eor imov r. ta matem. statist. Vip. 94, 6, stor. 93 5 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.
dy t = Y t y αy t dt + ky 3/ t dw t
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.
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 [,
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,
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
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 [,.
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 [,
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 <b Z σ p σ x = [ σ>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
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,
[ 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.
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
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 < 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
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 <.
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.
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 < b p> k 6k <b Zt Ψ,t L, Zt Ψ,t dt L,
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 < I Z exp Z +4q τ 3. Z C C t d dt dt Z t t Z d dt dt. ψ,t I C τ 3, Ψ,t C τ 3. p Z t Ψ,t Zt Ψ,t C Zt τ 6. sup t [, Z p t < τ 6 C> sup Z t Ψ,t C. t, [,
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
=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
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
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 <d< b 6k Z 3 s d > 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
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