Option Pricing When Changes of the Underlying Asset Prices Are Restricted
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1 Journal of Mahmaial Finan 8-33 doi:.436/jmf..4 Publishd Onlin Augus (hp:// Opion Priing Whn Changs of h Undrling Ass Pris Ar Rsrid Absra Gorg J. Jiang Guanzhong Pan Li Shi 3 Univrsi of Arizona uson USA Collg of Finan Yunnan Univrsi of Finan and Eonomis Kunming China Collg of Saisis and Mahmais Yunnan Univrsi of Finan and Eonomis Kunming China gjiang@mail.arizona.du Rivd April 8 ; rvisd Jun 3 ; apd Jul Ehangs ofn impos dail limis for ass pri hangs. hs rsriions hav a dir impa on h pris of opions radd on hs asss. In his papr w driv losd-form soluion of opion priing formula whn hr ar rsriions on hangs in undrling ass pris. Using numrial ampls w illusra ha vr ofn h impa of suh rsriions on opion pris is subsanial. Kwords: Opion Priing Rsriions on Ass Pri Changs Numrial Illusraion. Inroduion Convnional opion priing modls assum ha hr ar no rsriions on hangs of undrling ass pris. For ampl [] spif ha sok pris follow a gomri Brownian moion and sok rurns follow a normal disribuion. Basd on a porfolio rpliaion srag or quivalnl h risk-nural mhod opion pris ar drivd as pd paoff of h onra undr h risk-nural disribuion disound b h riskfr ra. In prai howvr ass pri hangs ma subj o rsriions imposd b hangs. For ampl CBO (Chiago Board of rad) and CME (Chiago Mranil Ehang) boh hav dail pri limis for fuurs onras p urrn fuurs. Dail pri limi srvs as a prauionar masur o prvn abnormal mark movmn. h pri limis quod in rms of h prvious or prior slmn pri plus or minus h spifi rading limi ar s basd on pariular produ spifiaions. For ampl h urrn limi of dail pri hangs on shor rm orn fuurs is 4.5%. In 996 Chins sok mark also inrodus rsriions on dail sok pri hangs. Spifiall p h firs rading da of nwl issud soks h limi of sok pri hang in a rading da rlaiv o prvious da s los pri is % and for soks ha bgin wih S S S*S lrs h limi is 5%. I is lar ha suh rsriions rdu h valu of opions sin rm rurns on a dail lvl ar ruld ou. Nvrhlss how o valua h pris of opion onras whn hangs on undrling ass pris ar rsrid? How muh is h a impa of suh dail pri limis on opion pris? hs qusions ar o b amind in h an liraur. In his papr w firs driv h opion priing formula whn hr ar rsriions on dail hangs of undrling ass pris. W prform h analsis undr h Blak-Shols-Mron modl framwork. hn w provid numrial omparisons bwn opion pris wih and wihou rsriions on undrling ass pri hangs.. Opion Priing wih Rsriions on Undrling Ass Pri Changs.. Risk-Nural Valuaion undr h Blak and Shols [] and Mron [] Framwork In his sion w firs rviw h risk-nural approah of opion priing undr Blak-Shols-Mron framwork. h sam approah will b usd in h n subsion o driv opion priing formula whn hr ar rsriions on undrling ass pri hangs. Blak and Shols [] and Mron [] assum ha sok pri S follows a gomri Brownian moion: ds Sd SdW () whr and ar pd rurn and volaili W is a sandard Brownian moion. I is also assumd ha h oninuousl ompoundd risk-fr inrs ra Coprigh SiRs.
2 G. J. JIANG E AL. 9 dnod b r is onsan. h k faur in Blak- Shols-Mron framwork is ha ass rurn volaili is onsan and mark is ompl. As suh in a risknural world pd rurn of h undrling sok is qual o risk-fr inrs ra. ha is ds rs d S dw. () whr W is a sandard Brownian moion undr h risk-nural probabili masur. Considr a Europan all opion wih srik pri K and mauri masurd in h numbr of rading das. h pri of suh opion an b ompud as r C E [ma( S K)] whr E indias paion undr risk-nural masur and is h im inrval of a rading da. As shown in man drivaivs books for ampl [3] h opion priing formula is givn as: r C S ( d ) K ( d ) (3) u whr ( ) du is h umulaiv disribuion funion (df) of sandard normal disribuion π and d S K r ln( / ) ( ) d d. his is h famous Blak-Shols-Mron opion priing formula. B onsruing a risklss porfolio wih opion and undrling sok and basd on no arbirag argumn [] and [] driv h abov opion priing formula as a soluion o a parial diffrnial quaion (PDE)... Closd-Form Opion Priing Formula wih Rsriions on Undrling Ass Pri Changs In addiion a and b ar wo posiiv onsans. runaing h lf ail of h normal dnsi b a blow h man and h righ ail b b abov h man h runad normal disribuion is illusrad in Figur. Normalizing h runad dnsi funion o mak sur h oal probabili is qual o w obain h pdf of a runad normal random variabl ~ r fr ( ) f( ) [ a b] whr f ( ) is h normal dnsi funion and b a (4) is h df of sandard normal disribuion. whr In prai pri rsriions ar ofn imposd in rms of dail simpl rurns. For ampl dail simpl rurns in absolu valu ar rsrid o b lss han hn for log rurns hs rsriions ar b ln( ). For h purpos of opion priing i is also onvnin o obain h hararisi funion (CF) of sok rurns. As drivd in h Appndi h hararisi funion of h runad normal variabl is givn b: a ln( ) and ( i) r ( ) ( ) (5) whr ( ) i is h CF of a normal random variabl and b a ( i) i i. Sin limis ar piall imposd on dail pri hangs and opion mauri an b mor han on da w nd o driv h disribuion of rurns ovr mulipl das. Dno ln S as h log pri Y ln S ln S as h dail log rurn. Suppos w hav rading das and h dail log rurns ar iid runad normal random variabls i.. Y ~ iid r( ). h log pri a h nd of da is As mniond in h inroduion man hangs impos rsriions on dail pri hangs of radd asss. As a rsul h rang of ass rurn (in logarihmi form) is no longr ( ) bu runad from boh blow and abov. h rsriion is pariularl imporan in opion priing sin h ail bhavior of ass rurns has a signifian ff on h paoff of opions onras. L us sar wih a normal random variabl ~ ( ) wih probabili dnsi funion: ( ) f( ). π Figur. runaing normal dnsi. Coprigh SiRs.
3 3 G. J. JIANG E AL. Y Y. ~iid r ( ) Similarl as drivd in h Appndi h CF of is givn b i ( i) ( ) ( ). (6) In h following w follow h sam risk-nural approah as oulind in h prvious subsion o pri opions whn hr ar rsriions on undrling ass pri hangs. As sn in h Blak-Shols-Mron framwork whn w mov from ral world ino risknural world volaili rmains h sam bu pd rurn is qual o risk-fr inrs ra. Opion pris ar hn alulad as pd paoff undr h risk-nural masur furhr disound b risk-fr inrs ra. In h following w firs driv h risk-nural disribuion of ass rurns whn dail rurns follow runad normal disribuions and hn driv a losd-form formula for Europan all opions. Lmma L Y ~ iid r( ) and h im inrval bwn obsrvaions is w hav i) Undr h risk-nural masur whr h pd rurn of h ass is givn b h risk-fr ra r w hav Y ~ iid r( ) wih () r ln. (7) ii) h pri of a Europan all opion wih srik pri K and mauri is givn b C S P( ln K) KP ( lnk) (8) r r P P h probabiliis and an b ompud numriall as p( i) ( ) P ( ) R d π i whr ( ) is h CF orrsponding o P. Proof: i) L b h im inrval of ah rading da and h fa ha pd rurn is qual o risk fr ra lads o: E [ S ] r S From h momn gnraing funion of a runad normal disribuion as drivd in h Appndi w hav () [ ] E S E S From h abov wo quaions w hav h prssion for. ii) Aording o risk-nural priing mhod h pri of a Europan all opion wih srik pri K and mauri is C E [ma( S K)] [ma( )] r r E K r K f ln K r r ln K ln K ( ) ( )d f ( )d K f ( ) d. whr f ( ) is h pdf of log pri a im. In addiion undr h risk-nural masur r r S E[ S] E r f ( )d So w hav S r. f ( )d Subsiuing his ino opion pri C w g f ( ) C S d K f ( )d Dno r ln K ln K f ( )d f ( ) g ( ). f ( )d Sin g ( ) g( ) is a pdf. hrfor w an wri h Europan all opion pri as r C S P( ln K) KP ( lnk). à End of proof. As shown in [4] ha h probabiliis in (8) an b ompud numriall b hir orrsponding CFs as follows. From h Fourir invrsion w hav p( i) ( ) P ( ) R d π i. o ompu h CF orrsponding o g( ) dnod as ( ) b dfiniion w hav f ( ) i i ( ) g( )d d f ( )d ( i ) ( f ( )d whr h numraor is givn b f )d ( i ) f ( )d M ( ) i ( i ) ( i ) M(( i ) ; ). Coprigh SiRs.
4 G. J. JIANG E AL. 3 and h dnominaor is givn b Hn () f ( )d M () (; ) M ( ) ( i ) () M ( i ) M(( i (; ) ); ) i ( i) M(( i); ). () M(; ).3. Numrial Illusraions In his sion w illusra numriall h diffrns of opion pris wih and wihou rsriions on undrling ass pri hangs. abl rpors h diffrns in Europan all opion pris undr diffrn snarios. h Blak-Shols- Mron pri dnod b CallBSM is h all opion pri wihou pri rsriion and is ompud from formula (3) h all opion pri wih pri rsriion dnod b Callr is ompud from formula (8) drivd in h prvious subsion. In Panl A w s h pri limi as 4.5% onsisn wih h urrn limi of dail pri hangs on shor rm orn fuurs a CBO iniial sok pri S $ srik pri K $ mauri das annualizd risk-fr inrs ra r 5%. h an- nualizd volaili rangs from 5% o 5%. As pd opion pris wih rsrid hangs in undrling ass pris ar lowr han h Blak-Shols-Mron pri. h rlaiv diffrn is highr as volaili inrass. For h a-h-mon opion onsidrd h rlaiv diffrn is 9.45% whn volaili is 4% whih is pial for individual sok rurns. abl. Europan Call Opion Pris wih and wihou Rsriions on Ass Pri Changs. Panl A: Variaion in Volaili Paramrs 4.5% S K r 5% s Callr CallBSM Diffrn (%).%.%.8% 5.77%.85% 9.45% 8.9% 37.44% Panl B: Variaion in Srik Pr i K Paramrs 4.5% S r 5%.4 5 K Callr CallBSM Diffrn (%). 7%. 7% 6. 7% 9.45% 47.5% 5.85% 36.97% Panl C: Variaion in Mauri Paramrs 4.5% S K r 5% Callr CallBSM Diffrn (%) 6. 3% 9. 3% 9.45% 9.8% 8. 6% 7.84% 6.76% Panl D: Variaion in Pri Limi Paramrs S K 5 r 5%.4 5 d % % 3% 4% 5% 7% % Callr CallBSM Diffrn (%) 685% 849% 96% 73.3% 3.73% 4. 97%.5% Coprigh SiRs.
5 3 G. J. JIANG E AL. As pd whn h opion srik p ri is high r h rlaiv diffrn also inrass. h rsuls ar illusrad in Panl B whr volaili is s as 4% pr annum and h srik pri rangs from $ 85 o $5. W no ha whn h srik pri K $ $5 h rlaiv diffrns ar mor han % and % rspivl. Panl C illusras h diffrns in opion pris wih diffrn mauriis. For h a-h-mon op ions onsidrd h rlaiv diffrns ar rahr onsisn vn whn mauri inrass o 6-monh ( 6 ) and - ar ( 5 ). Panl D illusras h ff of dail pri limis on opion pris whr is s o valus in a rang of % o %. As pd h rlaiv diffrn inrass as h absolu pri limi is lowr. For h ou-of-hmon opion onsidrd (srik pri K 5 ) h rlaiv diffrn in opion pri wih rsriion and wihou rsriion is mor han 3% whn h pri limi is s as 5%. 3. Conlusions In prai hangs ofn s dail limis for h pri hangs of radd asss. hs rsriions hav a dir impa on pris of opions radd on hs asss sin h rul ou dramai hangs in ass pris. In his papr w driv losd-f orm soluion of opion priing formula wih rsriions on undrling ass pri hangs. Using numrial ampl s w illusra ha vr ofn h impa of suh rsriions on opion pris an b subsanial. 6. Aknowldgmns h rsarh of Li Shi is suppord b NSFC (653). 5. Rfrns [] F. Blak and M. Shols h Priing of Opions and Corpora Liabiliis Journal of Poliial Eonom Vol. 8 No pp doi:.86/66 [] R. C. Mron hor of Raional Opion Priing Bll Journal of Eonomis Vol. 4 No. 973 pp doi:.37/3343 [3] J. C. Hull Opions Fuurs and Ohr Drivaivs 8h Ediion Prni Hall Uppr Saddl Rivr. [4] S. L. Hson A Closd-Form Soluion for Opions wih Sohasi Volaili wih Appliaions o Bond and Currn Opions Rviw of Finanial Sudis Vol. 6 No. 993 pp doi:.93/rfs/6..37 Coprigh SiRs.
6 G. J. JIANG E AL. 33 Appndi In h appndi w firs driv h momn gnraing funion (mgf) and hararisi funion (CF) of a runad normal random variabl. Rall ha h pdf of a runad normal random variabl ~ r ( ) fr ( ) f( ) [ a b] whr f ( ) is h normal dnsi funion and b a. () h mgf of a runad normal random variabl as ~ ( ) is drivd as r b Mr ( ) E fr ( )d a ( ) b a π b a d ( ) π d ( ) M ( ) whr M ( ) is h mgf of a normal random variabl and ( ) b ( ) d a π b ( ) ( ) a d d π π b( ) a( ) b a. Similarl h hararisi funion (CF) of a runad normal random variabl ~ r ( ) is givn b ( i) r ( ) Mr ( i) ( ) whr is givn b () and ( ) i is h CF of a normal random variabl b a ( i) i i. N w driv h CF of h log pri wih runad disribuion. Dno Z as h sum of h iid squn ~ r( ) hn h mgf of Z is ( ) r ) MZ ( ; a b) M ( ) M( and h CF of Z is ( i) r Z ( ; ab ) ( ) ( ) whr w mphasiz ha h hararisi funion Z also dpnds on normal random variabl paramrs and runaion paramrs ab. Dno S as h sok pri a h nd of da and ln S is h log pri. Suppos w hav rading das and h dail log rurns ar iid runad normall disribud in a risk-nural world i.. Y ~ r ( ) ( Y lns ln S ). h log pri a h nd of da is Y h mgf and CF of whr Y. ~ iid r ( ) ar drivd as ( ) M ( ) E M( ) i ( ) i i ( ) E ( ) b a b a ( ) M ( ) ( ) i normal mgf normal CF. Coprigh SiRs.
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