A motivation for conditional moment matching

Transcription

1 A motvaton for condtonal moment matchng Roger Lord Ths verson: February 8 th, 005 Frst verson: February nd, 005 ABSTRACT One can fnd approaches galore n the lterature for the valuaton of Asan basket optons. When the number of underlyngs s large one has to resort to bounds or approxmatons to value these optons. In ths respect, Curran [994 and Rogers and Sh [995 very successfully appled a condtonng approach. Recently, Lord [005 combned ther approach wth the tradtonal ad-hoc moment matchng approaches, to obtan an approxmaton whch s extremely accurate and has an analytcal bound on ts error. Here we revew ths approach and extend the results to multple condtonng varables, along the lnes of Vanmaele, Deelstra and Lnev [004. Keywords: Asan opton, average prce opton, basket opton, condtonng approach, moment matchng, lower bound, upper bound, analytcal approxmaton. The author s grateful to the organsers of the 3 rd Actuaral and Fnancal Mathematcs Day n Brussels for provdng hm the opportunty to speak at ther conference. Tnbergen Insttute, Erasmus Unversty Rotterdam, P.O. Box 738, 3000 DR Rotterdam, The Netherlands (e-mal: lord@few.eur.nl, Tel. 3-(0) ) and Modellng and Research (UC-R- 355), Rabobank Internatonal, P.O. Box 700, 3500 HG Utrecht, The Netherlands (e-mal: roger.lord@rabobank.com, Tel. 3-(0) ).

2 . Introducton Ths paper deals wth the prcng of European optons on arthmetc averages. If the average s a tme average of a sngle underlyng asset, these optons are referred to as Asan optons. Another possblty s that the average s taken over several assets at the same tme nstant; these optons are referred to as basket optons. Of course, mxtures of Asan and basket optons exst n the market. For nstance, the average could be taken over tme and over several assets, creatng what we wll refer to as an Asan basket opton. The reason for the exstence of such optons s clear. As far as basket optons are concerned, large companes may want to buy some downsde protecton on ther nvestments. One possblty to acheve ths would be to buy an opton on a basket that s representatve for the nvestments of the frm. What about Asan optons,.e. optons on a tme average of a sngle underlyng? A pure European opton on ths asset would exhbt a large dependence on the fnal value of the underlyng asset, and as such the opton s qute senstve to large shocks or prce manpulaton. To avod such ssues, many fnancal contracts often contan a so-called Asan tal, whch means that the fnal payoff s based on the average prce of the underlyng over a tme nterval before the expry date. Recently Schrager and Pelsser [004 have shown that unt-lnked guarantees contan rate of return guarantees, whch closely resemble Asan optons. Gven the fact that far value calculatons are currently the talk of the town, t s hghly mportant to be able to value these types of contracts. In the Black-Scholes model t s already not straghtforward to prce these types of contracts, the man reason for ths beng that no closed-form probablty law exsts for the sum of correlated lognormal random varables. As the valuaton of these optons s already mathematcally nterestng n the Black-Scholes model, many papers, ncludng ours, are based n ths lognormal settng. Wthn ths model many an approach has been used to value or approxmate these optons. Broadly speakng we can dvde these methods nto fve classes: approaches based on Monte Carlo smulaton, the numercal soluton of partal dfferental equatons (PDEs), ntegral transforms, analytcal approxmatons or analytcal bounds. We wll not attempt to gve an overvew of all these approaches here, we refer the nterested reader to Lord [005 and references theren. In prncple, the most flexble approach when consderng multple underlyngs s probably a Monte Carlo smulaton. Asde from the fact that t s very easy to mplement, a large advantage s that we can easly allow for more realstc dynamcs n the model. However, even though excellent control varates exst wthn the Black-Scholes model, the method can stll be somewhat computatonally ntensve, not to menton the addtonal problem of computng senstvtes wth respect to model and market parameters. Nowadays however, many structured products wth a basket as ther underlyng, use caps and floors on the performance of ndvdual underlyngs, so that Monte Carlo smulaton or the PDE approach s the only method that can be used. Here we gnore these addtonal features, and consder a smple European arthmetc Asan basket opton. Snce fnancal nsttutons demand quck and accurate answers for the value of a dervatve and ts Greeks, large parts of the lterature have focused on analytcal approxmatons and bounds. Probably one of the most wdely known and used approxmatons s that of Levy [99, who approxmates the arthmetc average wth a lognormal random varable, such that the frst two moments concde wth that of the true dstrbuton. A problem that ths approxmaton shares wth other ad-hoc moment matchng approaches, s that the sze of ther error s not known analytcally. Furthermore, most of these approxmatons only tend to work well for low to moderate volatlty envronments. The frst shortcomng has certanly motvated researchers to come up wth sharp lower and upper bounds on the value of these optons. A semnal paper n ths area s that of Rogers and Sh [995. In the context of Asan optons, they derved a sharp lower

3 bound and an upper bound on the value of an Asan opton. The technque used to derve the lower bound s remarkably smple, but very effectve they condton on a varable that s hghly correlated wth the basket, and then apply Jensen s nequalty to fnd a very sharp lower bound. Curran [994 arrved at the same lower bound, and was the frst to observe that the payoff of Asan basket optons can be splt nto two parts: one that can be calculated exactly, and one that has to be approxmated. Ths approach yelded Curran s so-called sophstcated approxmaton. Recently, Lord [005 showed that ths approxmaton of Curran actually dverges when the strke prce tends to nfnty. Combnng the deas of Rogers and Sh and Curran, he ntroduces the class of partally exact and bounded (PEB) approxmatons, whch are guaranteed to le between Rogers and Sh s lower bound, and a sharpened verson of Rogers and Sh s upper bound (due to Nelsen and Sandmann [003 and Vanmaele, Deelstra, Lnev, Dhaene and Goovaerts [005). In ths paper we wll, n the next secton, frst revew the condtonng approaches of Rogers and Sh and Curran. In Vanmaele, Deelstra and Lnev [004 t s shown how to extend the lower bound of Rogers and Sh so that we can condton on two random varables. Here we trvally extend ths lower bound, as well as the sharpened upper bound of Rogers and Sh, to allow for an arbtrary number of condtonng varables. In the thrd secton we revew the results of Lord [005, whch provde a clear motvaton for condtonal moment matchng. Buldng on the results of the second secton, we can extend the PEB approxmatons to allow for multple condtonng varables. Fnally, we show that a recent bounded approxmaton of Vanmaele, Deelstra and Lnev [004, whch also matches the frst two condtonal moments, satsfes all requrements of a PEB approxmaton. We end thepaper wth a bref numercal llustraton and some conclusons and recommendatons. The focus throughout the paper wll not be on exact expressons requred to calculate the varous bounds and approxmatons, but on the ratonale behnd the approaches. Before startng the next secton, we wll frst ntroduce some notaton. As mentoned, we wll base ourselves n the Black-Scholes framework. For notatonal convenence we wll work wth a constant parameter Black-Scholes model, although all results stll hold when these parameters are determnstc functons of tme, and the growth and spot rate are Gaussan. We assume the underlyng assets S, =,, N and the money market account M evolve accordng to the followng stochastc dfferental equaton (SDE): ds (t) = µ dt σdw (t) S (t) dm(t) = rdt M(t) (.) where all Brownan motons are correlated wth nstantaneous correlaton matrx R. Throughout the document we wll assume, wthout loss of generalty, that the current date s 0. The underlyngs of all optons we consder n ths paper wll be an Asan basket, whch at the maturty date T wll be defned as B(T) n: B(T) = A (T) = T 0 N = S (t) ρ (t)dt w A (T) (.) Here, the weghts w are postve and sum to, and smlarly all ρ are non-negatve functons, ntegratng to over [0,T. We wll only consder newly ssued, non-forward-startng call optons on ths Asan basket. Ths s no loss of generalty. Put optons can be prced va the Asan put-call 3

4 party, whereas runnng average optons can be treated as newly ssued ones, wth a correcton to the strke prce. Fnally, forward-startng optons pose no problems when nterest rates are determnstc or Gaussan. For ease of exposure we wll mostly deal wth forward prces n our analyss. The forward prce of the Asan basket call opton s equal to ts expected value under the rsk-neutral probablty measure Ð, condtonal upon all nformaton known at tme 0: ( B(T) K) c B (T, K) = Ä Ð 0 [ (.3) In the remander we wll leave out the superscrpt ndcatng the measure and the subscrpt ndcatng at whch tme the expectaton s evaluated, unless any confuson can arse. Havng ntroduced the notaton we wll use, we are now ready to turn to the next secton.. The condtonng approaches The most successful approxmatons and bounds all rely heavly on results frst derved by Rogers and Sh [995 and Curran [994. We brefly revew ther approaches here, whereafter we extend them to allow for multple condtonng varables, somethng whch was done for two condtonng varables by Vanmaele, Deelstra and Lnev [004. We wll here use Curran s dea of decomposng the Asan opton nto two parts: one that can be calculated exactly, and one that has to be approxmated. Suppose that we have a normally dstrbuted random varable Λ wth the convenent property that Λ λ(k) mples that B(T) K. Examples of such random varables wll be gven shortly. Followng Curran, we can then wrte: c B (T,K) = Ä[ = Ä[ ( B(T) K) ( B(T) K) ( B(T) K) ( B(T) ) c (T,K, Λ) c (T,K, Λ) K [ Λ λ(k) [ Λ λ(k) (.) As s shown n Curran [994, t s qute straghtforward to calculate the c -part, usng the convenent property that normally dstrbuted random varables are stll normally dstrbuted upon condtonng on a correlated normal random varable. We wll therefore refran from reproducng the exact formulae here. Ths leaves us wth the calculaton of the c -part, whch we can bound or approxmate. Let us frst however consder several possble random varables Λ, whch have the above property. A very natural canddate for such a Λ s the logarthm of the geometrc average, whch for the Asan basket wll be defned as: G(T) = N = G T G = (T) exp 0 (T) w lns ρ (t) (t)dt (.) An applcaton of the weghted Jensen s nequalty shows that B(T) G(T), wth equalty attaned f and only f all components of the average are equal. Defnng Λ GA = ln G(T), t s then obvous that when Λ GA ln K, we ndeed have B(T) K. Other possble condtonng varables, see e.g. Vanmaele, Deelstra and Lnev [004, arse from a frst order approxmaton of the Asan basket B(T) n ts drvng Brownan motons. In the settng of an Asan basket opton, we then obtan the followng condtonng varables and ther correspondng thresholds: 4

5 Λ Λ FA FA = = N = N = w w 0 T 0 T S (0) exp S (0) ( µ ) ( ) σ )t σw (t) ρ (t)dt λ FA(K) = K ( ( µ σ )t σ W (t)) ρ (t)dt λ (K) = K FA (.3) We note that the hgher the correlaton of Λ wth B(T) s, the larger the relatve contrbuton of c to the opton prce wll be. In practce, any of the above condtonng varables s qute hghly correlated wth B(T), provded that the volatltes of the underlyng assets are not too hgh. Ths s one of the key ponts as to why these condtonng approaches work so well c consttutes a large part of the opton prce, so that any approxmaton we make n c wll not have a large mpact. The larger the volatltes and maturtes are, the more mportant t becomes to have an accurate approxmaton to c. We now turn to the approxmatng part c. Both Rogers and Sh and Curran used Jensen s nequalty to fnd a lower bound on the value of these optons. A lower bound on c smply follows from: c (T, K, Λ) = Ä[ Ä[ Ä[ ( B(T) K) = Ä[ Ä[ ( B(T) K) ( B(T) K) Λ [ Λ<λ(K) Λ (.4) so that then the lower bound becomes the sum of ths lower bound and c : = Ä ( B(T) K) [ Λ<λ ( Ä[B(T) Λ K) LB(T, K, Λ) = Ä[ (K) c (t, K, Λ) [ (.5) Ths lower bound can n prncple be appled usng an arbtrary condtonng varable, not only condtonng varables for whch we have the aforementoned property. In Lord [005 t s shown how to calculate (.5) n closed-form for an arbtrary condtonng varable, and an arbtrary correlaton structure between the varous underlyngs. Ths greatly facltates the computatons requred for the lower bound, as otherwse we would have to resort to a numercal ntegraton over a dscontnuous ntegrand. Another approach to approxmate c wll be pursued n the followng secton. We wll now extend the lower bound so that we can condton on multple random varables. For two condtonng varables ths dea was frst pursued n Vanmaele, Deelstra and Lnev [004, so ths s merely a trval extenson of ther results. Suppose that we have a condtonng varable Λ and a set of condtonng varables Z, such that for any realsaton of the random varables n Z, Λ λ(k) mples that B(T) K. The lower bound on c then becomes: c (T, K, Λ) = Ä [( B(T) K) [ Λ<λ(K) [ [( B(T) K) [ Λ<λ(K) Λ, Z [ Ä[ ( B(T) K) Λ, Z = Ä Ä Ä (.6) so that the resultng lower bound s: [ Ä[ ( B(T) K) Λ, Z c (T, K, ) LB(T, K, Λ, Z ) = Ä [ Λ< λ(k) Λ (.7) 5

6 Note that the frst part n (.7) wll typcally have to be calculated va a multvarate numercal ntegraton, whereas the second part s the same as before, and can hence be done n closed-form. Let us now turn to an analyss of the error made by approxmatng the value of the Asan basket opton by the lower bound n (.7). Ths upper bound, based on the lower bound, was frst derved by Rogers and Sh [995. It s based on the followng nequalty: 0 Ä[X Ä Ä[X = ( Ä[ X Ä[X ) [ X Ä[X Var(X) More recently, Nelsen and Sandmann [003 and Vanmaele, Deelstra, Lnev, Dhaene and Goovaerts [005 sharpened ths upper bound consderably. We here extend ther sharpened verson to allow for multple condtonng varables. Proceedng as above, we fnd: (.8) 0 c B = Ä (T, K) LB(T, K, Λ, [ Ä[ ( B(T) K) Λ, Z Ä[ ( B(T) K) Λ, Z Ä [ Var( B(T) Λ, / ε (T, K, Λ, (.9) Ths yelds an upper bound whch agan has to be calculated va a multvarate numercal ntegraton. Nelsen and Sandmann and Vanmaele et al., usng only one condtonng varable, go one step further to derve a slghtly larger upper bound, that can be calculated n closed-form. Here t s equal to: ε (T,K, Λ, = Ä[Var Ä[Var ( B(T) Λ, / ( B(T) Λ, Ä[ ε (T,K, Λ, (.0) Both error estmates yeld an upper bound whch s equal to UB (T,K,Λ, = LB(T,K,Λ, ε (T,K,Λ,, for =,. It can be calculated n closed-form because we can wrte: [ ( B(T) Λ, = Ä Ä[ Var( B(T) Λ, Λ Ä (.) [ Var [ Λ< λ(k) As shown n Nelsen and Sandmann and Vanmaele et al., ths expresson can be calculated n closed-form. We do not reproduce the formulae here, as t only dstracts from the rest of the text and the calculatons are exactly the same as n the aforementoned artcles. Note that the varance of B(T) gven Λ and Z s zero f the set {Λ,Z} contans all random varables wthn B(T). Then the results above mply that the lower bound exactly concdes wth the true value of the Asan basket opton. Fnally, we menton that Rogers and Sh s upper bound corresponds to the lmt of UB for K tendng to nfnty. 3. The benefts of condtonal moment matchng As mentoned n the ntroducton, many orgnal approxmatons merely substtute the arthmetc average by a tractable random varable, whch has the same frst couple of uncondtonal moments. An example of ths s Levy s [99 approxmaton, whch fts a 6

7 lognormal random varable to the arthmetc average. These types of approxmatons typcally only work well when volatltes and maturtes are low. Furthermore, the sze of the error made can not easly be estmated. Here we show that condtonal moment matchng does yeld an analytcal error estmate. The proposed approxmaton follows from: ~ c (T, K, Λ, = Ä[ B ( B ~ (T) K) ( B(T) ) ~ c (T, K, Λ) c (T, K, Λ) K [ Λ λ(k) (3.).e. t agan exsts of an approxmatng part and an exact part. For Λ λ(k) we can take our approxmatng random varable B ~ (T) to be equal to B(T), yeldng the exact c -part. For Λ smaller than λ(k), we have to make an approxmaton. Gven certan crtera that B ~ (T) must fulfll, whch follow n the next theorem, we can fnd an analytcal error estmate as derved n Lord [005. Here we extend ths result to allow for multple condtonng varables. Theorem: If we mpose the followng two condtons on the approxmatng random varable B ~ (T) : Ä[B ~ (T) Λ = λ, Z = z = Ä[B(T) Λ = λ, Z = z Var[B ~ (T) Λ = λ, Z = z Var[B(T) Λ = λ, Z = z (3.) for λ (-,λ(k)), the resultng approxmaton n (3.) les between LB(T,K,Λ, and UB (T,K,Λ,. Proof: The proof follows along the same lnes as (.9)-(.0): 0 ~ c (T,K, Λ) LB(T, K, Λ, = Ä B [ ( B ~ ) ( B ~ Ä[ (T) K [ Λ<λ(K) Λ, Z Ä[ (T) K) [ Λ<λ(K) Λ, Z Var( B ~ Ä (T) Λ, / ε (T,K, Λ, Z [ ) (3.3) It s clear that the frst equalty holds, due to the constructon n (3.) and the fact that the condtonal moments are equal for Λ λ(k). The rest of the dervaton s smlar to (.9)-(.0). It mmedately follows that: LB(T, K, Λ, ~ cb (T, K, Λ, UB(T, K, Λ, (3.4) whch concludes the proof of the theorem. Ths theorem drectly motvates why t s good to match condtonal moments. Intutvely we can ndeed expect to obtan better results than by just matchng uncondtonal moments. The above theorem gves a rgorous (and typcally sharp) error bound for ths. Note that the moments do not have to be exactly matched the condtonal varance may actually be smaller. Approxmatons satsfyng (3.) and (3.) are dubbed partally exact and bounded (PEB) 7

8 approxmatons. The lower bound LB(T,K,Λ, s a specal case hereof. In Vanmaele, Deelstra and Lnev [004 another route s attempted. Wthout delvng nto detals, they construct an approxmaton va a (condtonally) convex combnaton of the lower bound and the partally exact and comonotonc upper bound (PECUB), the so-called LBPECUB approxmaton. The condtonal weghts are chosen by ensurng that the frst two condtonal moments are matched exactly. As such, t satsfes the crtera for t to be a PEB approxmaton, and hence t s bounded above by the UB as well of course the PECUB upper bound. We note that n practce the approxmatng part n (3.) wll have to be calculated va a numercal ntegraton. From a computatonal pont of vew one would therefore not lke to use too many condtonng varables. Typcally one condtonng varable may already be more than enough, as has been shown n Lord [005 for a pure Asan opton, and as we wll demonstrate for a pure basket opton n the next and fnal secton. 4. Numercal llustraton and conclusons To llustrate the effectvty of condtonal moment matchng we wll here provde a numercal example for a pure basket opton. The example has been taken from Mlevsky and Posner [998, and also features n Vanmaele, Deelstra and Lnev [004. The basket underlyng the opton s the weghted average of the normalzed G-7 stock ndces. Weghts, volatltes, dvdend yelds and correlatons can be found n the tables below. Country Index Weght Volatlty Dvdend yeld Canada TSE 00 0%.55%.69% Germany DAX 5% 4.53%.36% France CAC 40 5% 0.68%.39% U.K. FTSE 00 0% 4.6% 3.6% Italy MIB 300 5% 7.99%.9% Japan Nkke 5 0% 5.59% 0.8% U.S. S&P 500 5% 5.68%.66% Table : Weghts, volatltes and dvdend yelds of the basket Canada Germany France U.K. Italy Japan U.S. Canada Germany France U.K Italy Japan U.S. Table : Upper trangular part of the nstantaneous correlaton matrx between the varous assets As we use the normalzed values of the ndces, ths effectvely means we assume the ntal spot value equals for each ndex. In the followng table we compare the lower and upper bounds usng one or two condtonng varables to the true value obtaned from a Monte Carlo smulaton wth paths, usng antthetc varables and usng the geometrc basket as the control varate. Results are only shown for the most extreme example n Vanmaele et al., namely for a maturty of 0 years. Vanmaele et al. only consdered three strke prces, 0.95, and.05. However, the forward prce of the basket (the mean of ts uncondtonal dstrbuton) can be calculated as.57888, so that we found t mportant to nclude hgher strke prces n the table as well. The choces for condtonng varables are the same as n ths artcle the frst condtonng 8

9 varable s FA (cf. (.3)), the second s smlar to FA, apart from the fact that the sgn of the one but last Brownan moton s reversed. Strke MC (StdErr) LB FA UB FA LB FA,FA* UB FA,FA* (0.0036) (0.0037) (0.0038) (0.004) (0.0045) (0.0044) Table 3: Upper and lower bounds based on one or two condtonng varables The upper bounds are the UB upper bounds, see equaton (.0). Indeed, condtonng on more random varables sharpens the lower and upper bounds consderably, as was already demonstrated n Vanmaele et al., but s now also apparent from the new upper bound. To show that condtonal moment matchng actually works remarkably well, we compare varous condtonal moment matchng approxmatons to the true value of the opton. As mentoned earler, the LBPECUB approxmatons consdered n Vanmaele, Deelstra and Lnev [004 are convex combnatons of the lower bound and the PECUB upper bound. Results n ther paper were shown for usng the geometrc average as the condtonng varable. Two dstnctons can be made on the choce of the weghts for each bound: z(λ) ndcates that the frst two condtonal moments are matched exactly (yeldng a PEB approxmaton as noted n secton 3), whereas z u ndcates that the lower bound and the PECUB upper bound are weghted usng a global weght stemmng from another approxmaton n Vyncke, Goovaerts and Dhaene [003. The latter s not a condtonal moment matchng approxmaton, but t works rather well. The CurranM and Curran3M approxmatons are PEB approxmatons consdered n Lord [005 that ft a shfted lognormal random varable to the basket. The M approxmaton consdered here uses a shft equal to the condtonng varable Λ FA ; the remanng two parameters are chosen such that the frst two condtonal moments are matched exactly. The 3M approxmaton s a slght take on ths: the shft s now also consdered as a parameter, so that the frst three condtonal moments can be ft exactly. In both approxmatons we condton on Λ FA. Strke MC (StdErr) LBPECUB GA M 3M z(λ) z u (0.0036) (0.0037) (0.0038) (0.004) (0.0045) (0.0044) Table 4: Several approxmatons for the value of a basket opton We dd not get round to mplementng the LBPECUB GA approxmaton ourselves, so that we here only reproduce the values gven n Vanmaele et al. They only consdered strke values up to.05; as the forward prce of the basket s for a 0-year contract, we also found t mportant to consder slghtly hgher strke prces. As can be seen from the table, the 3M approxmaton seems to gve results that are very close to the true values and ths has only been acheved by usng one condtonng varable. The condtonal moment matchng approxmaton of Vanmaele et al., usng z(λ), seems to yeld too low values. However, ther approxmaton whch uses z u s a clear contender, yeldng results whch are comparable to those of the M approxmaton. Consderng the computatonal effort, whch has been nvestgated n Lord [005, we have a 9

10 slght overall preference for the M approxmaton, although ths s of course subject to dscusson. Concludng, n ths paper we revewed the condtonng approaches of Rogers and Sh [995 and Curran [994. Rogers and Sh s lower and (sharpened) upper bounds, as well as the PEB approxmatons of Lord [005, have been extended along the lnes of Vanmaele, Deelstra and Lnev [004 to allow for multple condtonng varables. Fnally, we have shown that the LBPECUB convex combnaton of the lower bound and the PECUB upper bound, consdered n Vanmaele et al., s ndeed a PEB approxmaton, and as such s bounded from above by the (sharpened) Rogers and Sh upper bound. In a numercal example the effectvty of condtonal moment matchng has been demonstrated. Bblography CURRAN, M. (994). Valung Asan and Portfolo Optons by Condtonng on the Geometrc Mean Prce, Management Scence, vol. 40, no., pp DEELSTRA, G., LIINEV, J. AND M. VANMAELE (004). Prcng of arthmetc basket optons by condtonng, Insurance: Mathematcs and Economcs, vol. 34, no., pp DHAENE, J., DENUIT. M, GOOVAERTS, M., KAAS, R. AND D. VYNCKE (00). The concept of comonotoncty n actuaral scence and fnance: Theory, Insurance: Mathematcs and Economcs, vol. 3, no., pp LEVY, E. (99). Prcng European average rate currency optons, Journal of Internatonal Money and Fnance, vol., pp LORD, R. (005). Partally exact and bounded approxmatons for arthmetc Asan optons, submtted, Erasmus Unversty Rotterdam and Rabobank Internatonal, MILEVSKY, M.A. AND S.E. POSNER (998). A closed-form approxmaton for valung basket optons, Journal of Dervatves, vol. 4, pp NIELSEN, J.A. AND K. SANDMANN (003). Prcng bounds on Asan optons, Journal of Fnancal and Quanttatve Analyss, vol. 38, no., pp ROGERS, L.C.G. AND Z. SHI (995). The value of an Asan opton, Journal of Appled Probablty, no. 3, pp SCHRAGER, D.F. AND A.A.J. PELSSER (004). Prcng rate of return guarantees n regular premum unt lnked nsurance, Insurance: Mathematcs and Economcs, vol. 35, no., pp VANMAELE, M., DEELSTRA, G. AND J. LIINEV (004). Approxmaton of stop-loss premums nvolvng sums of lognormals by condtonng on two varables, Insurance: Mathematcs and Economcs, vol. 35, no., pp VANMAELE, M., DEELSTRA, G., LIINEV, J., DHAENE, J. AND M.J. GOOVAERTS (005). Bounds for the prce of dscretely sampled arthmetc Asan optons, forthcomng n: Journal of Computatonal and Appled Mathematcs, to appear. VYNCKE, D., GOOVAERTS, M.J. AND J. DHAENE (003). An accurate analytcal approxmaton for the prce of a European-style arthmetc Asan opton, workng paper, Catholc Unversty Leuven and Unversty of Amsterdam. 0