This document presents a practical approach for pricing structured funds. It decomposes the value of senior shares into fixed income and embedded option components, modeled separately. The demand rate of return is modeled as a random process using the Hull-White model to deduce the bond price PDE and solution. The embedded option is modeled as a binary knock-in put using the Black-Scholes framework to derive a PDE solved numerically. Monte Carlo simulations validate the theoretical pricing. Junior share values are then easily calculated given the senior share values. This provides a more realistic pricing method compared to previous net asset value approaches.

1. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
©?Ä7 ´À ½d
Á‡
©?Ä7´˜«( .Ä7nã ¬§ÏL58aqÅ `k° 5•aq ¦ ºx°
?1 ]"duÙºx° UJøƒéíd…´ m§Ï 3Ú½ž É¥ ¬J,"Ù
`k° •k'½|úx|Ç•p ½Âç¿Œ±3½|e…žJøaquwOÏ Â
Ã§Ï •É Ø Å Ý]ö -À"ØÓuƒcýŒõê©zÏL•ÄÄ7] © Ý
鈰 ÀŠ?1½d§ © é ´ˆ° ´dŠ?1½d§äk•r • ¢S ´
Š^"
©ò©?Ä7`k° ´dŠ©•ÙÅ dŠ†Û¹wOÏ dŠ"b ½| ‡¦
ÂÃÇ•‘ÅÅÄ¿¦^Hull − White .éÙï §3db e¦^ºx¥5½d•{í Ñ
Å dŠÑl ‡©•§¿?1 nØ)í "`k° ¤Û¹ Ï ´˜‡ vwOÏ
§3AÛÙK$Äb e¦^Black − Scholes .?1nؽd§¿3ºx¥5eí ÑÙ÷
v ‡©•§¿¦^k• ©•{¦) ‡©•§" Xòü‡(J©O†|^„Akâ [
•{¦Ñ êŠ)?1' §©Û ) A5§ y nØ) O(5±9)†½|¢‚² ¬
Ü"ºx° ´dŠKŒ±d`k° dŠÏL©?Ä7 é=†Å› Ñ"
'''………cccµµµ©?Ä7§Black − Scholes .§Hull − White .§ºx¥5½d§êŠ [½d
I

2. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
A Practical Practice of Structured Fund Pricing
ABSTRACT
Graded fund, as known as structured fund, raises capital for equity-like junior shares by issuing bond-
like senior shares. Due to high accessibility and low cost of the leverage it provided, senior shares of
structured fund are madly sought after by individual investors in bull markets. The senior tranche promises
higher return comparing to risk free rate while also possessing an opportunity to gain from market falls.
Differing from most previous works, this article focuses on the pricing of each share’s transaction value
instead of the net value, which cannot be realized without discount or premium, and thus managing to be
more realistic and practical.
Decomposing the value of senior shares into the value generated from the fixed income and the value
contained in the embedded option allows us to consider them separately. Here we set the demanded rate of
return as a random character and depict it with Hull − White model. Under this assumption, we deduce
the partial differential equation of the bond price and deduct a theoretical solution. The embedded option
should be best described as a binary knock-in put and can be appropriately modeled under Black−Scholes
framework. The differential equation generated from it is numerically solved with finite difference method.
At last, Monte − Carlo simulations are applied to prove the validity of theoretical pricing. The value of
junior shares can be easily solved from the transformation mechanism given the value of senior shares.
Key words: Structured Fund§Black − Scholes Model§Hull − White Model§Risk Neutral Pricing§
Numerical Simulation Pricing
II

3. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
8¹
1 XØ································································································· 1
1.1 ©?Ä7{0 ················································································ 1
1.2 ykïÄnã ················································································ 1
1.3 ©M#ƒ? ················································································ 2
2 ÎÒ† ½ ························································································· 3
2.1 . Ä b ·············································································· 3
2.2 {¡†PÒ ½·············································································· 3
3 ¯K©Û ··························································································· 5
3.1 é©?Ä79ÙÝ]dŠ ?˜Ú ã ···················································· 5
3.1.1 é=†Å› ········································································ 5
3.1.2 ½ÏòŽÅ› ········································································ 5
3.1.3 ؽÏòŽÅ›······································································ 6
3.2 ©¦^ ½dg´†1©óä···························································· 6
4 ©?Ä7 nؽd··············································································· 8
4.1 `k° Å dŠ½d ······································································ 8
4.1.1 ‘Å|Ç . ········································································ 9
4.1.2 ‘Ň©•§¦) ··································································· 10
4.2 `k° Ï dŠ½d ······································································ 10
4.2.1 ‘Ň©•§ Ñ ··································································· 11
4.2.2 ‘Ň©•§¦) ··································································· 12
4.3 ºx° ½d ················································································ 13
5 ©?Ä7 „Akâ [½d ···································································· 15
5.1 „Akâ [ n········································································· 15
5.2 „AkÛ [ L§········································································· 15
5.3 „Akâ [ (J········································································· 16
5.4 †nØ) é'·············································································· 16
5.4.1 Å dŠnØ)†êŠ)é' ······················································ 16
5.4.2 Ï d‚nØ)†êŠ)é' ······················································ 17
5.5 'u)5Ÿ ?Ø ··········································································· 18
5.5.1 ©(J†yk©z(J' ······················································ 18
5.5.2 ) Å d‚†þï|Ç 'X ················································· 20
5.5.3 ) Å d‚†þï£E„Ý 'X ············································ 20
III

4. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
5.5.4 ) Ï d‚†Û¹ÅÄÇ 'X ·············································· 21
5.5.5 ) Ï d‚†Ãºx|Ç 'X ·············································· 22
6 o(†™5óŠÐ"··············································································· 22
ë•©z ······························································································· 23
` ···································································································· 24
IV

5. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
1 XØ
1.1 ©?Ä7{0
©?Ä7§½¡( zÄ7§´˜«3 Oƒ¥KÜ Å Vg†m‡ƒ M#.Ä7
¬"Ù O|©/ÏLéÄ7Ý]Âà © §òÄ7° ©•ü?"©O´aqÅ `k?
° †‘km ºx?° "1Ä7ò ÜÄ7] Ý]u ½ ºx] §`k?° •k
½|Ç ½Âçºx?° ›ÃK5g1Ä7Ý]¤ |G`k? E •{] "
g2007cIÝaÕu1 1˜|©?Ä7))aÕ©?–8§·I©?Ä7½|5 ±Y*
Œ" Ž2015c3 30F§ ½| k92|©?Ä7?u ´G §2015c Fþ¤ 7 ˆ
89.57· [1]
"†dÓž§þ ¤=òþ½˜1©?Ä7±‹» c ¤˜[ÕŒ Û¡§¿
JøÃXT+0×£ ••`þ ´^‡§ùæò•?˜ÚJ,©?Ä7Š•Ý]I áÚ
姕ٙ5 uÐFŒJø|±"
p*·I©?Ä7 ¬ uÐ{¤§UÙ ´á5þ «OŒNþŒ±8•n“"Ù¥1˜
“ ¬•µ4ªÄ7¶1 “ ¬U•m˜ª$1¿Ú½ÏòŽ¶1n“ ¬•Ø½Ïò
Ž m˜ªÄ7"
1.2 ykïÄnã
Š‘Ä7 ¬ M#§Æ.±9’.éu©?Ä7 ïÄY²•ÅÚJ," •{ü
ä•{ Ê [2]
3Ù©Ù5¥I©?Ä7½d•{ïÄ6±9=§[3]
35é/† ¦•ê! ©
ù ..Ä70 ^± O†½dïÄ6k¤ 9"1u[4]
3Ù2010 c ©Ù5˜a•ê©?
Ä7 ½d .9ÙmÇ ©Û6¥ éaÕaÚÄ7UìÄ7ÀŠ?uØÓ«m UCÙÂ
é ª§JÑ ^Black − Scholes .©Û"ê½°[5]
35¥I©?Ä7 O†½d•{ï
Ä6¥Ó $^Black − Scholes .§^îªÏ |Ü5 [dÏ"Ä7ÀŠØÓ¤‘5 ØÓ
|é •{‰ˆ° ‘5 ½kÂÃ"ë?ì[6]
3Ù2014 c ïÄ5¥I½|˜a•ê. ¦
©?Ä7½d•{ïÄ6¥˜t Black − Scholes . b §ÏLGARCH .Ú‘ÅÅÄ
Ç .¿•ª¦^Heston .?1¦)§J, nØ)†y¢œ¹ bÜÝ" é1n“©
?Ä7 ؽÏòŽ§yI—[7]
35©?Ä7½d•{ïÄ6¥JÑ^>9” æNÏ [ù
˜Å›"êŠ [½d• 2• ¦^u©?Ä7 +•§•)ëŒg[8]
2014c Ø©±9Á÷
S[9]
5¥I©?Ä7 ½dïÄ6"
–d§Ì‡ 'u©?Ä7½d ?Ø8¥uéuÙÀŠ ½dþ" ¯¢þ§½|þ¢S
´ ©?Ä7~„•3±Y-½ òÄdy–§ …ù«y–vk‘žmí£~f ª³"ƒ
ñ¸[10]
35©?Ä7 Âà Š•{ïÄ6ƒ¥¿£ ù˜¯K§ Ù?Øf}뎧™‰
ѽþ©Û"
8Bo(˜eyk ïÄg´§ŒNþÒ´òÄ7 Ü] w‰˜‡ N¿b Ù Ý
]uºx½|§Ä7o] CÄÑldºx½|‘Å$Ĥ°Ä ˜‡‘ÅL§"3ù‡‘Å
L§eUìÄ7 ¬ © KéÄ7o] ?1© "Ù?Ø Ø%•§(½3Ä7o] Œ
Uˆ ˆ«êŠe§Ä7ˆ ¤U© ÂÃ"¾wƒe§ù ©Ûq ÜœÜn§ ¯
¢þý Ý]L©?Ä7 Ý]öѬ²x§ù ©Û Ñ (Ø3¢SöŠ¥´éJ±A
24 • 1 1 •

6. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
^ "©?Ä7 ˜‡wÍA Ò´ÙòÄd •3"¯¢þ§©?Ä7`k° ~còd ´§
ºx° q•ÏÄd ´§ù3 ©é©?Ä7 ?˜Ú ãžk••[ 0 "òÄd •3
û½ éu˜‡`k° Ý]ö§Ùï¤ ´$uÙ ï] ÀŠ §ÙñÑd‚•´$uÙ
¤±] ÀŠ "3ù«œ¹e§ÏLOŽÄ7o] © 5 /nØdŠ0¿vk¢S¿
§ϕٴ،¢y "Þ~5`§˜‡`k° Ý]ö3½|œXpÞ!ºx Ð4rž±
pòdï`k° ¿3½|™•5àO!ºx r ž±$òd$–ÄdñѤ± `k°
"3DÚ ©Ûµee§duÙ±kÏS¿™?1L`k° ½Âé §ÙnØÀŠvk
u)UC§ÏdDÚ ©ÛµeÃ{)ºù‡Ý]ö ÂÃ" ¯¢þ§ù« ´üÑ ¡•/©
?AÓÄ0§®²•ˆŒÅ Ý]ö¤2•¦^§e© êŠ [Ü©Òù˜:Jø ˜‡äN
~f"
1.3 ©M#ƒ?
ØÓuyk ©z§ ©¤?Ø ©?Ä7½d§3uâ»þã ykØv§éÙ ´d‚
?1½d§½=édÙ½dLyÑ ´dŠ šÄ7ÀŠ?1½d§ù ½dâŒUk
• ¢S ´ Uå"‡‰ ù˜:§Ó I‡ë•é©?Ä7ˆ° òÄd ïÄ" ïÄ©
?Ä7òÄd§Äk‡²( ˜‡¯KÒ´`k° !ºx° Ú NòÄdƒm 'X" ê
²[11]
3Ù2010 c ïÄ5ETF Ä7@|Ŭ½d6¥•Ñ/¢y(JL²3 Ïm£2010 c2
4 F 2010 c2 23 F¤S§Šâ? Black − Scholes .ŽÑU100 ° ´ •@|z
°Ï dŠ•0.00002 §=3˜cS|^@|ŬŒ Œ±? ÂÕ20 0[11]
"ddα
wѧdu@|ö •3§©?Ä7 NòÄd´é §Œ±@•Cq•""?˜ÚŠâ©?
Ä7 é=†Å›Œ± (ا©?Ä7`k° Úºx° òÄdƒm•3pƒ@Ø
J§ÏÙü° òÄdƒÚAŒ— u""ù ˜5§e˜‡‡?Ø ¯K´`k° †ºx
° òÄdƒm´Xû½X ¯K"'uù˜:§êf[12]
35©?Ä7þ½° òÄd¯Kï
Ä6¥ Ñ(Ø@•´`k° òÄdÇû½ ºx° òÄdÇ" ‘ý•[13]
35©?Ä
7AB ° mòÄdÇ‹l•êÂÃÇÅÄ'X ¢yïÄ6¥|^õ GARCH .ïÄòÄd
dž‹l•êÅăm 'X `k° ƒé••-½§ ºx° Ké•êÅÄ••¯a
(ؽ†êf (جÜ"–u`k° òÄdK•Ïƒ§yk ©ż´ C[14]
ÏL¦
^¡†êâ£8 .?Ø K•òÄdÇʌσ 5·I©?Ä7òÄdÇK•Ïƒ ¢yï
Ä6±9 2+[15]
56Ä5éµ4ªÄ7òÄdK• Œëê£8©Û6"
24 • 1 2 •

7. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
2 ÎÒ† ½
2.1 . Ä b
• y?Ø î>5§y²( ©?Ø ©?Ä7´•µÄ7•YÏ•[Y §Ä7½|©a
• ¦.§+n•ª• Ä•ê.§ ©'~•1:1 §•¹ é=†Å› §•¹ ؽÏò
ŽÅ›…þòzŠ•1Ä7ÀŠ‡L½ˆ 1.5§eòzŠ•ºx° ë•ÀŠ$u½ u0.25 §
•¹ zc½ÏòŽÅ› §1Ä7° =ŒUÀŠ ×£ §©?° =ŒU½d|S ´
¥Iú5©?Ä7 ¬"
2.2 {¡†PÒ ½
Šâ½|±9êÆþéPÒ ¦^S.§ ©‰ÑXe{¡†PÒ ½µ
(1) ´ö:3 ©•@ ±kÄ7° ±Ï3˜‡$Š±c9±þ§ šÏòÄd@|!ºx°
ܿף I¦ ï©?Ä7° Ý]ö
(2) ´dŠ:•©?Ä7° é ´ö dŠ
(3) 1Ä:©?Ä71Ä7
(4) A° :©?Ä7`k° ¶ƒA/§`k° ½|d‚PŠAd‚¶`k° ÀŠP
ŠAÀŠ§Ù¦aÓ
(5) B° :©?Ä7ºx° ¶ƒA/§ºx° ½|d‚PŠBd‚¶ºx° ÀŠP
ŠBÀŠ§Ù¦aÓ
(6) t : cžm§3 ©¥±c•ü "
(7) T : Ä7 ˜‡$Š±c§t = 0•˜‡$Š±c m©§t = T•˜‡$Š±c (å§OŽ
Ü©PT = 1
(8) NVSUM :˜°©?Ä71Ä7ÀŠ
(9) NVSUM :©?Ä7eòzŠ
(10) NVA :˜°©?Ä7`k° ë•ÀŠ
(11) NVB :˜°©?Ä7ºx° ë•ÀŠ
(12) PA :˜°©?Ä7`k° ½|¤ d
(13) PB :˜°©?Ä7ºx° ½|¤ d
(14) VA :˜°©?Ä7`k° ´dŠ
(15) VB :˜°©?Ä7ºx° ´dŠ
24 • 1 3 •

8. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
(16) rR :Å ½|‡¦£ ǧ3 ©¥•éA° ‡¦£ ǧŠâ{¤² Cq Óu ´
¤AA?è’Å £ Ç
(17) rA :A° ½ÂÃÇ
(18) rI :3 cPAÚrAeA° Û¹ÂÃÇ
(19) r :úx|Ç
(20) dpA : A° òÄdǧdpA = PA
NVA
− 1 ,dpA• “LÄd§•K“Lòd
(21) dpB : B° òÄdǧdpB = PB
NVB
− 1 ,dpB• “LÄd§•K“Lòd
(22) S : ¦§3d?@•´1Ä7¤Ý] ¦|Ü
(23) V :Ï dŠ
(24) π : ï Ý]|ÜdŠ
(25) δ : ï Ý]|Ü¥ ¦ ±kêþ
24 • 1 4 •

9. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
3 ¯K©Û
3.1 é©?Ä79ÙÝ]dŠ ?˜Ú ã
©?Ä71Ä73 Ÿþ•˜‡ ] ¦.Ý]Ä7§`k° Ÿþ´˜«AÏ Å
ºx° KŒ±À•‘km ¦.Ä7"²(ù Ÿá5 Ä7Ü ^±=•µ1Ä7
ò ÜÄ7] Ý]u ½ ºx] §`k?° •k ½|Ç ½Âçºx?° ›
ÃK5g1Ä7Ý]¤ |G`k? E •{] "Ødƒ §,kn‡^±¬K• ·‚
é©?Ä7ˆ° ½d§§‚©O´ é=†Å›§½ÏòŽÅ›ÚؽÏòŽÅ›"
3.1.1 é=†Å›
©?Ä7 é=†Å›§Ù á8 3uïáåºx° †`k° m éXÓž››
NÄ7òÄdÇ3˜‡Ün ‰ŒS"Ù^±˜„ ½•µ3m½ ´Ïm§Ý]öŒ±3 ?
½|©OU½d ©?Ä7`k° †ºx° §¿‡¦÷+ ûU ½ ©'~£ ©¥
P•1:1§=˜°ºx° ˜°`k° òŽ•ü°1Ä7° ¤òÝ]ö¤±k ºx°
†`k° =†•1Ä7° "±k1Ä7 Ý]öŒ±‡¦Ä7úiUÙzFúÙ 1Ä7À
ŠNVSUM ×£1Ä7° "‡ƒ§Ý]ö•Œ±UÄ7úizFúÙ 1Ä7ÀŠNVSUM 1
Ä7° §¿‡¦÷+ ûU ½ ©'~òÙ¤±1Ä7° ©•`k° †ºx° "Ù
Ý]öBŒ©O3éA½|Uˆ° ´d?1 ´"
é=†Å› •3Jø ˜‡@|± §AO5¿ ©?Ä7 ^± ½=k1Ä7Œ±
UÀŠ ק٦° •U3 ?½| ´§Ï ©?Ä7,˜° ?½| ´dƒ'u
Ùë•ÀŠu)wÍòÄd ,˜° ™CĽCÄØŒž§Œ±ÏL° Ü¿ ×£½
° ©?1@|"ù˜^± •3Cq yeª¤áµ
PA + PB ≈ NVSUM (3.1)
5¿ © ½©?Ä7 ©'~•1µ1,=2 ∗ NVSUM = NVA + NVB , 5¿ òÄdÇ ½
µdpA = PA
NVA
− 1, dpB = PB
NVB
− 1Ï 3·‚ ½ ©'~eqkµ
dpA + dpB ≈ 0 (3.2)
3.1.2 ½ÏòŽÅ›
½ÏòŽ ˜8 ••`k° ±k<© Ù c ½|E§ÙäNöŠ•µ3z‡$Š±
c" ½ÏòŽ¥§z°`k° •Ù±k<† ˜°e˜±c ÀŠ•1 #`k° ±9À
Šƒ uNVA − 1 1Ä7[16]
"AO5¿ ´§ýŒõêƒc ©zò½ÏòŽÅ›{ü w‰
ÏGE§ù ?n¢Sþ¿ØÜn"Ï•`k° •Ï3 ?½|òd ´§Ï Œ±g,/
b Œõê`k° Ý]öÑ´òdï § zc ÏGEž%´±Ãòd 1Ä7/ª|G§
Ï `k° Û¹ÂÃÇ´puÙ ½ÂÃÇ "
½ÏòŽØK•ºx° Ý]ö ° êþ½dŠ"
24 • 1 5 •

10. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
3.1.3 ؽÏòŽÅ›
ؽÏòŽÅ› ˜8 ´ oˆ° Ý]ö Ã"ù˜:Œ±©•üÜ©w–§©Oé
AؽÏòŽÅ›¥ •þòŽÚ•eòŽ§•[Œë„Ä7 Æ[16]
"
•þòŽ´• oºx° Ý]ö çϕºx° éÝ]ö Ý]dŠÌ‡Ny3
Ùmá5þ§Ø”ëw2uy uÐïÄ¥% n eL[17]
µ
L 1: c½|þŒ¼ m'
©?Ä7 K]K Ï •ÏÀ
놀€ ü) ×Ê ±þ Ê› ±þ] Ê› ±þ] Ê› ±þ]
K]¤ `k° ½Âç $ K]8.6%§K 10.6% $ $
mI ½|•ê9º‚•ê üKI ETF nŒ •
m ê Щ•ü ˜ ü › Ê 8
lL¥·‚ØJwѧy3½|þˆ«Œ±¼ m 廥§·Ü¥ Ñr Ä =k©
?Ä7˜«§ ù•¤• ©?Ä7ºx° Ø% ´dŠ¿)º `k° ~cÄd ´
Ï" ŠâЩ ©'~1 : 1§Ð© `k° m•ü §ƒ u`k° Ý]ö z°Ý]
`k° Ý]ö ˜ K]" e½| Ч©?Ä7ÀŠ±YþÞ§džºx° Ý]ö
ÃÓ Ä7] '~ò ‡50%§@éu`k° Ý]ö ó§ÙÄd ï `k° m
'Çeü§Ý]dŠeü"Ï •‘±ºx° áÚå§Ú•þòŽÅ›[16]
§= 1Ä7À
ŠNVSUM ≥ 1.5ž§òNVB‡LNVA Ü©±1Ä7/ª|G‰ºx° Ý]ö§ yþò `
k° †ºx° ©'~£ 1 : 1§m£ ü "
•eòŽK´• o`k° Ý]ö Ã" ½|1œØ|ž§Ä7ÀŠëYº›§e
dª³±Y§NVB¬„eO†–•0§d `k° Ý]ö ES òÃ{ æ"• o
`k° Ý]öØÉÄ7Ý]›” K•§ ½ NVB ≤ 0.25ž§?1•eòŽ[16]
"òNVA‡
LNVB Ü©±1Ä7/ª|G‰`k° Ý]ö§ yeò `k° †ºx° ©'~£
1 : 1§m£ ü "d?•I‡AOJž5¿µ†½ÏòŽaq§Ø ƒc ïÄéeòÂ
à Ovk•Ä`k° •Ïòd ´ ¯¢§==reòwŠ˜‡ y ES öŠ5§"
¢Sþ§eò´¬•`k° Ý]ö‘5Âà §Ï•Ùƒ u•`k° Ý]öJø ˜‡
Ü©Ý]Œ±±ÃòdòÑ Å¬§Ï ÙÂà u
(NVA − PA) ∗ (NVA − NVB) (3.3)
§^òÄdÇL«,¿•Ä eò>u^‡•NVB ≤ 0.25§þª=•
−dpA ∗ NVA ∗ (NVA − 0.25) (3.4)
3.2 ©¦^ ½dg´†1©óä
dcã©Û§·‚Œ±ÚÑ © ½dg´"
24 • 1 6 •

11. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
òºx° †`k° ©m?n"é`k° §•âÙ^± OòÙdŠ©)•Ùzc ½
ÂÑ5 Å5dŠÚduؽÏòŽÅ›‘5 Û¹wOÏ dŠ§ò öƒ `k°
dŠ"2•â é=†5K§d`k° (J½dºx° "
e5 ©òÄkÏLºx¥5½dڑŇ©•§ •{©O‰Ñ©?Ä7`k° Úº
x° nؽd"¿ÏLêŠ [ •{5(½ .¥ ëê±9©ÛnØd‚†¢Sd‚ƒ
m 'X"
24 • 1 7 •

12. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
4 ©?Ä7 nؽd
VAdnÜ©|¤µ©O•dÙ ½ÂÑ5 Å dŠ(P•Vbond)§dؽÏòŽÅ›‘5
Û¹wOÏ dŠ(P•Vput)±9d é=†Å›‘5 é ´dŠ"Ù¥ é ´dŠ=
B° u)ŒÌòd½Äd=dpB ýéŠéŒžâ¬u)§·‚3d6žØ•Äù˜‘éVA
K•"Ïdk:
VA = Vbond + Vput (4.1)
4.1 `k° Å dŠ½d
`k° Å dŠ5gu ½Âç~„ ÂÃ/ª•3Õ1˜cÏ•±|ÇÄ:þO
nz‡Ä:§UcGE"GE•ª•§3½ÏòŽF§òƒ u cÝ|E 7 ± 1Ä7
/ª|G‰`k° Ý]ö§Ø€„ 7"3d·‚•Ä˜‡ålþgGE®L tžm `k°
"
Š 5¿ ´§·‚•Ä ©?`k° þ•[Y. "3Xd• žmƒS§ b |Ç
Y²ØCÄw,´ØÜn " ·‚òrA ˜•~ê"ù̇´•Ä rA ½•ª•3˜‡$
1±c ÐF§Šâ 1úÙ û’Õ1•±ÄO|Çþ2˜½ÌÝ(½§Ï XJòrA À•C
þ?1ï {§·‚¢Sþ´3éÄO|Ç?1ï " …ÙžmªÝ镧O( `·‚´
3éÄO|DZ˜c•m… lÑœ¹?1ï "ù«œ¹e§¦^CIR½´Hull − White ‘
Å|Ç .¿ØÜ·§¦^Oþ²LÆ |Çýÿ .qŒŒ‡Ñ © ?؉Œ§Ï ·‚
…òÙÀŠ~ê"•Ä rA– 3˜cžmƒS´ØC §ù˜b •k˜½ Ün5"
·‚òz˜ž•t ‡¦ÂÃÇrR§½=Å òy|ÇÀ•‘ÅCþ§òrR(t) À•‘Å
L§§¿b½ÙCÄdHull − White .•x"ù˜b •â3u§rRŠ•½| Ï"ÂÃǧ
w,•˜‡‘ÅCþ"¯¢þ§•Ä ©?Ä7`k° Ý]ö+Ṅ•x] Å Ý]ö§
†Å ½|˜—"Ï ·‚Œ±@•ù Ý]ö3?1Ý]ûüž§ ¬ò`k° †IÅ!
&^Å Å ˜3˜å' "ù ˜5§·‚Òkndb §Ý]öéu©?Ä7`k° Ï
"ÂÃdžŠ½| Ï"ÂÃÇ´˜— "¤±·‚¦^|ǽ|þ~^ ‘Å|Ç .£
ãrR(t)•Ò´Y ±¤ "
Vbond(t, rR(t)) = lim
n→∞
n
k=1
rA
k
t
rR(t)dt
(4.2)
ŠâÅ ½d ˜„•{§·‚¦^"EÅ 5E›`k° ÂçP˜‡rR(t)e §3T ž
•|G˜ "EÅ 3tž• dŠ•B(t, T)§KþªC•µ
Vbond(t, rR(t)) = lim
n→∞
n
k=1
rAB(t, k) (4.3)
Ï e5 ¯KC••Ä"EÅ dŠB(t, T)"Äk§·‚Ú~^ ‘Å|Ç ."
24 • 1 8 •

13. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
4.1.1 ‘Å|Ç .
|ÇL§˜„Œ±A^±e•§•x[18]
:
dR(t) = m(R(t), t)dt + σ(R(t), t)dW(t) (4.4)
Ù¥m(·), σ(·)•ü(½¼ê§{W(t) : t 0}•ÙK$Ä"|ÇL§ kþŠ£Eª³§=‘X
žmí£§|Ç¥yÑ3˜‡þŠNCþe{Ä ª³"~^ .kV asicek .[19]
!CIR
.[20]
±9Hull − White .[21]
"·‚3ùpÀ^ |Ç .´Hull − White .§ ˜m(·) X
eµ
m(R(t), t) = a(t) − b(t)R(t) (4.5)
ùpa(t), b(t)þ•‰½ Š¼ê"
dR(t) = (a(t) − b(t)R(t))dt + σ(t)dW (t) (4.6)
Ù¥a(t), b(t), σþ•š‘Å Š¼ê§W (t)•Å ½|þºx¥5ÿÝe ÙK$Ä"
duR(t)•‘Ň©•§dR(t) = (a(t)−b(t)R(t))dt+σ(t)dW (t) )§d5Stochastic Calculus
for Finance6 ½n6.3.2 [22]
•§R(t) •ê ‰ÅL§" ·‚¦)B(t, T)ž§·‚F" ÑÙ
÷v ‡©•§"
• B(t, T)§džT•½Š§B(t, T)d‘ÅL§ ´»R(s), t ≤ s ≤ T†Cþt (½" •Ä
R(t)•ê ‰ÅL§§Ï 3R(s), s ≤ t¥§k^ &E=•R(t)"nþ¤ã§·‚Œ± Ñ:
B(t, T) = f(t, R(t)) (4.7)
½ÂD(t)•byL§§÷vµ
D(t) = e− t
0
R(s)ds
(4.8)
·‚5¿ D(t)B(t, T) = D(t)f(t, R(t))•˜‡ L§§A^IT ˆOúª¦)Ù‡©/ª§ µ
d(D(t)f(t, R(t))) = f(t, R(t))dD(t) + D(t)df(t, R(t))
= −fDRdt + D(ftdt + frdR +
1
2
frrdRdR)
= D(t)[−Rfdt + ftdt + frdR +
1
2
frrdRdR]
= D(t)[−Rf + ft + (a(t) − b(t)R(t))fr +
1
2
σ2
(t)frr]dt + D(t)σ(t)frdW (t)
du·‚®•D(t)B(t, T) = D(t)f(t, R(t))´˜‡ L§§Ï Ùvkþ,†eüª³§=þ
ªdt‘ XêA •""dd·‚ f(t, R(t))·Ü •§•µ
ft(t, r) + (a(t) − b(t)r)fr(t, r) +
1
2
σ2
(t)frr(t, r) = rf(t, r) (4.9)
Ù>.^‡•µ
f(T, r) = 1, ∀r ∈ [0, ∞)
24 • 1 9 •

14. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
4.1.2 ‘Ň©•§¦)
e 5 · ‚ w « ¦ Ñ þ ã ‡ © • § )" ß ÿ ) / ª •f(t, r) = e−rC(t,T )−A(t,T )
,Ù
¥C(t, T), A(t, T)•[0, T]þ –½¼ê"dž·‚ µ



ft(t, r) = (−rC (t, T) − A (t, T))f(t, r)
fr(t, r) = −C(t, T)f(t, r)
frr(t, r) = C2
(t, T)f(t, r)
(4.10)
5¿ Tdž•½Š§ C (t, T)•C(t, T)ét ê§A (t, T)Ó"ò±þnª“•§ft(t, r) +
(a(t) − b(t)r)fr(t, r) + 1
2
σ2
(t)frr(t, r) = rf(t, r)§ µ
[(−C (t, T) + b(t)C(t, T) − 1)r − A (t, T) − a(t)C(t, T) +
1
2
σ2
tC2
(t, T)]f(t, r) = 0 (4.11)
dªé¤krþA ¤á§ r XêA •"§=µ
C (t, T) = b(t)C(t, T) − 1 (4.12)
“þª§ µ
A (t, T) = −a(t)C(t, T) +
1
2
σ2
tC2
(t, T) (4.13)
X•Ä •§ >.^‡f(T, r) = 1, ∀r§ A(t, T), C(t, T) >.^‡



C(T, T) = 0
A(T, T) = 0
ù ·‚ÒŒ±) µ



C(t, T) =
T
t
e− s
t
b(v)dv
ds
A(t, T) =
T
t
(a(s)C(s, T) −
1
2
σ2
tC2
(s, T))ds
(4.14)
•Ò´`§dž·‚)
Vbond(t, T) = B(t, T)e−R(t)C(t,T )−A(t,T )
(4.15)
Ù¥C(t, T), A(t, T)dþª‰Ñ"
4.2 `k° Ï dŠ½d
`k° Û¹wOÏ 5guÙؽÏòŽÅ›¥ •eòŽ§±e{¡eò§eòÌ
‡´• o`k° Ý]ö 7ج›”"Ù^± ˜X3.1.3!‰Ñ"du1Ä7 Ý]u
½§Ø”b ºx° ÀŠNVSUM CÄ„Ì ¦½|þ ÙK$ħ¿b eò>u^‡
•NVSUM ≤ NVSUM †NVB ≤ 0.25˜—"Š 5¿ ´d?ØU ˜NVB•ÙK$ħÏÙkX
Ø-½ m" ˜„Ä7 eò^‡dNVB‰Ñ§d?òÙ=†•NVSUM ‰Ñ§ù̇´Ñu
{z¯K •Ä Ñ `k° |Ç È§Ø,q‡Ú˜‡t Cþ"
dNVSUM
NVSUM
= rdt + σdW(t) (4.16)
e5·‚•ìBlack − Scholes•§ ï•{[23]
éNVSUM ïᇩ•§"
24 • 1 10 •

15. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
4.2.1 ‘Ň©•§ Ñ
• {zPÒ§3±eí ¥kPNVSUM •S§PÏ dŠ•V" ï|Ü•˜°V õÞ
∆°S˜Þ§ |ÜdŠ3tž• Lˆªµ
Πt = Vt − ∆St (4.17)
À ∆¦T|Ü3žmã(t, t + δt)S•Ãºx|ܧK3Ã@| Ke|Ü Ý]£ Ç ÓuÃ
ºx|Çr"
Πt+dt − Πt
Πt
= rdt (4.18)
“ Lˆª§ µ
dVt − ∆tdSt = rΠtdt (4.19)
Óž§A^IT ˆOúª µ
dVt = (
∂V
∂t
+
1
2
σ2
S2 ∂2
V
∂S2
+ rS
∂V
∂S
)dt + σS
∂V
∂S
dWt (4.20)
“þª§¿5¿ dS
S
= rdt + σd ˜W(t)§
(
∂V
∂t
+
1
2
σ2
S2 ∂2
V
∂S2
+ rS
∂V
∂S
)dt − ∆(rV dt + V σd ˜W(t)) + σS
∂V
∂S
dWt = rdt (4.21)
Šâ·‚ b §T|ܴúx §=dWt Xê•"§ µ
∆ =
∂V
∂S
(4.22)
“£þª •§µ
∂V
∂t
+
1
2
σ2
S2 ∂2
V
∂S2
+ rNVSUM
∂V
∂S
− rV = 0 (4.23)
2•Ä½)^‡µ
S>9eòzŠž§Ï á=‰1§z°Ï ¯k(½ ÂÃ0.75 ∗ (NVA − PA)"d
?·‚ Ñ ˜‡±ÏƒSeòŒUõgu) œ¹§ÏÙ4Ù¾„"
>u^‡µV (NVSUM , t) = 0.75 ∗ (NVA − PA)
e˜† Ý]Ï"T§Ï Ñvk ‰1§KÙdŠ•""
>Š^‡µV (NVSUM , T) = 0
òV †£V (NVSUM , t)§dd Ï dŠV (NVSUM , t)÷v •§



∂V
∂t
+
1
2
σ2
NVSUM
2 ∂2
V
∂NVSUM
2 + rNVSUM
∂V
∂NVSUM
− rV = 0
>Š^‡µV (NVSUM , t) = 0.75 ∗ (NVA − PA)
ªŠ^‡µV (NVSUM , T) = 0
(4.24)
24 • 1 11 •

16. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
4.2.2 ‘Ň©•§¦)
• {zPÒ§3±eí ¥kPNVSUM •S§PÏ dŠ•V"
ÄkŠCþ“†x = ln( S
S
)§ µ
∂V
∂S
=
∂V
∂x
1
S
(4.25)
∂2
V
∂S2
=
∂2
V
∂x2
1
S2
−
∂V
∂x
1
S2
(4.26)
|^±þ(J§ •§C•µ



∂V
∂t
+
1
2
σ2 ∂2
V
∂x2
+ (r −
1
2
σ2
)
∂V
∂x
− rV = 0
>Š^‡µV (0, t) = 0.75 ∗ (NVA − PA)
ªŠ^‡µV (x, T) = 0
(4.27)
2‰C†§-u = V − 0.75 ∗ (NVA − PA) •§C•µ



∂u
∂t
+
1
2
σ2 ∂2
u
∂x2
+ (r −
1
2
σ2
)
∂u
∂x
− r(u + 0.75 ∗ (NVA − PA)) = 0
>Š^‡µu(0, t) = 0
ªŠ^‡µu(x, T) = −0.75 ∗ (NVA − PA)
(4.28)
e5§ÏL¼êC†ò•§†•9D •§/ª§-µ
u = V eαt+βx
(4.29)
Kkµ
ut = eαt+βx
[Vt + αV ]
ux = eαt+βx
[Vx + βV ]
uxx = eαt+βx
[Vxx + 2βVx + β2
V ]
“ •§§ µ
eαt+βx
[Vt+αV ]+
1
2
σ2
eαt+βx
[Vxx+2βVx+β2
V ]+(r−
1
2
σ2
)eαt+βx
[Vx+βV ]−r(V eαt+βx
+0.75∗(NVA−PA)) = 0
(4.30)
z{ n§ü>ž eαt+βx
µ
Vt +(α+
1
2
σ2
β2
+(r −
1
2
σ2
)β −r)V +
1
2
σ2
Vxx +(βσ2
+(r −
1
2
σ2
))Vx =
0.75r ∗ (NVA − PA)
eαt+βx
(4.31)
‡¦Vx†V c Xê•"§ α§β÷v •§µ



α +
1
2
σ2
β2
+ (r −
1
2
σ2
)β − r = 0
βσ2
+ (r −
1
2
σ2
) = 0
(4.32)
24 • 1 12 •

17. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
) α§β" 


α = r +
1
2
(r − 1
2
σ2
)2
σ2
β = −
(r − 1
2
σ2
)
σ2
(4.33)
•§z•µ 


∂V
∂t
+
1
2
σ2 ∂2
V
∂x2
=
0.75r ∗ (NVA − PA)
eαt+βx
>Š^‡µV (0, t) = 0
ªŠ^‡µV (x, T) = ψ(x) =
−0.75 ∗ (NVA − PA)
eαT +βx
(4.34)
5¿ x = ln( S
S
)§ x ≥ 0o¤á"éψ(x)?1òÿµ
φ(x) =



ψ(x) x ≥ 0
−ψ(−x) x < 0
KV (x)3«•Ω = (x, t)| − ∞ < x < ∞, 0 ≤ t < Tþ÷v±e•§µ



∂V
∂t
+
1
2
σ2 ∂2
V
∂x2
=
0.75r ∗ (NVA − PA)
eαt+βx
V (x, T) = φ(x)
(4.35)
‰Cþ“†τ = T − t, µ



∂V
∂τ
−
1
2
σ2 ∂2
V
∂x2
= −
0.75r ∗ (NVA − PA)
eα(T −τ)+βx
V (x, 0) = φ(x)
(4.36)
dPossionúª[24]
µ
V (x, τ) =
∞
−∞
K(x − ξ, τ)φ(ξ)dξ +
τ
0
dυ
∞
−∞
K(x − ξ, τ − υ)
0.75r ∗ (NVA − PA)
eα(T −υ)+βξ
dξ (4.37)
Ù¥§K(x, t)•9D •§ Ä )§“a =
√
2
2
σµ
K(x, t) =



1
σ
√
2πt
e
−x2
2σ2t t > 0
0 t ≤ 0
(4.38)
• ò¤kƒc Cþ“†“£§ ©?Ä7`k° Û¹Ï nØ):
Vput =
V (ln( S
S
), T − t)
eα(T −t)+βx
+ 0.75 ∗ (NVA − PA) (4.39)
Ù¥V (ln( S
S
), τ)•þã9D •§ )"
4.3 ºx° ½d
3(½ `k° ÜndŠVAƒ §·‚•Ä¦^ é=†Å›5ïáºx° †`k°
m éX"Šâƒc ?ا·‚• 3@|ö•3 œ¹e§eªCq¤áµ
PA + PB ≈ NVSUM (4.40)
24 • 1 13 •

18. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
y3·‚‡(½ ´ˆ° Ün ´dŠ§Ï Œ±@•/ ( 0PAÒ´VA§/ ( 0
PBÒ´VB" 1Ä7ÀŠNVSUM ´zFúÙ §Ï Ò µ
VB = NVSUM − VA (4.41)
I‡5¿ ´§ºx° ØÓu`k° §Ù ´š~¹ …Ý]ö¥±ÝÅ•8 ÓýŒ
õê"3ù«œ¹e§˜‡/ ( 0 ´dŠ™7äk• ¿Â§ý¢ ´d‚É ½|œX†
]7¡G¹ K•¬ —Ù lnØdŠ"ùÒÐ'·‚Áã‰Ñ ¦ ½d§3Ä ¡©Û
ÙSºdŠƒ §„‡(ܽ| ŠY²!œX Ãõσ±? "¯¢þ§©?Ä7ºx°
´‘kØ(½5m ºx] §Ùɽ|ÅÄ K• Œu ¦§ÙØ(½5• Œu ¦"
3ù«œ¹e§‡ÏLOŽ˜‡nØd‚5• ´§JÝéŒ"
ƒé ó§`k° duÙÝ]öõ•n5 Å Ý]ö…Ù¤ä Å 5Ÿ§éuÙ/
( 0 ´d‚ ?Øw •k¿Â"
24 • 1 14 •

19. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
5 ©?Ä7 „Akâ [½d
3 !¥·‚2¦^„Akâ [ •{?1êŠ [½d[25]
¿†1ÊÜ©nؽd (J
?1' "
5.1 „Akâ [ n
¦^ [•{é7K] ?1½d•@©u¤|;· ž [26][27]
31977 c }Á"Ùg´•3
®•ÅÄL§ œ¹e§¦^OŽÅ‘Å)¤Ñl .‡¦VÇ©Ù ‘Åê5 [7K]
üz´»§z˜g ´»Œ±À•˜gÄ ¿éAј‡] d‚§ -Eù‡L§ÒŒ±
7K] d‚ ˜‡VÇ©Ù§âd·‚ÒŒ±¦Ñ7K] Ï"d‚"
±e(J•¦^MATLABŠ•óä)¤ §Ù¥Ü©“è/• ë•Ö7[28]
"
5.2 „AkÛ [ L§
·‚é©?Ä7`k° ½d?1êŠ [§dc¡ ?Ø• µVA = Vbond + Vput§·‚
©OéÅ dŠÚÛ¹Ï dŠ?1êŠ [§¿•ÄÙƒp K•"
Äk ÑrR£±e{P•R¤†NVSUM £±e{P•S¤÷v ëYCzL§:



dSt
St
= rdt + σSdWt
dRt = (a(t) − b(t)Rt)dt + σRdWt
(5.1)
e5éžm«m[0, T]?1lÑz§ ©•N°§τ = T
N
= ∆t
0 = t0 < t1 < . . . < tN = T
éAS, R•S0, S1 . . . SN †R0, R1 . . . RN džlÑ CzL§•µ



Si+1 = Si(1 + riδt) + σSSi
√
δtεi
Ri+1 = Ri + (a(ti) − b(ti)ri)δt + σR
√
δtεi
(5.2)
Ù¥
εi, εi ∼ N(0, 1)
εi = ρεi + 1 − ρ2εi
εi†εi pƒÕá§εi†εidƒ'XêρéX"
24 • 1 15 •

20. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
5.3 „Akâ [ (J
3ùpЫ˜Ü„Akâ [)¤ ´»±éÙ•{‰?˜Ú`²§d?ëêÀ •µ
S0 = 1, r = 0.04, σS = 0.4, a(t) ≡ 0.16, b(t) ≡ 2, R0 = 0.08, ρ = −0.05S“Úê250 Ú§-
Egê5g§Ù¦ëê ˜† ©ƒcé©?Ä7b ƒÓ" d?éuÅÄÇ O„Ì
JohnHull[29]
3ÙÍŠÏ ÏÀ9Ù¦û) ¬¥¦^ •{"ã¥ùÚIÑ ´»•Ä7>u
eò œ¹§3ÂBeòzŠ ±ð u0 L«Ù´»"
ã 1: „Akâ [´»
5.4 †nØ) é'
5.4.1 Å dŠnØ)†êŠ)é'
d?ëêÀ •µS0 = 1, r = 0.04, rA = 0.06, σR = 0.1, a(t) ≡ 0.16, b(t) ≡ 2, R0 = 0.08, ρ =
0, t = 0, T = 1"dž) µ 


C(0, 1) =
1 − e(
− 2)
2
A(0, 1) = 0.08 ∗ (
1
2
+
1
2 ∗ e2
)
(5.3)
Ïdkµ
24 • 1 16 •

21. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
Vbond = Vbond(t, T) = B(t, T)e−R(t)C(t,T )−A(t,T )
= 1.06 ∗ e−0.08∗ 1−e−2
2 −0.08∗( 1
2 + 1
2∗e2 )
= 0.9785
dž „Akâ [(JXeµ
L 2: „Akâ [ Å d‚† [gê'X
[gê Å d‚
5 0.9816
10 0.9807
100 0.9802
500 0.9803
1000 0.9803
dþLŒ±wÑ„Akâ [{ (JÂñ§Ù(J•Vbond = 0.9803"
5.4.2 Ï d‚nØ)†êŠ)é'
d?ëêÀ •µNVSUM (0) = 1, r = 0.04, σS = 5, NVSUM = 0.625, ρ = 0, t = 0, T = 1"
±e¦^k• ©{Š•nØ){"¤¦) •§•µ



∂V
∂t
+
1
2
σ2
NVSUM
2 ∂2
V
∂NVSUM
2 + rNVSUM
∂V
∂NVSUM
− rV = 0
>u^‡µV (NVSUM , t) = 0.75 ∗ (NVA − PA)
>Š^‡µV (NVSUM , T) = 0
(5.4)
±e{PNVSUM •S,¦^ lÑ‚ªXeµ
∂V
∂t
=
Vi,j − Vi−1,j
δt
∂V
∂S
=
Vi,j+1 − Vi,j−1
2δS
∂2
V
∂S2
=
Vi,j+1 + Vi,j−1 − 2Vi,j
δS2
“•§§ µ
Vi,j − Vi−1,j
δt
+
1
2
σ2
j2
δS2 Vi,j+1 + Vi,j−1 − 2Vi,j
δS2
+ rjδS
Vi,j+1 − Vi,j−1
2δS
= rVi,j (5.5)
òdªUVi,j−1, Vi,j, Vi,j+1 n 4킪§ •Ý /ª ^MATLAB¦)"
24 • 1 17 •

22. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
Ù>Š^‡†ªŠ^‡•µ 


Vi,M = 0
Vi,S = 0.15
VN,j = 0
Ù¥µi = 0, 1, 2 · · · , N j = 0, 1, 2 · · · , M
À δS = 0.1, δt = 0.0001§|^MATLABOŽ §t = 0, S0 = 1ž§Vput = 0.0198" dž„
Akâ [3S“1000gž µVput = 0.0203"
ã 2: Û¹Ï d‚†Ð©d‚ 'X
þãЫ k• ©{¤ )-¡3t = 0ž• 㔧=) Û¹Ï d‚†Ð©d
‚S0ƒm 'X"
5.5 'u)5Ÿ ?Ø
5.5.1 ©(J†yk©z(J'
Xc©¤J L §yk ïÄÑ´ éÄ7ÀŠ?1ï §ù nØØUéÐ £ãÃ
X/©?AÓÄ0ù ´üÑ"e¡§·‚‰Ñ˜‡äN ~f"b ·‚•ÄXe ˜Å
½1œµk´ëYþÞ§, ´˜ã²-ϧ; XH5˜ÅŒO§Xe㤫"
24 • 1 18 •

23. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
ã 3: b½ ½r³ã
¯¢þ§±þb r³†·‚A ½|2015c Lyš~aq§·‚ e5' 3ù«œ¹
e © ©Û•{†éÀŠ½d •{ƒm «O"du±þ b r³™â»Ø½ÏòŽzŠ§
e¦^ÀŠ½d{§ùãžmS©?Ä7`À° dŠA •²-O• §Ù•˜ û½Ï
ƒ==´zFÈ ½Âà ®§Xeㆤ«" e¦^·‚ ´dŠ½d{Œ±wѧ
˜m©3 ½þ ž§©?Ä7`k° dŠ´eü §ù´Ï•Ú½¥u)eò AÇé$§
Ï `k° Sº wOÏ dŠeü"3‘ ²-Ï¥§·‚ .(J†ÀŠ½d .˜
—§`k° dŠO•==dÈ |E°Ä" ‘X½| O§Œ±wÑÛ¹Ï dŠá
=wyÑ5"ùÙ¢Ò´/©?AÓÄ0üÑ Sº§Ù3Ú½¥½|œXpÞƒž¦Å$dï
`k° §–½|u)_=žKpd È"·‚ .Œ± •O( £±ù˜üÑ", Š
5¿ ´§·‚ .O( Ы `k° òd ´§Œ±w du½|‡¦ÂÃÇ´pu
½Âà §Ï † ½|ŒOÚåÛ¹Ï dŠþ,ƒc§ . Ž `k° ´dŠÑ$
uÀŠ§ù†½|¢SƒÎ"
ã 4: ÀŠ½d† ´dŠ½d é'
24 • 1 19 •

24. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
e5•Ä A‡¯K•)) Å dŠVbond†þï|ǃm 'X§†þŠ£E„݃
m 'X±9) Û¹Ï dŠVput† ¦ÅÄDZ9þï|ǃm 'X"
5.5.2 ) Å d‚†þï|Ç 'X
ã 5: Å d‚†þï|Ç'X
eã´3 ½Ù¦Cþe§CÄHull − White .¥ þï|Ç)Ñ ˜X Å d‚"d
㌱wѧþï|Ç pžÅ dŠ $§ …¥y C‚5 'X"ù´Ï•Hull −White|
Ç .•x |ÇCÄ þŠ£Eª³§Ïdþï|ÇÒƒ u . [Ñ |DzþŠ" ·
‚ .¥´ ½ ½|ǧ ˜‡¦ÂÃÇ£byǤ•‘ÅCÄ§Ï ‡¦ÂÃÇ p§Å
dŠ $" [(J†nØí ƒÎÜ"
5.5.3 ) Å d‚†þï£E„Ý 'X
ù´3 ½Ù¦Cþe§CÄHull − White .¥ þï£E„Ý)Ñ ˜X [Å d
‚ IO "d㌱wѧþï£E„Ý ¯§ [ Å IO "ù´Ï•þŠ£E
„Ý ¯§‘Å|Ç ´»Ò¬3˜‡ «•SCħêg„Akâ [) Å d‚
m ÅÄ•¬ " [(J†nØí ƒÎ"
24 • 1 20 •

25. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
ã 6: [Å d‚ IO †þŠ£E„Ý'X
5.5.4 ) Ï d‚†Û¹ÅÄÇ 'X
ã 7: Ï d‚†ÅÄÇ 'X
dþãwÑ‘ÅÄÇþ,Ï dŠJ,§†nØ(J˜—" (ÜêŠ(J•Œ±wѧ Å
ÄÇ ž§©?Ä7Û¹Ï dŠ´š~ "¯¢þ·I ½ ~œ¹eUFÅÄÇ0.05O
Ž§czÅÄÇ=•0.8†m§džÛ¹Ï dŠA •""ù•Ò‡Nѧ ½|²-$1ž§
©?Ä7eòA ØŒUu)" ‘XÅÄÇþ,§Û¹Ï dŠ×„J,§ù•Ò)º •
Ÿo34à1œ§cÙ´ O1œž§©?Ä7`k° ¤•½|J, 9:"
24 • 1 21 •

26. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
5.5.5 ) Ï d‚†Ãºx|Ç 'X
ã 8: Ï d‚†Ãºx|Ç 'X
dþãwѧ [(Jw«Ï d‚†Ãºx|ÇKƒ'"·‚ƒc•ÑÛ¹Ï •wOÏ
§ [(J†nØ(J˜—"
6 o(†™5óŠÐ"
©é©?Ä7½d ¯K?1 &?§ò©?Ä7 `k° dŠ©)•ÙÅ dŠÚÛ
¹Ï dŠ"éÅ dŠ¦^Hull − White .?1ï ¿?1 ½dúªí ÚOŽ"éÏ
dŠ¦^Black − Scholes .?1ï §|^k• ©•{?1 nØd‚OŽ"• ¦^„A
kâ•{éÅ ÚÏ dŠ?1êŠ [¿†nØ)?1 ©Û' § (Jw« © ½
d•{Ä Îܽ|¢S†Ù¦nØí (Ø"oN5`§ ©ºX é ¦Ï †|Çû)
¬?1½d L§§•)µd ½d ní Ù¤÷v•§§•§)Û) $Ž§k• ©•{
$^±9„Akâ [§Ä ºX cÌ6 ˜ û)¬½d •{"
3™5 󊥧·@•k±eŒ±U?ƒ?"˜´3éÛ¹Ï ?1½d L§¥§Œ±
•ÄaL§" X ©(ؤ•Ñ §©?Ä7Û¹wOÏ ý NydŠ´3ÅÄì
½|¥§ ù ½| Š‘X-ŒžE uÙ ŒUÚud‚âC σ"Ï aL§
¬•Ï .•O( [ù ½|" ´3éÅ dŠ?1½dž§Œ±•Ä˜t ½ÂÃ
Ç•~Š b §= b zc ½ÂÃÇd‘=ÅÄ ‡¦ÂÃÇ3 ccÐ êŠþ˜
‡ºx Å "
24 • 1 22 •

27. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
ë•©z
[1] àv. uMÅ ©?A Ô wlÄ Vg Ý]üÑ. þ°: uMy , 2015.
[2] Ê . ¥I©?Ä7½d•{ïÄ[D]. þ°: E ŒÆ, 2014.
[3] =§. é/† ¦•ê! ©ù ..Ä70 ^± O†½dïÄ[D]. ¤Ñ: >f‰EŒÆ, 2010.
[4] 1u, ?Ư. ˜a•ê©?Ä7 ½d .9ÙmÇ ©Û[J]. û’²L, 2010, 18: 34-38.
[5] ê½°. ¥I©?Ä7 O†½d•{ïÄ[D]. þ°: þ° ÏŒÆ, 2012.
[6] ëBì. ¥I½|˜a•ê. ¦©?Ä7½d•{ïÄ[D]. ¤Ñ:ÜH㲌Æ, 2014.
[7] yI—. ©?Ä7½d•{ïÄ[D]. ÜS: ÜSnóŒÆ, 2011.
[8] ëŒg. ÄuÏ ©) ¥I©?Ä7½d9Ý]üÑïÄ[D]. þ°: E ŒÆ, 2014.
[9] Á÷S. ¥I©?Ä7 ½dïÄ[D]. þ°: þ° ÏŒÆ, 2011.
[10] ƒñ¸. ©?Ä7 Âà Š•{ïÄ[D]. þ°: þ° ÏŒÆ, 2013.
[11] ê². ETFÄ7@|Ŭ½d[D]. þ°: E ŒÆ, 2010.
[12] êf. ©?Ä7þ½° òÄd¯KïÄ[J]. y ½| , 2014, 08: 64-70.
[13] ‘ý•. ©?Ä7A!B° mòÄdÇ!‹l•êÂÃÇÅÄ'X ¢yïÄ[J]. 7K ƆïÄ, 2015, 06:
53-61.
[14] C. ·I©?Ä7òÄdÇK•Ïƒ ¢yïÄ[D]. þ°: þ°nóŒÆ, 2015.
[15] 2+. 6Ä5éµ4ªÄ7ò(Ä)dK• Œëê£8©Û[J]. 㬠r, 2009, 12: 37-38.
[16] !&Ä7+nk•úi. !&¥y y 1’•ê©?y Ý]Ä7Ä7ÜÓ.
[17] êÊ…. ©?Ä7 #PŸÙ§ ¬M#;K wX ƒÊ.2uy uÐïÄ¥%,2015.
[18] ñrÿ, M«9, ?Ư . 7Kû) ¬½d êÆ .†Y~©Û[M]. p ˜Ñ‡ , 2013: 42-49.
[19] Vasicek Oldrich. An equilibrium characterization of term structure.Journal of Financial Economics[J], 1977, 5: 177-
188.
[20] Cox John, Jonathan Ingersoll and Stephen Ross. An intertemporal general equilibrium model of asset prices and a
theory of the term structure of interest rates[J]. Econometrica, 53, 1985: 363-407.
[21] Hull John C. and Alan White, Pricing interest rate derivative securities[J]. Review of Financial Studies. 1990: 573-
592.
[22] Steven E. Shreve. Stochastic Calculus for Finance[M]. Springer. 2012: 265-268.
[23] Black F. and Scholes M. The pricing of options and corporate liabilities. Econ. 81: 637-654.
[24] ñrÿ, •æú, 4܃ . êÆÔn•§ùÂ[M]. 1n‡. ®µp ˜Ñ‡ , 2007: 108-120.
[25] Üä . 7Kû) ¬½d §[M]. ¥I<¬ŒÆч , 2010: 239-245.
[26] Boyle P P, Schwartz E S. Equilibrium prices of guarantees under equity-linked contracts[J]. The Journal of Risk and
Insurance. 1977 : 639-660.
[27] Boyle P P. Financial instruments for retired homeowners[J]. The Journal of Risk and Insurance. 1977: 513-520.
[28] Misza Kalechman. Practical MATLAB basics for engineers[M]. Boca Raton, 2009: 34-44.
[29] John C. Hull. Option Future and Other Derivatives[M]. Å ó’ч , 2015: 404-421.
24 • 1 23 •

28. |
|
|
|
|
|
|
|
|
|
C
|
|
|
|
|
¾
|
|
|
|
|
‚
|
|
|
|
|
|
|
|
|
|
.’ O£Ø©¤
`
Ø©´3 IéP“ • e ¤ " P“@ýKI§ã+gC >Ežm$–´!b
F5• · Æ’"¨Ø=‡¦·Ýº˜‡½d ~f½ö•{§•3˜‡ÆÏ žmp‘X·
XÚ/ÆS Steven 7K‘Å©Û˜Ö§¦ ·••ß”/ ) lVǘm ï!ÙK$
Ä 5Ÿ ºx¥5½d •{Ú ‡©•§ Ñ ˜X ‘Å©Û•£"·3ïÄ)Ïm
òÒÖzxŒÆ7KêÆ‘8§ ÆÏ¤Æ ù •£˜½¬é· ïÄ)ÆSD–™5 …
’uÐkX-Œ •Ï"
Óž§ © „Akâ [Ü© M«9P“Ú•ƒŒP“ F%• §3déMP“
Ú•P“L«a "
24 • 1 24 •