PORTFOLIO REBALANCING ERROR WITH JUMPS AND MEAN REVERSION IN ASSET PRICES. By Paul Glasserman and Xingbo Xu Columbia University

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1 Stochastc Systems 2011, Vol. 1, 1 37 DOI: /10-SSY015 PORTFOLIO REBALANCING ERROR WITH JUMPS AND MEAN REVERSION IN ASSET PRICES By Paul Glasserman and Xngbo Xu Columba Unversty We analyze the error between a dscretely rebalanced portfolo and ts contnuously rebalanced counterpart n the presence of jumps or mean-reverson n the underlyng asset dynamcs. Wth dscrete rebalancng, the portfolo s composton s restored to a set of fxed target weghts at dscrete ntervals; wth contnuous rebalancng, the target weghts are mantaned at all tmes. We examne the dfference between the two portfolos as the number of dscrete rebalancng dates ncreases. Wth ether mean reverson or jumps, we derve the lmtng varance of the relatve error between the two portfolos. Wth mean reverson and no jumps, we show that the scaled lmtng error s asymptotcally normal and ndependent of the level of the contnuously rebalanced portfolo. Wth jumps, the scaled relatve error converges n dstrbuton to the sum of a normal random varable and a compound Posson random varable. For both the meanrevertng and jump-dffuson cases, we derve volatlty adjustments to mprove the approxmaton of the dscretely rebalanced portfolo by the contnuously rebalanced portfolo, based on on the lmtng covarance between the relatve rebalancng error and the level of the contnuously rebalanced portfolo. These results are based on strong approxmaton results for jump-dffuson processes. 1. Introducton. The analyss of a portfolo s dynamcs s often smplfed by assumng that the consttuent assets can be traded contnuously. For a tradng strategy defned by portfolo weghts, meanng the fracton of the portfolo held n each asset, contnuous tradng leads to an dealzed model n whch the actual weghts match the target weghts at each nstant. For hghly lqud stocks bought and sold on electronc exchanges, contnuous tradng s often a close approxmaton of realty. But for many other asset classes the practcal realty of dscrete tradng cannot be entrely gnored. A portfolo manager may not be able to mantan an deal set of portfolo weghts contnuously n tme; transactons costs and lqudty constrants may lmt the portfolo manager to rebalancng the portfolo to target weghts at dscrete ntervals. Ths research s supported n part by NSF grant DMS AMS 2000 subject classfcatons: Prmary 91G10; secondary 60F05, 60H35. Keywords and phrases: Jump-dffuson processes, portfolo analyss, strong approxmaton, Ito-Taylor expanson. 1

2 2 P. GLASSERMAN AND X. XU In ths paper, we analyze the error n approxmatng a dscretely rebalanced portfolo wth one that s contnuously rebalanced and thus more convenent to model. Our focus s on the effect of jumps and mean reverson n the dynamcs of the underlyng assets. For both features, we examne the lmtng dfference between the contnuous and dscrete portfolos as the rebalancng frequency ncreases. Our man results are as follows. Wth ether mean reverson or jumps, we derve the lmtng varance of the relatve error between the two portfolos. Wth mean reverson and no jumps, we show that the lmtng error, scaled by the square root of the number of rebalancng dates, s asymptotcally normal and ndependent of the level of the contnuously rebalanced portfolo; moreover, the lmtng dstrbuton s dentcal to the one acheved wthout mean reverson. In the presence of jumps, we show that the scaled relatve error converges to the sum of a normal random varable and a compound Posson random varable, based on an argument provded by a referee. For both the mean-revertng and jumpdffuson cases, we derve volatlty adjustments to mprove the approxmaton of the dscretely rebalanced portfolo by the contnuously rebalanced portfolo. These adjustments are based on the lmtng covarance between the relatve rebalancng error and the level of the contnuously rebalanced portfolo. The smpler case n whch the underlyng assets are modeled as a multvarate geometrc Brownan moton s analyzed n Glasserman [12]. The analyss there s motvated by the ncremental rsk charge (IRC) ntroduced by the Basel Commttee on Bankng Supervson [2, 3]. The IRC s ntended to capture the effect of potental llqudty of assets n a bank s tradng portfolo. It models llqudty by mposng a fxed rebalancng frequency for each asset class: some bonds, for example, mght have a lqudty nterval of two weeks, and tranches of asset backed securtes mght have lqudty ntervals of a month or even a quarter. The IRC s thus based on the dfference between dscrete and contnuous rebalancng. The possblty of jumps n asset prces s clearly relevant to portfolo rsk and to the modelng of less lqud assets. One would also expect jumps to have a qualtatvely dfferent effect on rebalancng error than pure dffuson addng jumps should cause the dscretely rebalanced portfolo to stray farther from the target weghts and ths s confrmed n our results. The potental mpact of mean reverson s less evdent: one mght expect mean reverson to offset part of the effect of dscrete rebalancng f t helps restore a portfolo s weghts to ther targets. We wll see that ths s the case, but only for the volatlty adjustment that comes from the covarance between the rebalancng error and the portfolo level. The dstrbuton of the relatve

3 PORTFOLIO REBALANCING ERROR 3 rebalancng error tself s, n the lmt, unaffected by the presence of mean reverson. Dscretely rebalanced portfolos arse n models of transacton costs and dscrete hedgng, ncludng Bertsmas, Kogan, and Lo[4], Boyle and Emanuel [5], Duffe and Sun [11], Leland [21], and Morton and Plska [22]. Sepp [24] examnes the asymptotc error of delta hedgng wth proportonal transacton costs under a jump-dffuson model wth lognormal jump szes. Guason, Huberman, and Wang [14] analyze the effect of dscrete rebalancng on the measurement of trackng error and portfolo alpha. In ther analyss of leveraged ETFs, Avellaneda and Zhang [1] examne the mpact of dscrete rebalancng and derve an asymptotc relaton between the behavor of the fund and the underlyng asset as the rebalancng frequency ncreases. Jessen [16] studes the dscretzaton error for CPPI portfolo strateges usng smulaton. Although these applcatons do not ft precsely wthn the specfcs of our settng, we nevertheless vew our analyss as potentally relevant to extendng work on these applcatons. In Glasserman and Xu [13], we use a contnuously rebalanced portfolo to desgn an mportance samplng procedure to estmate the tal of a dscretely rebalanced portfolo n a pure-dffuson settng, and the results we develop here suggest potental extensons to models wth jumps. The dstrbuton of the dfference between a dffuson process and ts dscrete-tme approxmaton has receved extensve study motvated by smulaton methods, as n Kurtz and Protter [20]. Jacod and Protter [15] study ths error for more general processes, ncludng processes wth jumps. Tankov and Voltchkova [26] apply the results of Jacod and Protter [15] to analyze the error n dscrete delta-hedgng, thus extendng the results of Bertsmas et al. [4] to models wth jumps. In ther analyss of dscretzaton methods, Kloeden and Platen [19] develop strong approxmaton results for stochastc Taylor expansons; Brut-Lberat and Platen [6, 7] derve correspondng expansons for jump-dffuson processes. These results provde very useful tools for our nvestgaton of rebalancng error. The rest of the paper s organzed as follows. Secton 2 ntroduces the mean-revertng and jump-dffuson models and states our man results on the lmtng rebalancng error. Secton 3 derves our volatlty adjustments for dscretely rebalanced portfolos. Numercal examples are gven n Secton 4. The rest of the paper s then devoted to provng our man results. In Secton 5, we provde background on strong approxmaton and then apply these tools to our results for the jump-dffuson model. Secton 6 covers the mean-revertng case. Proofs for the volatlty adjustments are gven n Secton 7. Secton 8 addresses complcatons that arse from the

4 4 P. GLASSERMAN AND X. XU possblty of portfolo values becomng negatve, whch we nterpret as a default. 2. Model dynamcs and man results. We begn by ntroducng two models of the dynamcs of the d underlyng assets n the portfolo, one wth mean reverson and one wth jumps. The frst model s as follows: Exponental Ornsten-Uhlenbeck (EOU) model: ds (t) = µ dt+du (t), = 1,...,d, S du (t) = β(θ U )dt+σ dw(t), U (0) = 0. For each = 1,...,d, the drft µ and volatlty vector σ = (σ 1,...,σ d ) are constants. The model s drven by W = (W 1,...,W d ), a d-dmensonal standard Brownan moton, and each U s a Ornsten-Uhlenbeck process. We recover geometrc Brownan moton as a specal case by takng β = 0. We also nvestgate portfolos under the followng dynamcs for asset prces: Jump-Dffuson (JD) model: ds (t) S (t ) = µ dt+ d σ j dw j (t)+d j=1 ( N(t) j=1 (Y j 1) ), = 1,...,d. Here, N s a Posson process wth ntensty 0 < λ <, and Yj > 0 s the jump sze assocated wth the th asset at the j th jump of N. The {Yj } are..d. across dfferent values of j. All of W, N and {Yj } are mutually ndependent. Each S s rght-contnuous, so the left lmt S (t ) s the value of S just pror to a possble jump at t. The two models could be combned to ntroduce both mean reverson and jumps n the asset dynamcs. However, our nterest les n analyzng the mpact of each of these features, so we keep them separate. To avod confuson between the two models, we underlne varables that are specfc to the EOU case. Gven a model of asset dynamcs, we consder portfolos defned by a fxed vector of weghts w = (w 1,...,w d ), such that d =1 w = 1. Interpret w as the fracton of value nvested n the th asset. The weghts could be the result of a portfolo optmzaton, but we do not model the portfolo selecton problem. In consderng only fxed weghts, we exclude portfolos n whch the weghts themselves change wth asset prces, and ths s a restrcton on the scope of our results. Kallsen [17] showed that under an exponental

5 PORTFOLIO REBALANCING ERROR 5 Levy model such as our JD model, constant weghts are n fact optmal for nvestors wth power and logarthmc utltes. There s a szeable lterature that argues the merts of rebalancng to fxed weghts. Km and Omberg [18] studed portfolo optmzaton wth mean reverson, but ther framework does not ft our settng. See, e.g., Chapters 4 6 of Dempster, Mtra, and Pflug [10] and the many references cted there. Wth contnuous rebalancng to target weghts w 1,...,w d, the value of the portfolo n the EOU model evolves as and thus dv(t) d V(t) = w µ dt+ =1 d w du (t), =1 (1) V(t) = V(0)exp { ( µ w 1 2 σ2 w ) t+ d w σ =1 t 0 e β(t s) dw s +(1 e βt ) θ }, where θ = w θ, µ w = w µ, Σ = (Σ j ) wth Σ j = d k=1 σ kσ jk and σ w = w Σw. In the jump-dffuson model, portfolo value evolves as dv(t) d V(t ) = ds (t) w S (t ) = µ wdt+ =1 d d w σ dw(t)+ w d =1 = µ w dt+σ w d W(t)+d ( N(t) =1 d j=1 =1 ( N(t) j=1 Y j 1 ) w (Y j 1) ), where W s a scalar Brownan moton, W(t) =,j w σ j W j (t)/σ w. Ths expresson assumes that V remans strctly postve, a requrement we wll return to shortly. The soluton to ths equaton s then gven by (2) {( V(t) = exp µ w 1 }N(t) 2 σ2 w )t+σ w W(t) j=1 [ d =1 w Y j We fx a horzon T over whch we analyze the evoluton of the portfolo. For the dscretely rebalanced case, we fx a rebalancng nterval t = T/N, correspondng to a fxed number N of rebalancng dates n (0,T]. Denote the value of the dscretely rebalanced portfolo by ˆV (or ˆV n the EOU case). Wth dscrete rebalancng, the portfolo composton s restored to the target ].

6 6 P. GLASSERMAN AND X. XU weghts at each rebalancng opportunty. Thus, the portfolo value evolves as ˆV((n+1) t) = ˆV(n t) d =1 S ((n+1) t) w, n = 1,...,N 1, S (n t ) and smlarly for ˆV. We normalze the ntal portfolo value to V(0) = ˆV(0) = ˆV(0) = 1. To ensure that the contnuously rebalanced portfolo preserves strctly postve value (.e., to rule out bankruptcy), we mpose the requrement that, almost surely, (3) d w Y > 0, =1 where Y 1,...,Y d have the dstrbuton of the jump szes assocated wth the d assets. That ths condton s suffcent can be seen from (2), and dfferentatng (2) reproduces the stochastc dfferental equaton that precedes t. Ths condton stll allows jumps to decrease portfolo value to levels arbtrarly close to zero. It holds automatcally f all portfolo weghts are postve. The condton s crucal for our analyss because we work wth the relatve error between the dscrete and contnuous portfolos, and the denomnator n the relatve error s the value of the contnuous-tme portfolo. We also make the followng techncal assumpton on the jump szes: Y k (4) w Y < and Y k < for k = 1,...,d; 3 and later, (5) ( ) Y k log w Y <. Here,. 3 ndcates the L 3 -norm of a random varable, and. ndcates the L 2 -norm. Assumptons (3) (4) wll be n force whenever we consder the jump-dffuson model; we use (5) n Secton 3. Even under these assumptons, we cannot rule out the possblty that the dscretely rebalanced portfolo value drops to zero and lower. We therefore adopt the conventon that the portfolo value s absorbed at zero f t would otherwse become less than or equal to zero; we refer to ths event as bankruptcy. We wll show (n Secton 8) that we can gnore the possblty

7 PORTFOLIO REBALANCING ERROR 7 of bankruptcy for our lmtng results because the effect becomes neglgble asymptotcally. Thus, n most of our dscusson, we treat the dscretely rebalanced portfolo as a postve process. We now proceed to state our man results for the EOU model. Our frst result approxmates the relatve error between the dscrete and contnuous portfolos wth a sum of ndependent random varables and dentfes the lmtng varance of the relatve error. Theorem 2.1. For the EOU model, there exst random varables {ǫ n,n, n = 1,...,N,N = 1,2,...}, wth {ǫ 1,N,...,ǫ N,N }..d. for each N, such that [( ) 2 ] ˆV(T) V(T) N (6) E ǫ n,n = O( t 2 ); V(T) n partcular, wth σ = d =1 w σ and (7) ǫ n,n = d n t w =1 (n 1) t Var[ǫ n,n ] = σ 2 L t 2 := s (n 1) t n=1 (σ σ) dw(r)(σ σ) dw(s), [ ] 1 2 (w (Σ Σ)w 2w ΣΩΣw+(w Σw) 2 ) t 2, where denotes elementwse multplcaton of matrces, Ω s a dagonal matrx wth Ω = w. Thus, [ ] ˆV(T) V(T) NVar σl 2 V(T) T2. The varance parameter n ths result can be understood as [ ( d ( d ) 2 σl 2 = Var 1 w (σ 2 Z)2 w σ )], Z =1 where Z N(0,I) n R d. We now supplement ths characterzaton of the lmtng varance wth the lmtng dstrbuton of the error: Theorem 2.2. As N, N (ˆV(T) V(T), ˆV(T) V(T) V(T) =1 ) (V(T)X,X),

8 8 P. GLASSERMAN AND X. XU where X N(0,σ 2 L T2 ) s ndependent of V(T), and denotes convergence n dstrbuton. The lmts n Theorems 2.1 and 2.2 concde wth those proved n Glasserman [12] for asset prces modeled by geometrc Brownan moton. Thus, we may paraphrase these results as statng that the presence of mean-reverson does not change the relatve rebalancng error, as measured by ts lmtng dstrbuton. The absolute error ˆV(T) V(T) does change. In both cases, ts lmtng dstrbuton s that of the ndependent product of the contnuous portfolo (V(T) or V(T)) and X, but the dstrbuton of the contnuous portfolo s tself changed by the presence of mean-reverson. A key feature of Theorem 2.2 s the asymptotc ndependence between the portfolo value and the relatve error. We wll see, however, that wth approprate scalng there s a non-trval covarance between these terms, and the strength of the lmtng covarance depends on the speed of meanreverson. We take up ths ssue when we consder volatlty adjustments n the next secton. We proceed to the lmtng varance of the relatve error n the jumpdffusonmodel. For each asset = 1,...,d, ntroducethecompoundposson process N(t) ( Jt = j=1 To smplfy notaton, we defne Ȳ j = Y j k w ky k j Y j k w ky k j 1 1, and then the compensated verson of Jt becomes J t = Jt λµy t, where µ y = E[Ȳ ]. Let J n = J (n t) J ((n 1) t) and W n = W(n t) W((n 1) t). Denote X N := (ˆV(T) V(T))/V(T). Theorem 2.3. For the JD model, under assumptons (3) and (4), [( ) 2 ] ˆV(T) V(T) N (8) E ǫ n,n = O( t 2 ), V(T) where (9) ǫ n,n = ǫ n,n + d =1 n=1 n t s ] w [b W n J n + d J (r)d J (s), (n 1) t (n 1) t ).

9 and b = σ σ, = 1,...,d. And Var[ ǫ n,n ] = σ 2 L t2 PORTFOLIO REBALANCING ERROR 9 = Var[ǫ n,n ]+ t 2 (w (b b M)w)+ t2 2 w M Mw, where Var[ǫ n,n ] s as n (7), b = [b 1,b 2,...,b d ], and M s the d d matrx wth entres (10) m j := λe[ȳ Ȳ j ]. Thus Var(X N ) σ 2 LT 2. In (9), the ǫ n,n are the error terms that arse n the case of geometrc Brownan moton (.e., wth λ = 0 n the JD model and, equvalently, wth β = 0 n the EOU model). As n the EOU model, the relatve error has a lmt dstrbuton. In the orgnal verson of ths paper, we showed that the lmt could not be normal. The followng result uses an argument due to a referee. Theorem 2.4. Under assumptons (3) and (4), f the jump part s not degenerate,.e. λ 0 and P(Y = 1, = 1,...,d) 1, then N ˆV(T) V(T) V(T) X, where X=X d + T N(t) d j=1 =1 w b ξ jȳ j and ξ j N(0,I) are..d. d- dmensonal standard normal vectors for j 1, ndependent of everythng else. The lmt does not hold n the L 2 sense. The jump-dffuson model produces a heaver-taled dstrbuton for the relatve error, resultng n the falure to converge to a lmtng normal dstrbuton. One can get some ntuton from the asymptotcs of ǫ n,n n (9), where the thrd term s nonzero only when there are at least two jumps n the perod. Though the thrd term n (9) converges to zero n probablty, t does contrbute to the lmtng varance as well as the thrd absolute moment, both of whch are of order Θ( t 2 ). Because of the presence of Ȳ n the lmt dstrbuton, we do not have an asymptotc ndependence result for the JD case, but logv(t) and X N are asymptotcally uncorrelated, as shown later n Proposton 3.2.

10 10 P. GLASSERMAN AND X. XU 3. Volatlty adjustments. We now apply and extend the lmtng results of the prevous secton to develop volatlty adjustments that approxmate the effect of dscrete rebalancng. To motvate ths dea, consder the contnuous-tme dynamcs of the portfolo value n (2), and consder frst the case wthout mean reverson, β = 0. In ths settng, V s a geometrc Brownan moton wth volatlty σ w, wth σ 2 w = w Σw, as defned followng (2). The parameter σ w s a useful measure of portfolo rsk under contnuous rebalancng. The correspondng parameter for horzon T n the EOU model s (the square root of) (11) σ 2 w,β := 1 T Var[logV(T)] = σ2 w 1 exp( 2βT), 2βT and, n the jump-dffuson model, under assumpton (5) ( ) 2 (12) σw,j 2 := 1 d T Var[logV(T)] = σ2 w +λe log w Y. In practce, σ w,β and σ w,j serve reasonably well for large N as an approxmaton for dscretely rebalanced portfolo. Our objectve s to correct these parameters to capture the mpact of dscrete rebalancng Volatlty adjustment wth mean reverson. From the defnton of X N, we can wrte value of the dscretely rebalanced portfolo as ˆV(T) = V(T)(1+X N / N), whch shows that ˆV(T) s the product of the contnuously rebalanced portfolo value and a correcton factor that s asymptotcally normal and ndependent of V(T). We would lke to calculate the volatlty of ˆV(T) the standard devaton of ts logarthm, normalzed by T but because ˆV(T) s potentally negatve, we cannot do ths drectly. Instead, we note that =1 V(T) := V(T)exp(X N / N) = ˆV(T)+O p (1/N), whch yelds a strctly postve approxmaton. The O p (1/N) error n ths approxmaton s neglgble compared to the O p (1/ N) dfference between the dscrete and contnuous portfolos, and we wll confrm that makng ths approxmaton does not change the lmtng varance.

11 (13) For V(T) we have Var[log V(T)] T PORTFOLIO REBALANCING ERROR 11 = 1 [ T Var logv(t)+ X ] N N = σw 2 1 e 2βT 2βT + Var[X N] TN + 2Cov[logV(T),X N] T N = σw,β 2 +σ2 LT t+o( t)+ 2Cov[logV(T),X N] T, N wth σ w,β as n (11) and σ 2 L the varance parameter n (7). Although X N s asymptotcally ndependent of V(T), the covarance term does not vansh fast enough to be neglgble. In the followng proposton, we fnd the lmt of the thrd term, and verfy the valdty of replacng ˆV wth V: Proposton 3.1. () The lmtng covarance s gven by NCov[logV(T),XN ] γ L T 2, where γ L = e β (γ L + w ( σ σ )β(θ θ)), wth (14) γ L = µ ΩΣw µ w σ 2 w +σ4 w w ΣΩΣw. () Moreover, E[( V(T) ˆV(T)) 2 ] = O(N 2 ), and N(Var[log V(T)] Var[logV(T)]) (σ 2 L +2γ L )T 2. Ths result appled to(13) suggests the followng adjustment to the volatlty for the dscretely rebalanced portfolo: (15) σ 2 adj = σ2 w,β +(σ2 L +2γ L ) t. At t = 0, we recover the volatlty for the contnuously rebalanced portfolo, but for small t > 0, the adjusted volatlty ncludes a correcton for dscrete rebalancng. The parameter γ L n (14) s the lmtng covarance derved n Glasserman [12] for assets modeled by multvarate geometrc Brownan moton; thus, at β = 0 we recover the volatlty adjustment derved there n the absence of mean reverson, as expected. The second part of the proposton confrms that the dfference between V(T) and ˆV(T) s neglgble. In Secton 4.2, we present numercal results llustratng the performance of the volatlty adjustment (15) n approxmatng the effect of dscrete rebalancng.

12 12 P. GLASSERMAN AND X. XU 3.2. Volatlty adjustment n the jump-dffuson model. We follow smlar steps n the jump-dffuson model. We set V(T) := V(T)exp(X N / N) wth X N = N 1 ( ˆV((n+1) t) N V((n+1) t) ˆV(b t) ), V(n t) and then (16) Var[log V(T)] T wth σ w,j as defned n (12). n=0 = σw,j 2 + Var[X N] TN + 2Cov[logV(T),X N] T, N Proposton 3.2. () The lmtng covarance s gven by NCov[logV(T),XN ] γ L T 2, where [ ] γ L :=γ L +λ w σ σ µ y +λ [ ( w (µ σ σ +λµy )E Ȳ log l w l Y l µ J )] and µ J = E [ log w Y j ]. () Moreover, E[(ˆV(T) V(T)) 2 ] = O(N 2 ) and (17) N(Var[log V(T)] Var[logV(T)]) ( σ L + γ L )T 2. The resultng volatlty adjustment s σ 2 adj = σ2 w,j +( σ2 L +2 γ L) t. Theasymptotcvaranceparametersfortherelatve error(σ 2 L and σ2 L )donot depend on the drft parameters µ, but, nterestngly, the drfts do appear n the asymptotc covarance γ L (and γ L and γ L ). We wll see that n a stochastc Taylor expanson of the relatve error, the µ appear only n those terms wth norms of order O( t 3/2 ). For the varance, t turns out that only terms wth norms up to order O( t) are relevant, but the covarance nvolves terms of norm O( t 3/2 ), and these nvolve the µ. Snce the volatlty adjustments are explctly related to the weghts, one could reverse the approxmaton as a gudelne for adjustng portfolo weghts to control the portfolo volatlty σ wth dscrete rebalancng.

13 PORTFOLIO REBALANCING ERROR 13 Table 1 Parameters estmated from S&P 500, FTSE 100, Nkke 225, DAX, Swss Market Index, CAC 40, FTSE Strats Tmes Index for Sngapore, Hang Seng, Mexco IPC, Tha Set 50 and Argentna Merval SP500 FTSE NIK DAX SSMI CAC STI HSI MXX SET50 MERV λ w µ µ J σ J Σ Numercal experments and further dscusson of the lmts Example for the jump-dffuson model. We begn wth the JD model model and examne the approxmaton for the relatve error provded by Theorem 2.4. We calbrated the JD model from the daly returns of global equty ndces based on themethod ntroduced n Das [9]. Theweghts are computed as the optmal weghts for power utlty wth rsk averson parameter γ = 2 followng the results of [9] 1. The data used s from March 2009 to March 2011, and the calbrated results are as n Table 1. Jump szes are modeled by Merton s jump model wth log(y ) N(µ J,σ J ). We calbrate the parameters by assumng the jump szes are perfectly correlated as n [9]. However, perfectly correlated jumps would have the same effect as constant jump szes because 1 The negatve weghts could cause defaults, even n the contnuous portfolo, though ths occurs very rarely wth out estmated value of σ J. In our numercal examples, we exclude paths wth defaults. We address ths ssue n Secton 8.

14 14 P. GLASSERMAN AND X. XU Dscrete portfolo Dscrete portfolo Theoretcal lmt Theoretcal lmt 1 Dscrete portfolo Theoretcal lmt Fg 1. Jump-dffuson model: QQ plots of X N versus X at N = 4 (upper left), N = 12 (upper rght), N = 360 (lower left). we are consderng relatve error. To make the example more nterestng, we smulate ndependent jumps szes nstead. Fgure 1 shows QQ plots of the value of dscrete portfolos versus the lmt as descrbed n Theorem 2.4, both smulated over 2500 replcatons. We choose N to be 4, 12 and 360 to represent quarterly, monthly and daly rebalancngs. As the number of steps N gets larger, the fgure ndcates convergence to the theoretcal lmt, though relatvely slower than n the EOU model. Snce the lmtng dstrbuton s not normal, we do not have an asymptotc ndependence result of the type n Theorem 2.2. But the numercal results n Table 2 stll show the correlaton between logv(t) and X N decreasng toward zero as N ncreases. Ths s to be expected because part () of Proposton 3.2 shows the covarance of logv(t) and X N convergng to zero at rate O(1/ N), and X N has a non-degenerate lmtng varance. In separate experments, we have found large dscrepances n the QQ plots when σj are doubled. Estmaton of m j n (10) becomes unstable, and condton (4) may be volated. Table 3 shows the error reducton of volatlty as (18) 1 σ adj ˆσ N σ w,j ˆσ N,

15 PORTFOLIO REBALANCING ERROR 15 Table 2 Correlatons for JD model and EOU model, between logv(t) (or logv(t)) and X N, wth 2500 replcates N JD 12% 13% 4% EOU 85% 61% 13% Table 3 Volatlty error reductons for JD model and EOU model, wth 50,000 replcatons. Formula (18) and (19) are used for JD model and EOU model, respectvely N JD 87% 46% 2% EOU 69% 55% 18% where σ adj s defned n (17). Ths measure shows the relatve mprovement acheved n approxmatng the volatlty usng the adjustment; a small value ndcates small mprovement, and a value close to 1 ndcates good mprovement. These estmates are based on 50,000 replcatons. When the correlaton between V(T) and X N s small, the error reducton tends to be unstable. As suggested by (16), when N s small and the covarance term n (16) s negatve, the error reducton can be small, or even negatve. In ths stuaton, numercal errors, especally from computng the requred expectaton of the Ȳ, can contamnate the results Example for the EOU model. For the purpose of llustraton, we use thesameparametersw,µandσfromsecton4.1.weusethemean-reverson rate β = 1 and long-run levels θ = 0.1 /d, = 1,...,d. Fgure 2 llustrates the convergence to normalty as N ncreases, usng 2500 replcates. Table 2 reports estmated correlatons between logv(t) and X N usng the same parameters as Fgure 2. As expected, the correlaton decreases toward zero as N ncreases. Table 3 evaluates the volatlty adjustment by reportng the estmated error reducton usng the adjustment, calculated as (19) 1 σ adj ˆσ N σ w,β ˆσ N, where σ adj s defned n (15) and ˆσ N s the volatlty of the dscretely rebalanced portfolo as estmated by smulaton. The results n Table 3 show apprecable error reducton, especally when the number of rebalancng dates N s small. When N becomes large, the denomnator σ w,β ˆσ wll become very small. The magntude of the reducton s not necessarly monotone n N. More examples for the dffuson case wthout mean reverson can be found n Glasserman [12].

16 16 P. GLASSERMAN AND X. XU Dscrete portfolo Standard Normal Quantles Dscrete portfolo Standard Normal Quantles Dscrete portfolo Standard Normal Quantles Fg 2. EOU model: QQ plots of X N/σ L T versus standard normal at N = 4 (upper left), N = 12 (upper rght) and N = 360 (lower left). 5. Asymptotc error va strong approxmaton. In ths secton, we develop tools for the strong approxmaton of jump-dffuson models whch we wll need to prove our results for that case. If X solves dx t = ã(x t )dt + b(x t )dw t + c(x t )dj t, and X N X 2 = O( t k ), then we call X N a strong approxmaton of order k. In the absence of jumps, Kloeden and Platen [19] show the same order then apples to almost sure convergence. Brut-Lberat and Platen [6] and [7] treat strong approxmaton for the jump-dffuson case. In followng ther approach t s convenent to thnk of dt as havng order 1, and dw and dj as each havng order 1/2, n terms of ther L 2 -norm. Approxmatons of orderk then nvolve keepng all terms of order k or lower. We use the followng representatons of the contnuous and dscrete portfolos. We set and V(1) = ˆV(1) = N n=1 N n=1 V(n t) N V((n+1) t) = n=1 ˆV(n t) N ˆV((n+1) t) = n=1 R n,n, ˆR n,n,

17 PORTFOLIO REBALANCING ERROR 17 where and ˆR n,n := = ˆV(n t) ˆV((n 1) t) {( d w exp µ 1 2 =1 d j=1 σ 2 j ) t+σ W n } N(n t) V(n t) R n,n := V((n 1) t) {( = exp µ w 1 } N(n t) 2 σ2 w ) t+ σ W n Then ˆR n,n R n,n = = d =1 N(n t) j=n((n 1) t)+1 ˆR n,n c R c n,n j=n((n 1) t)+1 j=n((n 1) t)+1 ( d =1 w Y j {( w exp µ µ w 1 2 σ } 2 σ2 w ) t+(σ σ) W n Y j d =1 w Y j d =1 w exp{(µ µ w 1 2 σ σ2 w ) t+(σ σ) W n } j Y j w exp{(µ µ w 1 2 σ σ2 w ) t+(σ σ) W n } j where ˆR n,n c /Rc n,n Glasserman [12], ˆR c n,n R c n,n = d =1 Y j ). w Y, s the rato of returns n the absence of jumps, as n {( w exp µ µ w 1 2 σ } 2 σ2 w ) t+(σ σ) W n Background on strong approxmatons. As n Kloeden and Platen [19] and Platen [23], we use the followng notaton. For a strng α = ( 1,..., k 1, k ) of ndces, let α := ( 1,..., k 1 ) and α := ( 2,..., k ), for k > 0. Thelength ofthestrngs gven byl(α) = k, andn(α) denotes thenumberof zeros n the strng α. Defne the herarchcal sets A l = {α l(α)+n(α) 2l}, and the correspondng remander sets B(A l ) = {α / A l, α A l }, for l = 1 2,1, 3 2,2,... For a predctable g satsfyng certan regularty and ntegrablty condtons n the man theorem of Platen [23], an terated ntegral I α s defned j

18 18 P. GLASSERMAN AND X. XU as follows: I α [g] t = g(t) f l(α) = 0; t 0 I α [g] z dz f l(α) = 0 and l(α) > 0; t 0 I α [g] z dw z f l(α) = > 0 and l(α) > 0; t 0 I α [g] z d J z f l(α) = < 0 and l(α) > 0. To have a better understandng of the notaton, one can nterpret the strng α = ( 1,..., k ) as the order for terated ntegraton, wth the drecton from left to rght correspondng to the order of ntegraton from nnermost to outermost ntegral. Each entry k ndcates the process aganst whch the ntegral s taken. For example, k > 0 ndcates an ntegral aganst the th k component of the Brownan Moton, whle k < 0 ndcates an ntegral aganst J k. The man result of Platen [23] shows that under our partcular settng where all coeffcent functons are lnear, we have the Ito-Taylor expanson f(t,x t ) = I α [f α (0,X 0 )] t + I α [f α (,X.)] t. α A l α B(A l ) Here we choose f(x) = x and coeffcents are defned by where f α (t,x) = L f(t,x) = x f l(α) = 0; ã(x) f l(α) = 1, 1 = 0 ; b1 (x) f l(α) = 1, 1 > 0 ; c(x) f l(α) = 1, 1 < 0 ; L 1 f α f l(α) > 1; f t +ã f x f x f = 0; j b 2 j f b x f > 0; f(t,x+ c(x)) f(t,x) f < 0. A more detaled treatment of strong approxmatons and ths notaton can be found n Platen[23]. For our applcaton, we need to approxmate w X ( t) := ˆR n,n /R n,n, where {( X,N (t) = exp µ µ w 1 2 σ ) }N(t) 2 σ2 w t+(σ σ) W(t) j=1 Y j wk Y k j.

19 PORTFOLIO REBALANCING ERROR 19 Each X,N satsfes the followng SDE: dx,n (t) (µ X,N (t ) = µ w 12 σ σ2w + 12 ) σ σ 2 dt +(σ σ) dw t +dj t = a dt+b dw t +d J t, where a = µ µ w 1 2 σ σ2 w σ σ 2 +λµ y and b = σ σ. For our analyss, we need some standard propertes of predctable quadratc varatons: < t,t >= 0, < t,wt >= 0 and < t, J j t >= 0 for all and j; < W,W j > t = δ j t, and < J j, J > t = m j t, for constants m j. To derve the approprate constants, we observe that E[ J t J j t ] = E[[ J, J j ] t ] = 1 4 E[[ J + J j, J + J j ] t [ J J j, J J j ] t ] [ = 1 4 E ( J s J s + J s j J j s )2 ] ( J s J s J s j + J j s )2 0<s<t [ ] = E (( J s J s )( J s j J j s )) 0<s<t = λte[ȳ Ȳ j ]. 0<s<t The thrd equalty s due to the fact that a compound Posson process N(t) =1 Z has quadratc varaton N(t) =1 Z2 (Cont and Tankov [8, Example 8.4]). Thus, we need m j = λe[ȳ Ȳ j ] Strong approxmaton for the jump-dffuson model. We now use the strong approxmaton scheme of order 3/2 to prove Theorem 2.3 and 2.4. Frst we wrte X,N ( t) = 1+ζ 1/2,N +ζ 1,N +ζ 3/2,N +r N, where ζ ị,n are defned as follows. Frst, t ζ1/2,n = b dw + 0 t 0 d J = b W + J. (From now on we drop the lmts of ntegraton for terated ntegrals taken over [0, t]. An ntegral of the form gd J should be understood as

20 20 P. GLASSERMAN AND X. XU (20) ζ 1,N = a + g(t )d J (t).) Contnung, we have dt+ d J b dw + b dwb dw + d J d J b dwd J and ζ3/2,n = a b dwdt+a dtb dw +a d J dt+a dtd J + b dwb dwb dw + b dwb dwd J + b dwd J b dw+ d J b dwb dw+ b dwd J d J + d J d J b dw + d J b dwd J + d J d J d J. (21) By observngthat w b = 0 and w J = 0, wefndthat w ζ1/2,n = 0. For the next term, we have w ζ1,n = w [ǫ n,n +b W J + d J d J ]. Here, ǫ n,n s the correspondng error term n the absence of jumps; the last two terms are the dfference between the contnuous and jump-dffuson cases. It s now easy to see that w ζ 1,N = O( t), and smlarly w ζ 3/2,N = O( t3/2 ). Now we need to show that the remander r N satsfes r N = O( t2 ). Lemma 5.1. (Modfed from Studer[25, Lemma 3.42].) Gven an adapted caglad (left contnuous wth rght lmts) process g(t), wth t 0 E[g(s)2 ]ds = K <, then tk, f M t = t; [( t ) 2 ] E g(s)dm s 0 K, f M t = W t ; m K, f M t = J t. (The ntegrand should be understood as g when M = W.)

21 PORTFOLIO REBALANCING ERROR 21 Proof. The result and proof are the same as n Studer [25]. To bound the error when we truncate a strong approxmaton, we can apply a result of Studer [25, Proposton 3.43], or s smlar result of Brut- Lberat and Platen [6, Theorem 6.1]. Out settng s smpler than thers because of the specal form of the dynamcs n the JD model. Lemma 5.2. (Modfed from Studer [25, Proposton 3.43].) Under our assumptons (3) and (4) for the JD model, there exst some constants C 1 and C 2 such that for any = 1,...,d E[(X,N (t) β A k I β [f β (0,X,N (0))]) 2 ] C 1 (C 2 t) 2k+1. Proof. Snce f(t, x) = x, the condtons n Studer [25, Proposton 3.43] (and those n Brut-Lberat and Platen[6, Theorem 6.1]) are satsfed. Thus, for any α B(A k ), we can fnd some constant C 3 sup E[(f α (t,x,n (t))) 2 ] C 3. 0 t T Denote n (α) bethenumberof components for J nα. By nducton andthe prevous lemma, we have for any α B(A k ), we can fnd some constant C 4 E[I α [f α (.,X,N (.))] 2 t] t n(α) (m ) n (α) C 3 t l(α) and B(A k ) (3d+3) k+1, therefore, E [( X t β A k I β [f β (0,X 0 )] ) 2 ] C 3 C 2k+1 4 t l(α)+n(α) ; ( α B(A k ) ( α B(A k ) C 1 (C 2 t) 2k+1. (E[I α [f α (.,X)]]) 1/2 ) 2 ) 2 (C 3 C4 2k+1 t l(α)+n(α) ) 1/2 As a consequence, for our settng we get Lemma 5.3. r N = X,N 1 ζ 1/2,N ζ 1,N ζ 3/2,N = O( t2 ).

22 22 P. GLASSERMAN AND X. XU 5.3. Correlaton between ζ1 and ζ 3/2. In ths secton, we show that the terms w ζ1,n and w ζ3/2,n are uncorrelated. Before specalzng to our settng, we derve some general propertes used extensvely n ths subsecton. To calculate the covarance between terated ntegrals, from Cont and Tankov [8, Proposton 8.11] we have (usng the notaton of Lemma 5.1) [ ] E[I α1 I α2 ] = E I α1 dm 1 I α2 dm 2 [ ] = E I α1 I α2 dm 1 + I α2 I α1 dm 2 + I α1 I α2 d[m 1,M 2 ] [ ] (22) = E I α1 I α2 dm 1 + I α2 I α1 dm 2 + I α1 I α2 d < M 1,M 2 >, where t f r(α ) = 0; M (t) = W t f r(α ) = 1; J t k f r(α ) = k < 0, wth r(α ) the rghtmost element of α. As before, when M = J k for some andk,weusetheleft-contnuousversonofthentegrand.whenm = W,we take ts transposen thentegrand. Hereweusethesquarebracket andsharp bracket to denote quadratc varaton and predctable quadratc varaton as ntroduced towards the end of Secton 5.1. When r(α ) 0 for both = 1 and 2, M r(α ) s a martngale, so after takng expectatons, the frst two terms n (22) vansh. Assumpton (4) mples square ntegrablty of these terated ntegrals, whch contan jump terms. Otherwse, when they consst of only dt or dw, ther ntegrablty s mmedate. Thus, we have the followng possble combnatons: When r(α 1 ) > 0 and r(α 2 ) = j < 0, M 1 and M 2 are uncorrelated martngales, so the expectaton of ther product s 0. Thus, we have: [ ] (23) E Iα 1 dw I α2 d J j = 0. When r(α 1 ) = r(α 2 ) = 1, [ E Iα 1 dw ] Iα 2 dw = E[I α 1 I α2 ]dt and when r(α 1 ) =, r(α 2 ) = j, [ ] (24) E I α1 d J I α2 d J j = E[I α1 I α2 ]m j dt.

23 PORTFOLIO REBALANCING ERROR 23 When r(α 1 ) = 0 and r(α 2 ) 0,the second and the thrd term n (22) vansh, leavng [ ] E I α1 dt I α2 dm 2 = E[I α1 I α2 ]dt. Now we apply these results to analyze the correlaton between ζ 1 and ζ 3/2. Let B l,n = {γ l(γ) = l,n(γ) = n}. All strngs n B l,n are of the same length l and have the same number of zeros n. We observe from (20) and (21) that ζ 1,N s a lnear combnaton of elements n B(A 1) and ζ 3/2,N s a lnear combnaton of elements of B(A 3/2 ). From here untl the end of ths subsecton, we let α and β be strngs wth l(α) = 1 and l(β) = 3/2, and we treat all possble combnatons of values of n(α) and n(β): (a) If n(α) = 0 and n(β) = 0 that s, nether contans dt ntegrals then (23) (24) show that E[I α I β ] equals to an ntegral aganst dt wth ts ntegrand ether zero or E[I α I β ]. Applyng the same argument agan, so we can say that E[I α I β ] s agan an ntegral aganst dt wth ts ntegrand ether zero or E[I α I β ], whch s zero, snce l(α) = 1. So E[I α I β ] = 0 for any α B 1,0 and β B 3/2,0. Hence any lnear combnaton of elements of {I α : α B 1,0 } and any lnear combnaton of elements of {I β : α B 3/2,0 } are uncorrelated. (b) If l(α) = n(α) = 1, but n(β) = 0, then I α s actually determnstc. So E[I α I β ] = I α E[I β ] = 0, snce I β s a martngale. Hence any lnear combnaton of elements of {I α : α B 1,1 } and any lnear combnaton of elements of {I β : α B 3/2,0 } are uncorrelated. (c) For the case n(α) = 0 and n(β) = 1, we observe that n our partcular settng, for any 0, I (,0) and I (0,) always appear n pars n ζ and have the same coeffcents. Usng ntegraton by parts we can consder them n pars, for 0, to get I (,0) +I (0,) = d(tm ) = t M, so E[I α (I (,0) +I (0,) )] = te[i α M ] = 0, the last equalty followng from the same argument as (a). Hence, any lnear combnaton of elements of {I α : α B 1,0 } and any lnear combnaton of elements of {I β : α B 3/2,1 } are uncorrelated. (d) If n(α) = 1, and n(β) = 1, then I α = t, whch s determnstc, and I (,0) + I (0,) = t M has zero mean. Hence any lnear combnaton

24 24 P. GLASSERMAN AND X. XU of elements of {I α : α B 1,1 } and any lnear combnaton of elements of {I β : α B 3/2,1 } are uncorrelated. To summarze, we have proved Lemma 5.4. w ζ 1,N and w ζ 3/2,N are uncorrelated Convergence Proofs. Usng our analyss of the strong approxmaton for the jump-dffuson case, we can now prove Theorems 2.3 and 2.4. Proof. (Theorem 2.3): We have ˆR n,n R n,n = 1+ w ζ 1,N + w ζ 3/2,N + w r N. We have shown that w ζ 1,N = O( t), w ζ 3/2,N = O( t3/2 ), w rn = O( t2 ), that E[ w ζ1,n ] = 0 and E[ w ζ3/2,n ] = 0, and that w ζ1,n and w ζ3/2,n are uncorrelated. We can now follow the argument used n Glasserman [12, Proposton 1] to prove (8). Next we calculate the varance of the relatve error. To condense (9), let A = w [b W J ], and B = w [ d J d J ]. By followng steps smlar to those used to prove Lemma 5.4, we can show that the parwse correlatons between ǫ n,n, A, and B are all zero. Thus, Var[ ǫ n,n ] = Var[ǫ n,n ]+Var[A]+Var[B]. We need to calculate the last two terms on the rght. For A, we have [( Var[A] = E[A 2 ] = E w b W J ) 2 ] = E[( W bω J) 2 ] [( ) ] = E J (Ωb bω) J W 2 = t 2 (w (b b M)w). For B, we have [( w ) 2 ] Var[B] = E[B 2 ] = E d J d J = w w j E[< J, J j > s ]m j ds,j = t2 2 w M Mw.

25 PORTFOLIO REBALANCING ERROR 25 Proof. (Theorem 2.4): Frst, from the expresson of the asymptotcs of the relatve error n (9), the contrbuton of the compensaton terms n the jump terms are of lower order, so we can replace J n and J wth Jn and J respectvely throughout (9) and (8) stll holds. That s, [( ) 2 ] ˆV(T) V(T) N E ǫ n,n = O( t 2 ) V(T) (25) where ǫ n,n = ǫ n,n + d =1 n=1 n t s w [b W n Jn + dj (r)dj (s)]. (n 1) t (n 1) t The last term n (25) s nonzero only when there are at least two jumps n the perod [(n 1) t,n t], whch has probablty O( t 2 ). Snce the number of jumps n dfferent perods are..d., the probablty that none of the tme ntervals has more than one jump s of order 1 O( t), so N N n=1 =1 d n t w (n 1) t s (n 1) t dj (r)dj (s) 0. For the same reason, we can gnore multple jumps n each t nterval n (25). More precsely, N(T) ( d ) d (26) N w b W n(j)ȳ n(j) w b W n(j) Jn(j) 0, j=1 =1 where n(j) s the ndex of the nterval when j th jump takes place. To analyze the lmt of (26), we rewrte t as (27) N n n(j) j=1,...,n(t) ǫ n,n + N n=n(j) j=1,...,n(t) =1 ǫ n,n + N N(T) j=1 d w b W n(j) Ȳn(j). Let N, notng that N(T) remans fxed. In (27), the frst term s ndependent of the other two terms, and t converges to X N(0,σL 2 T), as shown n Theorem 2.1. The second term n (27) converges to zero n L 2 and thus n probablty. Thus (27) converges n dstrbuton to N(T) X + j=1 d =1 w b ξ j Ȳ j, =1

26 26 P. GLASSERMAN AND X. XU where ξ j are..d. standard normal random varables ndependent of everythng else. The lmt does not hold n L 2, snce the L 2 -norm of the thrd term n (25) has order O( t 2 ), as shown n the proof of Theorem Strong approxmaton for the mean-revertng case. In ths secton, we prove Theorem 2.1. We buld on the strong approxmaton technque ntroduced n Secton 5.2, but the argument wll be somewhat smpler because we no longer have jump terms. Proof. (Theorem 2.1). The value of the dscretely rebalanced portfolo at t s gven by ˆV( t) = w exp {(µ 12 t σ ) t+σ 2 e β( t s) dw s 0 +(1 e β t )θ }, and the rato of the dscrete portfolo value to the contnuous portfolo value s gven by ˆR N = w exp{(µ 1 t 2 σ2 ) t+σ 0 e β(s t) dw s +(1 e β t )θ } R N exp{(µ w 1 2 σ2 w) t+ σ t 0 e β(s t) dw s +(1 e β t ) θ} = {( w exp µ µ w 1 t 2 ( σ 2 σw) ) t+(σ 2 σ) e β(s t) dw s 0 } +(1 e β t )(θ θ) =: w C ( t), where each C satsfes dc = C [(µ µ w 1 2 ( σ 2 σw 2 σ σ 2 ))dt+dū] dū = β(θ θ Ū)dt+(σ σ) dw. Usng strong approxmaton as ntroduced n Secton 5.2, we get (wth all terated ntegrals taken from 0 to t): ( C ( t) = 1+ µ µ w 1 ) 2 ( σ 2 σw 2 σ σ 2 ) ( t+ dūdt + dtdū )+ Ū + dū dū + dū dū dū +O( t 2 ),

27 PORTFOLIO REBALANCING ERROR 27 where t Ū = (σ σ) e β t e βs dw s +(1 e β t )(θ θ) 0 t ) = (σ σ) ( W β W s ds +β(θ θ) t+o( t 2 ). 0 Expandng the terated ntegrals of Ū and substtutng, we get [( C ( t) = 1+(σ σ) W + µ µ w 1 ) 2 ( σ 2 σw) 2 t + 1 ] [ 1 2 W B W + 6 W B W(σ σ) W +(µ µ w 1 2 ( σ 2 σw))(σ 2 σ) W t t )] (σ σ) (β W s ds 0 +β(θ θ) t+β(θ θ)(σ σ) W t+o( t 2 ), where we drop the term W t B 0 sdw s because ts L 2 -norm s O( t 2 ). Now takng the weghted sum of the C, we get [ ( ] w C ( t) = w σ 2 σw ) t W B W + [ 1 w 6 W B W(σ σ) W + (µ µ w β(θ θ) 12 ) ] ( σ 2 σ 2w ) (σ σ) W t +O( t 2 ) =: 1+ζ N 1 +ζn 3/2 +r where B = w B and r = O( t 2 ). Followng essentally the same arguments used n the jump-dffuson case, t s now easy to show that ζ N 1 = O( t) and ζn 3/2 = O( t3/2 ), and also that ζ1 N and ζn 3/2 are uncorrelated, leadng to ˆV(T) N V(T) 1 n=1 ζ N 1,n = O( t).

28 28 P. GLASSERMAN AND X. XU At the same tme, ζ N 1,n = 1 2 ( W B W Tr( B) t) = ǫn,n, concdes wth the ǫ n,n n the case of multvarate geometrc Brownan moton consdered n Glasserman [12]. The same lmt therefore apples here. Gven the representaton n Theorem 2.1, the proof of Theorem 2.2 s the same as that of Theorem 1 n Glasserman [12]. 7. Analyss of the volatlty adjustments The jump-dffuson case. Proof. (Proposton 3.2) Wth X N = N 1 ( ˆV((n+1) t) N V((n+1) t) ˆV(n t) ) V(n t) n=0 we can wrte Cov[logV(T),X N ] as Cov[logV(T),X N ] = [( N N 1 N E σ W k + k=1 n=0 N(k+1) j=n(k)+1 ( log )) ( ˆV((n+1) t) w Yj µ J V((n+1) t) ˆV(n t) ) ] V(n t) where, as before, µ J = E[log w Yj ]. If we nterchange the order of summaton and fx a value of n, we need to evaluate (28) [( E σ W k + N(k+1) j=n(k)+1 ( log for whch we have three cases: (1) k n+2. In ths case, we have [( ( E σ W k + )) ( ˆV((n+1) t) w Yj µ J V((n+1) t) ˆV(n t) ) ], V(n t) N(k+1) j=n(k)+1 ( ˆV((n+1) t) V((n+1) t) ˆV(n t) V(n t) log ) ] = 0, w Y j µ J ))

29 PORTFOLIO REBALANCING ERROR 29 because W(k) and N(k+1) j=n(k)+1 (log w Yj ) are both ndependent of (ˆV(n t),v(n t), ˆV((n+1) t),v((n+1) t)). (2) k = n+1. (28) becomes (29) [ ] ( ˆV(n t) E E σ W n+1 + V(n t) N(k+1) j=n(k)+1 ( log w Y j µ J )) ˆRn+1 R n+1. Multplyng the factors nsde the last expectaton produces two terms. For the frst, we have [ ] E σ ˆRn+1 W n+1 R n+1 = [ {( w E σ W n+1 exp µ µ w 1 2 σ ) 2 σ2 w t } ] N(n+2) +(σ σ) Yj W n+1 wl Yj l = j=n(n+1)+1 w ( σ σ σ 2 w ) texp{(µ µ w +σ 2 w σ σ) t+λ t(µy )} (30) = γ L t 2 + w σ σ λµ y t2 +O( t 3 ). For the other term, from (29) we have E [ N(k+1) = j=n(k)+1 ( log w Y j µ J ) ˆRn+1 R n+1 w exp{(µ µ w +σ 2 w σ σ) t} ] (31) E [( N(k+1) r=n(k)+1 (Ȳ +1) )( N(k+1) j=n(k)+1 ( log w Y j µ J ))], where E [( N(k+1) r=n(k)+1 )( N(k+1) ( (Ȳ r +1) log j=n(k)+1 w Y j µ J ))]

30 30 P. GLASSERMAN AND X. XU [ n n = e λ t(λ t)n (Ȳ E +1 )( log )] w l Yj l µ J n! n=1 j=1 k=1 l (32) = exp{λ tµ y } tλe[(ȳ +1)(log l w l Y l µ J )]. Substtutng (32) nto (31), we get [ N(k+1) ( E log ) ( ) ] ˆRn+1 w Yj µ J R n+1 j=n(k)+1 (33) = w exp{(µ µ w +σw 2 σ σ) t}exp{λ tµ y } [ ( tλe (Ȳ +1) log w l Y l µ J )]. l Applyng a Taylor expanson to the exponental part under assumptons (4) and (5), (33) becomes [ ( (34) w λ(µ σ σ +λµ y )E Ȳ log w l Y l µ J )] t 2 +O( t 3 ). l E Usng (30) and (34) we have for (29) [( ( σ W n+1 + N(k+1) j=n(k)+1 j=n(k)+1 log )) ( ) ] ˆRn+1 w Yj E = γ L t 2 +O( t 3 ). R n+1 (3) k < n+1. The same argument apples n ths case, and we have [( N(k+1) ( E σ W n+1 + log )) ( ˆV((n+1) t) w Yj µ J V((n+1) t) ˆV(n t) ) ] V(n t) = O( t 4 ). Hence we have N 1/2 Cov[logT(T),X N ] = γ LT 2 N +O(N 2 ). () For the second part of the proposton, we need to show that E[( V(T) ˆV(T)) 2 ] = O(N 2 ). By followng the steps of a smlar proof n Glasserman [12], t suffces to show E[V(T) 2 X 2 N ] <.

31 PORTFOLIO REBALANCING ERROR 31 We can wrte V 2 (T) = exp{2µ w T +σw 2 T}exp{2 σ W(T) 2σw 2 T}exp{ (λ λ)t} N(t) ( 2 exp{(λ λ)t} w Yj), j=1 and now we would lke to use the followng as a Radon-Nkodym dervatve: ( N(t) 2 exp{2 σ W(T) 2σwT}exp{(λ 2 λ)t} (35) w Yj). The frst exponental term s tself a Radon-Nkodym dervatve for the dffuson process. From assumpton (4), we have E[(Y ) 2 ] <, so we can choose an approprate λ such that f(y) = λy 2 f(y)/ λ s a well-defned densty functon, where f(.) and f(.) are the densty functons for w Y under the orgnal probablty and the new probablty measure, respectvely. Therefore, (35) s ndeed a Radon-Nkodym dervatve, and, under the probablty measure t defnes, each asset s drft s changed from µ to µ +2σ σ, and the w Y now have densty f. From Theorem 2.3, the convergence of the second moment of X N holds as long as the drfts and Posson rate are constant, and assumpton (3) and the frst nequalty of (4) hold under the new measure. Because of absolute contnuty, (3) wll stll hold. For (4) j=1 Ẽ[ Ȳ k +1 3 ] = exp{(λ λt)}e[ Y k 2 Ȳ k +1 ] exp{(λ λt)} Ȳ k +1 3 Y k 2 3 <. Hence we have proved the second part of the proposton The mean-revertng case. Proof. (Proposton 3.1): () Wth we have Cov[logV(T),X N ] (36) = N N N 1 k=1 n=0 X N = N 1 N n=0 ( ˆV((n+1) t) V((n+1) t) ˆV(n t) V(n t) k t ( ˆV((n+1) t) E [ σ e β e βs dw s (k 1) t V((n+1) t) ˆV(n t) )]. V(n t) ),

32 32 P. GLASSERMAN AND X. XU For k n+2, k t ( ˆV((n+1) t) E [ σ e β e βs dw s (k 1) t V((n+1) t) ˆV(n t) )] = 0, V(n t) For k = n+1, we have (n+2) t E [ σ e β = E [ ˆV(n t) V(n t) ( ˆV((n+1) t) e βs dw s V((n+1) t) ˆV(n t) )] V(n t) (n+2) t ]E[ σ e β e βs ˆRn+1 dw s ]. R n+1 (n+1) t (n+1) t (37) (n+2) t ] E [ σ e β e βs ˆRn+1 dw s (n+1) t R n+1 = w σ (σ σ)e β(1+ t) t 0 e βs ds exp {(µ µ w 12 ) ( σ 2 σ 2w ) t σ σ 2 e 2β t t 0 } e 2βs ds+(1 e β t )(θ θ)). We only need ts coeffcent on t 2, whch s w ( σ σ )e β (µ µ w 1 2 ( σ 2 σw) σ σ 2 +β(θ θ)) = w ( σ σ )e β (µ µ w +σw 2 σ σ +β(θ θ)) ( = e β γ L + ) w ( σ σ )β(θ θ). For the frst factor n (37), we have E So, we have [ ] ˆV(n t) = V(n t) n [ ] ˆR E n+1 = k=1 R n+1 n (1+O( t 2 )) = 1+O( t). (n+2) t ( ˆV((n+1) t) E [ σ e β e βs dw s (n+1) t V((n+1) t) ˆV(n t) )] = γ V(n t) L t 2 +O( t 3 ). k=1

33 PORTFOLIO REBALANCING ERROR 33 For the case k n, followng the same argument as n the proof of Glasserman[12, Prop. 4], we get (k+1) t ( ˆV((n+1) t) E [ σ e β e βs dw s k t V((n+1) t) ˆV(n t) )] = O( t 4 ), V(n t) and then (36) becomes N 1/2 Cov(logV(T),X N ) = γ L T2 N +O(N 2 ). The proof for part () follows the same lne as the one n Glasserman [12]. The only modfcaton needed s that now the Grsanov transformaton s a lttle more general, the change of measure now changng the standard Brownan moton W(T) to a Gaussan process T 0 eβs W(s). 8. Dealng wth defaults. As explaned n Secton 2, jumps n asset values can produce negatve portfolo values, even under contnuous rebalancng. Here we address ths ssue n greater detal. Assume that once a portfolo defaults (.e., drops to zero or below), t s absorbed at zero forever. It follows from (2) that such a default occurs n a contnuously rebalanced portfolo f and only f there s a jump before tme T wth w Y 0. Under assumpton (3), the contnuously rebalanced portfolo wll therefore never default. The dscretely rebalanced portfolo wll default at tme t n the n th tme nterval f and only f t s the frst tme that t [(n 1) t,n t] wth t = t t t t and ( ) ˆV(t) ˆR n,n (t) = ˆV((n 1) t) {( ) (38) = d w exp =1 µ 1 2 d j=1 σ 2 j t+σ W( t) } N( t) j=1 Y j 0. LetId n denotethendcatorofdefaultforthedscreteportfolo,wherein d = 1 means that the portfolo defaults n n th tme nterval, whle Id n = 0 f not. Lemma 8.1. Gven assumpton (3), P(I n d = 0) = O( t2 ). Proof. Under assumpton (3), frst we focus on the case of only one jump