Bas Peeters 1,2 Cees L. Dert 1,3 André Lucas 1,4

Transcription

1 TI /2 Tnbergen Insttute Dscusson Paper Black Scholes for Portfolos of Optons n Dscrete Tme Bas Peeters 1,2 Cees L. Dert 1,3 André Lucas 1,4 1 Faculty of Economcs and Busness Admnstraton, Vrje Unverstet Amsterdam, 2 Structured Products, IG Investment Management, The Hague, 3 Structured Asset Management, AB AMRO Asset Management, Amsterdam, 4 Tnbergen Insttute.

2 Tnbergen Insttute The Tnbergen Insttute s the nsttute for economc research of the Erasmus Unverstet Rotterdam, Unverstet van Amsterdam, and Vrje Unverstet Amsterdam. Tnbergen Insttute Amsterdam Roetersstraat WB Amsterdam The etherlands Tel.: +31(0) Fax: +31(0) Tnbergen Insttute Rotterdam Burg. Oudlaan PA Rotterdam The etherlands Tel.: +31(0) Fax: +31(0) Please send questons and/or remarks of nonscentfc nature to dressen@tnbergen.nl. Most TI dscusson papers can be downloaded at

3 Black Scholes for Portfolos of Optons n Dscrete Tme: the Prce s Rght, the Hedge s Wrong Bas Peeters, Cees L. Dert, André Lucas Ths verson: ovember 10, 2003 Abstract Takng a portfolo perspectve on opton prcng and hedgng, we show that wthn the standard Black-Scholes-Merton framework large portfolos of optons can be hedged wthout rsk n dscrete tme. The nature of the hedge portfolo n the lmt of large portfolo sze s substantally dfferent from the standard contnuous tme delta-hedge. The underlyng values of the optons n our framework are drven by systematc and dosyncratc rsk factors. Instead of lnearly (delta) hedgng the total rsk of each opton separately, the correct hedge portfolo n dscrete tme elmnates lnear (delta) as well as second (gamma) and hgher order exposures to the systematc rsk factor only. The dosyncratc rsk s not hedged, but dversfed. Our result shows that preference free valuaton of opton portfolos usng lnear assets only s applcable n dscrete tme as well. The prce pad for ths result s that the number of securtes n the portfolo has to grow ndefntely. Ths tes the lterature on opton prcng and hedgng closer together wth the APT lterature n ts focus on systematc rsk factors. For portfolos of fnte sze, the optmal hedge strategy makes a trade-off between hedgng lnear dosyncratc and hgher order systematc rsk. Key words: opton hedgng; dscrete tme; portfolo approach; preference free valuaton; hedgng errors; Arbtrage Prcng Theory. JEL Codes: G13; G12. Correspondence to: bpeeters@feweb.vu.nl, cdert@feweb.vu.nl, or alucas@feweb.vu.nl. Structured Products, IG Investment Management, Postbus 90470, L-2509LL The Hague, The etherlands Postdoctoral Program n Asset Management (FBA) and Dept. of Fnance, ECO/FI Vrje Unverstet, De Boelelaan 1105, L-1081HV Amsterdam, The etherlands Structured Asset Management, AB AMRO Asset Management, Postbus 283 (AP1010), L-1000EA Amsterdam, The etherlands Dept. of Fnance, ECO/FI Vrje Unverstet, De Boelelaan 1105, L-1081HV Amsterdam, The etherlands Tnbergen Insttute Amsterdam, Roetersstraat 31, L-1018 WB, Amsterdam, the etherlands

4 1 Introducton The portfolo approach to asset prcng and asset management has domnated large parts of the fnancal economcs lterature. Early work for lnear securtes lke stocks dates back to Markowtz (1952), Sharpe (1964), Lntner (1965), and Ross (1976). There are two key reasons for the wdespread acceptance of the portfolo approach to asset prcng. Frst, n practce securtes are hardly ever held n solaton. Any sensble analyss should therefore ncorporate the fact that asset prce movements may have a combned effect on ndvduals wealth levels. Second, fnancal securtes are subject to dfferent sources of rsk, such as common, economy-wde or systematc rsk and frm-specfc or dosyncratc rsk. By poolng a suffcent number of securtes nto a portfolo, the latter type of rsk can be elmnated, whereas the former cannot. Only systematc rsk factors are prced n equlbrum, and usually there s only a small number of them, see for example Chen, Roll, and Ross (1982), or Fama and French (1992,1993). Despte the early attenton for the portfolo approach to lnear assets, ts applcaton to non-lnear assets s of much more recent date, see Jarrow, Lando, and Yu (2003) and Björk and äslund (1998) for applcatons to portfolos of credt rsk nstruments and optons, respectvely. otwthstandng the mportant dfference between systematc and dosyncratc rsk followng from a portfolo perspectve, the domnant approach n the optons lterature has been based on asset prcng for ndvdual nstruments, or on a portfolo of optons wth a sngle underlyng securty, see Mello and euhaus (1998). For example, the formula of Black and Scholes (1973) and Merton (1973) for prcng a European call on a stock contans total volatlty of the stock,.e., both ts (prced) systematc rsk component and ts (unprced) dosyncratc rsk component. Also lookng at the hedgng portfolo followng from the Black-Scholes formula, we see that the total rsk rather than the systematc rsk only s hedged. Ths apparent ncongruence s addressed n the current paper, where we adopt the portfolo perspectve to opton prcng n dscrete tme. We have two reasons for consderng the dscrete tme framework. Frst, even though asset prces may move contnuously, tradng and re-balancng only takes place at dscrete tme ntervals due to for example montorng and transacton costs, see, e.g., Boyle and Emanuel (1980), Leland (1985), Glster (1990), Boyle and Vorst (1992), and Mello and euhaus (1998). Second, n the contnuous tme framework underlyng the Black-Scholes analyss wth asset prces followng standard dffusons, no gans are possble by takng a portfolo perspectve. In ths settng the optons are already replcated wthout rsk, and any devaton n prcng would lead to arbtrage opportuntes, see 2

5 also Björk and äslund (1998) and Kabanov and Kramkov (1998). Ths confrms the wdespread approach to opton prcng based on ndvdual nstruments. The portfolo perspectve does make a dfference f some form of market ncompleteness s ntroduced. Björk and äslund (1998) and Jarrow et al. (2003) do ths by havng asset prces follow jump-dffuson processes. In that sense, our approach s related as t ntroduces market ncompleteness through dscrete tradng moments whle asset prces move n contnuous tme. Ths ncompleteness results n non-zero hedgng errors for opton replcaton strateges, whch may even be correlated wth prced rsk factors, see for example Glster (1990). In ths paper we show that the ncompleteness ntroduced by dscrete tradng tmes can be overcome by adoptng a portfolo approach to asset prcng smlar to the APT lterature. By explotng the cross sectonal dmenson of the portfolo of underlyng values, a unque, preference-free prce can be establshed for a portfolo of optons usng only the underlyng stocks to construct a statc hedge portfolo. The prce of our hedge portfolo s equal to the sum of Black-Scholes prces as derved n the contnuous tme framework. In partcular, the prce does not depend on the preferences of the agent that hedges the optons. The correspondng hedge portfolo, however, s entrely dfferent from ts Black-Scholes counterpart. Whereas the typcal delta-hedge n the Black-Scholes framework lnearly hedges total rsk, the hedge portfolo n our framework hedges lnear (delta), second (gamma), and hgher order systematc rsk only. The dosyncratc rsk s dversfed and dsappears asymptotcally. Therefore t need not be hedged to obtan the correct portfolo prce. Ths provdes a closer te between the dfferent arguments n the standard optons prcng and APT lteratures as noted above. Statc hedgng of complex dervatves has been analyzed elsewhere n the lterature, see for example Carr, Ells, and Gupta (1998). The focus there, however, s on statcally replcatng complex dervatves wth smpler dervatves. Here, we concentrate on hedgng the (non-lnear) systematc rsk exposure by holdng a statc portfolo of stocks. The set-up of our paper s as follows. In Secton 2 we ntroduce the model and present the man results. Secton 3 gves some numercal llustratons and robustness checks of our fndngs. Secton 4 provdes concludng remarks. Proofs are gathered n the Appendx. 3

6 2 Model and man results Consder a set of securtes S, =1,...,, followng the multvarate contnuous tme processes ds = S (µdt)+s (Σ 1/2 d z), (1) wth S =(S 1,...,S ), µ =(µ 1,...,µ ),Σ=Σ 1/2 (Σ 1/2 ) a postve defnte covarance matrx, z =( z 1,..., z ) an -dmensonal standard Brownan moton, and a b =(a 1 b 1,...,a b ) for two -dmensonal vectors a and b. To study the dfferent effects of systematc and dosyncratc rsk n ths context, we mpose a factor structure on the covarance matrx Σ. In partcular, smlar to Björk and äslund (1998) and Jarrow et al. (2003), we set Σ=ββ + dag(σ 2 1,...,σ 2 ), (2) wth β =(β 1,...,β ). Ths mposes a one-factor structure on the movement of asset prces. Multple factors can also be accomodated n a straghtforward way, but lead to more cumbersome notaton. By focusng on a sngle systematc rsk factor only, we are able to pnpont n closed form the trade-off between hedgng systematc versus dosyncratc opton rsk at the portfolo level. Usng (2), we rewrte (1) for the th securty as ds S = µ dt + β dz 0 + σ dz, (3) where z = (z 0,z 1,...,z )san( + 1)-dmensonal standard Brownan moton wth z(t) (0,tI). For notatonal convenence we also defne σ 2 = σ 2 + β 2 as the th dagonal element of Σ from (2) correspondng to the total rsk of securty S. In (3), z 0 and z can be nterpreted as the systematc and dosyncratc rsk factor, respectvely. ext, consder a portfolo consstng of equally weghted short call optons on the dfferent securtes S. Other weghtngs for the optons n the portfolo or other types of opton contracts can be ncluded through straghtforward generalzaton. Björk and äslund (1998) show that n a perfect markets settng wthout jump rsk, (3) stll allows for perfect delta hedgng of a portfolo of contngent clams on the S s usng the standard Black-Scholes-Merton approach and hedgng portfolo. In dscrete tme however, the standard Black-Scholes hedge s no longer perfect, n the sense that the expected return of the hedge portfolo only vanshes n expectaton, and no longer almost surely. In ths paper we show that alternatve hedge portfolos can be found that provde a 4

7 lower hedge error varance n dscrete tme. These alternatve hedge portfolos focus on the hgher order exposure to the systematc rsk factor rather than on the lnear exposure to total rsk,.e., systematc plus dosyncratc rsk. We proceed n three steps, whch are formalsed n theorems 1 to 3. Frst, we derve an expresson for the hedge error varance when a standard delta hedge s mplemented for each ndvdual call poston n the portfolo. ext, we derve a smlar expresson for the case where an arbtrary portfolo n the underlyng values s held to hedge aganst fluctuatons n the opton portfolo s values. Ths allows us to demonstrate how hedge error varances can be reduced by an approprate choce of the hedge portfolo. Fnally, we provde our man theoretcal result showng that f the portfolo sze grows ndefntely, t becomes possble to construct a perfect statc hedge portfolo n fnte tme by concentratng on the lnear and hgher order exposures to the systematc rsk factor only. Let C (S,t) denote the usual Black-Scholes prce equaton for the prce of a call on securty S, and let CS(S,t) denote ts dervatve wth respect to S. The standard approach to hedge our opton portfolo over a dscrete tme nterval t s to construct a (hedge) portfolo consstng of a poston CS(S,t) n each of the underlyng securtes S, and a poston Q n cash, wth Q = 1 ( ) C (S,t) C S(S,t)S. (4) We defne the hedgng error H as H = C S(S,t)[S (t + t) S (t)] + Q [ e r t 1 ] [ C (S (t + t ),t+ t) C (S (t),t) ]. (5) For notatonal convenence, we wrte C (S (t),t)asc from now on. A smlar notaton s used for ts dervatves, e.g., C S for the delta of the opton. Usng the above hedgng strategy, the hedge portfolo s the sum of hedge portfolos for the ndvdual postons. We expand the hedge error (5) as a power seres n the length of the hedgng perod t, see also Leland (1985) and Mello and euhaus (1998). Under the present standard delta hedgng strategy, the expected hedgng error and ts varance take the followng form. Theorem 1 Usng delta hedgng of the ndvdual opton postons, the hedgng error H n (5) satsfes E[ H] =O( t 2 ), 5

8 and E [ ( H) 2] = 1 2 [ 1 ] 2 CSSS 2 β 2 t [ (C ) ] SSS 2 2 ( σ 4 β 4 ) t 2 + O( t 3 ). (6) The proof of ths theorem can be found from smple generalzatons of results n the lterature, and follows drectly as a specal case of Theorem 2. Trvally, for = 1, we recover the well-known expresson for the hedge error varance as the square of the opton s gamma. Clearly, ths explct expresson for the varance mples that the return of the opton portfolo s no longer replcated wthout rsk n dscrete tme followng the Black-Scholes hedgng approach. Consequently, t would seem that there s no unque preference free prce for the opton portfolo. The hedge error varance n (6) conssts of two terms. The frst term reflects the contrbuton of the systematc rsk factor z 0. Ths term s of order O( t 2 ), whch ndcates that the systematc component cannot be dversfed 1 n a large portfolo context,.e., s of order O(1) n. The second term, by contrast, s O( t 2 / ) and results from the dosyncratc rsk component of the securtes. The delta hedgng strategy thus clearly benefts from dversfcaton. The dosyncratc component of the varance s effectvely of order O(1/ ). In the lmt for large portfolo szes all dosyncratc rsk dsappears, and only rsk attrbutable to the common market factor remans. For fnte the varance s reduced as a result of the correlaton between the dfferent underlyng securtes. The correlaton between two ndvdual standard hedgng strateges (the off-dagonal terms n the above expresson) have also been derved n Mello and euhaus (1998). The varance reducton for large can, however, be taken a step further by optmzng over the choce of the hedge portfolo. In ths way, we are able to take a more explct advantage of the correlaton structure of the dfferent underlyng values. Consder an alternatve hedgng strategy, where the hedge portfolo contans a fracton D rather than CS of the th securty. Keepng the prce of the opton dentcal to the usual Black-Scholes prce, the cash nvestment follows drectly as Q = 1 ( ) C D S. (7) Usng ths non-standard hedgng strategy, we obtan the followng result for the hedge error varance. 1 Ths appears to be supported by the emprcal results n for example Glster (1990). 6

9 Theorem 2 Usng the hedge strategy wth D S nvested n securty, we can choose the hedge portfolo such that the hedge error H B satsfes E [ H B] = O( t 2 ). If only market rsk s prced,.e. µ = r + κ 0 β for all, wth κ 0 the prce of systematc rsk, then the hedge error varance s gven by E [ ( H B ) 2] = A 1 + A 2 + A 3, (8) wth A 1 = 1 ( ) X 2 σ 2 t, (9) [ 1 ( C SS S 2 X ) ] 2 t β 2 2, (10) A 2 = 1 2 A 3 = 1 2 [ 1 2 ( σ4 β 4 ) ( CSSS 2 X ) 2 +2µ σ 2 (X ) 2 2(µ r)σ 2 X C SSS 2 ] t 2, (11) where X =(D C S)S denotes the devaton from the standard (delta) hedge portfolo. All proofs are gathered n the Appendx. We restrct ourselves n Theorem 2 to the case where only market rsk s prced. Ths mples the absence of asymptotc arbtrage n the lmt, see Björk and äslund (1998). More formally, f µ = r + κ 0 β + κ σ, wth κ the prce of dosyncratc rsk, the excluson of asymptotc arbtrage mposes the restrcton that the set of { κ 0} has measure zero. By defnton, for the standard hedge portfolo X = 0 n Theorem 2 and we recover the expresson n (6). Theorem 2 states that the hedge error varance up to order O( t 2 ) conssts of three terms. The term A 2 reflects the systematc rsk component, whle the terms A 1 and A 3 reflect the dosyncratc part. It s agan easy to see that A 2 s O(1) n, whle A 1 and A 3 are O(1/ ). Moreover, the term A 1 s lnear n t, whereas A 2 and A 3 are quadratc n t. For fnte portfolo sze, n the lmt t 0 we should requre terms lnear n t to vansh n order to mnmze the hedgng rsk. Ths results n the usual allocatons D = C S through the expresson for A 1. In other words, n the contnuous tme lmt the optmal hedge portfolo s the sum of the ndvdual Black-Scholes hedge portfolos. However, ths s no longer the case f we consder the hedge performance for non-nfntesmal values of t. In such settngs there s a dfferent way to decrease the varance of the hedge portfolo. Ths s most clearly seen 7

10 by focusng on the lmtng large portfolo case,. We can then gnore the terms A 1 and A 3. Requrng E [ H B] = O( t 2 ) only mposes a sngle lnear restrcton on the set of allocatons D. The remanng flexblty n the set D can subsequently be used to reduce the varance at hgher orders n t. In partcular, we can select the remanng set of D s such that A 2 vanshes as well. Ths choce s possble as long as not all β s are dentcal. The ntuton for ths result s clear. In a large portfolo context, the dosyncratc rsk can be dversfed. As a result, n our one-factor model only an exposure to the sngle systematc rsk factor remans. The freedom n the choce of the portfolo composton can then be used to hedge aganst hgher order exposures to ths systematc rsk component. For example, settng A 2 to zero annhlates the systematc gamma exposure. By contrast, the standard hedgng approach a pror fxes the hedge portfolo composton such that the lnear delta exposure to the combned systematc and dosyncratc rsk factors are hedged. Consequently, no freedom remans to hedge the undversfable hgher order exposures to the systematc component. For fnte portfolo sze and fnte revson tme t there s a trade-off between hedgng all dosyncratc rsk (n the lmt t 0, the standard Black-Scholes approach) and hedgng market rsk only (n the lmt t ). Clearly, devatons from the standard hedge approach, both n hedge ratos and n reducton of varance, are larger for larger portfolo szes and larger revson ntervals. We can now establsh the generc result that n large opton portfolos hedgng hgher order systematc rsk s preferable to hedgng lnear dosyncratc rsk. Ths s done n the followng theorem. Theorem 3 (the hedge s wrong) Hedgng a portfolo of opton n the general settng defned earler, f only market rsk s prced, we can choose the allocatons D such that E [ H B] = O( t n+1 2 )+O(1/ ), (12) and E [ ( H B ) 2 ] =0+O( t n+1 )+O(1/ ), (13) f n and at least n of the parameters β are dfferent and not equal to zero. If dosyncratc rsk s prced, one can proove a smlar result. The number of securtes needed to construct the hedge portfolo n that case ncreases quadratcally wth n. Moreover the expected return of ths hedgng strategy 8

11 need no longer be zero, resultng n arbtrage opportuntes. Ths s to be expected, as ths market structure, already wthout any optons present, allows for the possblty of arbtrage. The result for the case wth dosyncratc rsk prced s presented n Theorem 4 n the Appendx. The result n Theorem 3 demonstrates that we can construct a rskless hedgng strategy for fnte tme t n the lmt f the β s are dfferent. In other words, the rsk that s ntroduced by gong to a dscrete tme set-up vanshes completely: the systematc rsk component can be elmnated to any arbtrary order of t by choosng the approprate non-standard hedgng strategy. At the same tme, the dosyncratc rsk component dsappears through dversfcaton. The key to the proof s that the hedge portfolo s chosen such that up to suffcently hgh order n t, ts expectaton condtonal on the systematc rsk component concdes wth ts uncondtonal expectaton, see (A10). Ths can be establshed by matchng the hgher order characterstcs of the systematc rsk exposure. To match the two types of expectatons, a set of constrants has to be mposed on the portfolo loadngs D. All of these constrants are lnear n the D s. The coeffcents of these constrants nvolve powers of the systematc volatlty β and hgher order dervatves of the Black-Scholes prce (to capture the approprate curvature), see the Appendx. For ths set of constrants to have a soluton, the number of securtes has to be suffcently large, n. Second, the system of constrants needs to be nonsngular. The latter s ensured by the requrement that at least n of the β s are dfferent and not equal to zero. Gven the entrely dfferent behavor of the standard and portfolo hedgng approach, one mght wonder whether the Black-Scholes prces are stll correct n the present dscrete tme framework. The next corollary states they are. Corollary 1 (the prce s rght) If only market rsk s prced and the number of securtes dverges ( ), and f the β s are dfferent such that we can set the approxmaton order n arbtrarly hgh (n ), then the only arbtrage-free prce of the optons s equal to ther Black-Scholes prce, except for a set of measure zero. Arbtrage opportuntes n Corollary 1 are defned as the possblty to gan a return hgher than the rskfree rate almost surely. By constructon, the hedge portfolo has the same prce as Black-Scholes, see (7). Gven the result from Theorem 3, ths mples that the sum of the optons prces equals the sum of Black-Scholes prces. As Theorem 3, however, also holds for every subseres of call optons and correspondng underlyng values, the set of optons wth prces dfferent from the Black-Scholes prce has measure zero. 9

12 3 umercal llustratons To further llustrate our results and provde more ntuton, we dscuss several numercal examples. We present results for a fnte, but ncreasng set of underlyng securtes, correspondng to Theorem 2, as well as results for the lmtng case, correspondng to Theorem 3. As already noted, n order for the result n Theorem 3 to hold, we need heterogenety n the securtes exposures β to systematc rsk. At the same tme, however, we want to lmt the number of free parameters n our numercal experments. We do so n the followng way. Frst, we set the rskless rate r to zero. Second, we only use a sngle ndcator for dosyncratc rsk,.e., σ 2 σ 2 = cσm, 2 where σ m s the systematc volatlty and c =0.5, 1.0, 1.5. We normalze all ntal stock prces S to unty, and we consder three month at-the-money call opton contracts and a hedgng frequency of one month, t = 1/12. Ths leaves us wth + 2 parameters: the market volatlty σm, 2 the prce of systematc rsk κ 0, and the β s. To retan comparablty across portfolo szes of the portfolo characterstcs, we model the β s as follows. For a gven cumulatve dstrbuton functon (cdf) F, we defne ( ) 2 1 β = F 1 σ m, (14) 2 wth F 1 the nverse of F, and = 1,...,. In ths way we allow for heterogenety across securtes wthout loosng comparablty over ncreasng portfolo szes. We set F to the normal cdf wth mean 1 and standard devaton 0.3. We use a market prce of systematc rsk κ 0 = 20% and a market volatlty of σ m = 25%. Smulatons for alternatve parameter settngs and dstrbutonal assumptons revealed smlar patterns. Fgures 1 through 5 present the key patterns. Fgures 1 through 4 hghlght Theorem 2 for fnte portfolo sze, whle Fgure 5 llustrates the asymptotc case of Theorem 3 and Corollary 1. Fgure 1 presents the hedge error varances for the standard delta hedge and the new portfolo hedge. The new hedge portfolo s optmzed by mnmzng the hedgng error varance as gven n Theorem 2 over the stock holdngs D. Two effects are clear. Frst, for the standard hedge, dversfcaton leads to a decrease n hedgng error varance for ncreasng portfolo sze. From about = 100 onwards, however, most dversfcaton benefts have materalzed and the varance remans stable. Ths remanng varance s caused by the second order exposure to the systematc rsk factor,.e., the systematc gamma rsk. Ths s also clearly seen n Fgure 2, whch plots the percentage of the hedge error varance due to the systematc rsk component. From about = 100 onwards, ths percentage les very close to 100%. In the 10

13 left-hand plot n Fgure 1, less dosyncratc rsk gves rse to hgher hedge error varance for large. Ths s due to the parameterzaton, where the total rsk σ 2 + β 2 s lower for lower values of σ 2 /σm. 2 Consequently, the gamma term C SS n (10) s hgher for smaller σ 2. As for large A 2 domnates A 3, ths explans the orderng of the curves for large. For small, A 3 s also mportant and the order of the curves s reversed. The left-hand panel n Fgure 1 further shows that, by contrast, the hedge error varance for the optmal hedge portfolo contnues to decrease for larger values of. As mentoned earler, ths s due to the fact that the optmal hedge portfolo also protects aganst hgher order exposures to the systematc rsk factor. Ths s seen n the rght-hand panel of Fgure 2, where the percentage of the varance due to systematc rsk exposure s plotted. These percentages decrease rather than ncreases n for suffcently large. In the lmt for dvergng to nfnty, the hedge error varance for the optmal portfolo even tends to zero. The percentage of systematc varance s smaller f there s less dosyncratc rsk for gven β s. Ths follows agan by lookng at (9) and (10). Smaller σ s, by defnton, gve rse to a smaller value of A 1.As a result, the scope for a reducton n A 2 by settng D CS wthout ncreasng A 1 too much s larger for lower dosyncratc rsk (and suffcently large ). The rght-hand plot n Fgure 1 clearly summarzes the results. Hedge error varances can be decreased sgnfcantly by adoptng a portfolo perspectve to hedgng optons. The benefts are larger f the systematc rsk component consttutes the more domnant source of rsk,.e., f σ 2 /σm 2 s smaller. Fgure 3 plots the stock holdngs D as a functon of β for varous portfolo szes. Ths allows us to see whether the optmal hedge portfolo overweghts small or hgh β stocks. As the results are very smlar for varyng ratos of σ 2 to σm, 2 we only present and dscuss the case σ 2 /σm 2 = 1. The frst thng to note s that the loadngs of the standard hedge, D = CS, are relatvely stable. They ncrease wth β, but ther varaton s neglgble compared to the varaton n holdngs for the alternatve hedge portfolos. The allocatons D of the alternatve hedge portfolo depend strongly on the market rsk exposure of the correspondng underlyng securty. Stocks wth zero systematc rsk exposure (β 0) receve a smlar loadng as n the standard hedge. Ths s ntutvely straghtforward as changng the holdngs n these securtes does not generate a large change n the systematc rsk exposure. The optmal hedge overweghts hgh β stocks and possbly negatve β stocks, and underweghts low and medum β stocks. The latter can even be shorted n substantal amounts for suffcently large. It may be less clear at frst sght why the portfolo loadngs take the shape they do n Fgure 3. We have argued n Secton 2 that the precse shape s due to the optmal hedge portfolo adaptng tself to hgher order exposure to 11

14 the systematc rsk factor. Ths effect can easly be vsualzed. In Fgure 4, we plot the condtonal expectaton of hedge errors E[ H B z 0 ], where the condtonng set contans the systematc rsk factor z 0. The result for the standard delta hedge portfolo shows that hedge errors are most extreme for extreme realzatons of z 0. For large realzatons of z 0, the standard hedge s unable to accomodate the convexty n the opton payoff. By overweghtng hgh β stocks and underwegthng low β stocks, the optmal hedge becomes more senstve to extreme realzatons of z 0. Ths s seen by the reducton n condtonally expected hedgng errors over a large range of z 0 outcomes. The ncreased ablty of the optmal hedge to capture the convexty n the opton payoffs becomes more apparent for larger portfolo szes. Ths s also evdent from Fgure 4. The condtonal expected hedgng error can be reduced further by consderng the true lmtng case. We do so n the followng way, makng explct use of the expressons used n the proof of Theorem 3. For approxmaton order n =1, 2,..., we consder a set of n dfferent β s. Each β can be consdered as representng a homogenous group of underlyng values. The β s are constructed as n (14). Usng these β s, we solve for the D 1,...,D n such that the hedge error varance s zero up to terms of order O(( t) n+1 ) and O(1/ ). Because we consder the lmtng case, we dscard the O(1/ ) terms. Ths approach yelds a lnear system of equatons, that can easly be solved numercally. The condtonal expectaton of the value of the opton portfolo s gven by (see Rubnsten (1984)) where E[ C z 0 ]= S ( d 1 ) e r(t t) K ( d 2 ), (15) S = S e (µ 1 2 β2 ) t+β z 0 t d 1 = ln( S /K )+r(t t)+ 1 2 Σ2 Σ d 2 = d 1 Σ Σ 2 = σ 2 (T t)+σ 2 t The condtonal expectaton of the change n the value of the hedge portfolo s calculated usng (A12). The top panels n Fgure 5 provde the results n terms of condtonally expected hedgng errors up to approxmaton order n =8. 2 ote that n = 1 corresponds to the standard hedge. There s a clear jump n hedge error sze when swtchng from n =1ton = 2. Smlar jumps 2 Ths nvolves the computaton of 8th order dervatves of the Black-Scholes prce equaton. 12

15 are seen at every pont where n ncreases from an odd to an even nteger, see the lower graph. Ths s due to the fact that odd moments of the normal dstrbuton are zero, see also (A13) and below. Also note the change n the scale of the vertcal axs between the top-left and top-rght panels. To obtan an ndcaton of the pattern of decrease for ncreasng n, we compute the expectaton of the squared curves n the top panel, augmented wth curves for hgher values of n. The natural logarthm of ths quantty s plotted n the bottom panel n Fgure 5. The decrease of the hedge error sze n pars of n s agan clearly vsble. Moreover, the rate of decrease appears exponental gven the roughly lnear pattern n the bottom panel. Ths agan llustrates the hedge portfolo s ablty to match the hgher order systematc exposures of the opton portfolo f dosyncratc rsk has been elmnated through dversfcaton ( ). The argument s very smlar to the approach taken n Arbtrage Prcng Theory: n the lmt of a large number of securtes, only the systematc sources of rsk n a portfolo of optons need to be hedged n dscrete tme. We end ths secton wth a few remarks on the computatonal aspects underlyng Fgure 5. To compute the optmal stock holdngs D for a specfc approxmaton order n, one needs the n-th order dervatve of the Black- Scholes formula. These dervatves enter the constant term n the lnear equatons for the D. Though these dervatves can easly be determned recursvely, some straghtforward manpulaton shows that they tend to ncrease n absolute sze wth n. In addton, the coeffcents of the D s n the n-th equaton of the lnear system are of the order β n, and thus decrease wth n. These two propertes gve rse to large D values for ncreasng approxmaton order n. Therefore also the contrbuton to the varance of the poston n each separate securty ncreases. As a result of the correlaton between these dfferent securtes, however, the total varance tends to zero. Usng standard double precson, ths cancellaton s only numercally tractable for n 9. The theoretcal results reman vald and can clearly be corroborated numercally for n 8, see Fgure 5. Moreover, the man practcal mplcaton of our results appears that there s already a sgnfcant reducton n error sze when swtchng from the standard delta hedge (n = 1) to a hedge ncorporatng the systematc gamma component (n = 2). 4 Conclusons In ths paper we have shown that n dscrete tme hedgng rsk can be reduced consderably compared to the standard Black-Scholes approach when takng a portfolo perspectve. For fnte portfolo sze one should not focus solely on 13

16 mnmzng all dosyncratc rsk exposures, but nstead consder the trade-off between reducng the lnear dosyncratc rsk exposures versus hgher order market rsk exposures. In the lmt of an nfnte number of securtes, market ncompleteness due to dscrete tradng may even be removed completely. Ths result s n lne wth results of Jarrow et al. (2003) who remove jump rsk by takng a large portfolo approach. In partcular, we showed that f the standard Black-Scholes framework s modfed to allow only for dscrete tradng dates, the Black-Scholes prces for a large portfolo of optons stll provde the correct prce except for a set of optons of measure zero. The correct hedge strategy n dscrete tme, however, s entrely dfferent from the standard Black-Scholes delta hedge: frst (delta), second (gamma), and hgher order exposures to the systematc rsk factors only are hedged at the expense of dosyncratc rsk. The latter s left unhedged, because t can be dversfed va the large portfolo context. Ths tes the lterature on opton prcng closer to that on APT. In the lmtng context of an nfnte number of securtes to dversfy dosyncratc rsk components, only systematc sources of rsk n portfolos of optons need to be hedged. Ths contrasts wth the contnuous tme Black-Scholes-Merton framework, where both systematc and dosyncratc sources of rsk are hedged. Our results have several mplcatons. Frst, our results suggest an alternatve approach to constructng hedges for portfolos of optons. Rsk factors should be dentfed at the portfolo level and t may be proftable to hedge hgher order exposures to systematc rsk factors at the expense of lnear exposures to dosyncratc rsk. Second, our results mply that agents managng portfolos of dervatves are at an advantage f ther portfolos comprse many dfferent underlyng values, subject to these underlyng values beng nfluenced by the same systematc rsk factors. In such a settng, the maxmum beneft can be obtaned from dversfcaton. Thrd, our results suggest further advantages of the portfolo perspectve f transacton costs are taken nto account. Though a complete analyss of the ssue of transacton costs n a portfolo context s beyond the scope of ths paper, some patterns are clear. For large portfolos of optons, t was shown to be better to hedge hgher order terms n the systematc rsk components than to hedge the lnear exposure to the total rsk of all underlyng values. If ndces are avalable that are hghly correlated wth the systematc rsk factor, our results may be renforced by the ncluson of transacton costs. These costs are typcally an order of magntude smaller for ndex related contracts lke optons and futures, than for ndvdual stocks. Ths s especally true f the effects of lqudty and prce mpact are taken nto account. As a result, the costs of settng up and adjustng the hedge portfolo s lkely to be smaller f part of the portfolo relates to ndex contracts. Ths suggests an nterestng lne 14

17 of future research. If systematc rsk s hghly correlated wth ndex nstruments, systematc gamma rsk can be explctly traded through the use of ndex optons. To assess whether or not t makes practcal sense to actually mplement opton hedgng and prcng decsons on a portfolo bass usng ndex nstruments, more nsght s needed nto the (numercal) balance between the dfferent determnants of the hedge error varance: the portfolo sze, the magntude of market rsk and dosyncratc rsk, the varaton n βs, and transacton costs. Especally the latter may be relevant n a portfolo context, as ndex nstruments usually ental sgnfcantly lower transacton costs. A carefull analyss of ths topc, however, requres a more nvolved mult-perod dynamc programmng approach, and s left for future research. Appendx A Proofs Proof of Theorem 2: Consder the portfolo consstng of holdngs D n each securty S, a cash holdng gven by Q n (4), and a short poston n each opton C. The change n value of ths portfolo between subsequent revson ntervals equals the hedgng error H and s gven by H = 1 D S + Q 1 C. (A1) Expandng ths expresson, and usng the fact that S s of order O( t 1/2 ), we obtan H = 1 [ D S + r ( C D ) S t C S S 1 2 C SS( S ) 2 Ct t 1 ] 6 C SSS( S ) 3 CSt S t + O( t 2 ). (A2) Snce the opton prces C are stll gven by the standard Black-Scholes expressons, we can use the Black-Scholes equaton and ts dervatve C t σ2 C SSS 2 r [ C C SS ] = 0 C St σ2 C SSSS 2 +(r + σ 2 )C SSS = 0 (A3) (A4) n order to substtute for Ct and CSt respectvely. Ths leads to H = 1 [ (D C ) S ( S rs t) 1 ( 2 C SS ( S ) 2 σ 2 S 2 t ) 1 ( 6 C SSS ( S ) 3 3 σ 2 S 2 S t ) +(r + σ 2 )CSSS S t] + O( t (A5) 2 ). 15

18 It s now straghtforward to compute the expected hedge performance and ts varance, usng the formulas n the second appendx. We fnd E[ H] = 1 ( D C ) S S (µ r) t + O( t 2 ). (A6) The varance of the hedge performance wth general hedge ratos D s gven by var[ H] = [ 1 ( ] 2 t D C 1 ( S) S β + D 2 C ) 2 S S 2 σ 2 t + 1 [ 1 [ (D C ) ] ] 2 S S C 2 SSS 2 β 2 t 2 [ 1 ( +2 D C ) ][ 1 ( S S β D j C j S) Sj µ j β j ] t 2 [ 1 ( 2 D C ) ][ 1 ] S S β C j SS S2 j β j (µ j r) t [ ( σ β 4 ) [ ( D CS ) ] 2 S CSSS 2 j j +2µ σ 2 ( D C S) 2S 2 2σ 2 (µ r) ( D C S) C SS S 3 ] t (A7) 2. These expressons can be smplfed consderably f we mpose the restrcton on the allocatons D that the total lnear market exposure of the hedge portfolo should vansh, 1 ( D CS ) S β =0. (A8) Ths restrcton can always be mposed, and s dentcally satsfed for the standard Black- Scholes hedge. The alternatve hedgng strategy we propose n ths paper also satsfes ths constrant. If only market rsk s prced,.e. µ = r + κ 0 β for all, wth κ 0 the prce of market rsk, ths condton ensures that the expected return on the hedge portfolo (A6) vanshes up to order O( t 2 ), or E[ H B ] = O( t 2 ), as stated n the theorem. Furthermore, the varance of the hedge portfolo now reduces to var[ H] = 1 ( D 2 CS ) 2 S 2 σ 2 t [ 1 [ (D CS [ 1 2 ) ] ] 2 S CSSS 2 β 2 t 2 4 ( σ β 4 ) [ ( D CS ) ] 2 S CSSS 2 +2µ σ 2 ( D C S) 2S 2 2σ 2 (µ r) ( D C S) C SS S 3 ] t (A9) 2. One mght be concerned about the postvty of ths expresson (and smlarly (A7)), snce t does contan terms that are not a pror postve. These terms, however, combne wth the terms lnear n t nto complete squares, up to terms of order O( t 3 ). 16

19 Proof of Theorem 3: Frst we show that the dfference between the expected value of the hedge error and ts expected value condtonal on the market process z 0 can be made arbtrary small. For any nteger n, we can choose the allocatons D such that E[ H] = E[ H z 0 ]+O( t n+1 2 ). (A10) where z 0 s (0, 1)-dstrbuted and now denotes the realzaton of the market process. In other words, we can choose the allocatons D such that the exposure to the common stochastc factor s cancelled to ths order. Moreover, t wll turn out that both expectatons vansh to ths order n t. We make use of an expanson of the condtonal expectaton E[ H z 0 ] n powers of z 0 and t to derve ths prelmnary result. Focussng on these parameters z 0 and t, we note that n the expanson of the hedgng error each factor of z 0 s always accompaned by a factor t 1/2. In addton, hgher powers of t arse n the expansons, whch means that we can wrte E[ H z 0 ] = h mn (z 0 t 1/2) m t n/2 m, n 0 m + n>0 = l h l kz0 k t l/2. (A11) l>0 k=0 The boundary condtons n the above expressons ensure that no constant (ndependent of t) term appears. The expresson on the second lne s more convenent for our purposes, and wll be used n the remander of ths appendx. Usng the fact that, for odd powers n of z,e[z n] = 0, t follows that hl k =0fl k s odd. Snce our am s to obtan allocatons D such that the exposure to the market stochastc factor z 0 n E[ H z 0 ] cancels, we need the coeffcents h l k to vansh. We therefore focus explctly on the terms n h l k proportonal to the allocatons D. These terms are equal to [ ] S E S z 0 = e βz0 t1/2 +(µ 1 2 β2 ) t 1. (A12) Expandng the above expresson n terms of z 0 and t, and denotng the contrbuton to h l k of the cash and dervatve component of the hedge portfolo by cl k, we obtan ( ) h l 1 1 k = m! k! D S µ 1 m 2 β2 β k c l k f l k s even, wth m = l k 2, =1 0 f l k s odd, (A13) where c l k s of course ndependent of the allocatons D. We can now choose the allocatons D such that all h l k vansh for k>0and l n, wth n gven. If only market rsk s prced, ths amounts to a set of n lnear constrants on the D. We show ths by defnng an operator K that vanshes when actng on E[ H z 0 ]. It s gven by K = t z 0 +2κ 0 t t z 0 2 t z0 2 r. (A14) ote that K does not depend on dosyncratc parameters (no ndex s present). We shall frst establsh that K ndeed dentcally vanshes on the condtonal expectaton of 17

20 the hedgng error. From the expresson for the condtonal expected stock values gven n euquaton (A12) we can nfer mmedtately that K E[S (t + t) z 0 ]=(µ r λβ )E[S (t + t) z 0 ]=0. (A15) Smlarly, we can show that K also vanshes when actng on the condtonal expectaton of the opton portfolo. Calculatng the condtonal expectaton and actng wth the operator K commute, whch mples that we can wrte (note that S, C, CS and C SS are all evaluated at tme t + t) K E[C z 0 ] = E[K C z 0 ] = E [C t + 12 β2 C SSS2 + C SS ( µ κ 0 β 1 2 σ2 + 1 ) σ z 2 t ] rc z(a16) 0. Usng the explct form of the condtonal expectatons n terms of an ntegral over the dosyncratc probablty densty, we can drectly establsh that E[C SS z z 0 ]=σ t E[C S S + C SSS 2 z 0 ]. (A17) Pluggng ths back n nto the prevous equaton, and usng the Black-Scholes equaton, yelds K E[C z 0 ]=(µ r κ 0 β )E [ CSS ] z 0. (A18) Fnally, t follows mmedately that K vanshes when operatng on the cash component of the hedge portfolo. As a result, we establshed that ndeed KE[ H z 0 ] = 0 (A19) f only market rsk s prced. ext we use the representaton of E[ H z 0 ] n terms of the expanson coeffcents h l k as gven n (A11), n order to derve the correspondng constrants on the coeffcents h l k. Those h l k that vansh dentcally as a result of (A19) do not mpose any restrctons on the allocatons D. We therefore rewrte (A19) as l [ 1 2 (l k)hl k rh l 2 k κ 0 (k +1)h l 1 k ] 2 (k + 1)(k +2)hl k+2 z0 k t l/2 1 =0, l>0 k=0 (A20) where we defned h 0 k, h 1 k, and hl k wth k>lto vansh. As a consequence, each coeffcent as denoted by the square brackets has to vansh separately. Usng these relatons, f follows that at each subsequent order n t 1/2, there s only one addtonal constrant, correspondng to h l l. Ths coeffcent s clearly not constraned by the above formula. Settng t to zero, n combnaton wth settng all h k k to zero for k<l, ensures that hm k vanshes for all m l. These remanng constrants, correspondng to h l l wth l n, can be satsfed by choosng the allocatons D f the system of constrants s non sngular, n other words, f the matrx B wth entres b l =(β ) l ( =1,...,, l =1,...,n) has rank(b) n. Ths s equvalent to the statement that out of the securtes we need at least n securtes wth dfferent β, and β 0. Ths establshes the prelmnary result n (A10). Moreover, settng h l l to zero for all l n mples that E[ H z 0] s zero to order O( t) n+1. In partcular also the coeffcents h 2l 0 need to vansh as a result of (A20). The 18

21 explct form of the frst few constrants that are not automatcally satsfed s gven by h 1 1 = 1 h 2 2 = 1 2 (D CS)S β =1 =1 [ (D CS)S CSSS 2 ] β 2. (A21) Clearly, these constrants correspond to the terms n (A7) that domnate n the large lmt, at lnear and quadratc order n t respectvely. In general, these constrants are [ ] h l l = 1 l D S a m m C l l! S m S m β l, (A22) m=1 where the coeffcents a m l are recursvely gven by a 1 l = 1, a m l = ma m l 1 + am 1 l 1 for m =2...l 1, (A23) a l l = 1, whch wll be useful for the numercal results n Secton 3. We next show that the varance of the condtonal expectaton vanshes up to hgher order terms n O(1/ ), or E[( H) 2 z 0 ]=E[ H z 0 ] 2 + O(1/ ). Ths can easly be establshed by wrtng ( H) 2 = 1 2 H H j j (A24) (A25) and notcng the H only depends on the stochastc realzatons z 0 and z. Takng expectatons of all dosyncratc processes z, we obtan E[ H H j z 0 ]=E[ H z 0 ]E[ H j z 0 ]+δ j f (z 0 ), (A26) where f (z 0 ) s a (smooth) functon of z 0. Insertng ths n (A25) proves (A24). Fnally we can combne the two results above to show that the varance of the uncondtonal expectaton of the hedgng error vanshes up to suffcently hgh order n t and 1/. We fnd E[( H) 2 ] = E[E[( H) 2 z 0 ]] = E[E[ H z 0 ] 2 ]+O(1/ ) = E[(E[ H]+E[ H z 0 ] E[ H]) 2 ]+O(1/ ) = E[ H] 2 + O(1/ )+O( t n+1 ), (A27) where we substtuted (A24) n the second equalty, and used (A10) n the last equalty. We next state an analogous theorem for the case when also dosyncratc rsk s prced,.e. µ = r + κ 0 β + κ σ. For large enough portfolo sze (), and suffcent heterogenety, now also n the expected returns (µ ) as well as n exposures to the common market process (β ), t s agan possble to construct a rskless hedge portfolo. Ths portfolo gves rse to (addtonal) arbtrage opportuntes. 19

22 Theorem 4 If both market and dosyncratc rsk are prced, we can choose the allocatons D such that for any nteger n the hedge error varance vanshes up to terms of order O( t n+1 ) and terms of order O(1/ ) f { 1 4n(n + 2) f n s even, mn = 1 4 (n +1)2 f n s odd, and rank(b) mn, where B s an mn -matrx wth elements b j, ( b j = µ j 1 ) m 2 β2 j βj k, (A28) (A29) wth j =1,...,, =1,..., mn, l = 3, k = l2, and m = l k 2. Proof of Theorem 4: Ths proof s mostly analogous to that of Theorem 3. The dfference s that, f dosyncratc rsk s also prced, t s no longer possble to construct an operator smlar to K that annhlates the condtonal expectaton of the hedgng error. As a result, we need a larger number (ncreasng quadratcally wth n) of securtes S n order to satsfy equaton (A10). More specfcally, whereas n the prevous theorem we only had to cancel the nontrval constrants correspondng to h l l wth l n, we now need to choose the allocatons D n order to cancel all h l k wth l n and k>0. ote that we do not need h l 0 to vansh n order for (A10) to be vald. The number of constrants can be found by straghtforwardly countng the number of nontrval h l k n (A13), leadng to the mnmal requred number of securtes to satsfy these constrants beng equal to { 1 4n(n + 2) f n s even, mn = 1 4 (n (A30) +1)2 f n s odd. In order for ths set of constrants to be non sngular, we need n addton that the matrx B wth elements b j =(µ 1 2 β2 )m β k, wth =1,...,, j =1,..., mn, l = j 3, k = 2j 1 2 l2, and m = l k 2, has rank(b) mn. Ths can be straghtforwardly deduced from (A13). The remander of the proof s dentcal to that of the prevous theorem. ote that n ths case there s no constrant on the coeffcents h 2l 0. Therefore the expected value of the hedge return no longer needs to vansh, gvng rse to arbtrage opportuntes. B Useful denttes The followng expressons are used throughout the computatons. They are gven wthout proof, but they can straghtforwardly be derved from equaton (1). The dscrete tme analogue of ths expresson s S S = e (βz0+σz) t 1/2 +(µ 1 2 σ2 ) t 1 (B31) where the stochastc realzatons z 0, z are ndependently (0, 1) dstrbuted. Takng expectatons of the product of powers of (B31) gves rse to [ ( S ) k ( ) ] l Sj k l ( )( ) [( k l E = ( 1) k+l+m+n exp m(µ 1 m n 2 σ2 )+n(µ j 1 2 σ2 j ) S S j m=0 n=0 20

23 + 1 2 (mβ + nβ j ) m2 σ ] 2 n2 σj 2 + mnσ 2 δ j ) t. (B32) Expandng the above expressons n powers of t up to the relevant order for the calculatons n the proof of Theorem 2, gves rse to [ ] S E = µ t S [ ] S S j E = [ β β j + σ 2 ] δ j t S S j [ + µ µ j β2 βj 2 + β β j (µ + µ j ) ] 1 +( 2 ( σ4 β 4 )+2µ σ )δ 2 j t 2 E E [ S S [ S S ( ) ] 2 Sj S j ( ) ] 3 Sj S j [ ( S ) 2 ( ) ] 2 Sj E S S j = [ µ σ j 2 + β 2 βj 2 ( +2β β j µj + σ j 2 ) + ( σ 4 β 4 +2σ 2 ( µ + σ 2 )) ] δj t 2 = 3 [ β β j + σ 2 δ j ] σ 2 j t 2 = [ σ 2 σ 2 j +2β 2 β 2 j +2( σ 4 β 4 )δ j ] t 2 for the expectaton of products of powers of these processes. References (B33) Björk, T. and B. äslund (1998). Dversfed portfolos n contnuous tme. European Fnance Revew 1, 361. Black, F. and M. Scholes (1973). The prcng of optons and corporate labltes. Journal of Poltcal Economy 81, Boyle, P. E. and T. Vorst (1992). Opton replcaton n dscrete tme wth transacton costs. Journal of Fnance 47 (1), 271. Boyle, P. P. and D. Emanuel (1980). Dscretely adjusted opton hedges. Journal of Fnancal Economcs 8, 259. Carr, P., K. Ells, and V. Gupta (1998). Statc hedgng of exotc optons. Journal of Fnance 53, Chen,.-F., R. Roll, and S. A. Ross (1982). Economc forces and the stock market. Journal of Busness 59, Fama, E. F. and K. French (1992). The cross-secton of expected stock returns. Journal of Fnance 47, Fama, E. F. and K. French (1993). Common rsk factors n the returns on stocks and bonds. Journal of Fnancal Economcs 33, Glster, J. E. (1990). The systematc rsk of dscretely rebalanced opton hedges. Journal of Fnancal and Quanttatve Analyss 25 (4),

24 Jarrow, R. A., D. Lando, and F. Yu (2003). Default rsk and dversfcaton: Theory and applcatons. Report, Cornell Unversty. Kabanov, Y. M. and D. Kramkov (1998). Asymptotc arbtrage n large fnancal markets. Fnance and Stochastcs 2, Leland, H. E. (1985). Opton prcng and replcaton wth transacton costs. Journal of Fnance 40 (5), Lntner, J. (1965). Securty prces, rsk and mxmal gans from dversfcaton. Journal of Fnance 20, Markowtz, H. (1952). Portfolo selecton. Journal of Fnance 7, Mello, A. S. and H. J. euhaus (1998). A portfolo approach to rsk reducton n dscretely rebalanced opton hedges. Management Scence 44 (7), 921. Merton, R. (1973). Theory of ratonal opton prcng. Bell Journal of Economcs and Management Scence 4, Ross, S. A. (1976). The arbtrage theory of captal asset prcng. Journal of Economc Theory 13, Rubnsten, M. (1984). A smple formula for the expected rate of return of an opton over a fnte holdng perod. Journal of Fnance 39, Sharpe, W. (1964). Captal asset prces: a theory of market equlbrum under condtons of rsk. Journal of Fnance 19,

25 log Varance Standard: σ 2 /σ 2 m =0.5 Standard: σ 2 /σ 2 m =1.0 Standard: σ 2 /σ 2 m =2.0 Optmal: σ 2 /σ 2 m =0.5 Optmal: σ 2 /σ 2 m =1.0 Optmal: σ 2 /σ 2 m = Varance rato σ 2 /σ 2 m =0.5 σ 2 /σ 2 m =1.0 σ 2 /σ 2 m = Fgure 1: Varances For a portfolo of at-the-money calls, the left panel depcts the hedge error varances of the standard delta hedge portfolo as well as the varances of the optmal hedge portfolo. The rght panel shows the rato of these varance. In both panels these results are shown as a functon of the portfolo sze. The systematc rsk exposures of the underlyng securtes are gven by β /σ m 1+0.3Φ 1 ((2 1)/(2)), Φ 1 the nverse standard normal c.d.f., and =1,...,. The dosyncratc varance s set to σ 2 = cσ 2 m for c =0.5, 1, 2, the market varance to σ m = 25%, the rskfree rate to r = 0, and the market rsk premum to κ 0 = 20%. The optons have a maturty of 3 months (T t =0.25) and the holdng perod s one month ( t =1/12). 23