A Skewness-Adjusted Binomial Model for Pricing Futures Options The Importance of the Mean and Carrying-Cost Parameters

Transcription

1 Journl o Mthmticl Finnc,,, Publish Onlin Fbrury ( A kwnss-ajust Binomil Mol or Pricing Futurs Options Th Importnc o th Mn n Crrying-Cost Prmtrs tor Johnson 1, Amit n, Brin Blyt 1 1 Dprtmnt o Finnc, Xvir Univrsity, Cincinnti, UA Dprtmnt o Economics, Xvir Univrsity, Cincinnti, UA Emil: {johnsons, sn, blytb}@xvir.u Rciv Octobr 16, 11; rvis Dcmbr, 11; ccpt Dcmbr 1, 11 ABTRACT In this ppr, w xtn th Johnson, Pwlukiwicz, n Mht [1] skwnss-just binomil mol to th pricing o uturs options n xmin in som til th symptotic proprtis o th skwnss mol s it pplis to uturs n spot options. Th rsulting skwnss-just uturs options mol shows tht or lrg numbr o subprios, th pric o uturs options pns not only on th voltility n mn but lso on th risk-r rt, sst-yil, n othr crrying-cost prmtrs whn skwnss xists. Kywors: Options Pricing; Blck-chols; kwnss 1. Introuction On o th intrsting, s wll s subtl, turs o th Blck-chols (B-) [] mol n th binomil option pricing mol (BOPM) with lrg numbr o subprios (n) is tht th mols pn only on th vrinc. In ths mols, th mn is not importnt in trmining th vlu o spot options n th mn n nt crry cost r not importnt or uturs options. Ths implictions o th mol, howvr, pn on th ssumption tht th logrithmic rturn o th unrlying scurity is normlly istribut. tuis by Johnson, Zubr, n Gnr [] n [4] hv shown tht in prios o incrsing stock prics or rts, th logrithmic rturn o stock inxs n intrst rts r otn chrctriz by positiv mn n signiicnt ngtiv skwnss, n in prios o crsing prics or rts, th logrithmic rturns r otn chrctriz by ngtiv mn n signiicnt positiv skwnss. Morovr, svrl rlir mpiricl stuis hv rport tht th B- mol consistntly unrprics options in th prsnc o skwnss; s or instnc, tin n tin [5], Wiggins [6], n Hston [7]. Jrrow n Ru [8] n Corro n Ti u [9] hv xtn th B- mol to ccount or css in which thr is skwnss in th unrlying scurity s rturn istribution. imilrly, Câmr n Chung [1] n Johnson, Pwlukiwicz, n Mht (JPM) [1] hv xtn th Cox, Ross, n Rubinstin (CRR) [11] n Rnlmn n Brttr (RB) [] binomil option pricing mol to inclu skwnss. In thir ppr, JPM lso show tht skwnss chngs th symptotic proprtis o th up (u) n own () prmtrs, lvting th rltiv importnc o th mn in vluing options. This proprty o thir skwnss mol suggsts tht whn istributions o logrithmic rturns r chrctriz by skwnss, th obsrv pricing biss ssocit with th B- mol my b u to not only th omission o skwnss, but lso th mn. Toy, th rivtiv mrkt or non-stock options (inics, currncis, bt scuritis, n commoitis) is omint mor by options on uturs contrcts thn options on spot scuritis. Th purpos o this ppr is to xtn th JPM skwnss-just binomil mol to th pricing o uturs options. In ition, givn tht on o th turs o th JPM skwnss mol or spot options is tht skwnss chngs th symptotic proprtis o th u n prmtrs, this ppr xmins in som til th symptotic proprtis o th skwnss-just binomil mol s it pplis to both uturs n spot options. Our rsults show tht th skwnss mol or uturs options hs similr symptotic proprtis s th mol or spot options. Howvr, in th cs o uturs options, th prsnc o skwnss lvts th importnc o th mn, s wll s th risk-r rt, th sst yil, n othr prmtrs tht r in by th crrying-cost mol.. Binomil Futurs Options Pricing Mol Th stnr binomil option mol vlus uturs op- Copyright cirs.

2 16. JOHNON ET AL. tions rcursivly by trmining th uturs option s intrinsic vlus t xpirtion n thn using th singlprio binomil mol t ch no to pric th uturs option qul to th vlu o its rplicting portolio: C P 1 n n! p j n j! j! * Rtn ( ) * Rtn ( ) j ( n j) Mx u X, 1 n n! p j n j! j! j 1 p 1 p j ( n j) Mx( X u, j ( n j) ( n j) whr: C = cll pric P = put pric R = nnul risk-r rt t = tim to xpirtion xprss s proportion o yr X = xrcis pric = currnt uturs pric u = th uturs up prmtr = th uturs own prmtr n = numbr o prios to th option s xpirtion t/n = lngth on th binomil prio = tim to xpirtion s proportion o yr (t) ivi by numbr o prios to th option s xpirtion p = risk-nutrl probbility For th cs o n option on inncil uturs contrct (.g., inx, currncy, or bt scurity) in which th unrlying scurity is just to rlct continuous sst yil (.g., ivin yil, orign risk-r rt, or coupon rt), th quilibrium uturs pric s trmin by th crrying-cost mol is: R n t (1) () () whr: = currnt spot pric ψ = nnul sst yil t = t/n = lngth o binomil stps s proportion o yr n = numbr o iscrt binomil prios o lngth t to th uturs xpirtion Th risk-nutrl probbility, p, or options on inncil uturs in in trms o th up n own prmtrs or th unrlying spot (u = u / n = /, whr u = u n = ) is: / R t n p u (4) I th crrying-cost mol hols, thn th up n own prmtrs or th uturs pric (u = u / n = /, whr u = u n = ) r givn s: u uu R ( n 1)( t n) u u u R n ( t n) R( t n) R ( t n) R ( n 1)( t n) Rn ( t n) R( t n) R ( t n) ubstituting Equtions (5) n (6) into Eqution (4), th risk-nutrl probbility or uturs cll n put option prics cn b ltrntivly in in trms o th uturs up n own prmtrs (u n ) : R ( t n) p u p u 1 p p u R( t n) R( t n) R( t n) R( t n).1. kwnss-ajust Formuls or u n In thir sminl 1979 ppr, Cox, Ross, n Rubinstin [11] riv th ormuls or stimting th u n prmtrs o th BOPM or spot option. Thy o this by stting th qutions or th xpct vlu n vrinc o th logrithmic rturn o th unrlying scurity qul to thir mpiricl vlus. Th rsulting qutions r thn solv simultnously or u n unr th ssumption tht th probbility o th unrlying scurity incrsing in on prio (q) is.5. By trting q s n unknown, JPM xtn th CRR binomil mol to inclu skwnss. pciiclly, in th JPM skwnss-just mol, th u,, n q vlus tht in binomil procss or spot scurity r oun by stting th qutions or th xpct vlu, vrinc, n skwnss qul to thir rspctiv mpiricl vlus n thn solving th rsulting qution systm simultnously or u,, n q. Tht is: n E g p g nq 1q ln t n nj nj j n 1 ln 1 1 ln V g nq q u tv k gn n q q q q u t, whr: g = logrithmic rturn o th unrlying spot pric = ln( 1 / ) (5) (6) (7) (8) Copyright cirs.

3 . JOHNON ET AL. 17 j = numbr o incrss in n prios p nj = probbility o j incrss in n prios, whr:, V, σ, = th nnuliz mpiricl vlus o th mn, vrinc, stnr vition, n skwnss o th spot pric s logrithmic rturn t = tim to xpirtion s proportion o yr n = numbr o prios Th vlus o u,, n q tht stisy this systm o qutions r: u ((1 ) )( ) ( ) q q tn tn ( q(1 q))( tn) ( tn) (9) (1) 1 14( tv ) q 1 n s, i, i. (11) For uturs options, th ormuls or stimting u,, n q r oun by stting th qutions or th popultion momnts or th uturs pric s logrithmic rturn qul to thir rspctiv mpiricl vlus, n thn solving th rsulting qution systm simultnously or u,, n q: n q 1q ln t nq q u tv 1 ln 1 1 ln nq q q q u t, () whr, V, σ, n r rspctivly th nnuliz mpiricl vlus o th mn, th vrinc, th stnr vition, n skwnss o th uturs pric s logrithmic rturn, ln( 1 / ), n t is th tim to xpirtion s proportion o yr. Th vlus o u,, n q tht stisy this systm o qutions r: u u ((1 q) q)( tn) ( tn) (1) ((1 q) q)( tn)) ( tn) (14) 1 14( tv ) q 1 nt ( ), i, i (15).. Rltions btwn Futurs n pot Mn n Voltility Th rltionships btwn th spot n uturs prmtrs ollow irctly rom th uturs n spot rltion in by th crrying-cost mol givn in Eqution (). pciiclly, rom Eqution (), on cn solv th rltionship btwn th spot n uturs momnts s ollows: ( R ) n t ( R )( n 1) t 1 1 ( R )( n 1) t 1 1 ( R ) n t ( R )( n 1) t 1 1 ln ( R ) n t ln 1 1 ( R ) t ln ln ln ln ln R t 1 1 (16) Tking th xpct vlu o both sis o Eqution (16) rsults in th mn ovr prio o tim t: ln ln 1 1 E E R t R t * * F * whr n * F r rspctivly th prioic mns o spot option n uturs option or prio o lngth t( = t/n). Multiplying both sis by 1/ t rsults in th nnuliz mns (µ n µ F ): 1 t 1 t R R * * t Tking th vrinc o both sis o (16) n nnulizing w obtin: 1 1 Vr ln Vr ln R t F F A similr proo shows tht δ = δ F. Thus, th rltionships btwn th mn, stnr vition, n skwnss on th uturs n spot logrithmic rturns r: s R (17) (18) (19) ubstituting (17) n (18) into Equtions (1) n (14), u n cn b xprss in trms o th spot mn n vribility, th risk-r rt, n th sst yil: Copyright cirs.

4 18. JOHNON ET AL. u ((1 q) q)( tn) ( tn) ((1 q) q)( t n) ( ( R))( t n) ((1 q) q)( tn) ( tn) ( q (1 q))( t n) ( ( R))( t n) () (1) Th irnc btwn th uturs up n own prmtrs (u n rspctivly), n th spot up n own prmtr (u n rspctivly (s Equtions (9) n 1)) is th nt cost o crry trm (R ψ). I R > ψ, thn th uturs mrkt is norml with th uturs pric (Eqution ()) xcing th spot. In this mrkt, u > u n >. On th othr hn, i R < ψ, thn th uturs mrkt is invrt with th uturs pric lss thn th spot. In this mrkt, u > u n >. Finlly, i th nt crry cost is zro, thn R = ψ n th uturs mrkt is nutrl; in this mrkt, th uturs up n own prmtrs will b qul to th spot up n own prmtrs, tht is, u = u n =. Th rltions btwn th prmtrs r illustrt in Tbl 1. Th tbl shows th uturs n spot up n own prmtrs clcult or numbr o scnrios: positiv mn n ngtiv skwnss css (incrsing pric cs) or norml, invrt, n nutrl uturs mrkt; ngtiv mn n positiv skwnss css (crsing pric cs) or norml, invrt, n nutrl uturs mrkt; n zro mn n skwnss cs (stbl pric cs) or norml, invrt, n nutrl uturs mrkt. Pnl A in th tbl shows th inputs or ch scnrio n Pnl B givs th corrsponing prmtrs.. Binomil Futurs Options Pricing Mol.1. Dcrsing Exchng-Rt Cs vrl mpiricl stuis hv shown tht prios o incrsing scurity prics r otn chrctriz by positiv mn n signiicnt ngtiv skwnss in th scurity pric s logrithmic rturn. As n xmpl o crsing pric scnrio, suppos th currnt U ollr/british poun xchng rt is = $1.6/BP, n thr is mrkt xpcttion o ollr pprcition ovr th nxt yr such tht th xpct istribution o logrithmic rturns or th xchng rt hs th ollowing nnuliz mn, vrinc, n skwnss: μ =.17597, V =.19555, n δ s =.86. Givn ths mpiricl momnt vlus, consir th pricing o cll n put options on British poun (BP) uturs contrct ch with X = $1.6/BP n tim to xpirtion o 7 ys, using thr-prio binomil mol. In pricing th options, ssum th ollowing: 1) Th spot $/BP xchng rt t tim is E = $1.6/BP ) Th nnul risk-r rt pi on U ollrs is R =.6 ) Th nnul risk-r rt pi on British pouns is ψ =.4 4) Th uturs contrct on th BP xpirs in on yr 5) Crrying-cost mol hols 6) Options on th BP uturs options xpir in 7 ys 7) 6-y yr Th lngth o th binomil prio in yrs is t = t/n = (7/6)/ =.5, with th cll n put options xpiring in (n option ) t = () (.5) =.75 yrs, n th BP uturs xpiring in n t = (4) (.75) = 1 yr. Th up n own prmtrs or th spot rt r u = n =.991, th up n own prmtrs or th uturs contrct r u = n =.9457, q =.5, th risk-nutrl probbility is p =.668, n th quilibrium uturs pric is $1.6/BP: u t q 1 nt V 1 q qt n( Rt/ n V q 1qt n Rt n ((1.5).5))(.75/) ( (.6.4))(.75 ) (.5/(1.5))(.75 ) ( (.6.4))(.75 ).9457 p R n t (.6.4)4(.5) u $1.6 $1.6 Th ngtiv mn n positiv skwnss in this cs yil u,, n q vlus tht rlct crsing xchng-rt scnrio in which th proportionl crs in th uturs rt ch prio is 1.%, xcing in bsolut vlu th proportionl incrs o 4.6%, n th probbility o th crs in ch qurtrly prio ( t =.5) is 1 q =.65: Copyright cirs.

5 . JOHNON ET AL. 19 Tbl 1. u n n u n vlus or irnt scnrios. Pnl: A cnrio Mrkt R ψ t/n Annuliz Annuliz Annuliz Mn Vrinc kwnss Positiv mn, Zro kwnss, Norml P ositiv mn, Ngtiv kwnss, Norml Positiv mn, Ngtiv kwnss, Norml Positiv mn, Zro kwnss, Invrt P ositiv mn, Ngtiv kwnss, Invrt Positiv mn, Ngtiv kwnss, Invrt Positiv mn, Zro kwnss tbl Positiv mn, Ngtiv kwnss tbl Positiv mn, Ngtiv kwnss tbl Ngtiv mn, Zro kwnss, Norml N gtiv mn, Positiv kwnss, Norml Ngtiv mn, Positiv kwnss, Norml Ngtiv mn, Zro kwnss, Invrt N gtiv mn, Positiv kwnss, Invrt Ngtiv mn, Positiv kwnss, Invrt Ngtiv mn, Zro kwnss tbl N gtiv mn, Positiv kwnss tbl Ngtiv mn, Positiv kwnss tbl Zro Mn, Zro kwnss, Norml Zro Mn, Zro kwnss, Invrt Zro Mn, Zro kwnss tbl Pnl B: cnrio Mrkt q pot u pot Futurs u Futurs Positiv mn, Zro kwnss, Norml P ositiv mn, Ngtiv kwnss, Norml Positiv mn, Ngtiv kwnss, Norml Positiv mn, Zro kwnss, Invrt P ositiv mn, Ngtiv kwnss, Invrt Positiv mn, Ngtiv kwnss, Invrt Positiv mn, Zro kwnss tbl Positiv mn, Ngtiv kwnss tbl Positiv mn, Ngtiv kwnss tbl Ngtiv mn, Zro kwnss, Norml N gtiv mn, Positiv kwnss, Norml Ngtiv mn, Positiv kwnss, Norml Ngtiv mn, Zro kwnss, Invrt N gtiv mn, Positiv kwnss, Invrt Ngtiv mn, Positiv kwnss, Invrt Ngtiv mn, Zro kwnss tbl N gtiv mn, Positiv kwnss tbl Ngtiv mn, Positiv kwnss tbl Zro Mn, Zro kwnss, Norml Zro Mn, Zro kwnss, Invrt Zro Mn, Zro kwnss tbl Copyright cirs.

6 11. JOHNON ET AL. ln ln.9457 ln % 4.6% Th binomil tr or th unrlying spot $/BP xchng rt, BP uturs contrct, Europn n Amricn uturs clls, n Europn n Amricn uturs puts r shown in Exhibit 1. In th thr-prio option cs, th binomil mol prics th Europn uturs cll t $.857 n th Europn uturs put t $.55. As shown in th xhibit, thr is n rly xrcis vntg or th Amricn uturs cll t th uppr no in prio, n n rly xrcis vntg or th Amricn uturs put t th lowr no in prio. As rsult, both th Amricn uturs put n cll options r pric slightly highr thn thir Europn countrprts. I th up n own prmtrs r not just or skwnss, thn q =.5 n th skwnss-just qutions or th up n own prmtrs or th spot n uturs simpliy to th CRR/RB ormuls: u u tn ( tn) tn ( tn) tn ( tn) ( tn) ( ( R))( tn) tn ( tn) ( tn) ( ( R))( tn) In this xmpl, th up n own prmtrs or th spot rt woul b u = n =.895, th up n own prmtrs on th uturs contrct woul b u = n = , n th risk-nutrl probbility woul b p = In this cs, in which skwnss is ssum to xist but is xclu in th stimts o th up n own prmtrs th binomil mol prics th Europn cll t $.786, 8.8% lss thn th skwnss-just mol, n th Europn put t $.477, 1.74% lss thn th skwnss mol. Aitionlly, thr woul not b n rly xrcis vntg E $1.6 / BP $1.6 $1.6 P $.55, P C $.857, C (.6.4 )(.5 )( 4) $.87 $.558 ue $ u u $ (.6.4)(.5)() u $ $ C u $.66, C u $.91 Pu $.1, Pu $.1 E $ $ (.6.4)(.5)() $ $ C $.85, C $.85 P $.71, P $.88 u E $ uu u $ (.6.4)(.5)() uu $ $ C uu $.1869 C uu IV $.1967 Puu, Puu ue $ u u $ (.6.4)(.5)() u $ $ C u $.1, C u $.1 Pu $.654, Pu $.654 E $1.4 $1.56 (.6.4)(.5)() $1.4 $1.56 C, C P $.591, P IV $.644 u E $ uuu u $ (.6.4)(.5)(1) uuu $ $ C uuu $.755,C uuu $.755 Puuu, Puuu u E $ uu u $ (.6.4)(.5)(1) uu $ $ C uu $.198, C uu $.198 Puu, Puu u E $ u u $1.989 (.6.4)(.5)(1) u $ $1.989 C u, C u Pu $.11, Pu $.11 E $1.14 $1.817 (.6.4)(.5)(1) $1.14 $1.817 C, C P $.918, P $.918 Exhibit 1. Binomil cll n put vlus unr crsing xchng rt cs: skwnss mol. Copyright cirs.

7 . JOHNON ET AL. 111 or th Amricn uturs cll n thr woul b only on rly xrcis vntg or th Amricn uturs put. As rsult, th Amricn uturs put option is pric slightly highr thn its Europn countrprt... Incrsing Exchng Rt Cs Prios o incrsing scurity prics r otn chrctriz by positiv mn n ngtiv skwnss in th scurity pric s logrithmic rturn. To illustrt option pricing unr this scnrio, suppos th mrkt xpcts th $/BP xchng rt to incrs ovr th nxt yr such tht th xpct istribution o th xchng rt s logrithmic rturn hs th ollowing stimt nnuliz momnts: μ =.17597, V =.19555, n δ =.86. In this incrsing xchng rt cs, th up n own prmtrs or th spot rt r u = 1.1 n =.95, th up n own prmtrs on th uturs contrct r u = n =.9456, q =.65, n th risk-nutrl probbility is p = Th positiv mn n ngtiv skwnss yil u,, n q vlus tht rlct n incrsing xchng-rt prio in which th proportionl incrs in th uturs rt ch prio is 9.%, xcing in bsolut vlu th proportionl crs o 5.6%, n th probbility o n incrs in ch qurtrly prio ( t =.5) is q =.65: ln ln ln % 5.6% In thr-prio option cs, th binomil mol or th incrsing cs woul pric th Europn uturs cll t $.86 n th Europn uturs put t $.55. Th Amricn uturs put n cll options r pric slightly highr thn th Europn option, givn rly xrcis vntgs or both th cll n put options. I th up n own prmtrs or this incrsing cs wr not just or skwnss, thn q woul b qul to.5, n th up n own prmtrs or th spot rt woul b u = n =.97447, th up n own prmtrs on th uturs contrct woul b u = n = , n th risk-nutrl probbility woul b p =.95. In this cs in which skwnss is ssum to xist but is xclu in th stimts o th up n own prmtrs, th binomil mol prics th Europn cll t $.84,.55% lss thn th skwnss-just mol, n th Europn put t $.51,.98% lss thn th skwnss mol. 4. Proprtis o th kwnss Mol In th cs o spot options, th xistnc o skwnss cts th rltiv contribution o th mn to th vlus o th up n own prmtrs n th symptotic proprtis, with th skwnss-just up n own prmtrs hving irnt symptotic proprtis thn th CRR/RB prmtrs. In th cs o uturs options, ths sm proprtis lso hol. In ition, with uturs options, th impct skwnss hs on lvting th importnc o th mn pns on th crrying cost vlu (R ψ) n whthr th uturs mrkt is norml, invrt or nutrl Rltiv Importnc o th Mn Trm In th cs o positiv mn, th mn bcoms mor importnt in trmining th vlu o th spot up prmtr vlu, th grtr th ngtiv skwnss (or quivlntly th mor q xcs.5). imilrly, or uturs options, th µ (R ψ) trm bcoms mor importnt in trmining th uturs up prmtr vlu (u ), th grtr th ngtiv skwnss. By contrst, in th cs o ngtiv mn, th mn bcoms mor importnt in trmining th vlu o, n th µ (R ψ) trm bcoms mor importnt in trmining th vlu o th uturs own prmtr ( ), th grtr th positiv skwnss (or quivlntly th mor 1 q xcs.5). To s th impct skwnss hs on incrsing th importnc o th mn or spot options n th µ (R ψ) trm or uturs options, consir th prvious thrprio incrsing xchng rt cs in which u = 1.1 n =.95 or th spot xchng rt. I skwnss wr zro, thn q woul b qul to.5, th xpct qurtrly mn woul b qul to.84 (=[(.5)ln(1.1) + (.5)ln(.95)]) n th nnuliz mn woul b.884. Th qurtrly vrinc woul b qul to.571 (=.5[ln (1.1).84] +.5[ln (.95).84] ), n th nnuliz vrinc woul b.149. I ths wr th ctul mpiricl vlus o µ n V, thn ln(u) woul b qul to.95 (u = 1.1), n th mn woul contribut % to th vlu o u n th vrinc woul contribut 77% to th vlu o u: q t n t n V 1q tn V 1 q qt n ln (1.1) ln (1.1) For th uturs option, th ln(u ) woul b qul to Copyright cirs.

8 1. JOHNON ET AL..995 (u = ) givn th nt cost o crry o. (=R ψ =.6.4). Th mn minus th nt cost o crry (R ψ) woul, in turn, contribut 19% to th vlu o u n th vrinc woul contribut 81% to th vlu o u : R t n 1 Rt n V 1 q qt n ln V q q t n ln (1.9451) I ngtiv skwnss wr prsnt such tht q =.65, thn th qurtrly mn woul b qul.44 (= [(.65) ln(1.1) + (.5) ln(.95)]) n th nnuliz mn woul b.176. Th qurtrly vrinc woul, in turn, b qul to (=.65 [ln(1.1).44] +.5 [ln(.95).44] ) n th nnuliz vrinc woul b Givn th mn n vrinc vlus n q =.65, th impli skwnss woul b δ =.86. I ths wr th ctul mpiricl vlus o mn, vrinc, n skwnss, thn ln(u) woul b qul to th nonskw vlu o.95 (u = 1.1), but with q =.65, th contribution o th mn to th vlu o u woul b 46% n th contribution o th vrinc to th vlu o u woul b 54%: t n V 1q q t n 1 tn V 1 q qt n ln ln For th uturs options, th ln(u ) woul likwis b qul to its non-skw vlu o.995 (u = ), but th contribution o th mn n th nt cost o crry (R ψ) trm to th vlu o u woul b 4%, n th contribution o th vrinc to th vlu o u woul b 57%: V q qt n Rt n R t n 1 [ 1 ] 1 V q q t n ln ln Thus, in th cs o positiv mn, ngtiv skwnss incrss th rltiv importnc o th mn on th up prmtrs or th spot n uturs rts 1. By contrst, in th cs o positiv mn, th mn trm hs n opposit irctionl impct on n thn th vrinc trm, with its ngtiv impct incrsing th grtr th ngtiv skwnss (or quivlntly th mor q xcs.5). For xmpl, in th no skwnss cs or th uturs option, ln( ) is qul to.569 ( =.9456). In this cs, th µ (R ψ) trm woul contribut % to th vlu o n th vrinc trm woul contribut 1% to th vlu o. In th skwnss cs, th µ (R ψ) trm woul contribut 69% to th vlu o, whrs th vrinc woul contribut 169% to th vlu o : Just th opposit rltionships hol in th cs o ngtiv mn with positiv skwnss. pciiclly or ngtiv mn, th mn bcoms mor importnt in trmining n th grtr th positiv skwnss, whrs th mn trm hs n opposit irctionl impct on u n u thn th vrinc trm, with its ngtiv impct on th up prmtr incrsing th grtr th positiv skwnss. Tbl summrizs th rltiv contributions o th mn n vrinc trms to th spot n uturs up n own prmtrs or th crsing xchng-rt cs n othr scnrios with irnt lvls o skwnss n irnt uturs mrkts. Pnl A tils th rltiv contributions or th spot n Pnl B tils th contributions or th uturs. 4.. Asymptotic Proprtis In th CRR/RB mol, s th numbr o subprios (n) 1 Not tht in this cs, µ > (R ψ) n th uturs mrkt is norml (R ψ > ) with th uturs pric xcing th spot pric. As rsult, ngtiv skwnss incrss th mn s impct on th up prmtr (46%) or th spot option mor thn th up prmtr or th uturs option (4%). In contrst, i th mrkt wr invrt (R ψ < ), thn ngtiv skwnss woul hv crs th impct o th mn on th uturs up prmtr mor thn th spot prmtr. Copyright cirs.

9 . JOHNON ET AL. 11 Tbl. () Rltiv contributions o th mn n vrinc trms to up n own prmtrs; (b) Rltiv contributions o th mn n vrinc trms to up n own prmtrs. Pnl: A Rltiv Contributions o cnrio Mrkt Mn to u Vrinc to u Mn to Vrinc to Incrsing: Positiv mn, Zro kwnss, Norml Incrsing: Positiv mn, Ngtiv kwnss, Positiv mn, Ngtiv kwnss, Norml Norml Incrsing: Positiv mn, Zro kwnss, Invrt Incrsing: Positiv mn, Ngtiv kwnss, Invrt Incrsing: Positiv mn, Ngtiv kwnss, Invrt Incrsing: Positiv mn, Zro kwnss Incrsing: Positiv mn, Ngtiv kwnss Incrsing: Positiv mn, Ngtiv kwnss Dcrsing: Ngtiv mn, Zro kwnss, tbl tbl tbl Norml Dcrsing: Ngtiv mn, Positiv kwnss, Dcrsing: Ngtiv mn, Positiv kwnss, Dcrsing: Ngtiv mn, Zro kwnss, Norml Norml Invrt Dcrsing: Ngtiv mn, Positiv kwnss, Invrt Dcrsing: Ngtiv mn, Positiv kwnss, Invrt Dcrsing: Ngtiv mn, Zro kwnss tbl Dcrsing: Ngtiv mn, Positiv kwnss tbl Dcrsing: Ngtiv mn, Positiv kwnss tbl tbl Mrkt: Zro Mn, Zro kwnss, Norml tbl Mrkt: Zro Mn, Zro kwnss, Invrt tbl Mrkt: Zro Mn, Zro kwnss tbl Pnl: B Rltiv Contributions o cnrio Mrkt µ (R ψ) Vrinc to u µ (R ψ) to u to Vrinc to Incrsing: Positiv mn, Zro kwnss, Norml Incrsing: Positiv mn, Ngtiv kwnss, Norml Positiv mn, Ngtiv kwnss, Norml Incrsing: Positiv mn, Zro kwnss, Invrt Incrsing: Positiv mn, Ngtiv kwnss, Invrt Incrsing: Positiv mn, Ngtiv kwnss, Invrt Incrsing: Positiv mn, Zro kwnss tbl Incrsing: Positiv mn, Ngtiv kwnss tbl Incrsing: Positiv mn, Ngtiv kwnss tbl Dcrsing: Ngtiv mn, Zro kwnss, Norml Dcrsing: Ngtiv mn, Positiv kwnss, Norml Dcrsing: Ngtiv mn, Positiv kwnss, Norml Dcrsing: Ngtiv mn, Zro kwnss, Invrt Dcrsing: Ngtiv mn, Positiv kwnss, Invrt Dcrsing: Ngtiv mn, Positiv kwnss, Invrt Dcrsing: Ngtiv mn, Zro kwnss tbl Dcrsing: Ngtiv mn, Positiv kwnss tbl Dcrsing: Ngtiv mn, Positiv kwnss tbl tbl Mrkt: Zro Mn, Zro kwnss, Norml tbl Mrkt: Zro Mn, Zro kwnss, Invrt tbl Mrkt: Zro Mn, Zro kwnss tbl Copyright cirs.

10 114. JOHNON ET AL. inc rss, th mn trm in th xponnt or spot options n th µ (R ψ) trm or uturs options go to zro s tr thn th squr root trm. Thus or lrg n, th CRR/RB mol pns only on th voltility. M orovr, with σ = σ, th up n own prmtrs or th uturs r qul to th spot, rgrlss o wh thr th uturs mrkt is norml or invrt: u tn tn u tn Using thss up n own prmtrs, th BO PM or uturs options convrgs to th sminl Blck uturs option mol [1] s n gts lrg. Th skwnss-just up n own prmtrs, though, o not hv th sm symptotic proprtis s th CRR/ RB prmtrs. In th skwnss mol, Eqution () or u inclus (1 q)/q trm, n Eqution (1) or inclus q/(1 q) trm, both o which chng th orr o mgnitu s n gts lrg. pciiclly, or th cs o ngtiv skwnss, th (1 q)/q trm cn b rwrittn s: 4 1 q V q n 4 V n 4V tn () Th xprssion (1 q)/q in Eqution (), in turn, is th sm orr o mgnitu s 1/n. This cn b sn by obsrving tht th trm [(1 q)/q]/[1/n] pprochs th constnt V s n gts lrg. Tht is: 1 q q 1 n 4V V n V n 4 4 n in th limit: 1 q q V 1 n s () With (1 q)/q hving th sm orr o mgnitu s 1/n, th trm [(1 q)/q*v /n] 1/ in Eqution (1) pprochs constnt multipli by [V ] 1/ /n, s n gts lrg. As rsult, or th cs o lrg n, th irst trm in th xponnt in Eqution (1) pproximts constnt ivi by n, which is in th sm orm s th scon trm, μ /n = (μ (R ψ))/n. Consquntly, both trms in th xponnt o qution (1) or u contribut qully, vn whn n is lrg. Thus, s n gts lrg, u pns not only on th vrinc n skwnss, but lso on th mn, risk-r rt, n sst yil. By contrst, Eqution (14) or is in in n (4) trms o q/(1 q). In this c s, s n gts lrg, th vrinc trm in th xponnt or pprochs th constnt δ /V n th μ /n trm pprochs zro. Thus, or th cs o ngtiv sk wnss with lrg n, p ns on th vrinc n skwnss, but not on th mn, risk-r rt n ivin yil, whrs u pns on ll thr prmtrs. Just th opposit symptotic rlt ionships occur whn skwnss is positiv. In this cs, th mn, vrinc, skwnss, risk-r rt, n ivin yil trm in (vn whn n is lrg), whrs just th vrinc n sk wnss trmin u. Not, similr symptotic rltions lso hol or th spot up n own prmtrs, with th s pot n uturs s- ymptotic rltions bing quivlnt w hn th nt cost o crry is zro (R ψ = ). To illustrt th rltiv importnc o th mn or th cs o lrg n, consir th prcing incrsing xchng-rt cs (option xpiring in.75 yr, R =.6 n ψ =.4) whr th stimt nnuliz mn, vrinc, n skwnss wr μ =.17597, V =.19555, n δ =.86. I w subivi th option prio (t =.75) into subprios (n = ), thn or th spot xchng rt th mn trm woul contribut % to th vlu o u n th vrinc trm woul contribut 5.86% to th vlu o u; or th uturs, th µ (R ψ) trm woul contribut 4.18% to th vlu o u n th vrinc trm woul contribut 56.8% to th vlu o u : pot: tn ln Futurs : Rt n [ ]*.75 ln I w subivi th option prio into nin monthly subprios (n = 9; t/n =.75/9 =.8; u = 1.94, =.9481; u = 1.767, =.9465), th mn trm woul contribut 7.95% n th vrinc trm woul contribut 6.5% or th spot rt, n or th uturs, th mn trm woul contribut 5.15% n th vrinc trm woul contribut 64.85%: pot: tn ln Futurs: Rt n *.75 9 ln Copyright cirs.

11 . JOHNON ET AL. 115 For lrgr n, th contribution o th mn is pproximtly th sm: whn n = 9 (wkly prios; t/n =.75/9 =.19), th rltiv contribution o th mn woul b 1.59% or th spot rt n 9.4% or th uturs; whn n = 7 (ily; t/n =.75/7 =.8), th contributions woul b 8.91% or th spot n 6.49% or th uturs; whn n = 1, th contributions woul b 8.51% n 6.11% or th spot n uturs, rspcttivly; whn n = 1, th contributions woul b 8.5% n 5.97%. By contrst, i skwnss wr zro (th CRR/RB mol with µ =.17597, V =.19558, n δ = ), th rltiv contributions o th mn trm or th spot rt n uturs woul b 8.6% n 5.8% or n = ((n = ; t/n =.75/9 =.8; u = 1.667, =.97447; u = , = ): pot: tn ln Futurs: ln R t n u *.75 ln Whn n = 9, th mn contribution or th spot n uturs, rspctivly, woul b 6.64% n 4.6%; whn n = 9, 14.86% n 14.4%; whn n = 7, 6.% n 5.55%; whn n = 1,.% n.96%; n whn n = 1,.1% n.1%. Figur 1 shows grphiclly th rltionship btwn th numbr o subprios n th mn s contribution to th up prmtr or th skwnss-just incrsing xchng-rt cs. Th grph in Figur 1 highlights th symptotic rltion, showing tht s n incrss, th mn s contribution to both th sopt n uturs up prmtrs crss symptoticlly with th symptot occurring t pproximtly n * = whr th minimum mn contribution is %. Figur summrizs th rltionship btwn th numbr o subprios n th mn s contribution to th up n own prmtrs or th CRR/RB cs in which skwnss is zro. Figur, in turn, shows similr symptotic rltion btwn th mn s contribution n n or th CRR/RB mol s th skwnss cs, but with th minimum mn contribution bing clos to zro. Figur shows grphiclly th rltionship btwn th numbr o subprios n th mn s contribution to th own prmtrs or th skwnss-just crsing xchng-rt cs chrctriz by ngtiv mn n positiv skwnss: µ =.17597; V =.19555; δ =.86. Th grph highlights th symptotic rltion, showing tht s n incrss, th mn s contribution to th own prmtrs crss symptoticlly with th symptot occurring t pproximtly n * = whr th minimum mn contribution is %. Figur 1. Mn contribution to u n u, μ = , V =.19558, δ = Figur. Mn contribution to u n u, μ = , V =.19558, δ =. It shoul b not tht sinc ll th prmtrs contribut to ithr lnu or ln s n gts lrg, th n vlu in which th skwnss-just mol pprochs continuous on pns on th rltiv vlus o µ, V, n δ. For th cs o δ <, th trm (1 q)/q pprochs constnt ivi by n in th limit. Th criticl vlu, n*, cn thror b oun by solving or th n tht mks (1 q)/q (Eqution ()) qul to lrg proportion (.g.,.99) o th limit (Eqution ()). Dining th proportion s 1 ε, whr ε is qul to th proportion o rror (.g., ε =.1), th n* tht is qul to 1 ε o th limit is: V n * Figur. Mn contribution to n, μ =.17597, V =.19555, δ = Copyright cirs.

12 116. JOHNON ET AL. Figur 4 shows similr symptotic rltion btwn th mn s contribution to th own prmtr n n or th CRR/RB cs, but with th minimum contribution bing clos to zro. 5. Dirncs in Futurs Option Prics btwn th CRR/RB Mol n th kwnss-ajust Mol Vlus or Europn uturs cll options on th British poun obtin using th skwnss-just mol n th CRR/RB Mol r prsnt in Tbl, n vlus or th Europn uturs put options on th British poun r shown in Tbl 4. Th cll n put uturs options ch hv xrcis prics o $1.6/BP n xpir in.75 yrs, n th British poun uturs contrct xpirs in on yr, with th uturs pric ssum to b qul to its crrying-cot vlu. Th binomil mol us or pricing is subivi into 6 prios o lngth 6 ys. Th tbls show thr uturs mrkts: norml uturs mrkt whr th nnuliz risk-r rt is ssum to b 6% on U ollrs n 4% on British pouns, n invrt mrkt whr th ollr rt is 4% n British poun rt is 6%, n nutrl mrkt whr ch rt is qul to 6%. Fi- nlly, th tbls show two xchng-rt scnrios: 1) An incrsing xchng-rt cs chrctriz by positiv mn n ngtiv skwnss: μ = , V =.19558, δ = ) A crsing xchng-rt cs chrctriz by ngtiv mn n positiv skwnss: μ = , V =.19558, δ =. A comprison o th uturs option vlus obtin using th skwnss-just mol with th CRR/RB mol illustrts th pricing irncs tht occur unr incrsing or crsing xchng-rt css chrctriz by skwnss. In gnrl, or both scnrios, th CRR/RB mol prics th Amricn n Europn uturs cll n th Amricn n Europn uturs puts lss thn th skwnss mol, with th grtst pricing irncs occu rring or out-o-th-mony options. pciiclly, th si- Figur 4. Mn contribution to n, μ =.17597, V =.19555, δ =. multions show unrpricing o th CRR/RB mol, rnging rom pproximtly 5% or in-th-mony options to pproximtly 5% or out-o-th-mony options. For th uturs cll option, in both th incrsing n crsing css, th CRR/RB mol unrprics mor or th invrt uturs mrkt cs (rng.56% % (incrsing); 9.4% - 5.9% (crsing)) thn th norml uturs mrkt cs (1.59% - 5.6% (incrsing); 4.79% - 4.1% (crsing)). For puts, th CRR/RB mol unrprics mor or th norml mrkt cs (rng 5.5% - 8.6% (incrsing); 9.14% -.8% (crsing)) thn th invrt mrkt cs (4.67% -.6% (incrsing); 4.85% - 1.5% (crsing)). Finlly, Tbl 5 comprs uturs cll n put prics with corrsponing spot cll n put prics or norml, invrt, n nutrl uturs mrkt unr stbl xchng-rt scnrio in which th mn n skwnss r zro. As shown, with zro skwnss, th skwnss mol n th CRR/RB mol or spot n uturs options r th sm. Th simultions, in turn, lso show tht this is th only cs in which Europn uturs n spot options r qul. 6. Conclusions A subtl tur o th B- mol n th BOPM or lrg n is tht ths mols pn only on th vrinc. Th mn is not importnt in trmining th vlu o spot options, n th mn n nt crry cost r not importnt or uturs options. This tur is consqunc o th ssumption tht th logrithmic rturn o th unrlying scurity is normlly istribut. In this ppr, w show tht in css whr skwnss xists, th skwnss-just up n own prmtrs or spot options pn mor on th mn thn non-skwnss-just prmtrs, n tht th skwnss-just up n own prmtrs or uturs options pn mor on th µ (R ψ) trm thn non-skwnss-just prmtrs. Furthrmor, th skwnss-just up n own prmtrs lso hv irnt symptotic proprtis such tht or lrg n, th mn or spot prics n th µ (R ψ) trm or uturs mintin thir rltiv importnc. Thus, th prsnc o skwnss srvs to ugmnt th rltiv importnc o th mn or spot options n th µ (R ψ) trm or uturs options. Using simultions, w show tht whn thr is n xpct incrsing pric trn chrctriz by positiv logrithmic mn n Th pricing irncs btwn th CRR/RB mol n th skwnss mol r consistnt with th ormntion mpiricl stuis o tin n tin [4], Wiggins [5], n Hston [6] who monstrt tht whn skwnss xists, th B- mol consistntly unrprics options. Also, s xpct, thr wr no signiicnt irnc in th prics or th Europn cll n put options obtin using th Blck uturs option mol (not shown) n th 6-prio CRR/RB binomil prics shown in Tbl 5. Copyright cirs.

13 . JOHNON ET AL. 117 Tbl. Comprison o skwnss-just uturs option mol with CRR/RB mol or norml, invrt n nutrl uturs mrkts n incrsing n crsing xchng-rt trns: Europn uturs cll options. Pnl A: Norml Rt Trn pot $/BP Futurs Pric kwnss Mol Europn CRR/RB Mol Europn % Dirnc CRR/RB-kw $1.45 $1.479 $.5 $ % Incrs Dcrs $1.45 $ % % % $.411 $ % % % % Pnl B: Invrt Rt Trn pot $/BP Futurs Pric kwnss Mol CRR/RB Mol % Dirnc Europn Europn CRR/RB-kw $1.45 $1.41 $.16 $.15.56% Incrs % % % $1.45 $1.41 $.46 $ % Dcrs % % % Pnl C: Nutrl Rt Trn pot $/BP Futurs Pric kwnss Mol CRR/RB Mol % Dirnc Europn Europn CRR/RB-kw $1.45 $1.45 $.76 $. 6.54% Incrs % % % $1.45 $1.45 $.19 $ % Dcrs % % % ngtiv skwnss or n xpct crsing pric trn chrctriz by ngtiv logrithmic mn n positiv skwnss, th CRR/RB mol or lrg n whn com- pr to th skwnss-just uturs options mol unrprics uturs options btwn 4% n %, with th lrgr unrpricing occurring or out-o-th mony options. Copyright cirs.

14 118. JOHNON ET AL. Tbl 4. Comprison o skwnss-just uturs option mol with crr/rb mol or norml, invrt n nutrl uturs mrkts n incrsing n crsing xch ng-rt trns: Europn uturs put options. Pnl A: Norml Rt Trn pot $/BP Futurs Pric kwnss Mol CRR/RB Mol % Dirnc Europn Europn CRR/RB-kw $1.45 $1.479 $.156 $ % Incrs % % % $1.45 $1.479 $.1565 $ % Dcrs % % % Pnl B: Invrt Rt Trn pot $/BP Futurs Pric kwnss Mol Europn CRR/RB Mol Europn % Dirnc CRR/RB-kw $1.45 $1.41 $.195 $ % Incrs % % % $1.45 $1.41 $.198 $ % Dcrs % % % Pnl C: Nutrl Rt Trn pot $/BP Futurs Pric kwnss Mol Europn CRR/RB Mol Europn % Dirnc CRR/RB-kw $1.45 $ Incrs $1.45 $ Dcrs Morovr, with th rltiv contribution o th mn n crrying cost vlus or lrg numbr o subprios bing t lst % or on o th up or own prmtrs in th skwnss mol n miniml in th CRR/RB, th unrpr icing cn b xplin by th chng in symptotic proprtis rsulting rom skwnss tht lvt th impor- Copyright cirs.

15 . JOHNON ET AL. 119 Tbl 5. Comprison o uropn spot n utur option prics or norml, invrt n nutrl uturs mrkts n stbl xchng-rt mrkt. Pnl A: Norml pot $/BP kwnss n CRR/RB pot Option Pric Europn Futurs Pric kwnss n CRR/RB Futurs Option Pric Europn %Dirnc Futurs pot Option Pric $ $1.48 $ % Put % % % $ $1.48 $ % Cll % % % Pnl B: Invrt pot $/BP kwnss n CRR/RB pot Option Pric Europn Futurs Pric kwnss n CRR/RB Futurs Option Pric Europn %Dirnc Futurs pot Option Pric $1.45 $.166 $1.4 $ % Put % % % $1.45 $.18 1 $1.4 $ % Cll % % % Pnl C: Nutrl pot $/BP kwnss n CRR/RB pot Option Pric Europn Futurs Pric kwnss n CRR/RB Futurs Option Pric Europn %Dirnc Futurs pot Option Pric $1.45 $. 6 $1.45 $.6.% Put % % % $1.45 $.164 $1.45 $.164.% Cll % % % tnc o th mn n crrying cost prmtrs. Finlly, it shoul b not tht th BOPM or uturs options convrgs to th sminl Blck uturs option mol s n gts lrg givn norml istribution or th logrithmic rturn. Th Blck uturs mol, in turn, irs rom th B- Mrton mol us or pricing spot options by th Copyright cirs.

16 . JOHNON ET AL. xclusion o th risk-r rt n th ss t yil in th qutions or 1 n. This irnc, though, only hols giv n th ssumption o no rmlity. I skwnss xists, thn th risk-r rt n sst yil, s wll s th mn, bcom importnt in pricing uturs options or th iscrt binomil mol, s wll s th Blck uturs option mol. REFERENCE [1] R.. Johnson, J. E. Pwlukiwicz n J. Mht, Bi- nomil Option Pricing with kw Asst Rturns, Rviw o Quntittiv Finnc n Accounting, Vol. 9, No. 1, 1997, pp oi:1.1/a: [] F. Blck n M. chols, Th Pricing o Options n Corport Libilitis, Journl o Politicl Economy, Vol. 81, No., 197, pp oi:1.186/66 [] R.. Johnson, R. A. Zubr n J. M. Gnr, Binomil pricing o Fix-Incom curitis or Incrsing n D- Intrst Rt Css, Appli Finncil Econo- crsing mics, Vol. 16, No. 14, 6, pp oi:1.18/ [4] R.. Johns on, R. A. Zubr n J. M. G nr, Pricing tock Options unr Expct Incrsing n Dcrsing Pric Css, Qurtrly Journl o Businss n Economics, Vol. 46, No. 4, 7, pp [5] E. M. tin n J. C. tin, tock Pric Distributions with tochstic Voltility: An Anlyticl Approch, Rviw o Finncil tuis, Vol. 4, No. 4, 1991, pp oi:1.19/rs/ [6] J. B. Wiggins, Option Vlus unr tochstic Voltility: Thory n Empiricl Estimts, Journl o Finncil Economics, Vol. 19, No., 1987, pp oi:1.116/4-45x(87)99- [7]. L. Hston, A Clos Form olution or Options n tochstic Voltility with Applictions to Bon n Cur- Vol. 6, No. rncy Options, Rviw o Finncil tuis,, 199, pp oi:1.19/rs/6..7 [8] R. Jrrow n A. Ru, Approximt Option Vlution or Arbitrry tochstic Procsss, Journl o Finncil Econo mics, Vol. 1, No., 198, pp oi:1.116/ 4-45 X(8)97-1 [9] C. J. Corro n T. u, kwnss n Kurtosis in &P 5 Inx Rturns Impli by Option Prics, Th Journl o Finncil Rs rch, Vol. 19, No., 1996, pp [1] A. Câmr n. Chung, Option Pricing or th Trnsorm-Binomil Clss, Th Journl o Futurs Mrkts, 6, Vol. 6, No. 8, pp oi:1.1 /ut.18 [11] J. C. Cox,. A. Ross n M. Rubinstin, Option Pricing: A implii Approch, Journl o Finncil Economics, Vol. 7, No., 1979, pp oi:1.116/4-4 5X(79)915-1 [] R. J. Rnlmn n B. J. Brttr, Two-tt Option Pricing, Journl o Finnc, Vol. 4, No. 5, 1979, pp oi:1.7/77 [1] F. Blck, Th Pricing o Commoity Contrcts, Journl o Finncil Economics, Vol., No. 1-, 1976, pp oi:1.116/4-45x(76)94-6 Copyright cirs.