_M_jump 0.4638 0.4758 _GARCH_EWMA 0.50 0.5 _TGARCH_jump 0.449 0.47 _GARCH_M 0.6774 0.699 _GARCH_ 0.636 0.6359 _EGARCH 0.684 0.6843 _GARCH_M_ 0.676 0.679 _TGARCH 0.667 0.6689 _TGARCH_ 0.6475 0.656 4 4. 3.9 6 6 S*=3.9 S*=4. S*=3.9 S*=4. 0.5487 0.6538 _GARCH_jump 0.4356 0.570 _GARCH 0.664 0.79 _GARCH_M_jump 0.494 0.5 _GARCH_EWMA 0.4806 0.5758 _TGARCH_jump 0.48 0.499 _GARCH_M 0.694 0.7507 _GARCH_ 0.5799 0.700 _EGARCH 0.633 0.758 _GARCH_M_ 0.5858 0.694 _TGARCH 0.65 0.7396 _TGARCH_ 0.5908 0.736 0. 0.05 5
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Models Wih Applicaions o Ineres Rae Term Srucures, Review of Financial Sudies, 8, 37-84. [] Nelson, D. B, 99, "Condiional Heeroskedasiciy in Asse Reurns: A New Approach." Economerica, 59, 347-370. [] Schulz, G.U. and Traumann, S., 989, "Valuaion of warrans - heory and empirical ess for warrans wrien on German socks", Working Paper, Universiy of Sugar (Germany), Ocober 989. [3]Schulz, G.U. and Traumann, S.,994, "Robusness of opion-like warran valuaion", Journal of Banking and Finance, page 84-859. [4] Vlaar, P. and F. Palm, 993, The Message in Weekly Exchange Raes in he European Moneary Sysem: Mean Reversion, Condiional Heeroskedasiciy, and Jumps, Journal of Business and Economic Saisics,, 35-60. [5] Zakoian, J-M, 994, Threshold Heeroscedasic Models, Journal of Economic Dynamics and Conrol, 8, 93 955. [6] 998 [7] 999-GARCH 4 [8] 998 (ARCH) 4 [9] 994 9 [0] 00GARCH 6 [] 000 ARMA-ARCH-M 5
GARCH EWMA TGARCH EGARCH GARCH-M GARCH Jump TGARCH Jump GARCH-M Jump GARCH TGARCH GARCH-M µ 9.50E-05 -.8E-04 3.45E-04 -.7E04 3.89E04-5.07E03 ** -6.89E04-7.95E04 ** -.09E03-4.76E04-5.3E04 -.3E03 σ 0.07 *** α 4.0E-05 *** 4.4E-05 *** -8.90E-06 *** 4.4E-05 *** 7.0E-06 6.40E-06 7.0E-06 6.70E-06 ** 6.60E-06 ** 6.80E-06** α 0.8 *** 0.7 *** 0. *** 0. *** 0.07 *** 0.05 ** 0.07 *** 0.05 *** 0.05 *** 0.06*** β 0.70 *** 0.70 *** -0.0 0.67 *** 0.83 *** 0.84 *** 0.83 *** 0.8 *** 0.8 *** 0.8*** ϕ 0.0 0.04 0.0 β 0.89 *** λ 0.074 *** δ 0.3 ** 0.04 0.0 ϑ 6.40E03 6.50E03 6.70E03 γ 0.03 *** 0.03 *** 0.03 *** c 0.3 *** 0.3 *** 0.3 *** ν 3.07 *** 3.09 *** 3.06 *** Log 864.77 903.069 830.60 903.4 898.05 905.98 308.58 309.45 308.64 3039.89 3040.9 3040.96 *** ** 5