Lecture 05 期权定价的二叉树模型.pdf

1Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.1Lecture 05ùƒçN¥=­Ž˜Pricing Stock Options with Binomial Trees ModelFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.2†„=­Ž˜L è ,í„B æq¹ 12%• "ÐCN¹ $20
ØñªM¹ $ 22 or  $18Stock Price = $22Stock Price = $18Stock price = $20Stock Price = $22Option Price = $1Stock price = $20Option Price=?• ³›Nì¹ $ 21aØù¥"Ð Ahùƒ
NìMĹ
Stock Price = $18Option Price = $0Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.3• Inƒ"BñF†
(""Ð
iBz Ahùƒ bh• í„BF†iÅ"
í‚"NÂ…MîF†¥N´ûB"
' 22( –1 = 18(
¤( = 0.2522 ( –118 (†„=­Ž˜L è ,í„B æq¹ 12%20( –c• 9ü^ª¾F†^
™Æ 0.25 "
i› bBz AhùƒbØñ[ª¥N´^ 22 €0.25 – 1 = 4.50C´¹ 4.5 €e – 0.12 €0.25 = 4.3670• ¾F†C´¥6BñVrT¹ 20( – c = 20 €0.25 – c
yN
 20 €0.25 – c = 4.3670¤žùƒNì¹
 c = 0.633Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.4/í„BF†pùƒNì• Inƒ"BñF† :™Æ(""Ð ,i› bBz3ý dSuSffdu−−=δS u( – ƒuS d ( – ƒdS uS dS•F†^í„B¥ ,’ S u( –ƒu= S d( –ƒd,'S ( -fƒuƒdƒFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.5/í„BF†pùƒNì (continued)• F†žù° T ¥N´^ : Su( –ƒuyN
F†C¥N´^ :(Su( –ƒu)e–rT• F†C¥N´Î V[VU¹
 S ( – f• î[
 ƒ= S( –(Su ( –ƒu) e–rT• |ë¥(}Æ
 V¤
pedudrT=−−ƒ= [ p ƒu+ (1 – p )ƒd]e–rTFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.6ùƒçN¥†„=­Ž˜• 3ý ¥NìÛ$'ÁNìMî7MîpedudrT=−−×1²‚
ƒ= [ p ƒu+ (1 – p )ƒd] e–rTSuƒuS dƒdSƒ
´E
Àq p
´E
Àq 1 – p 2Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.7׍9Ø-륆„=­Ž˜L èí„B æq¹ 12%¨T9ؤ
6523.09.01.19.00.250.12=−−=−−=×edudeprTSu = 22ƒu= 1Sd = 18ƒd= 0S=20ƒ=
$p(1–p)Su = 22ƒu= 1Sd = 18ƒd= 0Sƒ0.65230.3477ùƒ¥Nì¹
e–0.12 €0.25 [ 0.6523 €1 + 0.3477 €0 ] = 0.633Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.8„BÏŸçN Risk-Neutral Valuation• 3ý ¥Nì¹ ƒ= [ p ƒu+ (1 – p )ƒd]e-rT• "Ð¥çùNì¹
$E(ST) = pSu + (1- p)Sd =SerT• 9ü^ª
"Ð¥çùíq©¿——í„B æq rSuƒuSdƒdSƒÀqpÀq(1–p)dudeprT−−=Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.9„=­ŽL è
 xT Ahùƒ
X=21, r =12%
 6ù
Ä„ 3
"NÄ„6Ù 10%BÄ¥N´ = e–0.12 €0.25(0.6523 €3.2 + 0.3477 €0) = 2.0257AÄ¥N´ = e–0.12 €0.25 (0.6523 €2.0257 + 0.3477 €0) = 1.2823201.2823221824.23.219.80.016.20.02.02570.0ABCDEF6523.09.01.19.00.250.122=−−=−−=×edudepTrFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.10ùƒçN¥„=­Ž˜Su2SfSufdSudSd2Sdfufuufudfdd•žùHK TM
Ä„ T/2 M•ƒu= [ pƒuu+ (1 – p )ƒud] e-rT/2•ƒd= [ pƒud+ (1 – p )ƒdd] e-rT/2•ƒ = [ pƒu+ (1 – p )ƒd] e-rT/2dudepTr−−=2Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.11ùƒçN¥„=­Ž˜ cont’d•ƒ=[ p2ƒuu+ 2p(1-p)ƒud+ (1 – p )2ƒdd] e-rTSu2SfSufdSudSd2SdfufuufudfddÀq p2Àq 2p(1-p)Àq (1 – p )2dudepTr−−=2Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.12L è
 ¨=­Ž˜ó AxTÙùƒçNX=52, r =5%
 2Mù
Ä„ 1M
"NÄ„6Ù 20%ƒ=[ p2ƒuu+ 2p(1-p)ƒud+ (1 – p )2ƒdd]e-rT= e-2×0.05[0.62822 × 0 + 2 × 0.6282 × 0.3178 × 4 + 0.31782 × 20] = 4.19236282.08.02.18.0ee10.052=−−=−−=×dudpTr504.1923604072048432201.41479.4636ABCDEF3Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.13Generalization: n-Step[]dudepXdSuppCefnnTrnjjnjjnjjnrT−−=−⋅−⋅=∑=−−−where)0,max()1(so , years Tin steps are There0Financial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.14L=‹¨ÏÊ4 u „ d!"N¥Moîq¹ σ
Ä„¥„ɹ ∆tM
1/2
 0.25
 1/12

5"NÄB„¥œ/Ù¥–¹
V[Ê|€ õù
Cox, Ross„ Rubinstein üP¨ƒÕZEbuedett==−σσ∆∆t∆σtdtu∆−=∆+=σσ11dudptr−−=∆eFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.15In"У æ¥=­Ž˜• L è
• ¨ }9ð ŸçNTÏ¥• InòQ£ æ1q¹
 i=1,2,……n)( DPVS− S)1()1()(1 nffSDPVS −−=−LifFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.16ÅTùƒ¥=­ŽçN˜†¥«Ä
ÅTùƒ„xTùƒN´M©b† L-¥«Ä
ÅTùƒN´|[/Kv´
—]©f ƒ/xTùƒ¥N´
—4-³›î¤¥åmFinancial Engineering, Lingnan College,Master Courses Class, by Prof.NIU Hong, Spring 20088.17ÅTùƒ¥。