數(shù)理金融第八章二叉樹(shù)模型與美式期權(quán)的風(fēng)險(xiǎn)管理-wenkub
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【正文】 12212020112 2 2 222022202 2 202 l i m ( ) 1l i m ( ) 11 1 1l i m ( 1 ) ( 1 1 )2 2 2 1 ( 1 2 4 )l i m ( 1 ) ( 2 ) 2 2l i mrh rhhrh h h rhhhhhe u u ee e e erh r h h h h hrh r hrh h rhh rhh??? ? ? ????????????? ? ?? ? ? ?? ? ? ? ? ? ? ?? ? ? ?? ? ? ? ????Substituting for u and d, the terms of higher than 2 power are ignored. ,u e d e? ? ? ????From Cox， Ross and Rubinstein(1979) 54 ? 美式期權(quán)可以提前執(zhí)行，提前執(zhí)行從表面上看是一個(gè)非常微小的變化，但是歐式期權(quán)與美式期權(quán)（尤其是看跌期權(quán)）價(jià)值有很大的不同。 此期望值是期權(quán)的真實(shí)值嗎？ 28 For example: twostep binomial trees 22 2 2200 0 2 0 2 221 1 1 1 1 222 2 0 2 0 22222[ ( 1 ) m a x ( 0 , ) ] [ ( 1 ) m a x ( 0 , ) ]( 1 ) m a x ( 0 , ) ]( 1 ) m a x ( 0 , ) ] [ ( 1 ) m a x ( 0 , ) ]2 ( 1 ) m a x ( 0 , ) ]m a x ( 0j j j j j rhtjrhrhrhrrc C p p S u d X eC p p S u d X eC p p S u d X eC p p S u d X ep S d X ep p S X ep??? ? ???????? ? ?? ? ?? ? ?? ? ?? ? ?? ? ???2, ) ]rS u X e???29 CRR model: nstep binomial trees 0[ ( 1 ) m a x ( 0 , ) ]nj j n j j n j rtnjc C p p S u d X e ?? ? ??? ? ?? , , , j n jm n mif u p in c r e a s e s o w e c a n fi n d aj S u dS u dm X?? ?[ ( 1 ) ( ) ]nj j n j j n j rtnjmc C p p S u d X e ?? ? ??? ? ??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 rhrhpunm
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