Z = 1.2 X 1 + 1, 4 X 2 + 3, 3 X 3 + 0, 6 X 4 + 0, 999 X 5. X 1 X 2 X 2 X 3 X 4 X 4

X 5 X 4 X 4

Z = 0.717 X 1 + 0.847 X 2 + 3.107 X 3 + 0.420 X 4 + 0.998 X 5. X 5 X 4 Z = 6.56 X 1 + 3.26 X 2 + 6.72 X 3 + 1.05 X 4.

dv t = µv t dt + σv t dw t. V t t V 0 V t = V 0 ((µ 12 ) σ2 )t + σw t. (V t )

µ σ W t G = V t dg = ( G µv t + G V t t + 1 2 2 G 2 V σ2 2 V t t G V t = 1 G V t = 0 2 G t 2 V t = 1 2 V t. dg = (µ 12 σ2 ) dt + σdz. ) dt + G V t σv t dz, µ σ 2 G = V t ( µ 1 2 σ2) σ 2 lnv t T ( µ 1 2 σ2) T σ 2 T V T V 0 N ((µ 12 ) ) σ2 T, σ 2 T V T N ((µ 12 ) ) V σ2 T, σ 2 T 0 V T N ( V 0 + (µ 12 ) ) σ2 T, σ 2 T. T V t T ( µ 1 2 σ2) (T t) σ 2 (T t) V T V t N ((µ 12 ) ) σ2 (T t), σ 2 (T t)

V T V t V T N N ((µ 12 ) ) σ2 (T t), σ 2 (T t) ( V t + (µ 12 ) ) σ2 (T t), σ 2 (T t). D T T T T D T T P D = P r (V T < D T ). T σ 2

µ σ 2 /2 t ) x z Φ Φ(.) ( z (X) P r (X < z) = Φ σ(x) ( (z (x)) σ(x) ( DT V t ( µ 1 2 P D = Φ σ2) ) (T t) σ T t ). ( ) P D = Φ D T V t ( µ 1σ2) (T t) 2 σ. T t ( ) ln D T V t ( µ 1σ2) (T t) 2 σ T t T D

V t < D D t ( Φ d + σ ) T t ( ) ln D V Φ t (r 1 2 σ2 )(T t) σ. (T t) T V T < D V T µ σ µ r µ = r µ > r µ < r

( ) D V Φ t (µ 1 2 σ2 )(T t) σ. (T t) r µ T T T

T T D T = min(d, V T ) = D + min(v T D, 0) = D max(d V T, 0). T E T E T = max(v T D, 0). V t D T V T = E T + D T. T

V T < D V T V T > D V T D T D T t r B B = e r(t t) D. B D T D t D T

σ r D t = De r(t t) P t (V t, D, T t), P t D t = De r(t t) [De r(t t) Φ( d 2 ) V t Φ( d 1 )] D t = De ( r(t t)) Φ( d 2 ) + V t Φ( d 1 ). d 1 = ( V t) D d 2 = ( Vt) D σ2 ) + (r + )(T t) 2 σ T t σ2 ) + (r )(T t) 2 σ T t Φ d 2 d 2 = d 1 σ T t. L = (e( r(t t)) D) V t. D t = De r(t t) Φ( d + σ t) V t Φ( d), d = 1 + 1 L 2 σ2 (T t) σ. (T t)

B = B D t B = De ( r(t t)) Φ(d σ (T t)) + V t Φ( d) B = De ( r(t t)) [ Φ(d σ (T t)) + 1 L Φ( d) ]. E T = (V T D, 0) V t E t = V t Φ(d 1 ) De ( r(t t)) Φ(d 2 ), d 1 = ( Vt) D d 2 = ( V t) D σ2 ) + (r + )(T t) 2 σ T t σ2 ) + (r )(T t) 2 σ T t D t = 5.2526, E t = 17.7216

( ) P D = Φ ln D T V t ( µ 1σ2) (T t) 2. σ(t t) V t σ

σ V D V t E t σ σ V V t E t V t = E t + D

T t = 0 t = T P T P 0 = R 0, T = R 1 R 2... R T P R r 0, T = r 1 + r 2 +... + r T. (0, T ) V ar(r 0, T ) = V ar(r 1 ) + V ar(r 2 ) +... + V ar(r T ). V ar(r 0, T ) = T V ar(r t ). σ(r 0, T ) = T σ(r t ). E t T V t E t = V t Φ(d 1 ) De ( r(t t)) Φ(d 2 )

V t = E t + De ( r(t t)) Φ(d 2 ). Φ(d 1 ) d 1 = ( Vt D ) + (r + σ2 V 2 )(T t) σ V T t d 2 = ( V t D ) + (r σ2 V 2 )(T t) σ V T t. σ V V t V t σ V 10 10 E(R i ) = R + β i (E(R m ) R f )) R = (r) 1 R m

β i E t = V t Φ(d 1 ) De ( r(t t)) Φ(d 2 ). de t = µe t dt + σ E E t dw t. ( Et de t = µv t + E t V t t + 1 ) 2 E t 2 2 V σ2 2 V t dt + E t σv t dw t, t V t E t V t = Φ(d 1 ). σ E = σφ(d 1 ) V t E t σ = σ E E t, Φ(d 1 ) V t

d 1 d 2 σ E E t σ E D t (T t) 1 V t σ d 1 d 2 E t σ E

( ) 2 ( ) 2 ModelEt ModelσE 1 + 1. ObservedE t Observedσ E µ >

0.5 CL + 10 LL. L I D g D 0 T D = D 0 (1 + g) τ exp (r(t t)). τ=t+1 c T I = cdexp (r(t t)). τ=t+1 D L I

V t < D + I D V (D+I) t. L + D + I > V t > I + D D V t > L + I + D D + (V t L I D) = V t L I. D V (D+I) t, V t < D + I E T = D, L + D + I > V t > I + D. D + (V t L I D), V t > L + I + D D D+I ( D ) D + I V t D D+I D + I ( D ) D + I (V tφ(k 1 ) (D + I)e r(t t) Φ(k 2 )),

k 1 = ( V t) σ2 ) + (r + )(T t) D+I 2 σ T t k 2 = k 1 σ T t. L + D + I V t Φ(d 1 ) (L + D + I)e r(t t) Φ(d 2 ), d 1 = ( V t ) σ2 ) + (r + )(T t) L+D+I 2 σ T t d 2 = d 1 σ T t. E t = V t Φ(d 1 ) De ( r(t t)) Φ(d 2 ) E t = D ( D D + I V ( t+v t Φ(d 1 ) (L+D+I)e r(t t) Φ(d 2 ) Vt Φ(k 1 ) (D + I)e r(t t) Φ(k 2 ) )) D + I E t = D ( D D + I V ( t+v t Φ(d 1 ) (L+D+I)e r(t t) Φ(d 2 ) Vt Φ(k 1 ) (D + I)e r(t t) Φ(k 2 ) )) D + I E t = V t Φ(d 1 ) (L+D+I)e r(t t) Φ(d 2 )+ D ( Vt V t Φ(k 1 ) + (D + I)e r(t t) Φ(k 2 ) ). D + I d 1 = ( V t L+D+I σ2 ) + (r + 2 σ T t )(T t),

d 2 = d 1 σ T t, k 1 = ln( V t) σ2 ) + (r + )(T t) D+I 2 σ, T t k 2 = k 1 σ T t. σ E = σ V ( t Φ(d 1 ) + D ) E t D + I (1 Φ(k 1)). P D = Φ [ ( V t L+D+I ) ( + µ 1 σ2) (T t) 2 σ (T t) ] P D(1 y ) = 1 (1 P D) 1 (T t). T

10 10 µ R m 6 ρ ρ > 0, 9

σ E 6

V t E t V t E t V t = 1 E t V t (T t)

1 (1 pd) ( 1 (T t) )

ρ

Φ(0, 1) Φ(m, v) Φ(0, 2) N(0, 1) T 0

T Φ(0, T ) N(0, t) z = e t e Φ(0, 1) z = 0 z = t z = t

X t, t 0 µ R σ > 0 y 0 t > 0 X t+y X y σ 2 X t+y X y X u, 0 u y g ω (t) = X t (ω) X t1 X t0, X t2 X t1,..., X tm X tm 1 t 1 t 0 = t 2 t 1 =... = t m t m 1 X t σh 1/2 X t, t 0 X t X t, t 0 X t (ω), t 0 ω Ω ω Ω

σ h σ h S t, t 0 µ R σ > 0 X t+y X y (t, tσ 2 ). X t+y X y X u, 0 u y. ds = µsdt + σsdz. S µ σ z

C = SΦ(d 1 ) Xexp( rt )Φ(d 2 ) d 1 = 1 σ T + [ln( S X ) + rt ] + 1/2σ T, d 2 = d 1 σ T.