數(shù)理金融第八章二叉樹模型與美式期權(quán)的風(fēng)險(xiǎn)管理(編輯修改稿)
2025-02-07 09:49 本頁面
【文章內(nèi)容簡介】 30 [ ( 1 ) ( ) ] ( 1 ) ( 1 )nj j n j j n j nrhtnjmnnrh j j n j j n jnjmnnrh j j n jnjmc C p p s u d X eSe C p p u dXe C p p? ? ??? ? ?????? ? ????????rhedpud???rhp pu e???1 ( 1 ) rhp p d e??? ? ? ? ?31 () ( 1 ){ [ ( 1 ) ] [ ] }{ [ ( 1 ) ] [ ] }( 1 )( , , )nj j n j j n j nrhnjmnj rh rhnjmnj rh rhnjmnj n j jnjn j n j n jmj j jn j jS C p p u d eS C p d e p u eS C p de pueS C p pSB n m p? ? ?????? ? ?????? ? ???????????????[ 1 ( 1 ) ]rhrhp p d ep p u e????? ? ? ???( , , ) ( , , ) ( , , ) ( , , )n r htrc S B n m p X e B n m pS B n m p X e B n m p?????????: ( , , ) ( 1 ) ( , , ) ( 1 )nj j n jnjmnj n j jnjmhere B n m p C p pB n m p C p p??? ? ? ????????11 m in im a l in te g e r m n m m n mm is p o s iti v e w h ic hs a tis fie s S Su d X u d? ? ? ???rhp pu e???? How to pute u or d? 33 Choosing u and d ? One way of matching the volatility is to set hhu e d e?? ???where ? is the volatility and h is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein. Neutralrisk probability is rhedpud???34 Simplify first term ( , , ) ( , , )rtc S Xe B n m pB n m p ?? ???( 1 ) ( , , )[ ( ) [ ( 1 ) ]( 1 ) = ( ) [ ]( 1 ) ( 1 ) ( ) [ 1( 1 ) ]( 1 )nj j n j j n j nr hnjmnnr h j j n jnjmnj j n jnrmnjjjnC p p u d ee C pu p dpu p dCpu p d pu p dB n m pepu p dpu puCpu p d?? ? ??????????????????????????]( 1 )nnjjmpu p d?????=1 35 [ ( 1 ) ] [ ( 1 ) ] ( ) 1[ ( 1 ) ]rh rhn n rh n rrne d e dp u p d u d e eu d u dep u p d????? ? ? ? ? ? ???????Binomial equation 0[]nn j j n jnja b C a b ???? ? ( ) [ 1 ]( 1 ) ( 1 )[ 1 ]nj j n jnjmnj j n jnjmpu puCpu p d pu p dC y y?????? ? ? ?????( 1 ) rhp u p u yp u p d e???36 100 l i m ( 1 )l i m [ ( 1 ) ( 1 ) ]1[ 1 ( ]( 1 )1[ ( ]( 1 )( 1 )[ ( ]( 1 )nj j n jnnjmnmj j n j j j n jnnnjjrhC y yC y y C y ym nyNny yny mNny ynpu e mNnp p????????????? ? ? ?????????????????( 1 ) rhp u p u yp u p d e??? 11( 1 )( 1 )( 1 ) ( 1 ) ( 1 )( 1 ) ( 1 )()rhrh rhrh rh rhrhpunmny m eny y pu puneenp u e m np u e m np u e mnp u p d np pnp u e pu???????? ? ? ? ? ?? ? ????37 Simplify second term 100 l i m ( , , ) ( 1 )( 1 ) ( 1 )11 ( )( 1 )1()( 1 )nj j n jnnjmnmj j n j j j n jnnjjB n m p C p pC p p C p pm npNnp pnp mNnp p???????????? ? ? ?????????????38 Simplify all terms 1212( 1 ) 1[ ( ] ( )( 1 ) ( 1 ) ( ) ( )( 1 ) 1,( 1 ) ( 1 )rhrtrrhnp u e m np mc SN X e Nnp p np pSN d X e N dnp u e m np mddnp p np p????? ? ? ??????? ? ? ?????Next step, we must deduce d1 and d2 when n→∞ 39 deducing d1 and d2 (for m) l n ( ) l nl n ( / ) l n ( / )l n ( / ) l n ( / ) m n m n mnnuS u d X S d XdX S d X S dmmu d u d??? ? ?? ? ? ? ? ,hhu e d e?? ???l n [ ( ) / ] l n ( / )2l n ( / )hnhhe X S X S n hmhee?????????? ? ? ?l n ( / )l im2nX S n hmh??????40 deducing d1 and d2 (for p) 002222 l i m l i m( 1 ) ( 1 ) ( 1 ) ( 1 )1 / 2 22rhnhhedpudrh h hh h h hrh??? ? ? ???? ? ?????? ? ? ??? ? ? ? ????221 / 2 1 / 2( 1 ) ( ) ( )2 2 2 2 2r r nn p p n h h??????? ? ? ? ? ?41 122( 1 )( 1 )1 / 2 l n( / )( ) ( 1 ) ( 1 )22 2 12
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