The Pricing of Basket Options: A Weak Convergence Approach

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1 The Prcng of Baske Opons: A Weak Convergence Approach Ljun Bo Yongjn Wang Absrac We consder a lm prce of baske opons n a large porfolo where he dynamcs of baske asses s descrbed as a CEV jump dffuson sysem. The explc represenaon of he lmng prce s esablshed usng weak convergence of emprcal measure valued processes generaed by he sysem. As an applcaon, he closed-form formula of he lm prce s derved when he prce dynamcs of baske asses follows a mxed-double exponenal jump-dffuson sysem. AMS subjec classfcaons: 3E, 6J. Keywords and phrases: Baske opons, Jump dffuson, Weak convergence. Inroducon A baske opon s an opon whose payoff s a weghed sum or average of prces of he wo or more rsky asses ha have been grouped ogeher n a baske a maury. In general, s dffcul o prce baske opons explcly snce he jon dsrbuon of he underlyng baske asse prce process are unknown due o mul-dmensonaly, n parcular for he mul-dmensonal jump-dffuson baske asse prce dynamcs. Insead, some works have focused on he developmen of fas and accurae approxmaon echnques and esablshmen of sharp lower and upper bounds for baske opon prces, see heren hey revew and compare sx dfferen mehods for valung baske opons n a sysemac way, and dscuss he nfluence of model parameers on he performance of he dfferen approxmaons. Recenly 4 apply he asympoc expanson mehod o fnd he approxmang value of he lower bound of European syle baske call opon prces for a local volaly jump-dffuson model. In hs paper, we are concerned wh he prcng of a class of baske opons n a large porfolo of he underlyng baske asses when he prce dynamcs of baske asses s descrbed as a muldmensonal CEV-ype jump-dffuson process. Ths s relaed o he lm characerzaon problem of he rsk-neural expecaon for he baske opon payoff a maury. Dfferenly from and 4 revewed above, we here manly focus on he explc represenaon of he lmng prce of baske opons as a prcng problem for he large porfolo of rsky asses consdered n. In addon, he CEV-ype model consdered heren can also capure he local volaly, see also 8 and 7. In pracse, our explc lm prce presenaon can be also used o valuae baske opons when he number of rsky asses n he porfolo s relavely large, see also whch revews he VaR mehod appled o revalue complex opons n a large porfolo. The mehod we used n hs paper heavly depends on he echnque of weak convergence for emprcal measure-valued processes generaed by he weghed prce of rsky asses, parameer se and funcons of jumps sze of he prce dynamcs, see also and 3 whch apply he weak convergence mehod o sudy defaul cluserng and sysemc rsk. We wll prove ha he sequence of he above emprcal measure-valued processes s relavely compac on he Skorohod space and denfy he lm of he sequence of he emprcal Correspondng auhor. Emal: ljunbo@usc.edu.cn, School of Mahemacal Scences, nversy of Scence and Technology of Chna, and Wu Wen Tsun Key Laboraory of Mahemacs, Chnese Academy of Scence, Hefe, Anhu Provnce 36, Chna. Emal: yjwang@nanka.edu.cn, Busness School, anka nversy, Tanjn, 37, Chna.

2 measure-valued processes explcly n he dsrbuonal sense. We here pon ou ha our weak convergence analyss as a lmng marngale problem s no degenerae and hence he lm s also a measure-valued random process whch s no he deermnsc lm case as n and 3. also sudy he weak convergence of he emprcal measure-valued processes. They prove ha he densy of he lmng measure can be characerzed as he soluon o a nonlnear SPDE. However, hey do no apply hs densy o he prcng ssue. sng he lm as a measure-valued process, we characerze he lm prce of baske opons by employng Skorohod s represenaon heorem and Val s convergence heorem. Fnally we derve he closed-form lm prce when he prce dynamcs of he underlyng baske asses s modeled as a mul-dmensonal mxed-exponenal jump-dffuson he one-dmensonal case s frs consdered by 6 va Laplace ransform. The res of he paper s organzed as follows. Secon nroduces he mul-dmensonal CEVype jump-dffuson model for baske opons. Secon 3 provdes a dealed weak convergence analyss for emprcal measure-valued processes generaed by he weghed prce of rsky asses, parameer se and funcons of jumps sze of he prce dynamcs. Secon 4 proves he explc lm of baske opon prce usng he weak convergence obaned n Secon 3. A numercal example s presened n Secon 5. Baske Opons In hs secon, we descrbe he model and prce represenaon for baske opons. We consder underlyng rsky asses e.g. socks n he fnancal marke. The prce of a baske of hese rsky asses s hen defned as he weghed sum of he prces of rsky asses a me,.e., B = ws,, = where he weghs w, =,...,, are assumed o depend on he number of he underlyng rsky asses. Here we consder he weghs wh he form w = w for =,...,, where consans w R are ndependen of. If w =, hen B s he average of me- prces of rsky asses ha have been grouped ogeher n he baske. Le Ω, F, Q be a rsk-neural probably space, under whch, he prce dynamcs of he -h rsky asse s assumed o sasfy he followng CEV-ype jump-dffuson process, for =,...,, ds S = r v d j= σ j S β dw j l y Qdy, ds, S = S >, where r > s he neres rae, v > s he dvdend yeld of he -h rsky asse, σ j > s he volaly of he -h rsky asse subjec o he j-h rsky asse, and W = W ; =,..., d,, s a d-dmensonal Brownan moon. Here β, s he consan-elascy-of-varance CEV parameer. Snce β,, he exsence and unqueness of he srong soluon of SDE can be guaraneed by Theorem 9. on page 3 of 3 under he assumpon A below. Furher, f β =, corresponds o a CIR process wh jumps, whle f β =, s an exponenal Lévy process. For β,, s called a CEV process wh jumps. In parcular, f he CEV parameer β = and he jump funcon l y, hen he prce dynamcs s reduced o he mul-dmensonal geomerc Brownan moon model consdered n. Here denoes a opologcal space and ν s a σ-fne Borel measure on. Furher Qdy, ds denoes a Posson random measure whch s ndependen of d-dmensonal Brownan moon W = W ;. Correspondngly Qdy, ds := Qdy, ds νdyds denoes he compensaed Posson random measure wh compensaor νdyds. The measurable funcon of jump s sze l : R s assumed o sasfy l y > for all y,. Ths condon can guaranee ha he -h rsky asse prce s sll posve afer a common jump due o Posson random measure.

3 Le F = F ; wh F beng he rgh-connuous augmened flraon generaed jonly by W, Q up o me. Le T > be he maury of he baske opon and φ : R R be he payoff funcon. Here we assume ha φ s connuous and adms he growh condon φx C x κ wh κ >. We consder he payoff assocaed wh our baske opon gven by φ BT. Here, for a srke prce K >, φx = θx K for x R corresponds o he call baske opon f θ =, and corresponds o he pu baske opon f θ =. In hs case, he payoff funcon φ sasfes he lnear growh condon.e. κ =. Thus he prce of he baske opon s gven by V φ := e rt E φ w ST. 3 Here E denoes he expecaon w.r.. he rsk-neural measure Q. The man of he paper s o characerze he lm of he prce V φ analycally as by adopng he echnque of weak convergence whch wll be mplemened n he comng secon. = 3 Weak Convergence Analyss In hs secon, we provde a dealed analyss of weak convergence for he baske opon prce under he mul-dmensonal CEV-ype jump-dffuson model descrbed as. We collec he parameers assocaed wh he prce dynamcs of he -h rsky asse as follows, for, p = v, σ,..., σ d, w O p := R d R. 4 Le C denoe he se of all Borel measurable funcons on and O := O p C R. Then we can defne he sequence of emprcal measure-valued processes as, for, ν := = δ p,l,x, 5 on he Borel feld BO. Here δ denoes he Drac-dela measure on O and he -h weghed prce process, for =,...,, X := w S,. 6 Le S = PO,.e., he se of all Borel probably measures on O. Then ν = ν ; can be vewed as an S-valued rgh connuous wh lef lms r.c.l.l. sochasc process for each. For any smooh funcon fp, y, x C O defned on p, y, x O and ν S, defne he negral w.r.. he Borel probably measure ν by νf := O fp, y, xνdp dy dx. Then holds ha for f C O, ν f = f p, l, X,. = Moreover, follows ha from ha, for =,...,, X,, sasfes he followng SDE wh jumps, X = X = w S, and dx X = r v d j= σ j w β X βdw j l y Qdy, d. 7 We nex esmae he momen of he above weghed prce process whch wll be used o verfy he relave compacness of he sequence of emprcal measure-valued processes ν ; gven by 5. To hs purpose, we mpose he followng assumpon on he model parameers: 3

4 A The parameer se p, S ; and 4 consan C p,l >. Then we have l= l y l νdy; are domnaed by a Lemma 3.. nder he assumpon A, for any α, 4 and any T >, holds ha sup,,t X E α <. 8 Proof. Iô s formula mples ha, for all, E X α = E w S α δr v E X s α αα ds σ w β E E α l y α αl y νdy ds. Here σ = d follows ha j= σ j X s = X s βα ds for =,...,. oce ha βα α, α. Then usng Young s nequaly, E X s βα ds β α E α X s α ds C T,α, for some consan C T,α > whch depends on T, α. On he oher hand, here exss a consan C α > whch only depends on α such ha l y α αl y νdy α l y α l y νdy. C α l y l y α νdy. Then follows from A ha here exss a consan C p,l,t,α > whch s ndependen of so ha sup,t = E X α T C p,l,t,α C p,l,t,α sup s, Thus he momen esmae 8 follows form Gronwall s lemma. = E Xs α d. We nex wan o characerze he lmng generaor of he sequence of emprcal measure-valued processes ν ; gven by 5 as. To hs purpose, for any smooh funcon ϕ C R K wh K, and for any Borel probably measure ν S, we defne Φν := ϕνf. Here f = f,..., f K wh each elemen belongng o C O and he vecor νf := νf,..., νf K R K. Then we have Lemma 3.. Le he operaor A acng on he funcon Φν = ϕνf wh ν S be defned as AΦν := k= ϕνf rνl c f k νl c f k νl c3 f k k k,l= ϕνf k l j= ϕ νl yf ϕ νf νl dj f k νl dj f l k= ϕνf νl yf k νdy. k 9 4

5 Here he operaors are defned by, for p = v, σ,..., σ d, w O p, l C and x R, L c fp, l, x fp, l, x := x, L c fp, l, x fp, l, x := xv, L c3 fp, l, x := xβ w β σ fp, l, x, σ := σj, L yfp, l, x := fp, l, x ly, L yfp, l, x := xly and for j =,..., d, he operaor j= fp, l, x, y, L dj fp, l, x := x β β fp, l, x σ j w. Then for he sequence of measure-valued processes ν = ν ; gven by 5, holds ha m m lm Φν m Φν m AΦνs ds Ψ j ν j =, E m where < < m < and Ψ j BS all bounded measurable funcons on S wh j =,..., m. j= Proof. I follows from Iô s formula ha, for all f C O, fp, l, X = fp, l, X j= fp, l, Xs X sr v ds fp, l, X s where he F-local marngale, for, Thus we oban M := j= fp, l, X = = j= Then holds ha = = σjw β X βds s M fp, l, X s X sl y fp, l, X s fp, l, X s ν f = ν f fp, l, X s Xsl y νdyds, σ j w β X βdw j s s 3 fp, l, X s X s l y fp, l, X s Qdy, ds. fp, l, X = fp, l, X s = σjw β X βds s fp, l, Xs X sr v ds = M fp, l, X s X sl y fp, l, X s fp, l, X s rνs L c f νs L c f νs L c3 f ds 5 Xsl y νdyds. = M

6 whle for, M = = j= νs L yf νs f νs L yf νdyds, 4 ν s L dj fdw j s By vrue of Doob-Meyer decomposon 4, we also have ϕν f = ϕν f k= k,l= Then holds ha k= ϕν f = ϕν f ν s L yf ν s f Qdy, ds. 5 ϕνs f rνs L c f k νs L c f k νs L c3 f k ds k ϕνs f νs L yf k νs f k νs L yf k k ϕν s f k l j= νs L dj f k νs L dj f l ds ϕ ν s L yf ϕ νs f K Qdy, ds ϕ ν s L yf ϕ ν s f k,l= k= ϕν s f k l k= j= νdyds k= ϕνs f νs L dj f k dws j k ϕνs f νs L yf k νs f k νdyds. k ϕνs f rνs L c f k νs L c f k νs L c3 f k ds k j= ϕ ν s L yf ϕ ν s f νs L dj f k νs L dj f l ds M 6 k= ϕνs f νs L yf k νdyds. k Here M,, s an F-local marngale. Ths mples ha Φν Φν AΦν s ds, s an F-local marngale wh nal mean-zero. oce ha m j= Ψ jν j s bounded and F m - measurable. Then he desred resul follows from a local localzaon.. We nex prove ha he sequence of measure-valued processes ν ; defned by 5 are relavely compac, when vewed as a sequence of random processes on he Skorokhod space D S,. Ths can be mpled by he followng wo lemmas. Lemma 3.3. Suppose ha he assumpon A holds. Then for every T >, holds ha for any smooh funcon f C O, ν f R =. 7 lm sup P R sup,t 6

7 Proof. In lgh of Doob-Meyer decomposon 4, we have ν f = ν f A B M,, = where he erms for, A B := := = fp, l, Xs X sr v ds = Frs we have, for any T >, E A sup,t j= = fp, l, X s fp, l, X s X sl y fp, l, X s fp, l, X s = T = f C p,l fp, l, X E s X sr v ds T T fp, l, X E s = σ w β X β s ds E X s f ds C p,l oce ha β,. Then by vrue of Lemma 3., follows ha E T σjw β X βds, s Xsl y νdyds. = E Xs β ds. sup,t A C p,t,f for some fne consan C p,t,f > whch s ndependen of. sng he mean-value heorem and Lemma 3., we also have E sup,t B f T E X s = f T C p,l E X s ds C p,l,t,f, = l y νdy ds for some fne consan C p,l,t,f > whch s ndependen of. sng B-D-G nequaly and he local marngale 5, follows ha E sup,t = M T C T E νs L dj f ds j= T C T E ν s L yf νs f Qdy, ds. In erms of he operaor, we frs have T E νs L dj f ds f T C p,l j= E C T p,l,f E X β s X j β s ds,j= X β s ds = 7

8 C p,l,f E = C p,l,f T T,j= = X s β ds T E Xs β ds,,j= X j s β ds and by vrue of, he mean-value heorem and he assumpon A, follows ha T E ν s L yf νs f T Qdy, ds = E f T E X sl y νdyds f E C p,l,f T T = X s = X E s ds. = l y νdy ds ν s L yf νs f νdyds Thus from Cauchy s nequaly and Lemma 3., follows ha E sup M T C,T p,l,t,f E νs L dj f ds C p,l,t,f = j= T C p,l,t,f E ν s L yf νs f Qdy, ds C p,l,t,f, for some fne consan C p,l,t,f > whch s ndependen of. Hence he desred lm 7 follows form Markov nequaly. The followng lemma verfes he me regulary of he sequence of measure-valued processes ν ;. Lemma 3.4. Le he assumpon A hold and hx, y := x y for x, y R. Then here exss a posve random varable H γ wh γ > and lm γ sup E H γ = such ha for all, T, u, γ, and v, γ, holds ha E h ν uf, ν fh ν f, ν vf E H γ,. 8 Here E := E F wh, T. Proof. sng Doob-Meyer decomposon 4, for all, T and u, γ, ν uf ν f = A u A B u B = M u M. Frs we have for all, T and u, γ, A u A f C p,l C p,l,t,f u 4 u T = X s = ds C p,l f X s 8 ds T u = X s = X s 4β ds β ds

9 and holds ha C p,l,t,f γ 4 =: H γ, B u B T f u C p,l,t,f u C p,l,t,f γ 4 = X s ds = X s T X s ds = T Fnally we oban usng 5 ha E M u M u = E = j= u E = X s = X s 4β ds l y νdy ds ds =: H γ. νs L dj f ds ν s L yf νs f νdyds. Here we frs ge u E νs L dj f ds f u C p,l j= E X β s ds = T E C p,l,t,f γ 4 X s 4β ds =: E H 3 γ, and E u ν s L yf νs f νdyds f u E E C p,l,f γ 4 = X s = T = X s f E u l y νdy ds 4 ds =: E H 4 γ. Xsl y = νdyds Le H γ := H γ H γ H 3 γ H4 γ wh γ >. oce ha 4 β, 4. Then we have lm γ sup E H γ = usng Lemma 3., and for all, holds ha E h ν uf, ν fh ν f, ν vf 6E H γ, where we used h ν f, ν vf. Ths proves he esmae 8. ex we wll characerze he weak lm of he sequence of measure-valued processes ν ; defned by 5. To hs purpose, we defne he emprcal measures q := = δ p, η := = δ l and ψ := = δ X whch are respecvely relaed o he parameer se, jump funcons and nal weghed prce. Assume ha 9

10 A The lmng emprcal measures q = lm q, η = lm η and ψ = lm ψ exs n PO p, PC and PR respecvely. by For p = v, σ,..., σ d, w O p, l C, and x R, defne he followng measure-valued process ν B C D := B C D p, l, X p, l, x qdpηdlψdx,, 9 O where B BO p, C BC and D BR. Here he underlyng parameerzed process s gven by he unque srong soluon of he followng SDE wh jumps: X p, l, x = x r vx s p, l, xds σw β Xs β p, l, xd W s X s p, l, xly Qdy, ds, where W = W ; s a -dmensonal sandard Brownan moon whch s ndependen of he Posson random measure Qdy, d ds and σ := j= σ j. Then we have Theorem 3.5. Le assumpons A and A hold. Then he sequence of measure-valued processes ν ; defned by 5 weakly converges o he above measure-valued process ν = ν ; gven by 9 as. Proof. By vrue of he weak convergence of marngale problem as n Chaper 3 of 9, from Lemma 3., Lemma 3.3 and Lemma 3.4, follows ha ν weakly converges o ν as. We nex show ha he lm measure-valued process ν s ndeed gven by 9 n erms of he unqueness of marngale problems as n Chaper 3 of 9. In fac, usng 9, we have for f C O, ν f = f p, l, X p, l, x qdpηdlψdx. O Fx he parameers p, l, x O and applyng Iô s formula, we have f p, l, X p, l, x = f p, l, X p, l, x r vx s p, l, x f p, l, X s p, l, x ds σj w β Xs β p, l, x f p, l, X s p, l, x j= ds σw β Xs β p, l, x f p, l, X s p, l, x d W s f p, l, Xs p, l, x ly f p, l, X s p, l, x Qdy, ds f p, l, X s p, l, x ly f p, l, X s p, l, x X s p, l, xly f p, l, X s p, l, x νdyds. Take he negral on he boh sdes of he above dsplay w.r.. qdpηdlψdx. Then follows from ha ν f = ν f rν s L c f ν s L c f ν s L c3 f ds

11 j= ν s L dj fdw j s Ths resuls n usng Iô s formula ha ϕν f = ϕν f k,l= k= k= = ϕν f k,l= = ϕν f ν s L yf ν s f Qdy, ds ν s L yf ν s f ν s L yf νdyds. ϕν s f k l ϕν s f rν s L c f k ν s L c f k ν s L c3 f k ds k j= ϕ ν s L yf ϕ ν s f k= ν s L dj f k ν s L dj f l ds M k= ϕν s f ν s L yf k ν s f k ν s L yf k k ϕν s f k l ϕν s f νs L yf k ν s f k νdyds k νdyds ϕν s f rν s L c f k ν s L c f k ν s L c3 f k ds k j= ϕ ν s L yf ϕ ν s f AΦν s ds M, ν s L dj f k ν s L dj f l ds M k= where M,, s an F-local marngale. Ths yelds ha Φν Φν ϕν s f ν s L yf k νdyds k AΦν s ds, 3 s an F-local marngale. By vrue of he unqueness of marngale problems as n Chaper 3 of 9 and 3, he lm measure-valued process ν s ndeed gven by 9. 4 Prcng Baske Opons In hs secon, we urn o he man objecve of he paper on he prcng of baske opons n a large porfolo.e., as by employng Theorem 3.5 n he above secon. Recall ha he prce of he baske opon s gven by 3,.e., V φ = e rt E φ w ST = e rt E φ νt I, = where Ip, l, x = x for p, l, x O and he emprcal measure νt s defned by 5 wh = T heren. Here recall ha, for a srke prce K >, φx = θx K for x R corresponds o he call baske opon f θ =, and corresponds o he pu baske opon f θ =. Then we have he followng man lm resul on he baske opon prce gven by

12 Theorem 4.. Le assumpons A and A be sasfed. Then he prce of he baske opon adms he followng lm as, lm V φ = e rt E φ X T p, l, xqdpηdlψdx, 4 O where for p, l, x O, he parameerzed sae process X p, l, x,, s gven by. Here q, η and ψ are he lm emprcal measures assocaed wh he parameer se, jump funcons and nal weghed prce prces gven n he assumpon A. Moreover, f p p, l l ponwsely and X x as for some p, l, x O, hen we have lm V φ = e rt E φ XT, 5 where he sae process X := X p, l, x,, sasfes he followng SDE wh jumps gven by, X = X = x, and dx X = r v d σ w β X β d W l y Qdy, d. 6 d Here σ := j= σ. j Proof. Recall Ip, l, x = x for p, l, x O. For n, defne g n x := max n, mnx, n on x R. Then g n C b R and for all x R, g n x Ip, l, x as n. Furher g n x x for all n. Snce g n C b R for each fxed n, usng Theorem 3.5, has, for fxed n, νt g n weakly converges o ν T g n as. oce ha he payoff funcon φ s connuous. Then by employng he connuous mappng heorem, for fxed n, φ νt g n weakly converges o φ ν T g n as. We nex prove he unform esmae of he momen of φ νt g n ;. In fac, noce ha he payoff funcon φ adms he lnear growh condon. Then here exss a consan C K > whch may depend on he srke prce K such ha E φ νt g n C K E νt g n C K E g n XT = C K E g n XT g n X j T,j= C K E g n XT E g n X j T C K sup,j= = E XT. By vrue of Lemma 3., we have ha sup = E XT < and whch s also ndependen of, n. Ths mmedaely yelds ha sup E φ νt g n <. 7,n Snce for fxed n, φ ν T g n weakly converges o φ ν T g n as, by Skorohod represenaon heorem, here exss a probably space Ω, F, P and a sequence of random varables Y, Y,, on such ha Y = d φ ν T g n, Y = d φ νt g n, and Y Y P-a.s. as. Le Ẽ denoe he expecaon w.r.. P. From Y = d φ νt g n and he above unform esmae 7 under P, follows ha sup Ẽ φ Y <.,j=

13 Ths yelds ha Y ; s unformly negrable under P. Recall Y Y P-a.s. as, and hence Y P Y as. Then usng Val s convergence heorem, follows ha Y Y n L Ω, F, P, as. Ths mples ha Ẽ Y Ẽ Y as. Snce E φ ν T g n = Ẽ Y and E φ ν T g n = Ẽ Y, we oban for fxed n, holds ha lm E φ νt g n = E φ ν T g n. Also by he monoone convergence heorem, for fxed, we have E φ ν T g n E φ ν T I and E φ ν T g n E φ ν T I as n. Togeher wh he followng rangle nequaly E φ ν T I E φ ν T I E φ ν T g n E φ ν T I E φ ν T g n E φ ν T g n E φ ν T g n E φ ν T I, we oban E φ νt I E φ ν T I as. oce ha holds ha E φ νt I = e rt V φ, and E φ ν T I = E φ X T p, l, xqdpηdlψdx. O Ths proves he lm 4. If p p, l l ponwsely and X x as for some p, l, x O, hen he assumpon A s sasfed wh he lmng emprcal measures q = δ p, l = δ l and ψ = δ x. Thus we have O X p, l, xqdpηdlψdx = X p, l, x = X for. Then he lm 4 s reduced o he lm 5. Thus we complee he proof of he heorem. As an applcaon, we here consder a baske opon where he underlyng rsky asses are descrbed as a mul-dmensonal mxed-exponenal jump-dffuson model where he CEV parameer β =. amely, under he rsk-neural probably measure Q, he -h rsky asse prce sasfes he followng SDE gven by, for =,...,, ds S = r v λe e c Y d σ j dw j d e c Y k, 8 j= k= where,, a Posson process wh nensy parameer λ >, and Y k, k, s a sequence of..d. random varables whch have a common probably densy funcon gven by h Y y := l u m = l λ e λ y y l d n lj λj e λ j y y<, y R. 9 Here l u, l d = l u, l, l j R whch also sasfy m = l = n l j= j =, and λ >, λ > for all =,..., m and j =,..., n. If = d =, v = and c =, hen he above model 8 s reduced o he sngle asse prce model proposed by 6. We nex are concerned wh he explc lm represenaon of he followng baske opon prce as usng Theorem 4.: V φ = e rt E φ w ST = e rt E φ νt I. To hs purpose, we rewre 8 as he form gven by,.e., for =,...,, ds S = r v d = j= σ j dw j j= 3 l y Qdy, d.

14 Here l y = e c y for y := R \ and hence we have l y > for all, y. The characersc measure of Posson random measure Qdy, d s gven by νdy = λh Y ydy. I s easy o verfy ha νdy sasfes he assumpon A. We consder he case where he exended parameer se s gven by p = v, σ,..., σ d, w, c, S p = v, σ,..., σ d, w, c, s, as, and p ; s domnaed by a fne posve consan. Then s clear o have l y l y := e c y for all y as. Thus we can apply Theorem 4. o conclude ha lm V φ = e rt E φ XT, 3 where he sae process X := X p, l, x,, sasfes he followng SDE gven by, X = X = w s, and dx X = r v λe e c Y d σ d W d e c Y k. 3 Here σ := σ w β wh σ := d j= σ j. oce ha he lmng sae process X,, has a smlar srucure as o Eq. n 6. Then he quany on he rgh hand sde of he equaly 3 can be characerzed by s Laplace ransform explcly. We nex consder he call baske opon,.e. θ = for he pu baske opon case, s smlar. Hence e rt E φ XT = e rt E XT K. By nroducng a scalng facor ϖ > K, he prce e rt E XT K s equal o X Γ T z := ϖe rt E T ϖ e z, 3 where z := log ϖ K R. Thus Theorem 3.4 n 6 gves he Laplace ransform of Γ T z w.r.. z n he followng closed-form here we se c = for convenence, whch s gven by, for all γ, λ, k= ˆΓ T γ := Here he funcon Gx, x λ, λ, s gven by Gx := σ x r v λe e Y x λ l u e γz Γ T zdz = w s γ γγ ϖ γ ergγt. 33 m = l λ λ x l d n lj λj λ j x. 34 Thus we n fac have he Laplace ransform of he lm of he baske opon prce whch can be explcly expressed as 33. Then he correspondng lm of he prce can be obaned by applyng wo-sded Euler nverson EI algorhm, see also 4, 5. If λ = n 8.e., here s no common jumps n he prce dynamcs, s reduced o he mul-dmensonal geomerc Brownan moon model consdered n. Thus n hs case, he underlyng lmng sae process 3 s smplfed o he geomerc Brownan moon gven by j= dx X = r v d σ d W. 35 The funcon Gx gven by 34 s reduced o Gx = σ x r v x for x R. Hence he lm prce V φ of he baske opon as s equal o he followng Black-Scholes formula gven by e rt E X T K = d x e v T d Ke rt, 36 4

15 where x = w s, d, d R, denoes he dsrbuon funcon of he sandard normal random varable, and d := x σ log r v σ T, T K d := d σ T. 5 umercal Analyss In hs secon, we presen a numercal analyss for he example 3 nroduced n he above secon and es he qualy of our approxmaon for he call baske opon prce. More precsely, we compare our analyc lmng prce gven by 3 o he exac values esmaed hrough Mone- Carlo smulaons. We frs se he parameer p = v, σ,..., σ d, w, c, S,, as v =, σ j = σj k, j =,..., d, w =, c = c k, S = s k, where v = and w = ndcae ha we here consder he prce of he average of values of rsky asses whou dvdend yeld ha have been grouped ogeher n he baske. Here σ j >, j =,..., d, c R and s >. The convergence order of he lmng parameers s gven by k. Thus we have σ,..., σ d, c, S σ,..., σd, c, s, as. d Morevoer n lgh of Theorem 4., we defne he lmng volaly of he model as σ := j= σ, j and he lmng nal asse value s gven by x = w s = s usng he above seng of parameers. In addon, n he followng numercal analyss, we ake he common jump parameers see also 9 as l u =.6, l d =.4, m = n =, λ = λ =., λ =. and l = l =. We nex compare our approxmang prce for he call baske opon gven by V φ = e rt E ν ST = e rt E T I = usng he exac lm prce formula gven by 3 o he exac values esmaed hrough Mone-Carlo smulaons. As n, we wll be mplemenng hs comparson by consderng he wo prces wh respec o he srke prce K >, he lmng nal asse value x >, and he lmng volaly σ > respecvely. We sar analyzng he comparson of he wo prces wh respec o he srke prce K >. Here we fx he number of he rsk asses o = 3. From Fgure, we see ha he call baske opon prce usng he analyc lm formula and he esmang prce usng he Mone-Carlo smulaons we run he M = 3 smulaons are decreasng wh respec o he srke prce, whch s conssen wh he fndng on he call baske opon prce n. Fgure shows good agreemen beween he analycal lm prce formula n Theorem 4. and he Mone-Carlo esmae. Table compares he numercal values obaned usng he analycal lm prce formula n Theorem 4. agans he correspondng Mone-Carlo esmaes as we vary he srke prce of he call baske opon. The resuls ndcae ha analycal lm prce racks closely he correspondng Mone-Carlo smulaon prce, hence confrmng he effcency and accuracy of he analyc lm prce formula n Theorem 4.. The values n parenhess n Table gve he 95% Mone-Carlo confdence nerval,.e. for he suffcenly large number of smulaon M hs nerval wll conan he rue opon value for 95 smulaons ou of every. We nex analyze he comparson of he wo prces wh respec o he lmng nal rsky asse value x and he lmng volaly σ respecvely. As he lmng nal asse value ncreases, so 5

16 .6.4 Prce of call baske opon v.s. K Lm Prce MC Esmae Prce K Fgure : The dependence of he call baske opon prces on he srke prce K. We fx he oher parameers o σ =.4, T =, k = 6 and x =.4. K Lm prce lm V φ Mone-Carlo esmae , , , , , , , , , , ,.343 Table : umercal values of he lmng prce of he call baske opon usng Theorem 4. and he Mone-Carlo esmae for dfferen srke prces. We fx he parameers o σ =.4, T =, k = 6 and x =.4. The values n parenhess represen he 95% Mone-Carlo confdence nerval. does he prce of he call baske opon see he lef graph of Fgure. Ths s smlar o he case of he call baske opon wh respec o he lmng volaly from he rgh graph n Fgure. Fgure shows hese resuls when he number of rsky asses s fxed o = 3, where we have consdered he symmercal suaon for he lmng volaly, namely σj s se o he same value σ for j =,..., d as n. I s clear ha he lm prce gven by lm V φ wh φx = x racks closely o he Mone-Carlo esmae values under he above dfferen scenaros. Tables and 3 ls he numercal comparson resuls of he same maury T =, he srke prce K =.8, and he same convergence order k = 6 of model parameers. As n Table, he values n parenhess n Tables and 3 also correspond o he 95% Mone-Carlo confdence nerval, where he number of smulaons s se o M = 3 n hs numercal es. Acknowledgemens The research of L. Bo was parally suppored by SF of Chna o. 4754, 4636, Fundamenal Research Funds for he Cenral nverses o. WK3478 and The Key Research Program of Froner Scences, CAS o. QYZDB-SSW-SYS9. Y. Wang graefully acknowledges he suppor of SF of Chna o The auhors also graefully acknowledge he consrucve and nsghful commens provded by he anonymous revewer whch conrbued o mprove he qualy of he manuscrp grealy. 6

17 .5 Prce of call baske opon v.s. x * Lm Prce MC Esmae Prce.65 Prce of call baske opon v.s. σ Lm Prce MC Esmae Prce x * σ Fgure : We fx he parameers o K =.8, T = and k = 6. The lef graph shows he dependence of he call baske opon prce on he lmng nal asse value x. The rgh graph shows he dependence of he call baske opon prce on he lmng volaly σ. x Lm prce lm V φ Mone-Carlo esmae , , , , , , , , , , , , ,.53 Table : umercal values of he lmng prce of he call baske opon usng Theorem 4. and he Mone-Carlo esmae for dfferen lmng nal asse values. We fx he parameers o σ =.4, T =, k = 6 and K =.8. The values n parenhess represen he 95% Mone-Carlo confdence nerval. σ Lm prce lm V φ Mone-Carlo esmae , , , , , , , , , ,.7474 Table 3: umercal values of he lmng prce of he call baske opon usng Theorem 4. and he Mone-Carlo esmae for dfferen lmng volales. We fx he parameers o x =., T =, k = 6 and K =.8. The values n parenhess represen he 95% Mone-Carlo confdence nerval. 7

18 References C. Alexander, Marke Rsk Analyss, Prcng, Hedgng and Tradng Fnancal Insrumens Volume III, Wley, 8. L. Bo, A. Cappon, Blaeral cred valuaon adjusmen for large cred dervaves porfolos, Fnance Soch L. Bo, A. Cappon, Sysemc rsk n nerbankng neworks, SIAM J. Fnancal Mah Ca, On frs passage mes of a hyper-exponenal jump dffuson process, Oper. Res. Le Ca, Prcng and hedgng of quanle opons n a flexble jump dffuson model. J. Appl. Probab Ca, S.G. Kou, Opon prcng under a mxed-exponenal jump dffuson model, Manag. Sc P. Carr, V. Lnesky, A jump o defaul exended CEV model: An applcaon of bessel processes. Fnance Soch J. Cox, oes on opon prcng I: Consan elascy of dffusons. npublshed draf, Sanford nversy. A revsed verson of he paper was publshed by J. Porfolo Manag. n S. Eher, T. Kurz, Markov Processes, Characerzaon and Convergence, Wley, 986. G. Gesecke, R. Sowers, K. Splopoulos, Defaul cluserng n large porfolos: Typcal evens, Ann. Appl. Probab G. Gesecke, K. Splopoulos, R. Sowers, J. Srgnano, Large porfolo asympocs for loss from defaul, Mah. Fnance M. Krekel, J. de Kock, R. Korn, T.K. Man, An analyss of prcng mehods for baskes opons, Wlmo Magaz Ikeda, S. Waanabe, Sochasc Dfferenal Equaons and Dffuson Processes, orh- Holland, G. Xu, H. Zheng, Lower bound approxmaon o baske opon values for local volaly jump-dffuson models, Iner. J. Theore. Appl. Fnance