A Bias in the Volatility Smile

Transcription

1 A Bs h S D M. Chc Ths A. Hs Wg L J Muhusw Sb 9, 04 Absc s w-kw h k cs f s uc vs h v c h v s. W s h v f s wh Bck-Schs-M cg sg v, w wu bsv ss, skws, sks f h ghs us us. Wh s hc ssb s h b, s fsb cc. s v ffcu c h b wh ss s fc k fcs, s s c bv. W c s h w bu f hs ffc bu 8% f h bsv s. K ws: Bck-Schs-M, cg, v, v, v s, cu fc, s JEL Cssfcs: C8, C63, G3 Th uhs, scv, Js C. Fs Ew Ch f MBA Sus Pfss f Fc, D f Fc, Lus S Uvs, 909 BEC, B Rug, LA 70803, , chc@su.u; sg Asss Pfss f Fc, M Ahuj Cg f Busss, Cv S Uvs, Euc Avu, Cv, OH 445, ,, hs3@k.u; Ws Fcu Fw f Fc Pfss f Mhcs, D f Fc, Okh S Uvs, MSCS 56, Sw, OK, 74078, , w@h.ks.u; Assc Pfss f Fc, D f Fc, Cg f Busss As, K S Uvs, P.O. Bx 590, K, OH, 444, , juhusw@k.u. Ths ws s h 3 Au C-FDC-Hus Dvv Scus Rsk Mg Cfc, h R/Fc 03 Cfc, h HEC M 03 Mhc Fc Ds Cfc, h 03 OsMcs Cfc, h Suh Fc Assc, h Assc f Quv Fc, Bbg, b, Lus S Uvs, N Uvs f Sg, h Uvs f A, NYU-P. Th uhs c hfu cs suggss f Rb Bks, Jsh Zh, Ysg T, P Lck, H B-Au, Pu Kuc, Ph B, Jh Twbg, Psks Gbs, Tj Puh, Jsh Lgh, Mk S, Ov H, Sv Muzz, Rb Jw.

2 A Bs h S A s s h chs vhc f bgus. H Mv (P, 85). uc O cg s csuc wh h bjcv f vg h c f gv. Th s wk h s h Bck-Schs-M (BSM) (Bck Schs (973), M (973)), whch vs h vu f fcss k hugh h us f hg f bg-f cs. F ug ss h ks csh s cus cg css ug h s f, h vu s b h c f h ug, h xcs c, h skf, h x, h v f h ug. Of hs fv us h, h v f h ug s ubsvb. S f h ws uc, ws cgz h h h f vus f s v, c v h css. Obsv k cs f c s cu h b s h bck u s f h ubsv v f h ug. hs, cu f h v f h ug h ws bg us b k cs s h c s. Y hs css v c g: s h s ug wh h s x h s v. Ths sh f v css sks hs c b kw s h v s, wg h U-sh h s f bsv. s w w-sbsh h h ss s vs wh sc bh sk cs x, h cb ffcs f whch f kw s h v sufc. Y hs s s uzzg h. 3 Th fs vss f h w b Lé R (976) Schs T (978). E s css ssu h hs su ws fc f bsv ccucs h, such s gh ccu f h sck cs w schus f h w s g vu h h c wu b v sv c. Th c ch k b css jus f uqu vs hs b k vg v, s css wghg h vs b g vu, ss, h s vgs. Ths cs v s ss c h wgh s v. 3 Mhw v gs s f s s, T v k bu v ss s schzhc, (995,. 4).

3 Bcus h v ss h v f h ug, whch us b f b uqu, s h ff b sk shu hv h s v. 4 Rsch g xc hv s ucv h fc h s, f s, s ss. S skw s s sb sk. Whh h sh bw v xcs c sbs s, skw, sk, v f h hz s c h h s u vs f h ug ss. Th bvus x h hs b scuss xsv h u s h h s fc k ssus f f. Acu cs h k c h s s hs ssu, bu hs ssu s c fc s. Ah x s h vs f h ssc css fw b h ug cu b h suc f h v s. hs, w scb h x, s h bss f s x. Sus s fc ks. f f h ssus f h BSM, cu cs kw v cuus, csss g, wu h b v s? u ss h h vs f s wh ff sks wu qu h kw cs v us c h s h fs c. W shw h h hs sg xc su s u: v f vss c s ccc wh h Bck-Schs-M usg sg v, h vs css sks s ws v s qu b. Rkb, h s h f f ss, skws, sks h g sb h s bsv cc usg cs f ks h f f fc. W s shw h hs s uc h sh h u f h, whch s css wh c bsvs bs cu k cs. Of cus, hs su s uzzg, f shckg. Hw c b h c c wh kw v h b ub bck h v u f h c? x b b f u, h hs b susg bsc f h cs s f hw v s b. W w shw h h sh hg f svg f h v c kb ff vs f s h v b sk bu hv h cc ffcs. W s shw h h b bcs v ffcu cgz f h k 4 s vs css, ss kw s h sucu f v, cu gc hs fc. Th qus h v b cs v h f f h. Ths ssu s cu ff vs css ff hzs.

4 fcs h cus s s b c u hs, whch, f cus, s h s x f h xsc f h s. W k c h hs h buks h s. s qu g bv b v b h f k fcs h cu cg s. Y, s c h h b f h s cgz h b f kwg. A bvus qus s, hf, hw uch f h s s ccu f b hs b? W cuc c ss f h h w bu f hs ffc s 8% f h k-bsv s, bu h f s s s s, hs ffc c b v 90%. W x s ssb xs sus bu u f h hs s b wh c fx. Th f h s gz s fws. Sc w vw h xs f h v s h hv h u. Sc 3 vs f scfc f h b scuss f h kw chqus f vg vs h v s. Sc 4 scbs u cu scfcs vsgv. Sc 5 ss u sus, whch cus uc fgs bu s c ss v h Sc 6 cs u ccuss.. Pssb Exs f h S Fgu uss v s uc b h csg sk cs f B c s h Chcg B Os Exchg Ju 8, 03. Th s x Fbu 6, s h hv bu h g. 5 Th sck c s $54., h sk-f s 0.05%, bs x -h U. S. Tsu bs. Th cs k f h CBOE s wb s h cs f busss. EB s vs s hs Ac c s ffcv Eu c s. Th s s c sc gh b f s sk. g, w w us h s s u bu s s h hv b bsv. A c s, u sus w g sks skws, hs vs w b s wh v. Obsvs f v ss hv vk ub f ssb xs csb c wk. Much f h wk hs fcus fg sh fucs f h s sufc ccu s f h v f s wh 5 A f u sus w fcus c s. Th v s s f csuc wh u s cb f c u s. Th fcus f u s bs sg f us f h Bck-Schs-M f Eu c s h c f fc ks. S bs w s wh us. 3

5 sks xs cu. 6 Ou s ffs. BSM w, h xsc f s s h f ccug f h ffcs f h c f h ug, h xcs c, h sk-f, h x, h v, h c s wh shu b. 7 Thus, h s fc ccu f h ffcs h c. Th fc s, hf, bsb h v s. F x, U- sh s suggss h f cg h fv s fcs h ff cs, s wh h hghs ws sks s xsv h hs. Such x s css wh bg-f cg BSM w. Wh cuus csss g, c c c h h s ug. Thus, fc BSM w, sg c b v ss xsv h h h s ug. Ohws, h wu b bg u.. M Assus Pvus sch hs ssu h h v s s u v f f h s ssus. Ju csss scuss Rubs (994), M (976), Bs (99), Ds Su (999), wh schsc v s s b Hu Wh (987) su b Hs (993), Rubs (994), Dus, Fg, Wh (998), hs. sg, c su, Bksh, C, Ch (997) f h cg schsc s s, schsc v, schsc v wh jus s h s, g u sg v. Th ssb f h ffc f - s hs b s b C Su (996). Th ffcs f sc css (P, Rub, S (999)), sc ck sz (Ds Mhw (009)), bug ssu s bg (B Wh (004), Shu, P, L, Wu (00)), bhv (g (009)) hv s b cs. A w-cc bf s h uusu hgh vs w-sk us s f f csh suc, s b Rubs (994) bc fwg h csh f 987. F, h s h h ususg cs h s skw c su wh s c b, bu v s b f h, s us b Shw (998). H uss hw cs v 6 Sh v fucs cs, bcus h v cs h c qu f s h gh cu b, bu whch cu c. fc, s c f s qu cs, b sg h c c, bu b qug h v. A vs b s bu 0, g h h c h s s $0 bu s b sg 0% s h v h cg. 7 Rfc Bck-Schs-M w s w whch f h ssus f h cc, h ws, fc k. Hc, cs wu cf h. Of cus, h k ssu h s h h v s kw g u b vss. 4

6 b h cs sc f vc uc skw wh h v s bck u f h BSM. Th s s bv h hs xs cc. Y, h s s bv h cu fcs cu v s f h v. f cu fcs hv sgfc fuc h v, s ffcu h c f hs cc ss f h s., s w shw, h cs f hs ffcs wh cu s ks h b v ffcu c.. Cu Css A h ss sc f cu ffcs v s s Hsch (003), wh shws h su s h us c hv sgfc c h v. H css fss f qus, h xsc f b-sk ss, -schus cs f h ug. Ths sgh ss f c us c sgfc v v s. Ahugh Hsch s su s bb s cs us, h ffcs sgfc. Hsch s fcus s h --vbs bs h wu c bs us. cs, w ssu h us cc su fcus h chcs f hw us cv uus, kg ccu cvgc c, h sg, h -fg gh us. sh, w ss ssu h ks fc h ss h h BSM ssus h s c ccg hs bu h su csss us wh h s f hc kwg cu cbs. Th u v s kb v f scusss f cu s. F x, gv h h s c, cs-f su, ss f v ss subjc c scf b h us. Ths c s s scf xc h u. F sc, C Fgwsk (993) scb h css f ccug v s svg h cg f h v h ss h k cs qu (. 659). Lkws, Rubs (985) s f h h Nw-Rhs h cvgs quck (. 460). Chbs Nwkh (00) s h v c b s ccu usg v sch chqu (. 98). O fw sus xc scb h c qus. F sc, Lé R (976) qu h uc sch cu v su c wh ± $ f h s k c. Schs T (978) uz v cu h h wh us chg h s b ss h %., h s f 5

7 sg ss vs, whch qu b v fg chqu. Th vwhg j f s k f hw h v s v. 3. F Scfc f h Pb Th Bck-Schs-M fu s gv s wh c SN( ) K N( ) ( S / K) ( ( / )) (), whch N( ) ss h cuuv s fuc, c s h c c, S s h cu k c, K s h s sk c, τ s g x, s h cuus cu cs kw f s, σ s h v f h sck s u, whch s f h uz s v f h cuus cu u. Nw c M s h c f h s bsv cv qu k. Sg h c h k c, h v s h c su : wh c SN( ) K N( ) M ( S / K) ( ( / )), wh s h v, f s h v h fcs h k c qu h c. Of cus, s w-kw h s ssb c sv () f h v. 8 S schs hv x c ssus v cs-f xs (B Subh (988), Chc (996), Chbs Nwkh (00), Bh, Chsfs, Sk (996), Hbch (004), K (006), C M (996), L (005)). 9 Cs-f xs usfu ssh ggc cs bu hv bc ss cc g sc sch g bcus f h cs s f cus. 3. v Tchqus () 8 A c b Gh (0) shws f h h Bck-Schs-M s bg css f fucs f whch sus c b s fu. 9 S f hs chqus c b cs v h h us sg s qu s guss. A f cus, cu ws v chqu f h c cs-f x. 6

8 v hs v ccu sus b kg s v f ss, gussg h vu f h fuc h s v, gg w s, sg h css v, u hfu cvgg h f su. Cs h h s, f h v. Ths s ucs vu f h, c( ). Wh h bsv k c f h s c M, h ffc bw h k c f h s c( ) c SN( ) K N( ) c M ( ) ( ) M ( S / K) ( ( / )) ( ) wh Ths v chqus g vv h us f -fg cu f h cg qu, c( ) - c M = 0. Th cu gh sv hs b w ssss c qus f busss s f cvgc. Fuh, h c v, ccb, f ccug h w fuc h s f h v. As, s s h vv ccus f v f qu bqu h fg chqu. A c f hs s h h sc hs vk h v c f h cs f hs chc such fcs s sg s. W w shw h hs cu css hv sgfc fuc h xsc sh f h v s. Th b f hc kwg s wh -fg hs, fg s c b f sus f hcs cu scc. Ous f svg f h v, hwv, -fg bs v c fc g. 0 Thus, ususg, h hs b uch svg such bs h fc u. Kz (99) s w hs f vg h v, h Nw- Rhs h h bsc h. Bw (990) Chss (997) h bsc h. Oh w-kw fg hs, hugh h fc u, cu h sc h, h gu fs fs s h, h Dkk-B h.. (3) 0 Fg h f u c vs cs s -fg b h ss c fc. Ms fc bs cvgc, hwv, vv vvs. F x, cvgc ssus s svg f h b c, c sch b Muhusw Hs (0) suggss h h b gh cvg. sg, h gu fs chqu hs b cuf bs f c Bb hcs. S Chb (999). Rgu fs s scuss b Fg (0), bu u kwg hs b c swh h fc u. Dkk-B s us b Mb s h fu f. 7

9 Pbb h bs kw -fg v chqu s h f Nw- Rhs h. Gv s ws us, w x s. Ax B, w w v fs x f h v. W h us h x x h bhv f h Nw-Rhs h. Th fs x c b uc h fwg quc qu: q q q K K S q S K 3 K Of cus, h su s w-kw s q q S S c M 0 q q 4q q 3 q S K S K c c M M S S K K S K S K S K. (4) Th g s gv b S K S K c c M 0 S S K K S K S K S K, whch s h su s b C M (996). N h h s c b S K sv s w wh q < 0, whch s quv S K S K c c M M 0 c M, s gv b S S K K S K S K S K. Chss (997) ss h u bs wh h bsc h h us shs us fs h, kw s h Nw-Rhs h (. 330). 8

10 sg, hs h su hs b cs vus sch. Th uc x s h su F( ) SN ( S,, K,, ) K N ( S,, K,, ) c 0. (5) Th Nw-Rhs h s gh h s sv hs qu usg h fwg css, x x Fx ( ), F ( x ) wh chc f x 0 s. Ev f F(x) hs vvs vwh, hwv, h Nw-Rhs h. F x, Nw-Rhs s cvg f h s fuc, F(x) = x 3 x + wh x 0 = 0. Chsg g sg s ssu f Nw-Rhs. Ev wh sg g chc, h c f F(x ) F (x ) c sw cvgc cvgc h wg. F x, f h s xss, F(x) = s x x/, sg f x 0 = s h gs sv, sg f x 0 =.0 s h gs gv, sg f x 0 =.0 s gv f, sg f x 0 =.05 s sv f. Thus, sgh chgs h sg c c ff sus s css, sus. 3 Th fc s f h fuc F gv b f M (6) f S, K (7) whch ccu wh h 0,, 0. Ths fu ws f f scv b Ms-Kh (98). W x s s fuh. f 0, h h w css cs: h () f < < 0 wh f S K S, whch ccus wh 0 () 0 < < wh K f S K S, whch ccus wh 0. Whu ss f g, w K k h sc cs f scuss. Thf, F 0 v 0. f 3 Th gh cgz hs ssv cs css s chs h cs. Rc suvs f cc fc cs v b Hsh (99) B, Ss, Ss (006). 9

11 f, h h s w css: h () f 0 f wh f S K, () f wh f S K. h cs, w hv F 0 v. f F Equ (5), w bu h Nw-Rhs h wh x0 f f h xc fu gv b h s f h quc qu. W hv h vg v, F F S( ) 0, F wh fucs f h v wh ( ) ( ) ( ) ( ),. f x su xss, h h su s uqu b h sc sv vg. Ths su ucs h cvgc ssu f h Nw-Rhs h s : f h gh cvgs, us cvg h cc su. Ths sus f s h fwg h. Th. L b h fc f h v f h Bck-Schsf M fu. Th w chs h sg x0 f Fx ( ) 0 W hv x x. 0 F ( x ) () f 0, f h Nw-Rhs h. 0, h w hv x x f. Thus, h Nw-Rhs f f h vss h v. v, x () f, h f cvgc s quc v ( x ) ( ) ( ) ( ) f x. Th gh s cvg h 0, h w hv x x f. Thus, h Nw-Rhs f h uss h v. Th gh s cvg h v, x, h f cvgc s quc v 0

12 ( x ) ( ) ( ) ( ) f x. () f 0 f, h h Nw-Rhs h uss h v. Th gh s cvg h v cvgc s quc v ( x ) ( ) ( ) ( ) x. x Pf f hs h s v Ax A.. 4 f x. Mv, h f Th gh sf c gu cvgc s w chs sc sg Th vss uss u h cvx f h BSM fuc h v. N h s ssb h h f h Nw-Rhs h ucs sw cvgc squc f h v. Th cvgc cs h h gu h h f cvgc s quc. Wh w hv s g w f vwg h b, h cvgc cs c b sf f sc css. F x, f ( ) 0 cs (), w c c h x f b h cvgc f qu. Th c s u hc vv wh h v. cc, cvgc s bh css u h sc csg csg s. f v. f f, h w hv x 0 = x f, h sg s h f w sc ff sg s, h w wu hv h x 0 f x, whch cv h h. F h fs cs, w hv x > x 0 0 x. Thf, h w ssbs f h c f x : () wh h fs s x s x f f, w b h s su s h h; (b) wh h fs, w hv x < x c b c h s f. Thus, h Nw- Rhs h s ccusv uss x f f s. Th s h ccus f h sc cs, x uss x f s, whch s h s su s f 0 f cv u h. Th Nw-Rhs h c s f f h ss. Th bjcv fuc b ffb s. Wh hs b s xs f h cuus BSM, c ug, whb h vu f cv b s b 4 Hbch (004) s h h sc cs whch h sk s s h fw w gu cvgc. Thus, h v f -h- fw wu css cvg usg h Ms-Kh sg. Ths sc cs, hwv, s cv b h B-Subh fu.

13 gs bu h u h k, jcs g f scss h c -ffb. Th j f v sus h u h Nw- Rhs h. Th k ws us f Nw-Rhs s suz b Jck (006) wh ss h Cv ws hs h h bs -u h f chc f h -fg f sh fucs s Nw s gh. As, hs s ws s. (. ). Wh u fcus s Nw-Rhs, w cs h ghs. 3. Nuc Pcs Esg cg wh cu css vvs h us f sc, fg hc x cuus csss. R s hs ssu xc h u, bu cu sfw g s s ub cs, wh gus x v b L L (0). A sc sc f whch uc cs s v s h qu x f h Guss cuuv s fuc. Gv h h Guss s c b g, cu gh s css v hs cc u h cg, h su w b cc s g f c ccuc. Th sv w-kw xs, f h scb h cssc Abwz Sgu (97, ), s scuss bks s (. g., Chss (997, ), W (998,. 0), Hug (998, ), Wh (006,. 5)). Ax B w shw h fu ccu s h bb c s vs h v css sks. W susc h s cu gs s h bb suffc g hs b v. W x u w gh gh f hs cc f s susc h ccuc h bb s suc f cu u sus. Ths b h cs h s cs whch uss uw f hs cu ssus. 4. Cu Mhs A ccus sus f hs su cuc usg h ssc cug g R s cbu ckg, Rf (Mch (0)). Th ccus us s quu cs hc (8-b sg). Ths ckg ws fu c v h cs f h c s f ubs. 5 g, h hg s fu-s css. Fs, w k ssus bu h uc vus f h us hs us h BSM fu b h 5 Th R c s vb f h uhs u qus.

14 c f b v f cs. Th sfw ccs h s us h, s w s h ub f bs qus f uc sg, ucs h s vu vg. Ths uus cb wh ss f h fuc GBSO h cbu R ckg fos (Wuz. (03)). Sc, w v h xcs cs css w g f ss h ccus f BSM cs. Ths cs h k s h ssu k cs f s h s ug wh kw v, quv xs, vus sks. Th, w vs h css b ssug h h BSM cs h k cs us hs cs s h vs usg f h vus hs bv cjuc wh scf qu u c. Bcus h k cs fc h BSM ss, h sug vs shu b h s css sk cs. Fuh, w g gh f h vs css h g f sks cvg fu xs. W cs sv fcs cu ssus h fuc h sh f h v s. Fs, w v h uc cs f h gh. As scuss vus, h uc cu ccu v s wh h s s c h s wh s c f h k c. Bcus hs vsg c us fc BSM cs hgh g f cs, h c c b s f bw sg. W x cs f 0.0, 00, ch s f W s x sc fc u cs, h u f h Nw- Rhs h h v f h h chqus. Wh vus sch hs f hs, c su sgfc chgs h gh s uu. Th bs css us h s f Ms Kh (98) gv qu (7). Oh chqus qu v. W s wh - s h w f h v h Ms-Kh sg s h u f h v. 7 W s x h qu u. Ms sus f h v s hv b xchg-s qu s. P h 000, xchg-s qu s us f sxhs f s ss h $3, us f ghhs f s $3 6 Tc s gu cvgc, fc, s w vus scuss, h su c vg sgfc f h su. W cs h gh hv f f succssv s c v w sv gv f. W s s xu f,000 s, whch h gh ss h v s cs b h f s, v hugh h s gh b wh h scf c. 7 f cu sus v f z, h vus f s sv gv f, sug bbs f z, g ss. N h sg v wh bw z s b. f h v s w, s sws w h css. W b fw fus cvg wh gv. 3

15 . Au h 000, qu s bg g us f $0.05 f s ss h $3 $ f s $3. Sg 007, h CBOE w s g whch c s s. Ov, s hv b h g. Rc h h cu s h w ccu h BSM c h f h c bck h bck u h v. W ccu h c h u sxhs s. 8 W s f u cs bck h. Ths cs ffcv quu cs, s h c 3 c cs. 9 W w f hs -u cs s cuus, hugh sc skg h cuus u hc ss. 0 W cgz h c ug f g s x f h fc k ssus. W vw h f fc k s chcz f h cs f cuus, csss g wh xs, cs v sk-f. F, w s us h fv f chs svg - qus x hw h f h cs f v. Ths hs h Nw- Rhs, bsc, Dkk-B, sc, Rgu Fs. W vs h cg css g h v wh h cu cs h. Th sug v shu b h us c h s h fs c. Th s cc f hs x us s h w c u s u h fc k cs f h BSM w wh kw v h vsg h css v bck h v h ws us c h s. uv, wu s h su w wu v bck h g v. Bu fc, f w. Ax C vs s-b-s x shwg hw g u v f 0% cs u s v f 8%, vs f sgh h 40%, usg h Nw-Rhs h s v fuh ff wh s h hs. 5. Rsus 8 sg, wh cg s g g s v ucg b-sk ss, w w shw h cg, wh h fcs, c s v ss. 9 Ou hss s vs f cs xchg-s ks, whch scf h s g u. Cusz v-h-cu (OTC) ccs c us g u h s g. Du h g szs, OTC scs ss c u c cs, such s fu sx bu v h-w 0 fw css, c ug c ush h c bw s w bu. F x, 30- suck $8.50 hs BSM c f $7.77 wh w bu f $ Th BSM c wu b u $7.77, whch s bw h w bu. cu k, s c wu k b h w bu u u, hs cs, $7.78. Th vs 5.% (BSM -u c), 4.94% (c u bw w bu), 7.03% (c s w bu u u css ). Ths ffcs suffc g b f cc. Of h 84 cbs f xs sks x, hs w bu v ccus fv h css. css h vs ffc. 4

16 Th s c s s $00, wh h sk c s v f $75 $5 cs f $5. Th s s 4%, h s x 7 s, 30 s, 90 s, 8 s, csg s x f 0.09, 0.08, 0.466, , scv. Th v s s 0%. Rc h w g h cs bs hs s. 5. Ss U us Cs Ou fcus s h Nw-Rhs h, s s k b h s w us. Th sg vu f h Nw-Rhs h s h fu f Ms Kh (98), qu (7). W s cuc ss usg 00% s h sg, gv h v wu b bv 00% f qus. Ths sus ws fus cvg. Fgu shws h vs b sks usg h Nw-Rhs h u h vus ssus css bv. W s h wh c f 0.0, h qu hs shw s cu, h f v s. Th s s s uc f h shs us s vu -xs f h gs u, whch s css wh bsvs f sus usg k. Mv, v hugh h xcs cs sc u h -h- cs, h vs uch hgh f h u-f-h- s h h csg -h- s, whch s css wh c bsvs bs k cs. Wh h css cs f s 6hs, csg h c 00 ch s ucs sg c ghs. Y fc hs ghs s h chcscs f sks skws h f bsv wh k. Wh cuus cs, w c shws v, xc f ch s c whch cs h s ss. Bu f cus, s cc, scs c ccu cuus cs, ch s c s v uk b us s ghs. Thus, w s h h Nw-Rhs h s g u h sg v h ws us c h s css sks. fs u h v h ws us h fs c, h vs us shw f s., f h Nw-Rhs h w hv cuc ss bs h sg s f Bh, C M, L. W cu f ss h s sus. Usg u xs vs ugh x f h ffc f h sucu f v, hc, cb wh h s ffc, w cu x h v sufc. Ou fcus, hwv, s h c f ff xcs cs, s w w f csuc v sufcs, whch qu s g f shg g ub f xs. 5

17 As, w x fu h chqus f svg f h v. Ths g chqus qu sg v. F hs s f ss, w us - h Ms-Kh sg s h v. Th sus shw Fgus 3, 4, 5, 6. Th sus f h sc h Fgu 3 vu sgushb f hs f Nw-Rhs. Th bchk su f v f 0.0 css sks ccus wh cs qu cuus wh ch s c v h scc ccus f h ws sk. Th gu fs h, Fgu 4, ucs ss s skws sks f cs qu sxhs f gs f c, w c vs vu v. Sxhs cs uc skws sks f cs. P cs uc s f c f 0.0, bu h s us skws sks f w c. Cuus cs uc f css s gss f c. Th bsc h, us Fgu 5, ss skws sks, w c ss xcb h b. Cuus cs v b v uc h usu su f s vs bg w bw h u v. Fgu 6 shws h h Dkk-B h ucs skws sks f cs sxhs wh v wh w c. F cs s, ss ccu f c f 00, wh skws sks ccu f ch s c. Cuus cs g ss f c f , bu u cs v f ch s c. Ths sus s s Nw-Rhs h sc h. O gh sb sk f h v ss h s f hs ss bsv c, whch s s v -h-. Af g v f s, w f h s v s b ss h s u 5% b h usg Nw-Rhs wh v sh- s. Ess bu % f 0% kw v, f 0%. 5. Bsc Dscuss Ass f Nuc Rsus T c, w f v ss f g v h v f s c ccg h BSM. W x h s ssus s h b ug s sxhs. W f h h Nw-Rhs, h bsc h, Dkk-B uc v s vsu sus, bu h fg s h cfg. F cs qu 6hs s, h sc v ss, wh h hghs vs ccug wh h w sks. 6

18 O suc f cu cu b h sg. s w kw h chcs c sw w h s css h uc. Wh s ssb chcs, hwv, s ffcu ssss h c. Mv, h c x w cuc, w qu kw h su. Hc, u chc f h su cu bs u sg s. W csqu subjc v f h Hsbg Uc Pc: c su shg whu g. Thus, h sg cu w b v g b h w c ssss hs. Tc c hs ffc s w. Hg vhg s cs, s c c s f h s, bu c f. P c, whch w susc s c us sch cc, s bb s ugh. W sus ub h uss qu c f ch s, bu h bb shu. C h fc s c ug. Cs c c u g f cs hgh h h qu u. B fcg u c, c gv u c c v f -u cs. F x, sus cs qu s, sf, h hc ucs cs u h c cs. A c f $0.5 c v f -u (hc) cs f $0.45 $0.54. A c u u w hgh v, wh c u w w w v. Ths x c x f h s, hwv, sc wu h h vs wu b uf hgh w. Th vc shw h suggss h bs hgh vs uch c h bs w vs. Mv, h ffc sus f cuus cs u cs s cu g. S c ug c x f h ffc. 5.3 Fuh Ass: Th g Effc W x u ss s fuh b xg ssb suc f h b. f h suc c b f, cu c hw ss hs ssu wh s wkg wh cu cs h u v s kw. W fcus h sus f h Nw-Rhs h, such s s w us gvs v s sus sv h hs. Th vg f s h vv f h c wh sc h v. svg f h v usg Nw-Rhs, h v qu cvg s b h vg h sc f h fuc f z. Thus, h vg 7

19 v cu s wh h b s. Fgu 7 w shw sgh gu uc f h fs gh Fgu C, h cs f c ug c f $0.0. Th sks Fgu 7 sc cs ($.50 ) bu cv h s g, $75 $5. P A shws h s sf, wh P B shws h vgs css h csg gs f ss. K h s su f hs bg x, w kw h u vgs bcus w kw h u v. cc, kwg h vgs wu b b, s w ss. Phs susg, h vgs x w f h s gs f ss whch h vs qu hgh. Lw vgs suggs h wh g, g v chgs qu v c b s u. A csu x f h vgs shws h wh h bw 0.5, h vs v subs f h u vs. Thus, w hv sub usfu u f hub. O gh sb gu h wh vg < 0.5, h s v suffc cfc us h v. W gh s h h c v b fv (NR). A h s NR shu bb b sc. Th s c b fv (R). Nhss, h w -v bs wh such cuff. hs x h u v s kw, s w c s h u vgs. cc, h u v wu b kw, s w c h u vgs. Px vgs cu, hwv, b b f h vs f h hv b v. Fgu 8 shws h ffc bw h u vgs h x vgs f f h xs. Wh h xc f h -h- cs bg z, ffcs gv, g h h x vgs g. s css, such s h hghs ws sks, h csb g. Thus, g x vgs k ss h 0.5 cuff. Thf, w cs h sk h s h NR bs h ukw u v R bs h x vgs. Bu h s chc. Th u vgs wu b kw. Rgss f whch v s us ccu h vgs, h vu f cu f s NR bs s vg qus ccu sg s fws: S : Ccu h v. S : Ccu h vg usg h v. Th vg s us c h Nw-Rhs h, bu h h fu hs. Ths hs, hwv, c sc ss f h sh bw h c h v. Thus, h fuc b h cc cu b h vg. 8

20 S 3: f h vg s bw h cuff, ccu h h v s b fv (NR). Bu f h v s NR, hw c us cu h vg S? Th vg wu, hf, hv b NR. A f h vg s NR, h hw c ccu h h v s NR? A s. Thus, s g ccu h bcus hs b ccus wh v w vgs, cu us u vg s sg f wh hs b xss. Ths h s v us b OMcs, whch s v f h vg s bw W w kw h hs ch s ccu. F h bg, hwv, us g h ccu sg xs h, hs v cc, b. T, h u hs h v s s hugh s cg sccs h hugh s f k fcs c h BSM. Sus w s b c u u cs h c s hs f c scc. F x, us csvv jc v s u f % h c, u ss h h b-sk s. Fgu 9 s Fgu 7, bu h f h cs hv b f b fc f.0. Fgu 9 c ks x s sh Fgu 7, bu h gus f h vs vgs qu ff. F x, h 30- (x = 0.08) 75-sk. Whu h u, s v s 36.87% s vg s 0.36., hus, s NR. Wh % cg u, hwv, s v shs u 58.98% s vg s.47, whch wu k R. Nw, cs h c f hs su. W hv u-f-h- h w kw f hs c x s c v f 0%, wh h c u ff s wu b cc. s v 37% s s wc h u v,, sgg h ccu gc, s vg s s w h shu jc s bg NR. Bu f h w c s % u v BSM, s wu b f h fcs h c h s, s v s w s h s h u kw v s vg s suffc hgh h s R. s qu xg bv h fcs cu k c v b. A wh gh sb ssu h h s c u ccu f k fcs, c u cu s ccu u s us cg. Thus, w f ccu h c s v c b NR wh s fc k 3 s c hw OMcs jusfs g v wh vg s ss h 0.5, wh hs hv v s h vg. 9

21 s vu cs ccg, bu bcs NR fc k wh b cg ffcc. Mv, f b vg cuff w usfu u f hub, shu h u u ccu cus. Ufu, s. Ag h u f hub wh c f 00 sus h jc f uus css f s whch h v qus h kw v. F x, wh 90- s, w wu jc sks f , v hugh h vs 0.0. F 30- s, w wu jc sks f 85 bw v hugh h vs s cs 0.0. F 7- s, w wu jc s wh sks 90 bw v hugh sv f hs s hv vs f 0.0. Mv, sg b u cus cs h vs vgs s h h bc R, wh h w NR h bsc f b cg u. 4 A bvus qus s whh h s v h wu wk b. ss uk h s wh s, such s ju-ffus schsc v s, cu ssb b f f hs b. Phs ss-scv wu b b. s ssb h h -f v, whch svs f h v f h bbs whu ssg hugh f, cu b b v. Ufu, Jg T (005) f s cu ccs h s w. Mv, -bs -f vs bh qu h xsc f qu k b g f sks, whch c b b f vu qu s. 5.4 Ec Ess f h c T hs w hv shw hw u hhc s ssu b c fc k b h BSM wh kw v uc v s hugh h b f cu hs u h g v. wu b vub xcs s f w c s hw uch f h ss bsv cc ccu f b hs cu b. Th b, hwv, s h u hs w hv us hhc s wh kw v. Wh w u h w f s g, hwv, w u h b h h u v s kw. Sus w us h v f h h s css -h- s x f h kw v. A-h- s c h s w, s h 4 W s cs fuh ssb f uzg h b. s f ccug h v, gh b b ccu h vc. Ufu, h vc vg s s ws f - u-f-h-. qus h vg v b w s h v. Thus, f hgh v scks, h vc vg wu b v w. 0

22 f s kg hs s. Th vs shu s ccu fc h k cssus, v hugh h css s h u v. Rc h w us h c s f s f B s. Fgu 0 shws h vs bs h k cs f h B c s us cs h vs bs c f h B s. As, w us h v f h x -h- s h kw v. Th k vs hgh f h w sks, bu h shs kb s, cu h s whch h cuv ss ug uw bh bv bw h sk. W w hv w ss f v ss, s w c s h f h c s ccu f b cu ffcs. Sus h w f h gu f h s s h su f h squ vs f h vs f h kw v bs k cs. Th w f h gu f h s h ss f hs cu ffc s h su f h squ vs f h cu vs f h kw v bs cs. W h xss h f h h f b: N N c,, ( ) ( ) F hs B s, us u b x 40%. Thus, f hs qus, w s h h cu ffc ccus f bu 40% f h bsv s. W w uk f xsv c ss. D cc f h OsMcs s qu bss (Ju, A, Ju, Ocb) f f s h cu h scks h Dw us Avg s w s h S&P 500 Nsq xs, g f 896 ug-qu cbs. A f hs s hv x h g x. T hs, w hv ss Ac-s,.., xcs, ssus hw h gh ffc hs cu ccs. Nhss, s ssb v c sus whu ccug f h xcs fu Ac-s s, whch h f s vu scks. Thus, w cu h vus usg h b, whch cs h ffc f xcs h s v b vs. Of cus, h S&P 500 Nsq x s Eu-s. A cs bs h f h b-sk s.

23 T w f cb css h ug scks s w s v s cs u g f - u-f-h- s, h sk cs uc h cs css 0% bv bw h cus c f h ug sck. s css, s g w ugh ch h 0%, hs css h sck ws f h ss f h qu. Ths ss ccu jus 0 f h 896 css. W hv s bv h u s f s w bu. Rc h u hhc s, h gs cu ffc ccus w w f h -h- xcs c. Bu wh wkg wh c, f hs s suffc uc b cs, w c us h. Tucg u sks 0% bv bw w su ss f s bsvs f whch h cu ffc s k b h gs. Thus, u ss c w bus. Ag, w ssu h h v f h css -h- ss h u v. Ths v s ccu hugh h NR gh wh c f h MK s. A f h s h c usg b wh,000 ss. Ac xcs s k ccu, LBOR s us s h s. Ths cu vs -bs c f ch f h s. Th - f h csg b-sk s s us s h k c f h. Th v s h ccu f bh h k cs, g usg h NR gh wh c h MK s. Ths sus us ccu h -ssc v s f hw uch f h c v s s u cu ssus. Tb ss h, s v, xu -cs f ch ug. Th vg s 7.96% wh xu vus s ug v 80%. 6. Ccuss hs, w x sg sufuus qus. Sg k fcs s, c us kw v c h vs-g h g v f h c? Th sw wu bvus: c su cv h g v. As w shw h, hwv, cvg h g v s s sghfw. fc, cv vs s s css sks h kb s ss, sks, skws. Mv, hs s s uc f sh- s, s s h cs f h ss bsv sch cc. W shw h wh h chc f -fg chqu, c, sg v,

24 h h f c ug c uc hs s, sg x s h hs. s g gu h hs ffc s ch, c buc f cu css bu hvg gfu sgfcc. Y vw f h u shws kb bsc f cu scfcs, h ffc s hv b b schs. Suggss h cu s sc bsvs h uuswh u xcssv w vgs g h fc h us ccu gc: h v s qu h vg, whu wu h us h vg f h v c b us. Mv, s w shw, wh h fcs fuc h s, s wh uuswh cs w b, hb bg hs bs f h h fcs h v. A v f hs ccs w h, h s vc h u h hs v sc bsvs f hs s. A cs jc bsvs s such sch sg h su wu hv b. C h b b fx? W c css g hs s b cg h cs u quu cs g ch s c. Y, g h f qu wu qu g cs wh 3 cs. k h xchg-s s us v 30 s v f sxhs s, v wh cg, w c s bsv v ss wh s c b Bck-Schs- M. A su, g 3 c cs wu xc h x hgh css. W k c h h ffc shw h s h suc f h s. Ou c ss suggs h w bu s u 8%, bu w f csb vc h s hs ffc c b x hgh, s shw Tb. Ev u h s csvv s ssb, f x -ffh f su s v b css b h sch h s bvus fx, h csb gus f cc bh sch g. W f wh h sccg ccus h h v ss bsv cc cu b sgfc bs, whch c fs cg sgs, g sgs, bg hg scs h shu b sk-f bu fc c subg w z f uc. 3

25 Ax A. Pf f Th Usg h sc- T h f F(x), w hv wh s bw x x w hv f v = 0,,, F ( ) F x F x F x x x x x! ( ) ( ) ( )( ) ( ), x. Usg h Nw-Rhs fu (6), x F ( ) ( x ) F ( x ) Sus h. f w chs h x, h w hv f 0 f sc F(x 0 ) > 0 F (x 0 ) > 0. W s hv Fx ( ) x x x, x F ( x ) 0 F ( x ) F ( ) ( x ) 0, sc ssfs F"( f ) > 0. Thf, x x. Sus hs hs 0 f f, w v h + cs. Wh x f Fx ( ) x x x, F ( x ) x F ( ) F ( x ), w hv ( x ) 0, f wh s bw x f. Thf, x x hs f. Th Nw- Rhs h s cvg sc h squc s sc csg bu bw. Th v s h uqu su f h f F, s h gh us cvg. Th s x x F ( ) F ( x ) x ( ( )) ( ) ( ) ( ( x )) ( ( ) ( x ))/ x ( ) ( ), wh h sc fws f h vg v fus. F h csg f h fucs ( ) ( ), w b h qus, Ax -

26 x ( ( ) ( x ))/ ( ( x ) ( ))/ x, ( ) ( ) ( ) ( ) S f h cs < < 0 wh ( ) 0, h gh s cvg f f ( ( x ) ( ))/ ( ) ( ) f. Th f f h sc cs () s h s, wh s sv csg s csg gv. Ax B. Es Effcs f h N Pbb hs x, w s h h s f h bb c x f x s h c g -f cuv f h v gs h sk. L c M b h bsvb kw cu k c f Eu c. Ths vu s h k s u ccu h v h BSM cg. Th v,, s f f h fwg qu: c SN ( S,, K,, ) K N ( S,, K,, ), M S K ( S,, K,, ) ( S,, K,, ) ( S,, K,, ) (B.) whch h ukw s fuc f h sk c K f h us f hs. Th sk-f s s ssu b kw, h ug ss c S s bsv h s. Hc, h h s S, τ, s css hughu hs. Th qu (B.) c b sv f h v. Th vu f c s sc csg wh sc v, whch s s h h vg s c c csg, 0. Ths c gus h f su xss, w b uqu. f s c bg-f k, h h fwg bus kw f h vu f h c : (S K - ) + c M < S. Ccv, hs cs gu h xsc f uqu, gv vu f v, s h b s w f. Ax -

27 Th s fuc x h Guss cuuv sbu s gv b x k k z / ( ) x N( x) z. k k 0 k! (k ) (B.) Th ss (B.) s bsu cvg vwh. Thus, h qu (B.) c b w s f h w ss: k S K ( ) k k c S ( S,, K,, ) K ( S,, K,, ). (A.3) M k k 0 k! (k ) Th uc h sv h v s uc h w ss sv h v b h Nw-Rhs h f fg, s Ms Kh (98). Th h s us bw b vg h uc f h ss. B.. Th v f k = 0 0 Cs k = 0 h w ss (A.3) f h x f h v. Th fwg hs s x f h k c f h Eu c f h x f h v (B.), S K c M S( S,, K,, 0) K ( S,, K,, 0) ( ) 0 0 K K K K K ( ) 0. K K K Tkg h vv wh sc K bh ss f qu (B.4), w b K 0 K S K wh ( S K ) S,, K,, S,, K,, S 0 K 0 (B.4) (B.5) 0 Ex f h xss f (B.5) shws h b z, s c uc K -cs vs css sks. 5 Uk C M (996), w sv h v c s fuc f h sk c, sc u us us h v s. Hc, w s usg h sg bhv f h s f h v (.., h fc h h s hs u sk c h cs h uw cuv f h v s). Ax - 3

28 B.. Th v f k = Cs h w ss (A.3) f h sc x f h v. Th fwg hs s h sc (cubc) x f h k c f h Eu c f h cubc x f h v. S K c S ( S,, K,, ) K ( S,, K,, ) M 3 3 S ( S,, K,, ) K ( S,, K,, ) 6 (B.6) S h vus cs, w k h vv wh sc K bh ss f qu (B.6) b 3 ( S K ) S K 6 K K K wh S,, K,, S,, K,, S K S K S K. (B.7) Th xss f (B.7) vs h h sc (cubc) x f K h s Guss cuuv sbu fuc s cus s ffc h cu v, bcus h u (B.7) c chg sg f sv gv. Cs h u f (B.7), fu s fws: 3 S K S K K 6 K F hs, w hv h xc qu f h sk c K h s css ch h u f h v. Th s, K fwg qu: K 0 whch K s su f h 3 S K S K K 6 K.. (B.8) Ex f hs xss shws h h uc x f h Guss cuuv sbu fuc c uc -cs vs css sks. B.3. Th v f k = Ax - 4

29 Th w ss (A.3) f h -h x f h v. Th fwg hs s h -h (( + )-) x f h k c f h Eu c f h ( + )- x f h v (A.3): S K ( ) cm S S K K S K! ( ) k k k (,,,, ) (,,,, ) (B.9) k k0 k k B kg h vv wh sc K, w b ( ) ( )! ( )! k k k k k S K k k k 0 k k k 0 k K K wh S,, K,, Ax - 5 S,, K,,. Cbg K wh sfcs, w g K k ( ) k k S K k k 0 k! K (B.0) k k ( ) k ( ) k k S K. k k k0k! (k ) K k 0 k! us h s ffc, w us wh h s K * such h h K * s su f h fwg qu: * * k k ( ) * * * k ( ) k k * * S * * K 0 k k k 0 k! (k ) K * k 0 k! (B.) Ths s s u h v f h f x,. N h s, h f-h s f h bv qu chs * * k k ( ) * * * k ( ) k k S K * * * k * * k * * k 0k k K k 0k! ( )! N( ) 0. Thf, h xss cc v 0 such h wh > 0, qu (B.) hs su f K *. Hc, h b ss wh h x bcs csg ccu. Th hc sus suz h fwg h. Th B. L h v b f s h qu (B.), c SN ( S,, K,, ) K N ( S,, K,, ), M S K

30 Wh Nx () k 0 k ( ) x k k k! (k ). Th -h uc x f h v (B.) f h k c f h Eu c s gv b qu (B.9), k S K ( ) k k c (,,,, ) (,,,, M S S K ), k K S K k 0 k! (k ) S,, K,, S,, K,,. Th h fwg h ss h: S K () Th f chg f h -h v v wh sc h sk K s gv b h fu (B.0) () Th s cc v 0 f h ccuc f h x such h f v 0 = 0, s ffc ccus. Th s s, qu(b.7)). K 0 (h s su f (3) Th ssb sus f < 0 such h h s -h v x f h v hs s ffc. Th s s, K sk K *. 0 f s Ths h xs hw cu xs f h cuuv s c uc s ffc vs. A u, h c -cs vs css sks s css hs vs c u w h uw, h f s. Av, h c k skws sks. Th s h h s f h v wh sc sk b z u h ccuc f h x f h sbu. W hv s v hs shs wh vs h ug, bu h sus cb. Ax C. Ex f h Su f Nw-Rhs Cs 30- c suck 80 whch h ug s 00, h u v s 0%, h sk-f s 4%. Th cu w fw s c h usg h BSM fu. W h s h BSM c bck h fu usg v sch h, w v bck h 0% v. F hs us w us h Nw-Rhs h s g wh h ffc bw h BSM c b usg 0% h c uc b h s ss h 0.0. W us h Ax - 6

31 ss bw shw h sus usg c cs, hugh xc wh, w fu cs. S : Ob h BSM c usg v f 0%: Rsu: c = F h us w u hs h s, 0.6. W w s shw h sus wh hs vu u. S : D h sg f h Nw-Rhs h usg h Ms- Kh fu Ms T S K Kh (0.089) T Th Ms-Kh sg gus cvgc. H h sg s %. W cs f hs vu b ug. S 3: u h c f h h s wh h bsu vu f h c f 0.0. Th sus us h b bw. Th ubs bw u f s uss bu u h cus. Nw-Rhs, h c s b s h ffc v b h vv. Guss Pc Dffc Dvv c # As shw, h css cvgs sx s. Th su s 8.8%, whch s bu 4% hgh h h v us c h. H h g c b u, h su wu ff sgh 8.96%. Ax - 7

32 Ahugh shw h, h sus fuh ff 4.67% f h bsc h 34.7% f Dkk-B. Th gu fs sc hs cvg. Wh c ug, h sus 9.33% f h bsc h 4% f h Dkk-B h. Th s cvgc f h gu fs sc hs. Ax - 8

33 Fgu. S, B C Os, Ju 8, 03, xg Fbu 6, 03. Th sck c s $54. h sk-f s 0.05%. Th cs k f h CBOE s wb s h cs f busss Sk

34 Fgu. Ss, Nw-Rhs Mh A. O Pcs Sxhs 6hs cs, c = 0.0 6hs cs, c = 00 6hs cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s B. O Pcs Ps cs, c =0.0 cs, c = 00 cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s C. Cuus O Pcs

35 cuus cs, c = 0.0 cuus cs, c = 00 cuus cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

36 Fgu 3. Ss, Sc Mh A. O Pcs Sxhs 6hs cs, c = 0.0 6hs cs, c = 00 6hs cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s B. O Pcs Ps cs, c = 0.0 cs, c = 00 cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

37 C. Cuus O Pcs cuus cs, c = 0.0 cuus cs, c = 00 cuus cs, c ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

38 Fgu 4. Ss, Rgu Fs Mh A. O Pcs Sxhs 6h cs, c = 0.0 6hs cs, c = 00 6hs cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s B. O Pcs Ps cs, c = 0.0 cs, c = 00 cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

39 C. Cuus O Pcs cuus cs, c = 0.0 cuus cs, c = 00 cuus cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

40 Fgu 5. Ss, Bsc Mh A. O Pcs Sxhs 6h cs, c = 0.0 6hs cs, c = 00 6hs cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s B. O Pcs Ps cs, c = 0.0 cs, c = 00 cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

41 C. Cuus O Pcs cuus cs, c = 0.0 cuus cs, c = 00 cuus cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

42 Fgu 6. Ss, Dkk-B Mh A. O Pcs Sxhs 6hs cs, c = 0.0 6hs cs, c = 00 6hs css, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s B. O Pcs Ps cs, c = 0.0 cs, c = 00 cs, c = ME Sk Sk Sk 7 s 30 s 90 s 8 s 7 s 30 s 90 s 8 s 7 s 30 s 90 s 8 s

43 C. Cuus O Pcs cuus cs, c = 0.0 cuus cs, c = 00 cuus cs, c = ME Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s Sk 7 s 30 s 90 s 8 s

44 Fgu 7. s f Cs Sck c = $00, sk gs f $75 - $5 $.50 vs, sk-f = 4%, u v = 0%, xs f 80 s (0.4986), 90 s (0.466), 30 s (0.08), 7 s (0.09) s shw; su s b usg Nw-Rhs wh c f $0.0. O cs u. A. s B. gs

45 Fgu 8. Dffc bw Tu gs Px gs Px vgs b b svg f h vs usg h cu vs b h vgs.

46 Fgu 9. s f Cs wh % Pc Pu Sck c = $00, sk gs f $75 - $5 $.50 vs, sk-f = 4%, u v = 0%, xs f 80 s (0.4986), 90 s (0.466), 30 s (0.08), 7 s (0.09) s shw; su s b usg Nw-Rhs wh c f $0.0. O cs ccu usg h Bck-Schs-M f b % u. A. s B. gs

47 Fgu 0. S, Mk M Pcs, f B C Os, Ju 8, 03, xg Fbu 6, 03. Th sck c s $54. h sk-f s 0.05%. Th k cs k f h CBOE s wb s h cs f busss Sk Mk cs M cs

48 Tb. Ms, S Dvs, Mxu us f h Rv P f h S Accu f b Cu E Ess f h ssc f h Dw Js 30 scks h S&P 500 Nsq v D k qu vs wh x h u. Th v f h -h- s us s h u v. (b cus x g) Sck M S Dv Mxu AA AXP BA BAC CAT CSCO CX DD DS GE HD HPQ BM NTC JNJ JPM KO MCD MMM MRK MSFT NDX PFE PG SPX T TR UNH UTX

49 Z WMT XOM

50 Rfcs Abwz, M.,. A. Sgu (97). Hbk f Mhc Fucs wh Fus, Ghs, Mhc Tbs. Wshg, DC: N Buu f Ss. Bksh, G., C. C, Z. Ch (997). Ec fc f v cg s. Th Ju f Fc 5, B, W., A. Ss, D. Ss (006). N cx cs sss. Ju f N Scc Nuc Su 7, Bs, D. (99). Th csh f 87: Ws xc? Th vc f s ks. Th Ju f Fc, 46, Bh, M., Chsfs, N., Sk, G. (996). Cug h Bck-Schs-M v. Avcs Fuus Os Rsch, 8, 5 9. Bck, F., Schs, M. (973). Th cg f s c bs. Ju f Pc Ec, 8, B, N., Wh, R. (004). Ds bug ssu ffc h sh f v fucs? Th Ju f Fc, 59, B, M., Subh, M. (988). A s fu cu h s v. Fc Ass Ju, 5, Bw, S. (990). Esg v. Fgwsk, S., Sb, W., Subh, M. (Es.), Fc Os: F Th Pcc. Hw, L: Busss O w. C, L., Fgwsk, S. (993). Th f c f v. Th Rvw f Fc Sus, 6, Chb, J.-L. (.) (999). Mhs f Fs Ps. Ch. 3 A Hs f Aghs: F h Pbb h Mcch. Nw Yk: Sg-g. Chbs, D., Nwkh, S. (00). A v ch cug v. Th Fc Rvw, 38, Chc, D. (996). A gz s fu cu h v. Fc Rvw, 4, Chss, N. (997). Bck-Schs B: O Pcg Ms. Chcg: w Pfss Pubshg. C, C., M, T. (996). A s ccu fu cu v vs. Ju f Bkg Fc, 0, C, C., Su, T. (996). Skwss kuss S&P 500 x us b cs. Th Ju f Fc Rsch, 9, Ds, S., Su, R. (999). Of ss sks: A sucu scv. Ju f Fc Quv Ass, 34, 39. Ds, P., Mhw, S. (009). Mcsucu bss c ss f cg s. Rvw f Dvvs Rsch,, Dus, B., Fg, J., Wh, R. (998). v fucs: Ec ss. Th Ju f Fc, 53,

51 Fg, M (0). O g BSM v. Wkg, Uvs f S. G. Avb h:// Gh, S. (0). C Th b Exc Fu f? Wkg h://s.ss.c/s3/s.cf?bsc_=7994. Hbch, W. (004). A v Es f Bck-Schs-M, wkg vb h://s.ss.c/s3/s.cf?bsc_=5677. Hug, E. (998). Th C Gu O Pcg Fus. Nw Yk: McGw-H. Hsch, L. (003). Es v s. Ju f Fc Quv Ass, 38, Hs, S. (993). A Cs-F Su f Os wh Schsc wh Acs B Cuc Os. Th Rvw f Fc Sus 67, , Hsh, D. (99). Chs N Dcs: Ac Fc Mks. Th Ju f Fc 46, Hu, J., Wh, A. (987). Th cg f s sss wh schsc vs. Th Ju f Fc, 4, Jck, P. (00). B c. Wkg vb h:// Jg, G. J. Y. T (005). Th -f v s f c. Th Rvw f Fc Sus 8, K, M. (006). Fs vs v h c fuc h. Th Fc Rvw 4 ( ). Kz, M. (99). Wh cs kw bu sg v,. Fc Ass Ju, 47, Ju-Augus, 5. Lé, H., R, R. (976). S vs f sck c s cs. Th Ju f Fc, 3, L, S. (005). A w fu f cug v. A Mhcs Cu, 70, L, M., L, K. (0). A v succssv v-x h f cug h Bck-Schs-M v. Quv Fc,, Mch, M. (0). Pckg Rf: Mu cs fg- b., Aus: R Fu f Ssc Cug. Ms, S., Kh, G. (98). Th ccu f vcs f h Bck- Schs-M : A. Th Ju f Fc, 37, Mhw, S. (995). v. Fc Ass Ju, 5, Ju-Augus, 8 0. M, R. (976). O cg wh ug sck us scuus. Ju f Fc Eccs, 3, M, R. (973). Th f cg. Th B Ju f Eccs Mg Scc 4, Muhusw, J. T. Hs (0). Dscc bw Bck-Schs b. Wkg, K S Uvs.

52 P,., Rub, G., S, G. (999). Wh w s? O h s f h v fuc. Ju f Bkg Fc, 3, Rubs, M. (985). Nc ss f v cg s usg s qus h 30 s cv CBOE csss f Augus 3, 976, hugh Augus 3, 978. Th Ju f Fc, 40, Rubs, M. (994). b s. Th Ju f Fc, 49, Schs, R., T, R. (978). C sck v xcs b. Th Ju f Fc, 33, Shw, W. (998). Fc Dvvs wh Mhc. Cbg, U.K.: Cbg Uvs Pss. Shu, Y., P, G., L, S., Wu, T. (00). c f bug ssu chgs v: Bf f h s f h sub css. Th Ju f Dvvs, 7, g, G. (009). Th Bck-Schs-M s f h v s: A su su. Ju f Ecc Bhv & Ogz, 7, Wh, R. E. (006). Dvvs: Mks, u, Rsk Mg. Nw Yk: W. W, P. (998). Dvvs: Th Th Pcc f Fc Egg. Nw Yk: W. Wuz, D., hs. (03). Pckg fos: Bscs f vu., Aus: R Fu f Ssc Cug.