A Full Asymptotic Series of European Call Option Prices in the SABR Model with

Transcription

1 A Full Asymptotic Series of European Call Option Prices in the SABR Model with β = 1 Z. Guo, H. Schellhorn November 17, 2018

2 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula Exponential Formula Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] A Formula by Marc Yor Full Expression of Option Price Approximation of Option Price

3 SABR Model with β = 1 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula The SABR model is an extension of the Black Scholes model in which the volatility parameter follows a stochastic process: ds t = rs t dt + σ t S β t (ρdw t + 1 ρ 2 dz t ), (1) dσ t = ασ t dw t. (2)

4 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula Approximation for Implied Volatilities of SABR Model Hagan et al. derived, with perturbation techniques, an approximating direct formula for this implied volatility under the SABR model in [5]: σ BS (S 0, K) = [ ( (1 β) σ 0 (S 0 K) 1 β 2 [1 + (1 β)2 24 ln 2 S 0 K + (1 β) ln4 S 0 K + ] σ0 2 (S 0 K) 1 β +1 ρβασ 0 3ρ2 +2 α )τ+o(τ 2 2 ) 4 (S 0 K) (1 β)/2 24 ( ) where z := α σ 0 (S 0 K) 1 β 2 ln( S 0 K ) and x(z) = ln 1 2ρz+z 2 +z ρ 1 ρ. z x(z) ], (3)

5 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula Approximation for Implied Volatilities of SABR Model In the special case β = 1, the SABR implied volatility formula reduces to [ 1 + y σ BS (S 0, K) = σ 0 f (y) ( 1 4 ρασ ρ2 α 2) ] τ + O(τ 2 ), (4) 24 ( ) where y := α σ 0 ln( S 0 K ) and f (y) = ln 1 2ρy+y 2 +y ρ 1 ρ. European call: BS(t, x, σ BS ) = e x N(d + ) Ke r(t t) N(d ).

6 The Black-Scholes Theory Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula ds t = rs t dt + σs t dw t. (5) Let X t = ln S t denote the logarithm of stock price. The price of an European call option with payoff (X T K) + at time t satisfy the Black-Scholes-Merton equation: L BS (σ)bs(t, x, σ) = 0, (6) where L BS (σ) = L BS (σ) = σ2 + (r 1 t 2 x 2 2 σ2 ) r is x the Black-Scholes differential operator. And the closed-form solution of above PDE (6) is BS(t, x, σ) = e x N(d + ) Ke r(t t) N(d ). (7)

7 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula Generalization of Hull-White Formula Consider the model under a risk-neutral probability: ds t = rs t dt + σ t S t (ρdw t + 1 ρ 2 dz t ), t [0, T ] (8) Where W t and Z t are independent standard Brownian motions defined in a probability space (Ω, F, {F t } t 0, P), F t = F W t F Z t := σ{w s, Z s, s t}, and σ t is a square integrable process adapted to {F W t }.

8 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization of Hull-White Formula Generalization of Hull-White Formula Assume that hypotheses (H1) to (H4) in [1] by Alòs hold. Then, for all t [0, T ], V t = E[BS(t, X t, v t ) F t ] + ρ T ] e r(s t) Ft E [H(s, X s, v s )Λ s ds 2 t (9) where vs 2 = 1 T T s s σudu 2 is the future average volatility, and ( 3 H(s, X s, v s ) := x 3 2 ) x 2 BS(s, X s, v s ), ( T ) Λ s := σr 2 dr σ s. s D W s We denote V s,t = v 2 s (T s) = T s σ 2 udu and V t,s = s t σ2 udu.

9 Exponential Formula Exponential Formula by Jin, Peng & Schellhorn Theorem Suppose F D ([0, T ]) satisfies the following condition: [ ( ) ] (T t) 2n 2 (2 n n!) E sup (D 2 u 2 n... Du 2 1 F )(ω t ) 0, u 1,...u n (t,t ) n for fixed t [0, T ], then E[F F t ] = n=0 1 2 n n! [t,t ] n (D 2 s n... D 2 s 1 F )(ω t )ds n... ds 1. (10)

10 Freezing Operator Exponential Formula Definition Given ω Ω, a freezing operator ω t is defined as: { W (s, ω t W (s, ω), if s t; (ω)) = W (t, ω), if t s T. (11) The freezing operator ω t is a mapping from Ω to Ω.

11 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Apply Exponential Formula to F = H(s, X s, v s )Λ s Goal: E[H(s, X s, v s )Λ s F t ], recall that T V t = E[BS(t, X t, v t ) F t ]+ ρ e r(s t) E [ ] H(s, X s, v s )Λ Ft s ds. 2 t (12) Let F = H(s, X s, v s )Λ s, using iterated conditioning: [ E[F F t ] = E E [ ] ] H(s, X s, v s )Λ s FT W Ft Z Ft = E[Λ s G s F t ], (13) where G s = G(s, X s, v s ) = E [ H(s, X s, v s ) FT W F ] t Z depends only on Brownian motion {Z t } t 0.

12 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Apply Exponential Formula to F = H(s, X s, v s )Λ s Then the option price formula (9) becomes: where V t = E[BS(t, X t, v t ) F t ] + ρ 2 G s = Λ s := n=0 T s D W s σ 2 r drσ s = 1 2 n n! ωt Z Goal: E[G s Λ s F t ] T s T t e r(s t) E[Λ s G s F t ]ds, (14) 2ασ 2 r drσ s = 2ασ s V s,t, (15) [t,t ] n D 2n,Z τ n H(s, X s, v s )dτ n, t s. (16)

13 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Faà di Bruno s Formula for Malliavin derivative The Faà di Bruno s formula can be generalized to Malliavin derivative in the following way: If f and g are functions with a sufficient number of derivatives, then for a random variable F D N ([0, T ]) and n N, by chain rule and Faà di Bruno s formula we have D n t f (g(f )) = n k=1 f (k)( g(f ) ) ) B n,k (g (F ),..., g n k+1 (F ) Dt n F, (17) where B n,k (x 1,..., x n k+1 ) are the incomplete exponential Bell polynomials.

14 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Malliavin derivative of H s : D 2n,Z τ n H(s, X s, v s ) ( 3 ) H s = x 2 BS(s, X 3 x 2 s, v s ) = d e Xs d 2πVs,T Define two real-valued functions p( ) and q( ) such thatq ( p(s, X s, v s ) ) = H s, p(s, X s, v s ) = X s d ln( d ), (18) 1 q(x) = e x. (19) 2πVs,T

15 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Malliavin derivative of H s : D 2n,Z τ n H(s, X s, v s ) s Dτ Z X s = D τ σ u 1 ρ 2 dz u = σ τ 1 ρ 2 1 {τ s}, (20) Then by Faà di Bruno s formula, t D 2n,Z τ n H s = D 2n,Z τ n q(p(s, X s, v s )) = 2n k=1 q (k) (p(s, X s, v s )) B 2n,k ( b 1,..., b 2n k+1 ) D 2n,Z τ n X s = (1 ρ 2 ) n H s B 2n ( b1,..., b 2n ) Π n i=1 σ 2 τ i 1 {τi s} (21) where b k = p (k) (s, X s, v s ), k = 1,..., 2n

16 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Expression of G s G s = b j = = n=0 1 2 n n! ωt Z [t,t ] n D 2n,Z τ n H(s, X s, v s )dτ n (1 ρ 2 ) n 2 n H ω ( s B 2n b ω n! 1,..., b ω ) s 2n Π n i=1στ 2 i dτ n n=0 t = Hs ω (1 ρ 2 ) n 2 n V n ( n! t,sb 2n b ω 1,..., b2n) ω. n=0 { 1 ( 1) j+1 d (s, 2 X s, v s ), j = 1, 2; Vs,T d (s, X s, v s )) j ( 1) j+1 (j 1)!, j 3.

17 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Interpretation of Λ s G s Notice that Xs ω t := X t +r(s t) 1 2 V t,s + ρ s α (σ s σ t )+ωz t σ u 1 ρ 2 dz u = X t + r(s t) 1 2 V t,s + ρ α (σ s σ t ) (22) d ω ±(s, X s, v s ) := ω t Z d ±((s, X s, v s )) = d ± (s, X ω s, v s ) = X ω s ln K + r(t s) ± 1 2 V s,t Vs,T (23) and recall that Λ s = 2αV s,t σ s, thus Λ s G s is a function that depends on σ s, V t,s = s t σ2 udu and V s,t = T s σudu. 2

18 Joint Density of ( ) t 0 eσw s ds, W t Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Proposition ( 2 In [6] by Yor (1992): the joint density of ) t 0 eσws ds, W t has been derived for the case σ = 2, φ t,σ (x, y) := 1 ( t ) dxdy P e σws ds dx, W t dy for x > 0, y R, t > 0, where θ(r, t) = r 2π 3 t e π2 2t 0 = σ 2x e 2 σ 2 x (1+eσy ) θ 0 e ξ2 2t ( 4e σy/2 σ 2 x ), σ2 t, (24) 4 e r cosh ξ sinh ξ sin πξ dξ, r, t > 0. t (25)

19 Joint Density of Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price ( ) t 0 eσws µs ds, W t A straightforward application of the Cameron-Martin-Girsanov ( t theorem implies that the joint density of 0 eσws µs ds, W t ), σ > 0, µ R, which we denote by φ t,σ,µ (x, y), x > 0, y R, can be connected with the density φ t,σ (x, y) = φ t,σ,0 (x, y) through the formula φ t,σ,µ (x, y) = e µ σ y+ µ2 t 2σ 2 φ t,σ,0 (x, y µ t). (26) σ

20 Calculation of E[Λ s G s F t ] Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Define h(v t,s, v s, σ s ) = Λ s G s, then E[Λ s G s F t ] can be calculated as follows: = E[Λ s G s F t ] = E[h(V t,s, v s, σ s ) F t ] = E[E[h(V t,s, v s, σ s ) F s ] F t ] [ ] v = E h(v t,s,, σ s )F V T s s,t (v)dv F t 0 dx dy 0 0 dv h(σ 2 t x, where σ s (y) = σ t exp(αy 1 2 α2 (s t)). v T s, σ s (y))f V s,t (v)φ s t,2α,α 2(x, y)

21 Marginal Density of t 0 eσw s µs ds Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price The conditional density of V s,t is F V s,t (v) = 1 ψ σs 2 Vs,T ( v ), where σs 2 ψ Vs,T (v) = R φ T s,2α,α2(v, z)dz, and ) ( T F Vs,T (v) = P (V s,t v F s = P ( T = P s s σ 2 udu v σ s ) σ 2 s e 2α(Wu Ws) α2 (u s) du v σ s ) ( T s = P e 2α(Wu) α2u du v ) 0 σs 2, W T s < v σs = 2 φ T s,2α,α 2(x, z)dzdx. (27) 0

22 Marginal Density of t 0 eσw s µs ds Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price One straightforward application of (27) is using the conditional density of V t,t to obtain the first conditional expectation in (9): ( ) v E[BS(t, X t, v t ) F t ] = BS t, X t, F V 0 T t t,t (v)dv ( ) v 1 = BS t, X t, 0 T t σt 2 ψ Vt,T ( v σt 2 )dv 1 ( v ) v ) = 0 σt 2 BS t, X t, φ T t T t,2α,α 2( σt 2, z dzdv (28)

23 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price A Formula for European Call Option Price 0 T V t = E[BS(t, X t, v t ) F t ] + ρ e r(s t) E[Λ s G s F t ]ds. 2 t 1 ( v ) v = BS t, X t, φ T t,2α,α 2( T t σt 2 + ρα l( ) = e r(s t) σ s (y) f ( s, X x,y s, σt 2 T t 0 0 v T s f (s, X s, v s ) = V s,t H ω s ) ), z dzdv l(s, v, z, x, y)dydxdzdvds, (29) v ) φ T s,2α,α 2( σs 2, z φ s t,2α,α 2(x, y), ( ) (1 ρ 2 n )V t,s ( 2 n B 2n b ω n! 1,..., b2n) ω. n=0

24 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price Parameters of Approximation Results In the following tables we compare the values of the approximate option prices. We have chosen T t = 1, ln X t = 100, r = 0.1, σ t = 0.3, α = 1, ρ = 0, ±0.5 and varying values for the strike price K. Column 1: Strike price K; Column 2: Monte Carlo Simulation with number of simulation times N = 10 6 ; Column 3: Approximated prices obtained by Black-Scholes formula with volatility approximated by (4); Column 4: Approximated prices obtained by formula (29) with f ( ) approximated by (40).

25 ρ = 0 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price K Monte Carlo Hagan formula (29)

26 ρ = 0.5 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price K Monte Carlo Hagan 1st order approx

27 ρ = 0.5 Derivation for G s Conditional Expectation of Λ s G s : E[Λ s G s F t ] Full Expression of Option Price Approximation of Option Price K Monte Carlo Hagan 1st order approx

28 Second Approach to Calculate E[H s Λ s F t ] Recall that V t = E[BS(t, X t, v t ) F t ] + J, where J := ρ 2 = ρ 2 T t T e r(s t) t e r(s t) E[H s Λ s F t ]ds T = C 1 E 2α E [σ s E [ d e 2π t Xs d2 + 2 F W T F Z t [σ s E [ d e d2 2 F W T F Z t ] Ft ] ds ] Ft ] ds (30) for d ± evaluated at (s, X s, v s ), where C 1 = 1 2π ραke (T t).

29 Second Approach to Calculate E[H s Λ s F t ] Denote Q s := E [ d e d 2 can be written as J = C 1 T t 2 FT W F t Z ], ] then the correction term Ft E [σ s Q s ds. d (s, X s, v s ) = λ(v s,t )Z + γ(v t,s, V s,t, σ s ). (31) where Z = s t σ udz u is conditional normal with variance V t,s i.e. Z N (0, V t,s ), γ(v t,s, V s,t, σ s ) := κ + ρ α (σ s σ t ) 1 2 (V t,s + V s,t ) Vs,T, λ(v s,t ) := 1 ρ 2 V s,t.

30 Calculation of Q s Goal: E[R(s, X s, v s ) F t ] Q s = R (λz + γ)e (λz+γ)2 2 1 e z 2 2Vt,s dz = C 2 γe C 3γ 2 2πVt,s 1 where C 2 =, C (2 ρ 2 ) 3/2 3 = 1. Thus we have 2(2 ρ 2 ) J = C 1 T t T ] Ft Ft E [σ s Q s ]ds = C 1 C 2 E [R s ds. (32) t where R s := R(s, X s, v s ) = σ s γe C 3γ 2 only on Brownian motion {W t } t 0. is a random variable depends

31 Calculation of E[R s F t ] by Exponential Formula Goal: E[R(s, X s, v s ) F t ]. Now we can apply exponential formula (10) to R(s, X s, v s ) such that: [ ] E R(s, X s, v s ) F t = where r n (s, X t, v t ) = ω t W n=0 1 2 n n! r n(s, X t, v t ), t s, (33) [t,t ] n D 2n,W τ n R(s, X s, v s )dτ n.

32 [ ] First Order Approximation of E R(s, X s, v s ) F t Let f (x, y) = yxe C 3x 2, then R(s, X s, v s ) = f (γ, σ s ), and Dτ 2,W R s = f x (γ, σ s )Dτ 2,W γ +f xx (γ, σ s )(Dτ W γ) 2 +f y (γ, σ s )Dτ 2,W σ s By the structure of σ t for t [0, T ], we have the following results: D W τ σ s = ασ s 1 {τ s}, D 2,W τ σ s = α 2 σ s 1 {τ s}, D W τ V s,t = 2αV τ s,t, D 2,W τ V s,t = 4α 2 V τ s,t, D W τ V t,s = 2αV τ,s 1 {τ s}, D 2,W τ V t,s = 4α 2 V τ,s 1 {τ s}.

33 First Order Approximation for the Correction Term Therefore, J = ρ 2 T 1 C 1 C 2 t n=0 T t T e r(s t) E[H s Λ s F t ]ds = C 1 C 2 E[R s F t ]ds 1 2 n n! ωt W [t,t ] n D 2n,W τ n R(s, X s, v s )dτ n ds T T = C 1 C Dτ 2,W Rs ω dτds t 2 t = 1 [ T ] 2 C 1C 2 p 1 (s) + p 2 (s)ds + 2(T t). (34) where p 1 (s) := ωw t s t D2,W τ R s dτ, p 2 (s) := ωw t T t t s Dτ 2,W R s dτ

34 Conclusion Convergence Analysis Stochastic Volatility F.B.M

35 E. Alòs. A generalization of Hull and White formula with applications to option pricing approximation. Finance and Stochastics 10 (3) (2006) A. Lyasoff Another look at the integral of exponential Brownian motion and the pricing of Asian options. Finance and Stochastics, 20(4), (2016). J.-P. Fouque, G. Papanicolaou, K. R. Sircar Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, S. Jin, Q. Peng, H. Schellhorn A Representation Theorem for Expectations of Functionals of Brownian Motion. Stochastics, vol. 88(5), (2016). P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, Managing smile risk. Wilmott, pp (2002) M. Yor On some exponential functionals of Brownian motion. Adv. Appl. Probab., 24 (1992), pp

36 Lemma:Faà di Bruno s formula Faà di Bruno s formula. If f and g are functions with a sufficient number of derivatives, then d n dx f (g(x)) = ) n! n Π n i=1 m i! f ( n k=1 g mk) (g(x)) Πj=1( n (j) mj (x), j! (35) subject that all nonnegetive integers (m 1,..., m n ) satisfying the constraint n k=1 km k = n. A simpler formula expressed in terms of Bell polynomials B n,k (x 1,..., x n k+1 ): d n dx n f (g(x)) = n f (k)( g(x) ) ) B n,k (g (x),..., g n k+1 (x). k=1 (36)

37 Exponential Bell polynomials The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by B n,k (x 1, x 2,..., x n k+1 ) = n! ( xi ) ji Π n k+1 i=1 j i! Πn k+1 i=1, (37) i! where the sum is taken over all sequences j 1, j 2,..., j n k+1 non-negative integers such that these two conditions are satisfied: n k+1 i=1 j i = k and n k+1 i=1 i j i = n.the sum n B n (x 1,..., x n ) = B n,k (x 1, x 2,..., x n k+1 ) (38) k=1 is called the nth complete exponential Bell polynomials.

38 1st order approximation of f ( ) and option prices Let m > 0, define L ω s = vs 2 (T s)hs ω and ( ) m (1 ρ f m (s, X s, v s ) := L ω 2 n )V ) t,s s 2 n B 2n (p (Xs ω ),..., p 2n (Xs ω ) n! n=0 then the first order approximation f 1 (s, v s, X s ) is then calculated as following: ( f 1 (s, X s, v s ) = L ω s 1 + (1 ρ2 )V [ t,s (p (1) (Xs ω ) ) 2 + p (2) (Xs )] ) ω 2 = d ω e X ω s 2π d ω ( 1 + (1 ρ2 )V t,s d ω2 3) 2 V s,t (39) (40) for d ω ± evaluated at (s, X s, v s ).

39 Convergence Analysis Conditions on the convergence of the series (T t) 2n [ ( ) ] 2 (2 n n!) 2 E sup (Du 2 n... Du 2 1 F )(ω t ) 0, u 1,...u n (t,t ) n c 2n [ ( n! 2 E ( sup H s B 2n b ω 1,..., b2n) ω Π n i=1 σ 2 ) ] 2 τ i 1 {τi s} 0, τ i (t,t ) n where c = (T t) 1 ρ 2, and b j = p (j) (s, X s, v s ) for j = 1,..., 2n.

40 Full expression of p 1 (s) s p 1 (s) := ω t W D 2,W τ R ω s dτ t [ ( )[ = R ω s α 2 1 (s t)+ γ ω +2C 3γ ω ρα 2 (s t) 1 e α2 (T s) 2α 2( σ 2 t α 2 (e α2 (s t) e α2 (T t) )+γ ω) 1 2ασ t ( 1 α 2 (1 e α2 (s t) ) (s t)e α 2 (s t) ) e α2 (s t) e α2 (T t) α 2 (1 e α2 (s t) ) (s t)e α 2 (T t) e α2 (s t) e α2 (T t) ] +(6C 3 +4C 2 3 γω ) ( 1 ραe 2 α2 (s t) + σt e α2 (s t) ) 2 (s t) 2 ρ 1 α σt (e 2 α2 (s t) 3 e 2 α2 (s t) σ 2 ) + t 2α (1 e 2α2 (s t) ) e α2 (s t) e α2 (T t) 2σ 3 ρ 1 t α e α 2 2 (s t)( 1 2 (s t)) α (1 2 (T t) ω e α (s t)e )A α σ4 [( t 1 2 (s t))+(e α 4 2 (1+e 2α α 2 (T t+s t) α 2 (s t) α 2 (T t) 2 α 2 ) (T t+s t) e e )+α e (s t) A ω 2 ( (1 e 2α2 (s t) α 2 (T t+s t) α 2 (T t) 2 2α 2 ) (T t) ) + 2(e e ) + α e (s t) A ω ] ] 3 (41)

41 Full expression of p 2 (s) T T p 2 (s) := ωw t Dτ 2,W Rs ω dτ = Dτ 2,W Rs ω dτ s s [ ( )( 1 ) = Rs ω γ ω + 2C 3γ ω 2α 2 ( Vs,T ω + γω ) where B ω 3 = R ω s [ ( 2α 2 2C 3 (γ ω2 + Vs,T ω γω ) A ω 3 (2αV ω s,t )2 ](T s) Vs,T ω γ ω ) ] + α 2 B3 ω (T s) (42) = 4V s,t ω 2 A ω 3 (C = γω3 +(2C3 2 Vs,T ω +3C 3)γ ω2 +(C3 2 V s,t ω +4C 3 Vs,T ω )γω) + 6C 3 V ω s,t V ω s,t γ ω. (43)

42 A ω 1, Aω 2, Aω 3 4C 2 A ω 1 := ωt W A 3 γω2 + (4C3 2 Vs,T ω + 8C 3)γ ω + 6C 3 Vs,T ω 1 = + Vs,T ω 3 C 3 (2C 3 γ ω2 + (V ω A ω 2 := ωt W A s,t + 2C 3 Vs,T ω + 3)γω + 3 Vs,T ω ) 2 = + Vs,T ω 3 C A ω 3 := ωt W A 3 2γω3 + (2C3 2 3 = V ω s,t + 3C 3)γ ω2 + (C 2 3 V ω s,t + 4C 3 + 6C 3Vs,T ω + 3 4Vs,T ω 2 + V ω s,t Vs,T ω 3 γ ω 1, Vs,T ω 3 γ ω 1 2 V, s,t ω γω V ω s,t )γω