Pricing power options in a jump diffusion model

2011 c 5  Journal of East China Normal University (Natural Science) May 2011 ©©©ÙÙÙ???ÒÒÒ: 1000-5641(2011)03-0012-09

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¥ã©aÒ: F224.7; F224.9 ©zI£è: A DOI: 10.3969/j.issn.1000-5641.2011.03.002

Pricing power options in a jump diffusion model

SU Xiao-nan, WANG Wen-sheng

(College of Finance and Statistics, East China Normal University, Shanghai 200241, China) Abstract: Under the assumption that the underlying asset prices obey jump diffusion processes and the market interest satisfies Vasicek model, and when the interest is correlated with the asset prices, by the way of change of measure, a closed solution of pricing of power option was given. Moreover, some special situations were considered. Key words: jump diffusion model; power options; stochastic interest rate; change of measure

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ʇÃÞ7K½|, ½|¥¤kØ(½5dVǘm_{(Ω, F, P )}5£ã, {Ft}t>0´¹tžƒc½|¥¤k&Eσ-6. -{St}t>0L«Iºx]d ‚L§_{, {rt}t>0}L«½|¥Ãºx|ÇL§_{, {B}t}t>0L«Õ1±âr,§‚©O÷vXe ‘Ň©§ dSt St− = µtdt + σtdfW1t− λβdt + d N_X(t) i=1 Ui, drt= (ea − ebrt)dt + σrdfW2t,dBt= rtBtdt, B0= 1,

Ù¥β = E[Ui] , ea, eb, σr~ê, σt´'užmt(½5¼ê, fW1t, fW2t´½Â3Vǘ

m_{(Ω, F, P )}þü‡IOÙK$Ä_,…Cov(dfW1t,dfW2t) = ρdt, Ui´˜ÕáÓ©Ù‘Å Cþ, Ui>−1 yStšK, f (y)ln(1 + Ui)VǗݼê, N (t)´rݏλ  ÑtL§_{, N (t), Ui}†ÙK$ÄWf1t, fW2tƒpÕá. ˜ªwÞÏÏÂÏCT = (STα− Kα)+, Ù¥ST´ÏFI]d‚, K´Ád‚, 0 < α 6 1´ëê.

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K2.15‰Ñ˜‡dÿÝ,(JXe. ÚÚÚnnn2.1 -_ηtL«_{Radon-Nikodym}êL§ ηt= dQ dP    Ft = expn Z t 0 ϑ1(u, ru) p 1 − ρ2dfW1u+ Zt 0 ϑ2(u, ru) − ρϑ1(u, ru) p 1 − ρ2 ! dfW2u+ −1₂ Zt 0 ϑ2₁(u, ru)du − 1 2 Zt 0 ϑ2₂(u, ru)du o , (1) Ù¥ ϑ1(u, ru) = ru− µu σu p 1 − ρ2 − ρϑ2(u, ru) p 1 − ρ2 , ϑ2(u, ru) = a_{− ea + (eb − b)r}u σr ,

Kd§₍₁₎¤½ÂÿÝQ´˜‡dÿÝ,=by]d‚L§3VÇÿÝQe ´ _. u´,d_Girsanov½nŒ W1t = Wf1t− Zt 0 ru− µu σu du, W2t = Wf2t− Zt 0 a− ea + (eb − b)ru σr du ´ü‡IOQÙK$Ä_,…Cov(dW1t,dW2t) = ρdt, KŒW1t = ρW2t+ p 1 − ρ2_W 3t Ù ¥W3t´†W2tƒpÕáIOÙK$Ä. duEÜÑtL§ N(t)_P i=1 ln(1 + Ui)´†ÙK$ÄWf1t, fW2tƒpÕá,lÿÝPº x¥5ÿÝQÿÝC†Ø¬UCEÜÑtL§rÝÚln(1 + Ui)VÇ©Ù.l ,º x]d‚L§StÚ|Çrt3ºx¥5ÿÝQekXeL«. dSt St− = rtdt + σtdW1t− λβdt + d N(t) X i=1 Ui, (2) drt= (a − brt)dt + σrdW2t. (3) -C(t, T )L«_tž˜ªwÞÏd‚,dºx¥5½dŒ C(t, T ) = EQ  Bt BT (Sα T − Kα) +_F t  . Bå„_,P I1= EQ  Sα TBt BT I{Sα T>Kα}  Ft  , I2= KαEQ  Bt BT I{Sα T>Kα}  Ft  . ÏdŒ C(t, T ) = I1− I2. (4) OŽC(t, T ),I‡e¡ü‡Ún. ÚÚÚnnn2.2 I2 = Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! Z∞ −∞ N_{(d2(t, T, y)) f}n_(y)dy +e−λ(T −t)N_{(d2(t, T, 0))}  , Ù¥ d2(t, T, y) = lnSt K + Λ(t, T ) − λβ(T − t) − RT t  σ∗2_{(s, T ) + ρσ} sσ∗(s, T ) +σ 2 s 2  ds + y qRT t ∆ 2_(s)ds , σ∗(t, T ) = σr(1 − e −b(T −t)₎ b , ∆ 2_{(s) = σ}∗2_{(s, T ) + 2ρσ} sσ∗(s, T ) + σs2,

Λ(t, T ) =rt− a b  σ∗_{(t, T )} σr +a(T − t) b , D(t, T ) = 1 − e −b(T −t) b , A ′ (t, T ) = −aD(t, T ) + 1₂σ_r2D2(t, T ), fn_{(y)ln(1 + U} i)—ݼêf(y)n­òÈ, N(·)L«IO‘ÅCþ\È©Ù¼ê.

yyy ²²² du|Ç´‘Å,·‚Ú\ÏÿÝQT_{,=±"EÅ P(t, T )ŠOd}

ü , QT_{'uQRadon-NikodymêXe.} dQT dQ = P(T, T ) P(0, T )BT = 1 P(0, T )BT . (5) Ϗ ½ | | Ç ÷ vVasicek ., ¤ ±Tž   Ï, ¡ Š 1 " E Å 3tž   d ‚_{P(t, T )}kXe•/ª[12]_. P(t, T ) = e−rtD(t,T )−A(t,T )_, ¿…P(t, T )÷vdP (t, T ) = rtP(t, T ) − σ∗(t, T )P (t, T )dW2t.l ,d§(5)Œ dQT dQ = exp ( − ZT 0 σ∗(u, T )dW2u− 1 2 ZT 0 σ∗2(u, T )du ) . Šâ_Gisranov½n_{, W2t= W2t+}Rt

0σ∗(u, T )du´QTIOÙK$Ä.

u´,dItˆoúªÚ§(2),k ST = Stexp    ZT t rsds + ZT t σsdW1s−1 2 ZT t σ_s2_{ds − λβ(T − t) +} N(T ) X i=N (t)+1 ln(1 + Ui)   . (6) d§(3),¦rs= eb(t−s)rt+a_b(1 − eb(t−s)) + σre−bs Rs tebudW2u,Ïd ZT t rsds = Λ(t, T ) + ZT t σ∗(s, T )dW2s. (7) 2ÏL§₍₆₎ڐ§_(7),U ST = Stexp n Λ(t, T ) + ZT t σ∗(s, T )dW2s+ ZT t σsdW1s− 1 2 ZT t σ2_sds −λβ(T − t) + N(T )_X i=N (t)+1 ln(1 + Ui) o = Stexp n Λ(t, T ) + ZT t ∆(s)dWs− ZT t  σ∗2(s, T ) + ρσsσ∗(s, T ) +σ 2 s 2  ds − λβ(T − t) + N(T )_X i=N (t)+1 ln(1 + Ui) o , (8) ùp ∆(s)dWs= [σ∗(s, T ) + ρσs]dW2s+ p 1 − ρ2_σ(s)dW 3s.

Šâ_Bayes{Kڐ§₍₅₎Œ I2 = KαP(t, T )EQ T I{Sα T>Kα}  Ft  = KαP(t, T )EQT  EQT  I(Sα T>Kα)|Ft∨ σ  N(T )_X i=N (t)+1 ln(1 + Ui)  Ft  . d ,d§(8)Œ¯‡_{Sα T >Kα}du n ZT t ∆(s)dWs+ N_X(T ) i=N (t)+1 ln(1 + Ui) > ln K St− Λ(t, T ) + λβ(T − t) + ZT t  σ∗2(s, T ) + ρσsσ∗(s, T ) + σ2s 2  dso. 2ÏL†OŽŒ I2 = KαP(t, T )EQ T N  d2  t, T, N(T )_X i=N (t)+1 ln(1 + Ui)  Ft  = Kα_e−rtD(t,T )−A(t,T )_EQT  N  d2  t, T, N_X(T ) i=N (t)+1 ln(1 + Ui)  = Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)λn_{(T − t)}n n! Z∞ −∞ N_{(d2(t, T, y)) f}n_(y)dy +e−λ(T −t)N_{(d2(t, T, 0))}  , K¤Ún2.1y². OŽI1, -dQα dQ = Sα T BT EQ_[SαT BT] . Š â_Girsanov½ n_, 3 V Ç ÿ ÝQα_e, N(T )P i=1 UiE ,´ E Ü Ñ t L §, Ù r Ý eλ = λEQ_eαln(1+Uj)_{, ln(1 + U} j)—ݼêfe(y) = e αy_f(y) EQ_[eαln(1+Uj )_].

e5,òOŽI1,Ù(JXe.

ÚÚÚnnn2.3 I1= Stα _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! Z∞ −∞

N_{(d1(t, T, y)) ef}n_{(y)dy + e}−eλ(T −t)N_{(d1(t, T, 0))}  , Ù¥ d1(t, T, y) = d2(t, T, y) + α sZT t ∆2_(s)ds.

yyy ²²² d§₍₆₎ÚN (t) P j=1 ln(1 + Uj)†W1t, W2tƒpÕጠdQα dQ = SαT BT EQ_[STα BT] = expn Z T 0 ασsdW1s+ ZT 0(α − 1)σ ∗_{(s, T )dW} 2s−(α − 1) 2 2 ZT 0 σ∗(s, T )ds −α 2 2 ZT 0 σ_s2ds − ρα(α − 1) ZT 0 σsσ∗(s, T )ds o exp ( α N(T )_P j=1 ln(1 + Uj) ) EQ " exp ( α N_P(T ) j=1 ln(1 + Uj) )# , (9) , ,ÏLBayes{KŒ I1 = BtEQ  Sα T BT I{Sα T>Kα}  Ft  = Sα tEQ α [I(Sα T>Kα)|Ft] = S_tαEQα  EQα  I(Sα T>Kα)  Ft∨ σ  N(T ) X i=N (t)+1 ln(1 + Ui)  |Ft  . qd§₍₉₎Ú_Girsanov½nŒ,3Qα_e, c W1t = W1t− Zt 0 ασsds − ρ Zt 0(α − 1)σ ∗_{(s, T )ds,} c W2t = W2t− Zt 0(α − 1)σ ∗ (s, T )ds − ρ Zt 0 ασsds, ü‡IOÙK$Ä_.Ïd ST = Stexp  Λ(t, T ) + ZT t ∆(s)dcWs+ α ZT t ∆2(s)ds (10) − ZT t  σ∗2(s, T ) + ρσsσ∗(s, T ) + 1 2σ 2 s  ds − λβ(T − t) + N_X(T ) i=N (t)+1 ln(1 + Ui)  , ùp_{∆(s)dcWs= σ}∗_{(s, T )dcW} 2s+ σsdcW1s. ÏL§(10)Œ¯‡{STα>Kα}du n ZT t ∆(s)dcWs+ N(T )_X i=N (t)+1 ln(1 + Ui) >lnK St − Λ(t, T ) + λβ(T − t) − α ZT t ∆2(s)ds + ZT t  σ∗2(s, T ) + ρσsσ∗(s, T ) + 1 2σ 2 s  dso. u´,ÏL NP(T ) i=N (t)+1 ln(1 + Ui) 'uσ N_P(T ) i=N (t)+1 ln(1 + Ui) ! Œÿ9ÙK$ÄÕáOþ5Œ

± I1 = StαEQ α N  d1(t, T, N_X(T ) i=N (t)+1 ln(1 + Ui))  Ft  = S_tαEQα  N  d1(t, T, N_X(T ) i=N (t)+1 ln(1 + Ui))  = S_tα _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! Z∞ −∞

N_{(d1(t, T, y)) e}fn(y)dy + e−eλ(T −t)N_{(d1(t, T, 0))}  . dÚn_2.2!Ún_2.49§(4)Œ˜ªÏ3tždŠ,(JXe. ½½½nnn2.1 ˜ªwÞÏ3_tžd‚ C(t, T ) = Sα t _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! Z∞ −∞

N_{(d1(t, T, y)) e}fn_{(y)dy + e}−eλ(T −t)_N(d

1(t, T, 0))  −Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! Z∞ −∞ N_{(d2(t, T, y)) f}n_(y)dy +e−λ(T −t)_N(d 2(t, T, 0))  , Ù¥ d1(t, T, y) = d2(t, T, y) + α sZT t ∆2_(s)ds, d2(t, T, y) = lnSt K + Λ(t, T ) − λβ(T − t) − RT t  σ∗2(s, T ) + ρσsσ∗(s, T ) +σ 2 s 2  ds + y qRT t ∆2(s)ds . 5552.1 _{˜ªwOÏÏÂÏ(K}α_{− S}α T)+,KÙtžd‚ P(t, T ) = Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! Z∞ −∞ N_{(−d2(t, T, y)) f}n_(y)dy +e−λ(T −t)N_{(−d2(t, T, 0))}  −Sα t _X∞ n=1 e−eλ(T −t)_λen_{(T − t)}n n! Z∞ −∞ N_{(−d1(t, T, y)) ef}n_(y)dy +e−eλ(T −t)N_{(−d1(t, T, 0))}  . e¡ò‰Ñ˜AϜ¹,=|ÇØ´‘ű9I]d‚ÑlAÛÙK$Ä ž˜ªÏd‚.

5552.2 |Ǐ~ê_rž_,˜ªwÞÏ_tžd‚ C(t, T ) = S_tα _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! Z∞ −∞

N_{(fd1(t, T, y)) ef}n_{(y)dy + e}−eλ(T −t)N_{(fd1(t, T, 0))}  −Kαe−r(T −t) _X∞ n=1 e−λ(T −t)λn_{(T − t)}n n! Z∞ −∞ N_{(fd2(t, T, y))f}n_(y)dy +e−λ(T −t)N_{(fd2(t, T, 0))}  , Ù¥ ed1(t, T, y) = ed2(t, T, y) + α sZT t σ2 sds, ed2(t, T, y) = − lnK St + r(T − t) − 1 2 RT t σ 2 sds − λβ(T − t) + y qRT t σs2ds . ƒA˜ªwOÏtžd‚ P(t, T ) = Kαe−r(T −t) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! Z∞ −∞ N_{(−fd2(t, T, y))f}n_(y)dy +e−λ(T −t)N_{(−fd2(t, T, 0))}  −Sαt _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! Z∞ −∞ N_{(−fd1(t, T, y)) ef}n_(y)dy +e−eλ(T −t)N_{(−fd1(t, T, 0))}  . 5552.3 _|Ǐ~êr,I]d‚ÑlAÛÙK$Ğ_,˜ªwÞÏtž d‚ C(t, T ) = S_tαN_{(d1(t, T )) − K}α_e−r(T −t)N_{(d2(t, T )), (11)} Ù¥ d1(t, T ) = d2(t, T ) + α sZT t σ2 sds, d2(t, T ) = − ln K St + r(T − t) − 1 2 RT t σ 2 sds qRT t σs2ds . ƒA˜ªwOÏtžd‚ P(t, T ) = Kαe−r(T −t)N_{(−d2(t, T )) − St}αN_{(−d1(t, T )). (12)} éëêαØÓŠŒ±ØÓ˜ªÏd‚,AO,α= 1 ž§(11)Ú(12)© OIOîªwÞÚwOÏd‚úª_.

½½½nnn2.2 _{ln(1 + Ui)}Ñl©ÙN_{(µ, σ}2₎ž_,˜ªwÞÏ3_tžd‚ C(t, T ) = Sα t _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! N(d ∗ 1(t, T, n)) + e−eλ(T −t)N(d∗1(t, T, 0))  −Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! N(d ∗ 2(t, T, n)) +e−λ(T −t)_N(d∗ 2(t, T, 0))  , Ù¥ d∗1(t, T, n) = d∗2(t, T, n) + α sZT t ∆2_{(s)ds + nσ}2_, d∗2(t, T, n) = lnSt K + Λ(t, T ) − λβ(T − t) − RT t  σ∗2_{(s, T ) + ρσ} sσ∗(s, T ) +σ 2 s 2  ds + nµ qRT t ∆2(s)ds + nσ2 . yyy ²²² ÄkOŽI2. ÏLÿÝC†k I2 = KαEQ  Bt BT I{Sα T>Kα}  Ft  = KαP(t, T )EQTI{Sα T>Kα}  Ft  = Kαe−rtD(t,T )−A(t,T ) _X∞ n=1 e−λ(T −t)_λn_{(T − t)}n n! N(d ∗ 2(t, T, n)) +e−λ(T −t)_N(d∗ 2(t, T, 0))  . 2 Š âGirsanov½ n_, 3 V Ç ÿ ÝQαe, N(T )_P i=1 UiE ,´ E Ü Ñ t L §, Ù r Ý eλ = λEQ_eαln(1+Uj)_{, ln(1 + U} j)—Ý¼ê e f(y) = e αy_f(y) EQ_[eαln(1+Uj)] = 1 2√πσe −(y−µ−ασ2 )2 2σ2 , =ln(1 + Uj)Ñl©ÙN(µ + ασ, σ2) . l Œ I1 = BtEQ  S_Tα BT I{Sα T>Kα}  Ft  = Sα tEQ α [I(Sα T>Kα)|Ft] = Sα t _X∞ n=1 e−eλ(T −t)_eλn_{(T − t)}n n! N(d ∗ 1(t, T, n)) + e−eλ(T −t)N(d∗1(t, T, 0))  .  2d₍₄₎ªC(t, T ) = I1− I2Œ±(J.

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As before, when Bi(a) = 0, i = 1, 2, · · · , 9, one has F (α(x)) = F (x). The conclusion follows for n= 5 by taking a0=  0, 0, 3 2g−2 a−₃, 3 2g2− a−₃,g + 2 g−2 a−₃, a−₃,0, 0, 0, 0  . This ends the proof.

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(þ1 20 )

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