3 17 ( ) Journal of Eas China Normal Universiy (Naural Science) No. 3 May 17 : 1-641(17)3-9-1 - ( 18) : -. Iô.... : -; ; : O11.6 : A DOI: 1.3969/j.issn.1-641.17.3.3 Pricing Asian opion under mixed jump-fracion process GENG Yan-jing ZHOU Sheng-wu (Deparmen of Mahemaics China Universiy of Mining and Technology Xuzhou Jiangsu 18 China) Absrac: This paper mainly sudied he geomeric average Asian opion pricing on he condiion ha he underlying asse followed mixed jump-fracion process. The general Iô s lemma and he self-financing dynamic sraegy were obained by using he parial differenial equaion of such opion pricing in he mixed fracional environmen wih jump. Wih he combinaion of boundary condiion an analyic formula for he geomeric average Asian opion was derived by solving he parial differenial equaion. The numerical experimens were showed o discuss he influence of differen parameers on he opion value. The resuls were he generalizaion of some exising resuls which was closer o he real financial marke. Key words: mixed jump-fracion process; geomeric average Asian opion; parial differenial equaion. Kemna Vors [1 : 16-6-3 : (13XK3) :. E-mail: gengyanjing ah@qq.com. :. E-mail: zswcum@163.com.
3 ( ) 17 ; Wong Cheung [ ; Ching-Sung Chou [3 -. ( ) ;. Markov Iô. Cheridio [4-. Kuznesov [6 Zähle [7 Mishura [8 Black-Scholes. [9-1 Black-Scholes. Poisson. Farshid Mehrdous [11 ; Nisha Rambeerich [1 -.. [13 - ; [14 - Iô. [1 [16 - Iô. FoadShokrollahi [17 -. - - - Black-Scholes. 1. ds = (µ q )S d + σs (dm + dn ) (1) µ q σ M H = B + B H B B H H ( 1) ; N = Q λ Q λ Q B B H. 1( - Iô ) W = B + B H + N f( x) C 1 (R + R R) f( W ) τ (τ W τ)dτ f x (τ W τ )dτ f x (τ W τ )τ H 1 dτ L (P)
3 : - 31 f( W ) =f( ) + x (τ W τ)dw τ + [ τ (τ W τ) + 1 (1 + λ + HτH 1 ) f x (τ W τ) dτ. () N i i. W = B + B H + Q λ i = 1 W ( ) 1 ( 1 ) ( 1 ) Iô f( 1 W ) = f( ) + 1 1 [ τ + 1 (1 + HτH 1 ) f 1 x λ dτ + x x db τ + [ f( W ) = f( 1 W 1 ) + 1 τ + 1 (1 + HτH 1 ) f x λ dτ + x f( W ) 1 f( 1 W 1 ) f( 1 W ) 1 [ f( W ) =f( ) + τ + 1 (1 + HτH 1 ) f i ( ) f( W ) =f( ) + x λ x 1 x db τ + dτ + x db τ + x dbh τ + f( 1 W 1 ) f( 1 W ). 1 [ τ + 1 (1 + HτH 1 ) f x λ dτ x + x db τ + x dbh τ + f(τ W τ ) f(τ W τ ). τ g(φ) C (R R) (dq dq ) = λd g(q ) Iô [14 τ [ g(q τ ) g(q τ ) = g (Q τ )dq τ + λ g (Q τ )dτ. 1 x dbh τ 1 x dbh τ. W = B + B H + Q λ τ [ f(τ W τ ) f(τ W τ ) = x dq τ + λ f x dτ. f( W ) =f( ) + + x db τ + =f( ) + [ τ + 1 (1 + HτH 1 ) f x λ x dτ x dbh τ + x dq τ + λ f x dτ [ τ + 1 (1 + λ + HτH 1 ) f x λ dτ + x x dw τ.
3 ( ) 17 (1) { S = S exp (µ τ q τ )dτ σ (H + λ + ) + σw }. (3) f( W ) = S exp { (µ τ q τ )dτ σ (H + λ + ) + σw } 1. 3 S - (1) K T ( T) V c ( J S ) : + (r q )S + 1 S σ S V S + J (lns lnj ) J = r V (4) T < S < + < J < + σ = σ (1 + H H 1 + λ) V c (T J T S T ) = (J T K) +. V = V c ( J S ) J J = e 1 [ ( dj = J 1 lns τ dτ + 1 ) lns lns lnj = J. ln Sτdτ : dp = r P d ds = (µ q )S d + σs (db + db H + dn ) θ = (θ θ 1 ) V = θ P + θ 1 S 1 dv = θ dp + θ 1 ds + θ 1 q S d dv = θ dp + θ 1 ds + θ 1 q S d = (V θ 1 S )r d + θ 1 (µ q )S d + θ 1 σs dw + θ 1 q S d = (V θ 1 S )r d + θ 1 µ S d + θ 1 σs dw. [ dv = + (µ q )S + lns lnj S J J d + 1 σ S (1 + H H 1 + λ) V S d + σs dw. S
3 : - 33 θ 1 = S µ S S d + r V d r S S d + σs S dw = d + [(µ q )S d + σs dw S + dj + 1 J V S [σ S (d + HH 1 d + λd) r V = + (r ln S ln J q )S S + J J + 1 σ S (1 + HH 1 + λ) V. σ = S σ (1 + H H 1 + λ). r V = + (r q )S + lns lnj S J + 1 J σ S V S. 4 S - (1) K T ( T) V c ( J S ) T V c ( J S ) =(J ST ) 1 T exp {r (T ) T r (r θ q θ ) T θ = T dθ T σ H = r θ dθ + (σ H ) (T H H ) + (σ λ ) (T ) }N(d 1 ) Ke T rθdθ N(d ) () (λσ + σ )(T ) 4T σ (T H H ) (T ) + Hσ (T H+1 H+1 ) (H + 1)(T ) [ 1 4H(T H+1 H+1 ) (H + 1)(T H H ) + H(T H+ H+ ) 1 σ T (H + 1)(T H H σλ = T ( λ + 1σ) ) 3T (σλ ) (T ) + (σh ) (T H H ) + 1 d 1 = T ln J S T K T + r (T ) (σ λ ) (T ) + (σh ) (T H H ) d = d 1 (σλ ) (T ) + (σh ) (T H H ) N(x) = 1 x e d. π 3 ( T) V c ( J S ) (4). ξ = 1 T [ lnj + (T )lns V c ( J S ) = U( ξ )
34 ( ) 17 = U + U lnj lns ξ T = U T = U S ξ TS J ξ TJ V S = ( U T ) ( T ) U = S ξ TS TS ξ T TS U ξ (4) U ( + r q 1 ) T σ U + 1 ( T ) U T ξ σ T ξ U(T ξ T ) = (e ξt K) +. = r U (6) α() β() γ() τ = γ() η τ = ξ + α() W(τ η τ ) = U( ξ )e β() U = e β()[ W τ γ () β ()W + W α () η τ U ξ = e β() W η τ U ξ = ( e β() W ) = e β() W ξ η τ ητ (6) γ () W [ τ + α () + (r q 1 σ ) T W T η τ + 1 σ( T ) W T ητ (r + β ())W =. (7) γ () + 1 σ( T ) ( = α () + r q 1 T σ) T = r + β () = T α(t) = β(t) = γ(t) = β() = α() = T T r θ dθ γ() = σ + λσ 6 (r θ q θ )dθ σ + λσ (T )3 T 4T (T ) + Hσ (T H+1 H+1 ) T(H + 1) [ + Hσ T H H + (T H+ H+ ) H T (H + 1) σ (T H H ) 4(T H+1 H+1 ) T(H + 1)
3 : - 3 (7) W τ = W ητ W( η ) = (e η K) + (8) (8) W(τ η τ ) = 1 πτ = 1 πτ = I 1 + I. + + (e y K)e (y η τ ) 4τ dy e y e (y η τ ) 4τ dy K πτ + e (y η τ ) 4τ dy I 1 = 1 πτ = e τ+η τ + 1 + π y η τ τ τ = e y e (y η τ ) 4τ dy = e τ+η τ η τ +τ τ 1 + π e (y η τ τ) 4τ dy e d = e τ+η τ N ( ητ + τ lnk τ ). I = K πτ + y η τ τ = e (y η τ ) 4τ dy = K + π η τ π e ( ητ lnk d = KN ). π (8) ( W(τ η τ ) = e τ+η ητ + τ lnk ) ( ητ lnk ) τ N KN τ τ d 1 = τ + η τ lnk τ = d = η τ lnk τ = = e τ+η τ N(d1 ) KN(d ). (9) (σλ ) (T ) + (σh ) (T H H ) + 1 T ln J ST (σ λ ) (T ) + (σh ) (T H H ) 1 T ln J S T K T + r (T ) (σ λ ) (T ) + (σh ) (T H H ) = d 1 (σλ ) (T ) + (σh ) (T H H ). K T + r (T )
36 ( ) 17 1 S - (1) K T ( T) V p ( J S ) T V p ( J S ) = (J ST ) 1 T exp {r (T ) 4. r θ dθ + (σ H ) (T H H ) + (σ λ ) (T ) }N( d 1 ) + Ke T rθdθ N( d ). (1) V (T J T S T ) = (K J T ) + 3 (4) V p ( J S ). 3 4. - (1). S = 8 K = 8 σ =.4 r =. q =.1 S = 8 = T = 1 r =. q =.1 σ =.4 K = 8... 1. 1.. /V 4 4 3 3 1 1 H =.1 H =.3 H =. H =.7 H =.9 3 4 6 7 8 9 1 11 1 /S /V P 6 4 3 1 1 H H =.1 H =.3 H =. H =.7 H =.9 3 4 6 7 8 9 1 11 1 /S Fig. 1 Asian opion pricing corresponding o differen H 3 4. 3 4.
13Ï ñò : Ü êa-*ñ.e æªï ½d 4 6 lambda = lambda = lambda = 4 lambda = 6 ⴻ ᵏᵳԧ /V 3 lambda = lambda = lambda = 4 lambda = 6 ⴻ䏼ᵏᵳԧ /V 3 1 1 4 3 1 3 4 6 7 8 9 1 11 1 㛑 ԧṭ /S 4 6 Fig. Asian opion pricing corresponding o differen λ ⴻ䏼ᵏᵳԧ /Vp 1 ⴻ ᵏᵳԧ /Vc 1 1 1. ᵏ. 䰤 T. ã3 Fig. 3.8.6.4 H. ᮠ 䎛ᯟ 1 1. 1. ᵏ. 䰤 T..8.6.4. ᮠ H 䎛ᯟ 1. âda ê! Ï múæªï d 'X The relaion of Hurs exponen expiry dae and Asian opion ⴻ䏼ᵏᵳԧ /Vp 1 8 1 1 14 16 18 㛑 ԧṭ /S éaøó λ æªï d ã ⴻ ᵏᵳԧ /Vc 37 1 1 1 8 8 6 T ã4 Fig. 4 4 ( bda Lam ᕪᓖ 䐣䏳 6 T bda Lam ᕪᓖ 䐣䏳 a rý! Ï múæªwþï d 'X The relaion of jump inensiy expiry dae and Asian opion Ø b d v Ü êa-*ñl ÏL Iˆo ÚnÚgK üñí Ñ Ü êùk$äe a æªï ½d. $^CþO {é½d.?1 )
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