Pricing Asian option under mixed jump-fraction process

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Ελλεν Ασπάσιος
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1 3 17 ( ) Journal of Eas China Normal Universiy (Naural Science) No. 3 May 17 : 1-641(17) ( 18) : -. Iô.... : -; ; : O11.6 : A DOI: /j.issn 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 : : (13XK3) :. gengyanjing ah@qq.com. :. zswcum@163.com.

2 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 [ 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 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( ) [ 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.

4 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

5 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 )

6 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)

7 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 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 )

8 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 ) (1). S = 8 K = 8 σ =.4 r =. q =.1 S = 8 = T = 1 r =. q =.1 σ =.4 K = /V H =.1 H =.3 H =. H =.7 H = /S /V P H H =.1 H =.3 H =. H =.7 H = /S Fig. 1 Asian opion pricing corresponding o differen H

9 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 㛑 ԧṭ /S 4 6 Fig. Asian opion pricing corresponding o differen λ ⴻ䏼ᵏᵳԧ /Vp 1 ⴻ ᵏᵳԧ /Vc ᵏ. 䰤 T. ã3 Fig H. ᮠ 䎛ᯟ ᵏ. 䰤 T 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 㛑 ԧṭ /S éaøó λ æªï d ã ⴻ ᵏᵳԧ /Vc 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 )

10 38 ( ) 17.. [ [ 1 KEMNA A G Z VORST A C F. A pricing mehod for opions based on average asse values [J. Journal of Banking and Finance (1): [ WONG H Y CHEUNG H L. Geomeric Asian opions: Valuaion and calibraion wih sochasic volailiy [J. Quaniaive Finance 4 4(3): [ 3 CHOU C S LIN H J. Asian opions wih jumps [J. Saisics & Probabiliy 6 6(14): [ 4 CHERIDITO P. Regularizing fracional Brownian moion wih a view owards sock price modelling [D. Zürich: Swiss Federal Insiue of Technology 1. [ CHERIDITO P. Arbirage in fracional Brownian moion models [J. Finance and Sochasics 3 7(4): [ 6 KUZNETSOV Y A. The absence of arbirage in a model wih fracal Brownian moion [J. Russian Mahemaical Surveys (4): [ 7 ZÄHLE M. Long range dependence no arbirage and he Black Scholes formula [J. Sochasics and Dynamics (): 6-8. [ 8 MISHURA Y S. Sochasic Calculus for Fracional Brownian Moion and Relaed Processes [M. Berlin: Springer 8. [ 9 WANG X T. Scaling and long-range dependence in opion pricing I: Pricing European opion wih ransacion coss under he fracional Black Scholes model [J. Physica A 1 389(3): [1 WANG X T. Scaling and long-range dependence in opion pricing V: Muliscaling hedging and implied volailiy smiles under he fracional Black Scholes model wih ransacion coss [J. Physica A 11 39(9): [11 MEHRDOUST F SABER N. Pricing arihmeic Asian opion under a wo-facor sochasic volailiy model wih jumps [J. Journal of Saisical Compuaion and Simulaion 1 8(18): [1 RAMBEERICH N. A high order finie elemen scheme for pricing opions under regime swiching jump diffusion processes [J. Journal of Compuaional and Applied Mahemaics 16 3(): [13 XIAO W L. Pricing currency opions in a fracional Brownian moion wih jumps[j. Economic Modelling 1 7(): [14 PENG B. Pricing Asian power opions under jump-fracion process [J. Journal of Economics Finance and Adminisraive Science 1 17(33): -9. [1. [J. ( ) 13 3(6): 9-9. [16. [J. 13 3(11): [17 SHOKROLLAHI F. Acuarial approach in a mixed fracional Brownian moion wih jumps environmen for pricing currency opion [J. Advances in Difference Equaions 1 7(1): 1-8. (: )

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