Derivative Securities: Lecture 4 Introduction to option pricing

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Transcription:

Divaiv cuiis: Lcu 4 Inoducion o oion icing oucs: J. Hull 7 h diion Avllanda and Launc () Yahoo!Financ & assod wbsis

Oion Picing In vious lcus w covd owad icing and h imoanc o cos-o cay W also covd Pu-all Paiy which can b viwd as laion ha should hold bwn uoan-syl us and calls wih h sam xiaion Pu-call aiy can b sn as icing convsions laiv o owads on h sam undlying ass Wha oh laions xis bwn oions and sads on h sam undlying ass?

all ad all ad: Long a call wih sik sho a call wih sik L (L>) L inc h ayo is non-ngaiv h valu o h sad mus b osiiv L all alll L all alll ad maks mony i h ic o h undlying gos u

Pu ad Pu ad: Long a u wih sik L sho a u wih sik L (L>) L inc h ayo is non-ngaiv h valu o h sad mus b osiiv P L Pu PuL L PuL Pu ad maks mony i h ic o h undlying gos down

Buly ad Buly sad: Long call wih sik long call wih sik L sho calls wih sik (+L)/ (+L)/ L inc h sad has non-ngaiv ayo i mus hav osiiv valu B L ( L) / L all alll all Bulis mak $ i h sock ic is na (+L)/ a xiaion.

addl addl: Long call and long u wih h sam sik - + addls mak mony i h sock ic movs away om h sik and nds a om i

angl angl: Long u wih sik long call wih sik L L>. L angl is also non-dicional lik a saddl bu maks $ only i h sock movs vy a away. addls and sangls a on usd o xss viws abou volailiy o h undlying sock and a non-dicional.

Risk-vsal Risk-vsal : Long u wih sik sho call wih sik L L>. L Dicional sad. an b sn as inancing a u by slling an usid call.

alnda ad alnda ad: ho call wih mauiy Long call wih mauiy < am sik Mauiy Mauiy I h undlying ays no dividnds bwn and hn h long mauiy call is abov ininsic valu a im. alndas hav osiiv valu. Fo Amican oions calndas always hav osiiv valu

Rconsucing all ics om Buly ads Assum o simliciy a counabl and ha h sock ic can only ak valus on h laic all n all all n numb o siks n n n n 3... A call can b viwd as a oolio o call sads A call sad can b viwd as a oolio o buly sads m m 3... Bi i i i +

alls as su-osiions o buly sads n n n n n n i i i i n w B w w B n B n B all h wighs cosond o valus o Buly sads cnd a ach. In aicula hy a osiiv

Fom wighs o obabiliis max max. ; ) valus (assuming ha h sock can only ak PV $ all w B w w

Fis momn o is h owad ic all F q q A call wih sik is h oion o buy h sock a o a im Is valu is ho h sn valu o h owad ic (ay now g sock la). I ollows ha h is momn o is h owad ic. I also ollows ha u ics a givn by a simila omula namly Pu ( ) all q F

Gnal Payos Any wic diniabl uncion () can b xssd as a combinaion o u and call ayos using h omula (his is us aylo xansion) hus a uoan-syl ayo can b viwd as a sad o us and calls. By linaiy o icing dy Y Y dy Y Y F dy Y Y F F F F '' '' '' ' ' F F F F dy Y Y dy Y Y F dy Y Y all dy Y Y Pu '' '' ' '' '' ) ( ) ( ) ( ) ( claim wih ayo a Fai valu o

Fundamnal hom o icing (on iod modl) An abiag oouniy is a oolio o divaiv scuiis and cash which has h ollowing ois: - h ayo is non-ngaiv in all uu sas o h mak - h ic o h oolio is o o ngaiv (a cdi) Assum ha ach scuiy has a uniqu ic (i.. assum bid-o). I h a no abiag oouniis hn h xiss a obabiliy disibuion o uu sas o h mak such ha o any uncion () h ic o a scuiy wih ayo () is P onvsly i such a obabiliy xiss h a no abiag oouniis

Pacical Alicaion o uoan Oions A icing masu is a obabiliy o uu ics o h undlying ass wih h oy ha F I w dmin a suiabl icing masu hn all uoan oions wih xiaion da should hav valu givn by all Pu h main issu is hn o dmin a suiabl icing masu in h al acical wold.

Wha dos a icing masu achiv in h cas o oions? all(;) F Oion ai valu is a smoohd ou vsion o h ayo h icing masu givs a modl o comu h oion s ai valu as a uncion o h ic o h undlying ass h sik and h mauiy

h Black-chols Modl Assum ha h icing masu is log-nomal i.. log-uns a nomal ~ - ~ N Z q Z X q q dy N X y y X

all icing wih h Black-chols modl ln / / / d d d d d d d d q A d d d all A A q A A q A A q A A q A A q q

Black-chols Fomula x q q d x N F F d F d d N d N q Ball ln ln cumulaiv nomal disibuion

Black-chols Fomula a wok =$48 =$5 =6% sigma=4% q=

Muli-iod ass modl Divaiv scuiis may dnd on mulil xiaion/cash-low das. Fuhmo h -iod modl dscibd abov is igid in h sns ha i canno ic Amican-syl oions. W consid insad a mo alisic aoach o icing basd on h saisics o sock uns ov sho iods o im (.g. day). W assum ha h undlying ic has uns saisying W also assum ha succssiv uns a uncolad. ~ N

Modling h un o a ic imsis (OHL) Ln Modl closing ics o xaml. h % changs bwn closing ics a nomal and uncolad = closing ic o iod (-)

Paamiaion annualid sandad dviaion annualid xcd un % daily sandad dviaion => 5.9% annualid sandad dviaion 5.9 5 5

Picing Divaivs L us modl h valu o a divaiv scuiy as a uncion o h undlying ass ic and h im o xiaion... hang in mak valu ov on iod : o o V V

h hdging agumn onsid a oolio which is long divaiv and sho socks. Assum divaiv dos no ay dividnds q q V q V q V V PNL including inancing and dividnds : Poi and loss

Analying h sidual m o o o o h sidual m has ssnially o xcd un (vanishing x. un in h limi D_>.) ondiional xcaion o silon

h ai valu o ou divaiv scuiy is h PNL o h long sho oolio o divaiv and ba shas has xcd valu his oolio has no xosu o h sock ic changs. ho i () sns h ``ai valu o h divaiv h oolio should hav o a o un (w alady ook ino acc is inancing). hus: his is h Black-chols aial dinial quaion (PD). ) ( ) ( o q o PNL q

Amican-syl calls & us onsid a call oion on an undlying ass aying dividnds coninuously. inc h oion can b xcisd anyim w hav max. () h minal condiion a = cosonds o h inal ayo max. hus h uncion () should saisy h Black-chols PD in h gion o h ()-lan o which sic inqualiy holds in () and i should b qual o max(-) ohwis. h soluion o his oblm is don numically and will b addssd in h nx lcu.

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