Teor imov r. ta matem. statist. Vip. 94, 2016, stor

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Teor imov r. ta matem. statist. Vip. 94, 2016, stor"

Σουσάννα Παπακώστας
7 χρόνια πριν
Προβολές:

1 eor imov r. ta matem. statist. Vip. 94, 6, stor Abstract. e article is devoted to models of financial markets wit stocastic volatility, wic is defined by a functional of Ornstein-Ulenbeck process or Cox-Ingersoll-Ross process. We study te question of exact price of uropean option. e form of te density function of te random variable, wic expresses te average of te volatility over time to maturity is establised using Malliavin calculus. e result allows us to calculate te price of te option wit respect to minimum martingale measure wen te Wiener process driving te evolution of asset price and te Wiener process, wic defines volatility, are uncorrelated.

2 dy t = Y t y αy t dt + ky 3/ t dw t

3 Ω, F, F = F W,W t,t, P W t, W t, t B t =e rt r> S t, t ds t = μs t dt + σy t S t dw t, dy t = αy t dt + kd W t. S Y X t =,X t =,e rt S t W W σ : R R + c σx q + x l x R c q l α k S t = S exp μt X t = S exp μ rt σ Y s + σ Y s + σy s dw s. σy s dw s dx t =μ rdt + σy s X s dw s.

4 S + M t + A t μ σt X t = M t = σy sx s dw s A t = [Y t =Y e αt, Var[Y t = k α e αt. Y t = Y e αt +k e αt s d W s. B t =e rt r> ds t = μs t dt + Z t S t dw t, Z > dz t =b Z t dt + k Z t d W t. A b k k < b S t = S exp μt + Zs dw s. [Z t =Z e t +b e t, Z Var[Z t =Z k e t e t + bk e t. k < b X t NA g g, X t g t [,

5 Q P Q NA g Q F t dq dp Ft =exp r μ/σy s dw s + r μ /σ Y s +ν s, ν s d W s ν =ν t t ν s < P ν Q Q S t,y t ds t = rs t dt + σy t S t dw Q t, dy t = αy t kνt dt + kd W Q t, Wt Q μ r = W t + σy s, W t Q = W t + νs, Q νs P X X = X + M + A M P A Q P Q = P F P M Q X = S M t = σy sx s dw s A t =μ rt Q ds t = rs t dt + σy t S t dwt Q, dy t = αy t dt + kd W Q t,

6 W Q μ r t = W t + σy s, W Q t = W t, Q F t dq =exp dp Ft r μ Zs dw s + ν,s d W s Q r μ + ν,s, ν =ν,t t ν,s < P ν,s = dq r μ = L,t := exp dw s r μ, dp Ft Zs [L,t = Q X t =exp Zs d W s Q Q NA g Q S t,z t ds t = rs t dt + Z t S t dw Q t, dz t = b Z t dt + k Z t d W Q t, W Q μ r t = W t +, Zs W t Q = W t, Q V C C =S K + K V C =e r Q S Q K+ =e r Q Q S Q K+ Y s, s. Y s, s

7 σ := σ σ σ := Q S Q K+ Y s, s ln = S e r S +r + Φ σ ln K σ ln S +r KΦ σ ln K σ, σ Y s Φ σ σ := σ Z s σ σ W = W t,t [, Ω, F F = F W t,t [,, P Ω=C[,, R Ĉ R F = fw t,...,wt n, f = fx,...,x n Ĉ R n t,...t n [, S t D t F = n i= F S f x i W t,...,wt n [,tit, D, F t [,. D : L Ω L [,, R F D, F, G, = FG+ DF, DG H, L Ω S F, =[ F + DF H /. D L Ω H = L [,.

8 i L [,, R δ u L [,, R δ L Ω D DF, u H CF /, F D, C u ii u δ δu L Ω F D, Fδu = DF, u H δ Dom δ L, = L [,, D, L, u L = u, t dt + D s u t dt. u L, δu u u t dw t u L, δu = δu u t dw t. D, DF DF H F [ DF px = F>x δ DF. H F δ ut,, ω L [, Ω [ ut, Dom δ δut, dt < ut, dt, [, Dom δ ut, dtdw = ut, dw dt. t [,

9 σ = Z s σ = σ Y s νx =σxσ x σ σ Y σ [ p σ x = σ> x η t e αs dw s dt e α D η t d dt, [ η t = α k e αt νy t [ e α t t e αt+t νy t νy t dt dt, D η t 6k x k Zt Ψ,t dw dt Ψ,t ψ,t d dt, ψ,t := exp t b t k 8 [ t Ψ,t = ψ,t Zt Zt, ψ,t ψ,t d dt dt. D Y t = α t D Y s + k <t. D Y t = k e αt <t. σ σ D σ Y t =σy t σ Y t D Y t =k e αt νy t <t, I σ = σ Y t dt

10 σ = I σ D I σ = D σ Y t dt =k e αt νy t dt,. Dσ δ := δ Dσ, H Dσ H = = 4k = k α D σ = k t D σ d = 4k e αt νy t dt, e αt νy t dt d e αt νy t e αt νy t ddt dt [ e α t t e αt+t νy t νy t dt dt. ζ =e α η t dt = ζ := D σ Dσ ut, dt, ut, :=η t e α <t. ζ L, ζ δ ut, [ u t, ddt ut,, ω L [, Ω [ u t, d + [ = η t e α <t d + eα α [ eα α ηt dt <, t [, D s ut, d e α <t D s η t d ηt + [ D s η t C. ut, t [, [ [δut, u t, d + D s ut, d C,

11 [ δut, dt < δ = e α η t <t dtdw δ = = δ = η t η t e α η t <t dw dt. e α <t dw e α D η t <t d dt e α dw e α D η t ddt. dt σ D Z t = k exp [ b k Zt 8 = k exp t b t k Zt = kψ,t Zt. 8 D I Z t = I Z t = Z t dt D Z t dt = k ψ,t Zt dt,. D σ δ := δ D σ. D σ = k ψ,t Zt dt, D σ = = k D σ d = k ψ,t Zt dt d t Zt Zt ψ,t ψ,t ddt dt. ζ := D σ D σ ζ = k Zt Ψ,t dt = k ũt, dt, ũt, := Z t Ψ,t.

12 ζ δ [ ũt,, ω L [, Ω [, ũt, Dom δ ũ t, dtd = Z t Ψ,tddt < t = [ [ δũt, dt ũ t, dt + D s ũt, D t ũs, dt <, = k δ = k Zt Ψ,t <t dtdw = Zt Ψ,t <t dw dt k Zt Ψ,t <t dw Ψ,t D Zt <t d dt = k = k Zt Ψ,t dw dt Zt Ψ,t dw dt D Z t Ψ,t ddt Z t Ψ,t ψ,t ddt. K V C = C =S K + ln S +r + x ln K S Φ x ln K e r S +r x ln K Φ pxdx, x px = p σ x, p σ x, p σ x p σ x

13 Y Z Y R x [Y a, Y + a σ x σ Y σ Y τ =inft >: Y t Y a τ = τ Z > τ =inft >: Z t Z Z τ = τ τ p < τ p K> X = X t,t X t = X + M t + A t, M t = αs A t = βs αs K βs K a> τ a [X a, X +a a λ, K τ N >τ a Pτ a <λ 4 exp a. πa 8Kλ Y t = Y α Y s + kw t W N>a+ Y τ N =inft >: Y t N Ŷ t := Y t τn = Y α Y s τn s τn + k s τn dw s, C C 3 Ŷ t K = N α k τ N a =inft >: Ỹt Y a τ N a = τ a τ C Pτ a <λ C exp C, <λ<c 3, λ τ D η t

14 D η t = α [ [ k e αt e αt <t ν Y t e α t t e αt+t [ [ νy t νy t dt dt νyt e α t t e αt+t [ νy t νy t dt dt e α t t e αt+t νy t e αt <tν Y t +νy t e αt <tν Y t dt dt. η t D η t = α k e αt [ [ D νy t e α t t e αt+t νy t νy t dt dt [ = α k e αt D νy t [ +νy t D = D α k e αt Y t ν Y t νy t [ [ e α t t e αt+t νy t νy t dt dt [ e α t t e αt+t νy t νy t dt dt [ e α t t e αt+t νy t νy t dt dt [ e α t t e αt+t νy t νy t dt dt [ e α t t [ e αt+t νy t D Y t ν Y t +νy t D Y t ν Y t dt dt. D Y t =e αt <t [, η t <t L, η t L, [ η t L = η, t dt + D η t dtd <.

15 A = x R : σ x σ Y σ Y / τ =inft : Y t A τ = τ [ e α t t e αt+t νy t νy t c> νy t c σ Y t,τ η t = α k e αt νy t σx c> x e α t t e αt+t νy t νy t dt dt α k e αt νy t [ τ τ [ e α t t e αt+t νy t νy t dt dt 4α kcσ Y e αt νy t C x> C [ τ τ x x e α t t e αt+t dt dt. e α t t e αt+t dt dt ψx := = α x x d e αs = x e α x e α x + d α = e αx + + e αx + x α α α = C4 e αx e αx +αx 3. ψ = ψx = C4 e αx e αx +αx 3/4α R e αx αx e αx x ψ : ψ x = α e αx Cx e αx. ψx = x ψ s C x e αs s > C x 3. σ x e α t t e αt+t νy t νy t dt dt Cτ 3 ν x C + x m C> m σx η t C e αt νy t τ 3 C + Y t m τ 3.

16 sup t ηt C sup + Y t m τ 6 C. t [ ηt dt <. D η t C τ 3 ν Y t +τ 6 ν Y t m C + Y t m τ 3 + τ 6 C> sup D η t C t η t L, νy s ν Y u du D η t dtd <, ζ L, [ ζ L = e α η, s d [ + t D e αt η s dtd <. [ e α η s d [ e α ηs d <, = [ D e αt η s dtd t [ e αt D η s t<s dtd [ e αt D η s dtd <. sup [, Z p t k 6k <b Zt Ψ,t L, Zt Ψ,t dt L,

17 Zt Ψ,t dt L, = Zt Ψ,t dt d + D l Zt Ψ,t dt dld <. Zt Ψ,t = Z t ψ,t [ t t Zt Zt ψ,t ψ,t d dt dt. I := I Z = Z = t Zt Zt ψ,t ψ,t d dt dt t Zt Zt exp t τ τ t τ τ t exp t τ := inf t : Z t Z > Z q> exp t Z t < exp t b k t 8 b k t d dt dt. 8 τ = τ q := b k 8 exp t Z exp t sup Z t Ψ,t C. t, [,

18 Zt Ψ,t dt d Z t Ψ,t dtd C. = D l Zt Ψ,t dt dld D l Z t Ψ,t dt dld D l Z t Ψ,t dt dl d. D l Z t Ψ,t =D l [ Z t ψ,t Zt Zt t ψ,t ψ,t d dt dt Zt = D l exp t [ [ t exp Zt Zt exp t t = D l Zt exp t t t d dt dt t Zt Zt exp t exp t d dt dt + Z t exp t D l [ t Zt Zt exp t exp t d dt dt

19 =exp t [ Z t exp t [ Dl Z t + q Z t Z t t t Zt Zt exp t exp t Zt Zt Zt exp D l Zs t exp t t exp t D l Z t + q Z t Z t + D l Z t Z t D l Z t ψ,t I D l ψ t l,t Z t Ψ,t =kψ,t + q Z t k Z t ψ,t I t d dt dt exp t t D l Z t + q Z t ψ l,s Z 3 s I Zs + d dt dt D l Zs Zt ψ l,t ψ,t ψ,t + q Z t + Z t ψl,t + q Z t ψ l,s Zs d dt dt. ψ l,s Z 3 s + d dt dt. [ sup D l Z t Ψ,t <. l,,t [ D l Z t Ψ,t ψ t l,t = [kψ,t + q Z t k Z t ψ,t I t ψ l,s Z 3 s I Zt ψ l,t ψ,t ψ,t + q Z t + Z t ψl,t + q Z t ψ l,s Zs ψ l,s Z 3 s + d dt dt

20 k +k ψ,t [ Zt ψ,ti ψ t l,t + q Z t ψ l,s Z 3 s t ψ,t < I ψ,t ψ,t Zt ψ l,t + Z t ψl,t [ D l Z t Ψ,t k + q Z t +k [ Zt I t + Z t + q Z t + q Z t I Z 3 s + q Z t ψ l,s Zs Zt + q Z t ψ l,s Z 3 s + d dt dt. + Z 3 s d dt dt =k I +k I. Z s a + b n n a n + b n I := t 4 / + q Z t I t Z 3 + q I Z t 4 / s Z 3 s C Zt 4 t +8q4 τ. τ Z 3 s Zt 4 t Zt 4 Z 3 s 7 t Z 4 t C I C I I := [ Zt I t 8 t Zs, Z 3 s Zt t + q Z t + Z 3 s + Z t + q Z t [ Zt Z 3 s 4 d dt dt Z s

21 Zt I 8 [ 9 Z 4 4 t I 6 4 C Z 4 4 t τ 48 4 C = 4 9 Z 4 [ Zt Zt t exp 4 Z+4q Z + q Z t + q Z t 3 Z s Zt t + q Z t + Z t + q Z t t t [ Zt + Z t + q Z t [ Zt + Z t + q Z t + Z t + q Z t 4 Zt 4 Z 3 s Z s + Z 3 s t + q Z t d dt dt 4 Z s t + q Z t Z s + Z 3 s 4 d dt dt + Z 3 s 4 d dt dt, τ 48 Z 4 t 4 Zs + q Z t 8 Z 3 s Zt 4 Zt 8 t + 8 q8 6k b k. p p + d = Zt 8 t Z 3 s 8d d sup Zt d <. t [, Z 4p t I C Z 4p t p p 8d t Z 3 s Zs d d <. d

22 Ø

Σχετικά έγγραφα

P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:

P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example: (B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds