The Black-Scholes Formula

Transcription

1 lack & chols h lack-chols Fomula D. Guillmo Lópz Dumauf dumauf@fibl.com.a Copyigh 6 by D. Guillmo Lópz Dumauf. No pa of his publicaion may b poducd, sod in a ival sysm, o ansmid in any fom o by any mans lconic, mchanical, phoocopying, coding, o ohwis wihou h pmission of D. Guillmo Lópz Dumauf his documn povids an oulin of a psnaion and is incompl wihou h accompanying oal commnay and discussion.

2 Hdging If you wiing a call opion, you a xposd o h isk of h sock pic ising abov h sik pic a xpiaion In od o cov h call, you nd a xpiaion, cash qual o h diffnc bwn h cun pic of h sock and h sik pic of h opion -K In lack-chols, h hdg is a pofolio compoundd wih som shas of h undlying sock and som isklss asss U -onds How much of ach o pu in h hdg is h ky o dmining h opion s valu.

3 Pfc hdg A pfc hdg always pays xacly h amoun ncssay o cov h opion. Payoff plicaion: if h opion xpis in h mony, h hdg povids xacly h amoun ncssay o cov h call if h opion xpis ou of h mony h hdg will b woh nohing

4 Dynamic hdging Dynamic hdging is a ading sagy which plicas h payoff of h opion houghou is lif caing a pfc hdg I has a fixd and known oal cos Dynamic hdging is a pocss fo managing h isk of opions and i is impoan fo xplaining an opion s valu.

5 Dynamic hdging lack-chols hdging is dmind by h wighs of h hdging pofolio. caus h paams of h fomula a consanly changing, h hdging pofolio mus b consanly updad o flc h nw wighs h pofolio which dosn flcs h cun lack- chols wighs is ou of balanc. h pocss o kping h pofolio in balanc is calld balancing If h hdging sagy is pfomd cocly, h valu of h pofolio will b qual o h valu of h opion a all ims

6 Coss of h hdging. -up cos iniial cash flows associad wih h hdging sagy. Mainnanc coss: infusion and ansacion coss coss associad wih balancing h hdg,.g. buy mo sha of sock o bonds In addiion, h a: fs and axs associad wih making h ansacions coss suling fom h bid-ask spad and h inabiliy o xcu ads a xacly h pic spcifid by h sagy

7 lf-financing Dynamic Hdging A hdging sagy whos oal cos a any im xcluding ansacion coss is qual o s-up coss is calld slf-financing. h coin-ossing xampl A gambl dcids making a b of $. H nds 3 facs o win.5!5. 5.!5. 5.!.

8 h Dla of an opion h Dla of an opion is h a of h chang in h opion pic wih spc o h chang in h sock pic Whn w look a changs in h valu of an opion, w a insd in changs laiv o h undlying sock pic. If h pic of h sock changs $, wha will b h chang in h opion s valu?

9 h Dla of an opion Opion is dp in h mony and na xpiaion: h dla is clos o on, so h is a dolla fo dolla laionship bwn movmns of h spo pic and movmns of h opion s valu Opion is dp ou of h mony and na xpiaion: h dla is clos o zo bcaus h opion s valu is absoluly insnsiiv o changs in h sock pic and i will b woh almos nohing Implici in h & modl h is only faco of uncainy in h valu of an opion: volailiy In uh, h uncainy implici in an opion is mo complicad han his.

10 h Dla of an opion Ral wold is mo complicad: h dynamic of sock pic is mo complicad han a gomic bownian moion Jump isk: if h is a pobabiliy of sock pic will unxpcdly jump downwad, his can chang h opion s valu vn whn h spo pic dos no chang ownian moion modl dos no includ his ffc In pincipl, w can obsv his jump compaing h spo pic and opion valu a som im, doing h sam a a la im caus h pic of h opion is no conolld by h sock pic alon, his may no giv an accua sima of h dla of h opion Evn if w igno pobabiliis of jumps and changing volailiy, h valu of h opions dos chang as im lapss s paam ha

11 h Dla of an opion In h & wold, sock pic a supposd o follow a GM wih consan volailiy and h opion s valu changs fo only wo asons:. im chang and. ock pic chang W can us an appoximaion o h dla of h opion: h acual a of chang in h valu of h opion wih spc o h chang in h sock pic...

12 h Dla of an opion Exampl: =,87; c =,59 =,89; c =,6 = C,6,89,47/ 365,59,87,, = =,47/ 365 C,5 h valu of h dla is,5 h numao says h im adjusd chang in h pic of h opion was $, whil h im adjusd chang in h sock pic was $, h aio says ha fo vy $ chang in h sock pic, h was an appoximaly $,5 in h valu of h opion How h fomula woks in h al wold?

13 Call voluion fo ansn Da ock pic Call Days o mauiy Va Va Call Dla Call fomula 4/3/6,8,4 38 /3/6,89,6 3,7,8,57 /3/6,87,59 3 -, -,,5,55 3/3/6,89,6 9,,,5,6 3/3/6,87,53 9 -, -,7,35,59 5/3/6,86,5 7 -, -,3,3,5 5/3/6,9,55 7,4,5,5,6 7/3/6,9,68 5,,3,65,58 7/3/6,99,6 5,9 -,8-8,889,69 7/3/6,9,69 5 -,9,9 -,86,57 7/3/6,95,75 5,4,6,5,36 7/3/6,96,84 5,,9,9,77 7/3/6,97,85 5,,,,93 8/3/6,95,7 4 -, -,5,75,83 8/3/6,97,83 4,,3,65,85 8/3/6,96,83 4 -,,,77 8/3/6,95,85 4 -,, -,,83 8/3/6,97,9 4,,5,5,8 8/3/6,96,87 4 -, -,3,3,88 9/3/6,97,85 3, -, -,,9 9/3/6,98,85 3,,,83 3/3/6,99,9,,5,5,85 3/3/6,3,3,4,33,85, 3/3/6,, -, -,3,65,7 3/3/6,,,,,,7 /4/6,97,75 9 -,5 -,45,9,7,5,95,9,85,8,75,7,8,6 ock pic Call Appoximad fomula,4,,,8,6,4,,4,, Call,8,6,4, /3/6 /3/6 4/3/6 6/3/6 8/3/6 3/3/6 /4/6 3/4/6

14 h appoxima fomula h fomula is a good appoximaion bcaus:. If w choos a im vy clos o im and apply h fomula, hn w obain valus fo h dla ha a vy clos o h acual valu of h dla a im. Each im w subsiu h appoxima valu of h dla fo h al valu, w poduc a small o. u hs small os do no accumula, and do no mak ou appoximaion uslss!!

15 Dla appoxima fomula Wha maks h fomula an appoximaion is h fac ha w compa h chang in h opion pic o h chang in h sock pic ov small piod of im. If all ims a infinisimally clos, w g a fomula ha is no long appoxima bu is pcisly coc... Wha if h dnominao of h fomula is zo? his dos no invalida h fomula, bcaus accoding o h M modl, h pobabiliy ha h sock pic a im will b xacly qual o is zo h pobabiliy of any givn vn is always zo, only angs of vns ma in pobabiliy hoy Of cous, in aliy i can happn.

16 h lack-chols hdging sagy If w know h dla of a uopan call o pu opion, w can hdg h opion and dmin is hoical valu h assumpions bhind h lack-chols fomula play a ky ol in dmining h valu of an opion. Idnifying how laxing hm is an impoan pa of undsanding h lack-chols mhod

17 Hdging sagy in lack-chols A vy impoan pa is o undsand how h fomula anslas ino a hdging sagy ha plicas h payoff of h opion hn, no-abiag assumpions nsus ha h valu of h opion is qual o h cos of h hdging sagy

18 Hdging sagy in lack-chols h valu of h hdging pofolio a im, is h AME as h & Fomula fo h opion s valu Onc h hdging is s up, h nx sp is o mainain h hdg In od o kp h pofolio in balanc, w will hav o coninuously monio h valus of and, buying o slling shas In od o nsu h sagy is slf-balancing w wan h balancing coss a vy sp o b as clos o zo as possibl

19 Hdging sagy in lack-chols In aliy, h hdging sagy is usd o hdg an opion no on a singl sha of sock bu on som lag numb of shas In Agnina, a sandad opion conac is on shas of h undlying inc dla is a numb bwn and, i psns h pcnag of h lo of shas ha should b puchasd

20 h hdging sagy is slf-financing In wold h is no ansacions coss, so up coss=valu of opion=oal cos of hdging hfo, s-up coss qual h oal cos of hdging, which implis ha h mainnanc cos of hdging mus qual zo and h pofolio is slf-financing If h fomula holds, hn h hdging sagy mus b slf-financing. I would b mo convincing if w sad wih h hdging sagy and hn showd ha i plicas h payoff of h opion and is slf-financing

21 Rbalancing h pofolio Whn w s up h hdging pofolio a im is valu is C = uppos ha a im w dcid o balanc h pofolio. h amoun spn a and a im Amoun spn caus w nd o compa h valu of all puchass a, h mony spn a im mus b discound by a faco of --.

22 Rbalancing h pofolio = h mony spn on hdging a im is And h mony spn in sing up h hdg a im and balancing a im is + Raanging his a bi, w obain + Rplacing in h las xpssion fo h appoxima dla, w hav C C +

23 Rbalancing h pofolio C C + C C C = + & fomula fo h call a im adjusd fo im valu Opion s valu a im adjusd fo im valu Opion s valu a im his says if w s up h pofolio a and balanc a, h oal cos of h opion is h cos of h opion a im If holds a im, i holds a im N-; h sam agumns imply ha i holds a im N-, N-3.so fomula mus hold a im

24 h lack-chols fomula fo and h fomulas a givn in ms of cumulaiv nomal disibuion funcion: d = ln + f X + d = d C = N d X N d

25 lack-chols and h ownian Moion Modl Compuing h pobabiliy ha C xpis in h mony is quivaln o compuing h pobabiliy ha K h un on fom im o im is givn by ln his un is a andom vaiabl wih man n And sandad dviaion - hfo, h andom vaiabl is givn by ln his is nomally disibud, wih man zo and sandad dviaion

26 lack-chols and h ownian Moion Modl Now, w ansfom h quaion K dividing boh sids by, aking logaihms, subsacing - / - and finally, by dividing by - K ln ln caus w wan an xpssion wih lss han o qual o us h cumulaiv nomal disibuion w nga his quaion and obain: K + + ln ln

27 lack-chols and h ownian Moion Modl Fo h oiginal samn o b u K hn h andom vaiabl on h lf-hand sid of quaion mus b lss han o qual o h valu on h igh-hand sid h pobabiliy h inqualiy holds is xacly givn by h cumulaiv nomal disibuion of h igh-hand sid, so h pobabiliy ha C will xpi in h mony is K + + ln ln + = K N K P ln

28 Excciss. Using & fomula, calcula h fai valu fo a call opion on Pobas, fo diffn sik pics. oday is 7/4/6 and h annual isk f a is 4,9%. hn compa his valu wih h opion mak valu.. Calcula h implid volailiy fo K=4 Opion mak valu ock pic