Pricing MultiAsset Cross Currency Options


 Esmond Bennett
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1 CIRJEF844 Prcng MultAsset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212 CIRJE Dscusson Papers can be downloaded wthout charge from: Dscusson Papers are a seres of manuscrpts n ther draft form They are not ntended for crculaton or dstrbuton except as ndcated by the author For that reason Dscusson Papers may not be reproduced or dstrbuted wthout the wrtten consent of the author
2 Prcng MultAsset Cross Currency Optons Kenchro Shraya, Akhko Takahash Frst Verson: March 3, 212, Ths Verson: November 23, 212 Abstract Ths paper develops a general prcng method for multasset cross currency optons, whose underlyng asset conssts of multple dfferent assets, and the evaluaton currency s dfferent from the ones used n the most lqud market of each asset; the examples nclude cross currency optons, cross currency basket optons and cross currency average optons We also demonstrate that our scheme s able to evaluate optons wth hgh dmensonal state varables such as 2 dmensons, whch s necessary for prcng basket optons wth 1 underlyng assets under stochastc volatlty envronment Moreover, n practce, fast calbraton s necessary n the opton markets relevant for the underlyng assets and the currency, whch s also acheved n ths paper Furthermore, we nvestgate the mpled correlatons n the cross currency markets on the dates before and after the events, Lehman Shock and Tohoku Earthquake 1 Introducton Ths paper presents a general framework for prcng multasset cross currency optons under a broad class of multdmensonal dffuson models We notce that the underlyng assets of a multasset cross currency opton are related wth multple underlyng asset markets as well as at least one currency market Moreover, t s necessary for practcal use of a prcng model to take the nformaton of each underlyng asset s opton market nto account Then, calbraton to each opton market needs a more complex model than BlackScholes model such as stochastc volatlty models to reflect the skew/smle and term structure of mpled volatltes observed n the opton market Thus, a multdmensonal dffuson model should be appled to prcng a multasset cross currency opton and relevant calbratons, where an analytcal valuaton method s necessary for fast computaton On the other hand, t s almost mpossble to obtan a closedform opton prcng formula under a multdmensonal dffuson settng An effectve method for overcomng ths problem s an asymptotc expanson scheme whch s a unfed method n order to acheve accurate approxmatons of opton prces and Greeks n Forthcomng n the Journal of Futures Markets Graduate School of Economcs, Unversty of Tokyo Graduate School of Economcs, Unversty of Tokyo
3 multdmensonal models (For nstance, please see [37], [28], [29], [21], [32], [33], [3], [31] for the detal We also remark that the Mathematcal foundaton of ths method reles on Watanabe theory n Mallavn calculus (For nstance, please see [34], [36], [22] for the detal Applyng the scheme, ths paper derves an approxmaton formula for prcng multasset cross currency optons, and presents several practcal examples wth numercal analyss whch takes actual opton markets nto consderaton As the frst example, we evaluate cross currency basket optons n stochastc volatlty models based on the calbraton to the relevant opton markets of USD quoted currency pars Moreover, for examnaton of the accuracy of our approxmaton method, we also evaluate basket optons whose underlyng asset conssts of 1 dfferent assets under BlackScholes [3], Constant elastcty of varance (CEV [8] and λsabr [23] models Partcularly, we demonstrate that our scheme can be appled to prcng optons wth hgh dmensonal state varables such as 2 dmensons, whch s necessary for prcng basket optons wth 1 underlyng assets under stochastc volatlty envronment Ths feature s an advantage of ths method comparng to other analytcal (approxmaton schemes There are several exstng lteratures (eg [5], [6], [18], [25] and [38] that derve approxmate formulas for prcng basket optons where each underlyng asset prce follows BlackScholes model [14] derves an approxmaton formula n jumpdffuson model, and [35] provdes an approxmaton formula n a local volatlty and jumpdffuson model Also, [4] derves closed form formulas for the opton prce and the Greeks of Asan(average basket optons n energy markets under lognormal (BlackScholes model of each underlyng asset prce wth momentmatchng method [1] shows the analytc bounds for Asan basket optons under BlackScholes model wth comonotonc approach [28] and [29] proposes a new prcng formula for basket optons under general dffuson settng by applyng the asymptotc expanson scheme Recently, [24] develops a new symbolc algorthm for the asymptotc expanson scheme, and apples t to prcng optons on VIX under the Gatheral double lognormal stochastc volatlty models, where the underlyng asset s expressed as squareroot of a lnear combnaton of a stochastc varance and ts stochastc mean reverson level [19] approxmated basket opton under SABR model usng Markovan projecton Moreover, [1] derves a very accurate formula for prcng basket optons under a general class of local volatlty models ncludng CEV and BlackScholes models, and demonstrates the accuracy usng 1 underlyng assets (For nstance, see [2] and [15] for the related artcles Our work s the frst one whch derves an approxmaton formula and mplements numercal experments n stochastc volatlty envronment for prcng basket optons wth 1 underlyng assets, as well as prcng currency basket optons based on calbraton to the real vanlla opton markets The second example s cross currency average optons [2] proposes approxmatons of average opton prces under BlackScholes model However, t s almost mpossble for one parameter set under the BlackScholes opton prcng model to reproduce market prces wth varous strkes for a gven maturty Then, [37] [28], [29], [27] apply an asymptotc expanson method to prcng average optons under the general dffuson process of the underlyng asset prce; they consder a contnuous average of an asset prce, whch does not represent the underlyng prce of a contract n the real world precsely 2
4 Hence, the formula they derved can not be used drectly for valuaton of the optons traded n actual markets Thus, [26] develops an approxmaton scheme wth stochastc volatlty models that takes specfc features of commodty average prce optons nto consderaton (See the paper for the detal [9], [11] and [13] derve approxmaton formula for cross currency average basket optons under BlackScholes model Ths paper extends [26] to the cross currency correspondents whch s very useful for frms outsde Unted States mportng energes such as crude ols traded manly wth the US dollar Especally, we evaluate average optons on Japaneseyen based West Texas Intermedate (WTI futures, where SABR model s appled to WTI futures prce processes whle an extended λsabr model s used for the JPYUSD spot foregn exchange rate 1 process We show ts numercal examples based on the calbraton to the WTI futures opton market as well as to the USDJPY currency opton market To the best of our knowledge, ths s the frst work for prcng cross currency average optons ncludng numercal analyss whch reflects the actual opton markets Furthermore, we nvestgate the mpled correlatons n the varous cross currency markets In partcular, we apply SABR model to each USD quoted foregn exchange rate, as opposed to models such as Double Heston model (eg [7], [16], n whch the two volatlty processes for the USD quoted currency pars relevant wth a cross currency par are perfectly correlated Clearly, the performance of calbraton by our model seems much better, whch s confrmed n our numercal analyss As a related work, [12] analyzes the currency opton markets n farly detal by applyng hs ntrnsc currency framework under stochastc volatlty envronment As for the data of the numercal analyss, we use the ones on the dates before and after the events such as Lehman Shock and Tohoku Earthquake Moreover, the senstvtes of mpled volatltes wth respect to the correlaton parameters are also examned The organzaton of the paper s as follows: Secton 2 dscusses a general dffuson model used for prcng multasset cross currency optons and shows several examples ncluded n ths class of optons Secton 3 derves an approxmaton formula for multasset cross currency optons under a multfactor extenson of the λsabr stochastc volatlty model Secton 4 presents prcng cross currency basket and cross currency average optons wth numercal examples Secton 5 nvestgates the mpled correlatons n the varous cross currency opton markets Appendx shows the dervaton of the prcng formula and the calbraton results assocated wth Secton 5 2 MultAsset Cross Currency Optons Let th asset S k,l be a spot foregn exchange rate that stands for the prce of the unt amount of the currency k n terms of currency l Wthout loss of generalty, we assume that k denotes Japanese yen (JPY whle l denotes US dollar (USD, and wrte S for S k,l that s, 1 Japanese yen = S US dollars Also, let S ( = 1,, n denotes the prce of the asset n terms of USD 1 prce of Japanese yen n terms of US dollars 3
5 Next, suppose that the dynamcs of S ( =, 1,, n are expressed under the USD rskneutral measure as follows: ds (t = S (t[α (tdt ˆσ (t, S dz(t], (1 dσ (t = f (t, σ dt ν (t, σ σ (tdz(t, (2 ds (t = S (t[α (tdt ˆσ (t, S ]dz(t, (3 dσ (t = f (t, σ dt ν (t, σ σ (tdz(t, (4 where α (t = r USD (t r JP Y (t, α (t = r USD (t δ (t( = 1,, n, and Z denotes the 2(n 1dmensonal Brownan moton under the USD rskneutral measure; r USD, r JP Y and δ denote the rskfree nterest rates of USD, that of JPY and the dvdend rate of th asset, respectvely Moreover, 2(n1dmensonal parameters, ˆσ (t, S and ν (t, σ ( =, 1,, n are defned by ˆσ (t, S := σ (tg (t, S (c, (t, c,1 (t,, c, (t,,, (5 ν (t, σ := (ν, (t, σ,, ν,2n1 (t, σ := ν (t, σ (c n1, (t, c n1,1 (t,, c n1,n1 (t,,,, (6 where f (t, x, g (t, x and ν (t, x, ( =,, n are some [, T ] R R functons c,j (t, ( j 2n1 are [, T ] R functons whch are obtaned by the Cholesky decompostons of the relevant correlaton matrces Next, let Y, ( = 1,, n denote the prce of the asset n terms of JPY: Y (t = S (t S (t (7 Then, the dynamcs of S, σ and Y under JPY rskneutral measure are gven as follows: ds (t = S (t[(α (t ˆσ (t, S ˆσ (t, S dt ˆσ (t, S dw (t] (8 dσ (t = [f (t, σ ˆσ (t, S σ (t, S ν (t, σ ]dt ν (t, σdw (t, (9 dy (t = Y (t[(α (t α (t dt ( ˆσ (t, S ˆσ (t, S dw (t], (1 where W denotes the 2(n 1dmensonal Brownan moton under the JPY rskneutral measure Under ths settng, n the followng sectons we wll consder approxmatons for prcng optons whose underlyng asset prce process X defned by X(t = n w (ty (t, (11 =1 where w (t s a determnstc functon of the tme parameter t; for nstance, the payoff of a call opton wth strke prce K and maturty T s expressed as max {X(T K, } Hereafter, we wll call ths type of optons multasset cross currency optons We remark 4
6 that settng S 1, we can treat USD denomnated optons as a specal case We are also able to consder quantotype products when the underlyng asset s gven by n X(t = w (ts (t, (12 =1 under the JPY rskneutral measure We also assume that the nterest rates r USD, r JP Y and dvdend rates δ are nonrandom just for smplcty, whch mples that α (t and α (t n the above equatons become determnstc 2 Next, let us see several wellknown opton products whose underlyng asset prces are descrbed by (11 wth specfc w (t 1 Cross Currency Optons The smplest example may be cross currency optons such as EUR/JPY optons, where S 1 stands for the exchange rate USD/EUR Then, we set the weghts w (t as w 1 (t = 1 and w (t = ( = 2,, n 2 Spread Optons The underlyng asset prces of spread optons are the dfference of futures prces/nterest rates wth dfferent maturtes, or the dfference of the prces of dfferent assets In ths case, settng the weghts as w 1 (t = 1, w 2 (t = 1 and w (t = ( = 2,, n, we have X(t = Y 1 (t Y 2 (t (13 3 Basket Optons The underlyng asset of a basket opton s the weghted average of the prces of dfferent assets, where the weghts are typcally some prespecfed (postve constants, that s w (t w > for all : X(t = n w Y (t (14 =1 4 Average Optons Average optons are one of popular products especally n the commodty markets; the futures contracts wth several consecutve maturtes may become the underlyng assets of an average opton as n OTC ol market(eg WTI market (See [26] for the detal of the structure of products More generally, let us ntroduce new processes Z (t defned by m ( Z (t = Y j=1 t ( j I ( {t ( j t}, 2 When the underlyng asset s a futures contract (denomnated by USD wth no dvdends, the asset prce process s a martngale under the USD rskneutral measure Then, α = n (3 5
7 where t ( 1 < < t m ( T, m denotes the number of the cross currency prce Y to whch the average opton refers, and the dynamcs of each Y s descrbed by the stochastc dfferental equaton (1 wth (9 Then, we can deal wth a cross currency(eg Japanese yenbased average opton whose underlyng asset prce X(t s gven by the followng: X(t = 1 M n Z (t = =1 n m =1 j=1 1 ( ( M I {t ( j t} Y t ( j, (15 where M = ( n =1 m Note that f each Y t ( j, = 1,, n, j = 1,, m s regarded as a dfferent asset s prce, ts weght functon n (11 s gven by w (j (t = 1 ( M I {t ( j t} 3 Approxmaton formula Ths secton wll derve an approxmaton formula whch s useful for analyss of multasset cross currency optons In partcular, we take an extended λsabr model as the underlyng asset prce dynamcs, whle the smlar formula can be derved for the general model, (1(4 n the same procedure Frst, we brefly explan the extended λsabr model that we apply to our analyss n the followng sectons We specfy the coeffcent functons for (1(6 n the prevous secton as follows: f(t, σ = λ (θ (t σ (t, (16 g(t, S = S β 1, (a constant β [, 1], (17 ˆσ (t, S = σ (tg(t, S (c,, c,1,, c,,,, (18 ν (t, σ = ν (t = (ν, (t,, ν,2n1 (t = ν (c (n1, (t, c (n1,1 (t,, c (n1,(n1 (t,,,, (19 where λ and ν, =, 1,, n are some postve constants Also, c,j (t, ( j 2n 1 are [, T ] R functons whch are obtaned by the Cholesky decompostons of the relevant correlaton matrces Then, n sum we obtan the dynamcs of S and σ for =, 1,, n as follows: ds (t = α S (tdt ˆσ (ts (tdz(t; S ( gven, α s a constant (2 dσ (t = λ (θ (t σ (tdt ν (tσ (tdz(t; σ ( gven, (21 where ˆσ (t and ν (t are defned by (18 and (19, respectvely; λ s a postve constant, and θ(t s a determnstc functon of the tme parameter t; Z denotes 2(n 1 dmensonal Brownan moton under the USD rskneutral measure Then, the dynamcs of S, σ and Y under the JPY rskneutral measure are gven by (8(1 6
8 Next, we wll present the prcng formula under the λsabr model For the SABR model [17], t s enough to set λ = n the λsabr model In partcular, the followng theorem shows a formula for a Europeantype call opton whose payoff s gven by where C(T = max {X(T K, }, (22 X(t = n w (ty (t =1 Although the detal of the dervaton of the formula s qute lengthy and hence s omtted, let us brefly explan the dea In the frst place, n order to get perturbatve random processes of (8(1, we replace ˆσ, ˆσ and ν n (8(1 by ϵˆσ, ϵˆσ and ϵ ν, respectvely to obtan S (ϵ, σ (ϵ and Y (ϵ As a result, we obtan X (ϵ (T, a perturbatve random varable of X(T, and then expand X (ϵ (T around X ( (T whch s X (ϵ (T evaluated at ϵ = (Due to the zero volatltes of S (, σ ( and Y (, the process {X ( (t : t T } has no volatlty term, and hence X ( (T corresponds to the ntal prce of the T  maturty forward contract for the underlyng asset Next, we mplement an asymptotc expanson of the densty functon for a normalzed random varable, ˆX(ϵ (T := X(ϵ (T X ( (T ϵ of whch lmtng densty s Gaussan For example, the asymptotc expanson up to the ϵ 2 order for the densty functon of ˆX (ϵ (T denoted by f ˆX(ϵ(x s obtaned as follows: { H f ˆX(ϵ(x 3 (x; Σ = n[x;, Σ] 1 ϵc 1 Σ 3 ( } ϵ 2 H 6 (x; Σ H 4 (x; Σ H 2 (x; Σ C 2 Σ 6 C 3 Σ 4 C 4 Σ 2 o(ϵ 2, (23 where n[x;, Σ] s the normal densty functon wth mean and varance Σ, that s n[x;, Σ] = 1 2πΣ exp The coeffcents C 1,, C 4 are some constants Moreover, ( x 2 2Σ H k (x; Σ denotes the kth order Hermte polynomal, H k (x; Σ = ( Σ k e x2 /2Σ d k e x2 /2Σ dx k Also, notce that the call payoff on X (ϵ wth maturty T and strke K can be expressed by { } {( } max X (ϵ (T K, = ϵ max ˆX(ϵ (T y,, (24 where y = X( (T K ϵ Then, by usng the expanson of the densty and the call payoff s expresson, we obtan an approxmaton of the call opton prce through an expanson of the rght hand sde of the equaton below: [ { }] [ {( }] e rt E max X (ϵ (T K, = ϵe rt E max ˆX(ϵ (T y, (25 Partcularly, settng ϵ = 1 n ths approxmaton, we are able to get an approxmaton for the call opton prce wth payoff (22, C(T = max {X(T K, } Consequently, we obtan the followng theorem 7
9 Theorem 31 Let C( be the tme prce of the European call opton wth payoff (22 Then, an approxmaton formula of C( obtaned by an asymptotc expanson up to the ϵ 3 order s gven by settng ϵ = 1 n the followng expresson: ( ( y e {ϵ rt yn Σn[y;, Σ] Σ ( ϵ 3 H 4 (y; Σ H 2 (y; Σ C 2 Σ 4 C 3 Σ 2 C 4 n[y;, Σ] ϵ 2 H 1 (y; Σ C 1 n[y;, Σ] Σ }, (26 where r s a constant rskfree rate, y = X( (T K ϵ, X ( (T stands for the ntal prce of the T maturty forward contract for the underlyng asset, n[x;, Σ] s the normal densty functon wth mean and varance Σ, and N(x denotes the standard normal dstrbuton functon The coeffcents C 1,, C 4 are some constants Moreover, H k (x; Σ denotes the kth order Hermte polynomal, and partcularly, H 1 (x; Σ = x, H 2 (x; Σ = x 2 Σ and H 4 (x; Σ = x 4 6Σx 2 3Σ 2 The detal of the dervaton as well as the coeffcents C 1,, C 4 are n Appendx A 4 Numercal Examples After showng an approxmaton result n a smple BlackScholes model, ths secton wll descrbe numercal examples of basket optons and cross currency average optons We remark that n applyng the approxmaton formula (26 n Theorem 31 to all the numercal examples, the perturbaton parameter ϵ n (26 s set to be 1 Generally speakng, our expanson formula s able to approxmate opton prces very well for all the examples In partcular, Subsecton 42 wll show that hgh dmensonalty s not a bg ssue n a sense that the same formula (26 can be appled to prcng optons wth hgh dmensonal state varables such as 2 dmensons, whch s necessary for prcng basket optons wth 1 underlyng assets under stochastc volatlty envronment Ths feature s an advantage of ths method comparng to other analytcal (approxmaton schemes On the other hand, because the expanson s made around ϵ =, that s zero volatlty, the approxmaton s expected to become less accurate as the underlyng asset volatlty and the volatlty on volatlty are larger Moreover, as we remark n the end of the prevous secton, snce the lmtng densty of the normalzed random varable ˆX (ϵ (T s Gaussan, as the dstrbuton of the underlyng asset prce s closer to a normal dstrbuton, the approxmaton s expected to be better and vce versa Furthermore, n opton prcng, the dfference of the underlyng asset prce dstrbuton has the larger effect for prcng more OTM optons because the shape of the tal n the dstrbuton becomes more mportant Hence, an approxmaton for OTM opton prces s expected to become more dffcult when the underlyng dstrbuton s less close to normal 8
10 These ponts wll be shown or/and dscussed n detals wth concrete numercal experments n the followng subsectons, whch wll ndcate a robustness of our method n terms of accuracy even under dffcult stuatons for ths approxmaton scheme 41 PlanVanlla Opton under BlackScholes Model The frst example s on the BlackScholes model wth no drft: ds(t = σs(tdw (t; S( = x > (27 In ths smplest case, Σ = σ 2 x 2 T, and C, = 1,, 4 n (26 are easly obtaned as follows: C 1 = σ4 x 3 T 2 2, C 2 = σ8 x 6 T 4 8, C 3 = 2σ6 x 4 T 3 3, C 4 = σ4 x 2 T 2 (28 4 The next table compares the approxmaton result for call optons wth the exact result by BlackScholes formula (BS n the table, where the parameters are set as r =, x = 1,, σ = 15 and T = 1 We observe that the thrd order approxmaton (denoted by AE 3rd n the table s rather well Table 1: European Call Opton (BlackScholes model Strke AE 3rd 2,41 1, BS 2,44 1, Dfference Relatve Dfference (% % % % % 2% 42 Basket Optons wth 1 Underlyng Assets Frst, we examne the accuracy of our method by prcng basket call optons wth 1 underlyng assets Due to ts hgh dmensonalty (eg 2 dmensons under the stochastc volatlty model used for Table 4, ths product s hard to be evaluated by other exstng analytcal methods especally under stochastc volatlty models Even numercal methods such as Monte Carlo smulatons are very tmeconsumng to obtan accurate prces We also note that our method s fast enough for practcal usage 3 For nstance, by usng one core of Xeon X GHz, t takes 38 seconds by our approxmaton formula (26 to obtan the result reported n Table 4 below whle t does 7,3 seconds by Monte Carlo smulaton That s, the computatonal speed of our method s around 1,9 tmes faster than that of Monte Carlo smulaton Moreover, we remark that when the dmenson of the underlyng state varables s lower, our method has more advantage than Monte Carlo smulaton n terms of the computatonal speed (For prcng a plan vanlla call opton under λsabr model, t takes only seconds by the method whle 184 seconds by Monte Carlo smulaton, that s 438, tmes faster 3 Please see for nstance, [1] for other fast accurate analytcal method for prcng basket optons 9
11 Let us consder the payoff: for n = 1, {( n } max S (T K,, =1 where we do not consder cross currency optons that s, S 1 As for the underlyng asset prces S ( = 1,, n, we take λsabr, CEV and BlackScholes models Specfcally, λsabr model s expressed n the followng: for = 1, 2,, n, ds (t = α S (tdt σ (ts (t β c dz(t; S ( gven, α s a constant (29 dσ (t = λ (θ σ (tdt ν σ (tdz(t; σ ( gven, (3 where λ and θ are postve constants, and Z s a 2ndmensonal Brownan moton; c and ν are defned by c = (c,1,, c,,,, (31 ν = ν (c (n,1,, c (n,(n,,,, (32 where ν s a postve constant, and c,j, (1 j 2n are obtaned by the Cholesky decomposton of the relevant correlaton matrx In λsabr model, the detals for parameters specfcaton n the equatons above are as follows 1 The sum of the underlyng asset s ntal prce s 1; Each ntal asset prce s generated by the followng procedure: Frstly, we generate 99 random varables, a S ( = 1,, 99 from a unform dstrbuton n [, 1], that s U[, 1] Then, we arrange them n ascendng order and relabel those by b S so that b S 1 < < bs 99 Next, we take the dfference of the 1 ordered couples to defne c S = b S bs 1 wth settng bs = and bs 1 = 1 Fnally, we defne S ( = 5c S 5 to get S ( ( = 1,, 1 2 We set α wthout loss of generalty, The average of ntal volatlty σ ( and that of the constant meanreverson level θ (t θ are 15%, and the average of volatlty on volatlty ν s 3% We defne each ntal volatlty and volatlty on volatlty by σ ( = 5c σ 1 and ν = 5c ν 25, where cσ and c ν are obtaned n the smlar way as c S above 3 The parameters λ and β are set as λ = 1 and β = 5, respectvely 4 The correlaton between two dfferent asset prces s 8; the correlaton between an asset prce and a volatlty s 4; the correlaton between two dfferent volatltes s 3 CEV and BlackScholes models have no stochastc volatlty processes (21, of course Moreover, n BlackScholes and CEV models the parameters are specfed as those correspondng ones n λsabr model, except that n BlackScholes model, we set β = 1, and determne each asset prce volatlty as σ BS = σ ( 1 by usng σ ( n λsabr model 1
12 By applyng the 3rd order expanson formula n Theorem 31, we evaluate the basket call optons wth 1 year maturty and strke prces 8, 9, 1, 11 and 12 Then, we compare those wth the prces calculated by Monte Carlo, where the random number generator s Mersenne Twster, the number of trals s 3 mllon wth the antthetc varable method and tme steps are 64/year We remark that n Monte Carlo smulatons the convergence becomes very slow for ths large number of the underlyng assets, and hence, we provde the standard errors n the tables The results of Black Scholes, CEV and λsabr models are gven n Table 2  Table 4, respectvely, where AE3 stands for the approxmate prce by the thrd order asymptotc expanson based on the formula (26 n Theorem 31 Table 2: Basket Opton (BlackScholes model Strke AE 3rd 2,64 1, Monte Carlo 2,66 1, Dfference Relatve Dfference (% % % % 3% 11% MC Std Error Table 3: Basket Opton (CEV model Strke AE 3rd 2,7 1, Monte Carlo 2,71 1, Dfference Relatve Dfference (% % % % 2% 5% MC Std Error Table 4: Basket Opton (λsabr model Strke AE 3rd 2,38 1, Monte Carlo 2,377 1, Dfference Relatve Dfference (% % % % 1% 7% MC Std Error From these results, we observe that the 3rd order expanson can approxmate basket opton prces rather well 11
13 43 Currency Basket Optons Ths subsecton wll examne currency basket optons usng actual data In partcular, we evaluate basket optons wth three month maturty, whose underlyng assets consst of fve currency pars; JPYUSD, EURUSD, AUDUSD, GBPUSD and CADUSD, where the quote currences are USD Moreover, the model parameters are obtaned by the calbraton to planvanlla opton prces wth three month maturty as of the frst busness day of Nov 211; partcularly, we remark that the correlaton parameters among currency pars should be estmated through the relevant 1 cross currency opton markets whose procedure wll be descrbed below Also, for prcng vanlla and basket optons we apply our formula n Theorem 31 to (the extended SABR model wth β = 1: for =, 1, 2, 3, 4, ds (t = α S (tdt σ (ts (t c dz(t; S ( gven, α s a constant (33 dσ (t = ν σ (tdz(t; σ ( gven, (34 where c and ν are defned by c = (c,, c,1,, c,,,, (35 ν = ν (c (n1,, c (n1,1,, c (n1,(n1,,,, (36 where ν s a postve constant, and c,j, ( j 2n 1 are obtaned by the Cholesky decomposton of the relevant correlaton matrx In the followng, let us brefly descrbe the calbraton procedure: (Calbraton Procedure (Step 1 Suppose that each USD quoted exchange rate follows (33(34 under the USD rskneutral measure Then, gven α and S ( by observaton of the market, through the calbraton to each European planvanlla currency opton market as of the frst busness day of Nov 211, we estmate σ (, ν and the correlaton between the exchange rate and ts volatlty, that s ( c ν (Step 2 Each exchange rate of 1 cross currency pars 4 follows the stochastc dfferental equaton for S /S j ( j whch s obtaned through (1 wth (9, where we replace S j wth S for the defnton of Y n (7 and specfy the parameters by usng (33(34 nstead of (1(4 By calbraton to each European planvanlla cross currency opton market, we estmate the remanng 4 correlatons that s, ( c c j ( c ν j, ( c j ν, ( ν ν j, whle ( c ν and ( c j ν j are already obtaned n (Step 1 above; note that the estmaton s subject to satsfyng the postve defnte of the relevant 4 4 correlaton matrx The results are lsted n Table 5 and 6 However, we fnd that the 1 1 correlaton matrx n Table 6 does not satsfy the postve defnte condton In order to fx t, 4 EURJPY, GBPJPY, AUDJPY, CADJPY, EURCAD, GBPCAD, AUDCAD, EURAUD, GBPAUD, EURGBP 12
14 through smultaneous calbraton to fve cross currency opton markets, we reestmate the correlaton parameters subject to satsfyng the postve defnteness of the 1 1 correlaton matrx The result s gven n Table 7 We note that the fttng to the market prces based on the correlaton matrx n Table 7 becomes worse than before σ ( ν JPYUSD EURUSD GBPUSD AUDUSD CADUSD Table 5: Volatlty Parameters Table 6: Correlaton Matrx I JPY EUR GBP AUD CAD JPY Vol EUR Vol GBP Vol AUD Vol CAD Vol JPY EUR GBP AUD CAD JPY Vol EUR Vol GBP Vol AUD Vol CAD Vol Table 7: Correlaton Matrx II JPY EUR GBP AUD CAD JPY Vol EUR Vol GBP Vol AUD Vol CAD Vol JPY EUR GBP AUD CAD JPY Vol EUR Vol GBP Vol AUD Vol CAD Vol Next, we evaluate the 5 currency basket opton wth three month maturty, where the weghts of the exchange rates other than JPYUSD are equal to 1, whle the weght of JPYUSD s 1 so as to be of the same order The weghted average of the ntal forward prces of fve currences as of the frst busness day of Nov 211 s We compute the prces of put optons wth strkes 6 and 615, as well as of call optons wth strkes 625, 635 and 65 We use the parameters gven n Table 5 and Table 7 For comparson, the benchmark prces are calculated by Monte Carlo smulatons, where the random number generator s Mersenne Twster, the number of trals s 5 mllons wth the antthetc varable method and tme steps are 25 per year 13
15 The result s gven n Table 8, where AE 3rd stands for the approxmate prce by the thrd order asymptotc expanson based on the formula (26 n Theorem 31: we observe that the approxmatons are less accurate than those for λsabr model n Table 4, whch s partly because the calbrated volatlty parameters on the volatltes n Table 5 are much hgher than 3 % for Table 4 Also, we set β = 1 n (33, whle β = 5 n (29, from whch we recall the followng observaton: because the asymptotc expanson method s based on an expanson around a normal dstrbuton and the dstrbuton of the underlyng asset prce s closer to a normal when β s closer to zero, the smaller s β, the approxmaton becomes more accurate n general Table 8: Currency Basket Opton Strke Monte Carlo AE 3rd Dfference Relatve Dfference (% 23% 4% 1% 7% 37% 44 Cross Currency Average Optons Ths subsecton descrbes the evaluaton of cross currency average optons In partcular, we evaluate average optons on Japanese yen(jpybased WTI future prces, whch refers to a JPYbased WTI futures prce every busness day Let us brefly descrbe the specfc features of the WTI average prce opton: the underlyng prce of an average opton s the average of the settlement prces of the frst nearby WTI futures contract durng the last one month pror to the maturty of the average opton Note also that the expraton of an average opton s the last busness day of a calendar month, whle tradng of a WTI futures contract usually ceases on the thrd busness day pror to the twentyffth calendar day Hence, as the expraton of tradng a futures contract s about a week before the end of a calendar month, the futures contracts wth two consecutve maturtes become the underlyng assets of an average opton More precsely, let the reference dates t (1 1,, t(1 m 1 for the frst contract and t (2 1,, t(2 m 2 for the subsequent contract (t (1 1 < < t (1 m 1 < t (2 1 < < t (2 m 2 = T Also, denote the relevant two underlyng futures prces as S ( = 1, 2, and the spot (USD quoted JPYUSD exchange rate as S Then, the JPYbased average prce at opton s maturty T s expressed as follows: X(T = 1 m 1 ( Y 1 t (1 m 2 ( j Y 2 t (2 j, (37 M j=1 j=1 where M = m 1 m 2 and Y = S /S ( = 1, 2 Thus, the payoff functons of average call and put optons wth strke prce Kand maturty T are gven by max {X(T K, } and max {K X(T, }, respectvely 14
16 We take the followng specfcatons for our numercal examples wth the data on July 1, 29 The maturtes of average optons: two month and four month The relevant WTI futures traded on the NYMEX dvson of the CME Group: SEP9(untl August 18 and OCT9(from August 19 for the two month maturty, and NOV9(untl October 19 and DEC9(from October 2 for the four month maturty The maturtes of the relevant currency optons: one month, two month, three month and sx month As for the JPYUSD exchange rate process S, we apply the λsabr model wth β = 1 because we mplement smultaneous calbraton to currency optons wth 1,2,3,6 month maturtes, n whch we need to take the term structure of the JPYUSD exchange rate volatlty process nto account For the WTI futures prce processes S ( = 1, 2, We take SABR model wth β = 5, 1, snce our prevous analyss n [26] has found that SABR model can acheve good calbraton to the WTI futures opton market (See [26] for the detal In sum, the relevant stochastc processes are descrbed by the solutons to the followng stochastc dfferental equatons: ds (t = α S (tdt σ (ts (t c dz(t; S ( gven, α s a constant (38 dσ (t = λ (θ σ (tdt ν σ (tdz(t; σ ( gven, (39 For = 1, 2, λ and θ are postve constants ds (t = σ (ts (t β c dz(t; S ( gven, (4 dσ (t = ν σ (tdz(t; σ ( gven (41 Here, c ( =, 1, 2 and ν ( =, 1, 2 are defned by c = (c,, c,1,, c,,,, (42 ν = ν (c (n1,, c (n1,1,, c (n1,(n1,,,, (43 where ν s a postve constant, and c,j, ( j 2n 1 are obtaned by the Cholesky decomposton of the relevant correlaton matrx Note also that the futures prce processes have no drfts n (4 Each JPYbased WTI prce Y = S /S ( = 1 or 2 follows the stochastc dfferental equaton for Y whch s obtaned through (1 wth (9, where we specfy the parameters by usng (38(41 nstead of (1(4 The correlatons between the JPYUSD rate and WTI futures prces, as well as those between dfferent futures contracts prces are estmated from the Hstorcal estmaton data of the prevous two month(four month for the twomonth(fourmonth maturty optons, whch are reported n Table 9 and Table 1 below 15
17 Table 9: Correlaton I (2M JPYUSD Sep9 Oct9 JPYUSD Sep Oct Table 1: Correlaton I (4M JPYUSD Nov9 Dec9 JPYUSD Nov Dec The other correlatons necessary for the evaluaton of cross currency average optons are gven n Table 11  Table 14, whch are determned n the followng way: The correlaton between the NOV9 prce and the DEC9 volatlty s set to be the same as the one between the DEC9 prce and DEC9 volatlty whch s obtaned by the calbraton and s reported n Table 15 and Table 16 below The same rule s appled to the correlaton between the DEC9 prce and the NOV9 volatlty, the correlaton between the SEP9 prce and the OCT9 volatlty, and the correlaton between the OCT9 prce and the SEP9 volatlty The followng correlatons are set as : the JPYUSD rate s volatlty and the futures prces volatltes; the JPYUSD rate s volatlty and the futures prces; the JPYUSD rate s volatlty and the futures prces volatltes All the correlatons between the futures prces volatltes wth dfferent contracts are set as 999 Table 11: Correlaton II (β = 1, 2M JPYUSD Sep9 Oct9 vol of JPYUSD vol of Sep9 vol of Oct9 JPYUSD Sep Oct vol of JPYUSD vol of Sep vol of Oct
18 Table 12: Correlaton II (β = 5, 2M JPYUSD Sep9 Oct9 vol of JPYUSD vol of Sep9 vol of Oct9 JPYUSD Sep Oct vol of JPYUSD vol of Sep vol of Oct Table 13: Correlaton II (β = 1, 4M JPYUSD Nov9 Dec9 vol of JPYUSD vol of Nov9 vol of Dec9 JPYUSD Nov Dec vol of JPYUSD vol of Nov vol of Dec Table 14: Correlaton II (β = 5, 4M JPYUSD Nov9 Dec9 vol of JPYUSD vol of Nov9 vol of Dec9 JPYUSD Nov Dec vol of JPYUSD vol of Nov vol of Dec The results of calbraton are reported n Table 15 and Table 16 Table 15: Calbrated Parameters (β = 5 S( α σ( λ θ ν ρ JPYUSD SEP OCT NOV DEC
19 Table 16: Calbrated Parameters (β = 1 S( α σ( λ θ ν ρ JPYUSD SEP OCT NOV DEC Gven the parameters above, we fnally evaluate cross currency average optons that s, put optons wth strkes 6 and 65, and call optons wth strkes 7, 75 and 8; also, the ATM prces for the twomonth maturty and for the fourmonth maturty are gven by and 6969, respectvely In addton, n order for nvestgaton of the accuracy of our approxmatons, benchmark prces are computed by Monte Carlo smulatons, where the random number generator s Mersenne Twster, and the number of trals s 25 mllon wth the antthetc varable method and wth 256 tmesteps for calculaton of each prce The results of the thrd order approxmate prces are reported n Table 17  Table 2, where AE 3rd stands for the approxmate prce by the thrd order asymptotc expanson based on the formula (26 n Theorem 31 We observe that our thrdorder formula (26 provdes very accurate approxmatons Also, we remark that the approxmatons wth β = 5 are more accurate than those wth β = 1 especally for OTM optons, snce as noted at the end of the prevous subsecton, our asymptotc expanson s made around the normal dstrbuton, and the underlyng prce s dstrbuton under β = 5 s closer to a normal dstrbuton than the one under β = 1 Moreover, we would lke to stress that the computatonal speed n calbraton and prcng s very fast 5, whch mples our formula s so useful n practce especally for hgh dmensonal prcng models Table 17: Thrdorder Approxmate Prces (β = 5, 2M Strke Monte Carlo AE 3rd Dfference Relatve Dfference (% 7% 2% 1% 1% 1% 5 The advantage of our method n computatonal speed s demonstrated for prcng basket optons wth 1 underlyng assets n Subsecton 42 18
20 Table 18: Thrdorder Approxmate Prces (β = 1, 2M Strke Monte Carlo AE 3rd Dfference Relatve Dfference (% 12% 3% 1% 1% 8% Table 19: Thrdorder Approxmate Prces (β = 5, 4M Strke Monte Carlo AE 3rd Dfference Relatve Dfference (% 5% 2% 1% 1% % Table 2: Thrdorder Approxmate Prces (β = 1, 4M Strke Monte Carlo AE 3rd Dfference Relatve Dfference (% 13% 5% 2% % 2% 5 Correlatons n the Currency Opton Market Ths secton nvestgates the correlatons mpled n the currency opton market A correlaton between dfferent currency pars s wdely traded as a correlaton swap n the current Overthecounter(OTC market On the other hand, the correlatons such as between currency pars volatltes are not explctly observable n the market In ths secton, through calbraton to the currency opton market, we extract the mpled correlatons ncludng the drectly unobservable ones n the market In calbraton, we use SABR model wth β = 1 for the dynamcs of each USD quoted currency par and apply the same method as n Secton 43 to obtan that of a cross currency par For emprcal nvestgaton, we use the currency pars, EURUSD, USDJPY and EURJPY for hghly lqud ones and USDKRW, JPYKRW, USDSGD, SGDJPY for relatvely llqud ones Moreover, we take the dates before and after the events such as Lehman Shock and Tohoku Earthquake n Japan In partcular, as for Lehman Shock, we use the data on September 8th and 22nd 28 whle we use the data on March 3rd and 17th 211 for Tohoku Earthquake; For JPYKRW, as the smle data of ts opton market n 28 s not avalable, we mplement the analyss only for the data n 211 On the other hand, as the smle data for the SGDJPY opton market s not 19
21 avalable for the perod around the March 11th, 211 was not avalable, we take the data of March 1st and 29th, 211, nstead In addton, we use the data for currency optons wth maturtes 1M, 2M, 3M and 6M for our analyss Furthermore, although we manly apply SABR model wth ts own volatlty process for each USD quoted par as n (33(34, we also test the model, n whch the volatlty processes for the USD quoted currency pars relevant wth a cross currency par are perfectly correlated as n Double Heston model(eg GautherPossamare (21 In the followng tables, the former model s denoted by (4f whle the latter one s denoted by (3f For (4f, the way of calbraton s exactly the same as (Calbraton Procedure descrbed n Secton 43 As for (3f, whle (Step 1 n the calbraton procedure s the same, the only remanng correlaton used for the calbraton to a cross currency opton market s the one between two relevant USD quoted foregn exchange rates Table 21  Table 23 show the calbraton results for cross currency opton markets, from whch we observe that the calbraton by (4f s much more accurate than that by (3f as expected We also report the results for the calbraton to the 3 month opton markets of the USD quoted currency pars n Table 31  Table 33 of Appendx, whch shows farly good performance n general; the results for the calbraton to the other maturtes optons are qute smlar, and thus they are omtted Table 21: Volatlty Smle (% 28/9/8 28/9/221D 25D ATM 25D 1D 1D 25D ATM 25D 1D EURJPY Calbrated Vol (4f Calbrated Vol (3f Market Vol Dfference (4f Dfference (3f SGDJPY Calbraton Vol (4f Calbrated Vol (3f Market Vol Dfference (4f Dfference (3f Table 22: Volatlty Smle (% 211/3/3 211/3/171D 25D ATM 25D 1D 1D 25D ATM 25D 1D EURJPY Calbraton Vol (4f Calbrated Vol (3f Market Vol Dfference (4f Dfference (3f JPYKRW Calbraton Vol (4f Calbrated Vol (3f Market Vol Dfference (4f Dfference (3f
22 Table 23: Volatlty Smle (% 211/3/1 211/3/291D 25D ATM 25D 1D 1D 25D ATM 25D 1D SGDJPY Calbrated Vol (4f Calbrated Vol (3f Market Vol Dfference (4f Dfference (3f Table 24 and Table 25 show the correlaton senstvtes assocated wth EURJPY and JPYKRW optons as examples, where each correlaton senstvty s defned by the change n the mpled volatlty(% caused by 1 change of a correlaton nsde our model(4f Frstly, we frst observe that the correlaton between two foregn exchange rates has the largest mpact as expected, and the drecton of the mpact s the same for each moneyness On the other hand, the correlatons between an exchange rate and the volatlty of another exchange rate have mpacts manly for the skewness of the mpled volatlty curve Although the correlaton between volatltes has the smallest mpact, t has a reasonable effect on the smle shape of the mpled volatlty curve snce ts (absolute senstvty at ATM s the largest among the dfferent moneynesses Note also that those observatons are unchanged between the dates before and after the events Table 24: Correlaton Senstvtes (EURJPY 211/3/3 211/3/17 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 1M 1D 747% 176% 93% 19% 935% 251% 87% 3% 25D 83% 35% 89% 62% 974% 128% 5% 58% ATM 919% 76% 3% 78% 185% 3% 3% 77% 25D 876% 13% 86% 66% 116% 231% 74% 63% 1D 76% 113% 25% 37% 1141% 47% 137% 33% 2M 1D 767% 187% 18% 17% 618% 273% 43% 7% 25D 823% 41% 96% 73% 782% 15% 86% 63% ATM 911% 78% 28% 92% 126% 68% 6% 9% 25D 889% 145% 12% 77% 117% 262% 65% 71% 1D 744% 139% 23% 4% 11% 397% 22% 27% 3M 1D 757% 26% 11% 17% 662% 285% 59% 14% 25D 831% 38% 11% 83% 795% 12% 96% 71% ATM 945% 93% 4% 16% 121% 8% 6% 13% 25D 912% 162% 114% 88% 1154% 283% 84% 79% 1D 718% 141% 267% 44% 171% 415% 254% 24% 6M 1D 719% 239% 15% 1% 445% 344% 4% 6% 25D 83% 4% 126% 96% 72% 84% 126% 14% ATM 992% 111% 6% 126% 176% 139% 125% 145% 25D 976% 24% 127% 12% 1193% 346% 78% 112% 1D 743% 189% 321% 41% 881% 395% 333% 31% 21
23 Table 25: Correlaton Senstvtes (JPYKRW 211/3/3 211/3/17 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 1M 1D 688% 383% 257% 56% 135% 259% 493% 196% 25D 184% 368% 51% 93% 1164% 192% 237% 39% ATM 142% 248% 79% 15% 1123% 96% 3% 354% 25D 625% 2% 69% 65% 15% 6% 122% 215% 1D 479% 345% 32% 15% 1119% 235% 299% 62% 2M 1D 975% 5% 263% 72% 788% 359% 52% 264% 25D 1162% 48% 78% 113% 1117% 321% 232% 418% ATM 162% 248% 48% 127% 1131% 22% 5% 477% 25D 742% 18% 65% 75% 917% 68% 113% 262% 1D 697% 375% 73% 22% 1111% 4% 25% 11% 3M 1D 183% 56% 27% 74% 1287% 463% 519% 244% 25D 1176% 413% 11% 116% 135% 277% 266% 442% ATM 169% 232% 9% 131% 1181% 93% 57% 512% 25D 815% 38% 6% 73% 117% 114% 19% 259% 1D 843% 393% 12% 29% 1224% 352% 32% 158% 6M 1D 193% 469% 316% 72% 1272% 541% 573% 98% 25D 121% 174% 169% 126% 136% 329% 295% 18% ATM 1128% 67% 44% 144% 1238% 117% 72% 26% 25D 897% 29% 63% 71% 149% 136% 11% 97% 1D 124% 342% 19% 41% 1412% 419% 33% 58% Fnally, the calbrated correlaton parameters for cross currency opton data are gven n the rows, Calbraton(4f and Calbraton(3f of Table 26  Table 3 Also, the calbrated parameters assocated wth opton markets of the USD quoted currency pars are reported n Table 34  Table 36 of Appendx In addton, hstorcally estmated correlatons are shown n Table 26  Table 3 for comparatve purpose In order to estmate hstorcal correlatons, we use the spot prces and ATM optons volatltes, where the reference perod s the same as the correspondng opton s maturty Generally speakng, t s observed that a correlaton between two foregn exchange rates(s 1 S 2 moves smlarly for the mpled and hstorcally estmated ones, whle a correlaton between a exchange rate and another exchange rate s volatlty(s 1 V 2 or V 1  S 2 does not For EURJPY, the mpled correlatons between the foregn exchange rates(s 1 S 2 after Lehman Shock went up; t was consstent wth the hstorcally estmated ones, whch was caused by the rse of both EUR and JPY aganst USD after the shock After Tohoku Earthquake, the mpled and hstorcally estmated correlatons between two foregn exchange rates(s 1 S 2 fell down, whch s consstent to the rse of both EUR and USD aganst JPY after the earthquake On the other hand, the mpled correlatons between two volatltes(v 1 V 2 for the one month(1m rose up to 9%, whle the correspondng hstorcally estmated correlatons fell down: ths dfference may be caused by the serous accdents of the Fukushma nuclear plant rght after the earthquake, whch leads to the substantal rse n the shortterm 22
24 mpled volatlty much qucker than n the hstorcally estmated one For the correlaton between a exchange rate and another exchange rate s volatlty(s 1 V 2 or V 1 S 2, there seems lttle relaton n sze and behavor between the mpled and hstorcally estmated ones As for SGDJPY, wth no data of 1 delta, we have to calbrate four parameters based on three volatlty ponts( for ATM and 25 delta/25delta, whose results are not stable except the correlatons between two exchange rates(s 1 S 2 The correlatons between two foregn exchange rates(s 1 S 2 do not move smlarly for the mpled and hstorcally estmated ones; also, the levels of the correlatons are rather dfferent among the mpled and hstorcal ones Note also that we need to care about the low lqudty of the currency par SGDJPY, especally after bg events, whch makes the opton prces move quckly wth few trades JPYKRW volatlty s data s not avalable for the perod around the Lehman Shock, so that we examne the data n March 211 The mpled correlatons between two exchange rates rose after the earthquake, that s JPY rose up and KRW fell down aganst USD, whch was consstent wth the behavor of the hstorcally estmated ones However, the levels were dfferent among the mpled and hstorcal correlatons Fnally, we remark that for the case of the model (3f, the correlatons between two exchange rates(s 1 S 2 are lower than those for the case of the model (4f Table 26: S 1 = EUR, S 2 = JP Y 28/9/8 28/9/22 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 1M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton
25 Table 27: S 1 = EUR, S 2 = JP Y 211/3/3 211/3/17 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 1M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton Table 28: S 1 = SGD, S 2 = JP Y 28/9/8 28/9/22 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 S 1 S 2 S 1 V 2 V 1 S 2 V 1 V 2 1M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton M Calbraton(4f Calbraton(3f Hstorcal estmaton
An asymptotic FX option formula in the cross currency Libor market model
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