Exotic FX Swap. Analytics. ver 1.0. Exotics Pricing Methodology Trading Credit Risk Pricing

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1 Exoic FX Swap Analyics ver 1. Exoics Pricing Mehodology Trading Credi Risk Pricing

2 Exoic FX Swap Version: ver 1. Deails abou he documen Projec Exoics Pricing Version ver 1. Dae January 24, 22 Auhors Deparmen Mehodology Trading Organizaional Uni Credi Risk Pricing Conac person Ioannis Rigopoulos, Filename ExoicFXSwapAnalyics.ex Disribuion lis Name Posiion / Funcion Deparmen Lis of amendmens Version Dae Reason for and exen of changes Auhors January 22 Original version Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 2 of 29

3 Exoic FX Swap Version: ver 1. Conens Conens 1 A word on Symbols 4 2 Inroducion 8 3 Vanilla CMS: Domesic Index Paid in Domesic currency Special Case: European Swapion Special Case: Caple Special Case: European Opion on a Forward Swap Special Case: European Opion on a Forward Libor Special Case: Libor in Advance Floaing Coupon Special Case: Forward Libor Floaing Coupon Special Case: Exoic Forward Libor Floaing Coupon Special Case: Libor in Arrears Floaing Coupon Special Case: Forward Libor in Arrears Floaing Coupon Special Case: Forward CMS Swap Floaing Coupon wih Cap and Floor Quano CMS: Foreign Index Paid in Domesic currency. 25 Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 3 of 29

4 Exoic FX Swap Version: ver 1. 1 A word on Symbols 1 A word on Symbols The following rules apply in he usage of symbols in his aricle o represen various mahemaical and financial quaniies: 1. A unique leer or combinaion of leers ligaure is reserved for represening a family of quaniies ha are "srongly" associaed o each oher. Ses of subscrips, superscrips and various accens like, ˆ, ec serve o provide addiional clarificaion when needed. For example, he leer R is used o represen various noions of he concep of ineres rae. These include he shor rae i.e. insananeous spo rae, he forward shor rae, he spo and forward simply compounded rae and he spo and forward generic swap rae. Noe ha all hese quaniies are measured in he same uni, namely he 1 ime. The fac is ha hese quaniies are only apparenly differen, since all of hem reduce o special cases of he generic forward swap rae. The laer can be fully represened by: where crv is he curve associaed wih he rae, is he observaion ime of he rae, is he sar of he associaed swap, 1,..., N is he series of he coupon paymen daes wihin he swap and N is he number of he coupons periods of he associaed swap. Obviously his noaion is no paricularly pleasan o he unrained eye, so we could as well wrie: R or R or even R o R crv;, 1,..., N mean he same hing when he res is assumed known from he conex. 2. There are cases where he same symbol may refer o wo quaniies which differ from each oher in a more fundamenal sense han he one described above. Take for example he R. There are hree possible inerpreaions a hand: a R refers o some specific realizaion of he rae a ime. In oher words i is a single number. b R refers o all possible realizaions of he rae a ime. In oher words i is a random variable. c R refers o all possible realizaions of he rae a all possible imes. sochasic process. In oher words i is a Neverheless using he same symbol provides for enhanced readabiliy. The appropriae inerpreaion should be implied by he conex. 3. There migh arise cases where we need o use separae symbols o disinguish beween poenially differen valuaions of he fundamenally same quaniy. For example, le CF be he symbol reserved for represening a generic cash flow amoun paid or received. In paricular we wrie CF for a cash flow occurring a ime. Le s furher suppose we are dealing wih wo separae cash flows, boh occurring a he same ime, e.g. he firs cash flow is paid in he form of a coupon by a reasury bond, he oher is paid in he form of a dividend by a sock. How do we express hem symbolically? There are hree approaches: a By using some accen, i.e. CF for he firs cash flow and CF or CF for he second. This approach is very readable bu only convenien for a small number of differing quaniies. b By appending an ineger index as subscrip, i.e. CF 1; for he firs cash flow and CF 2; for he second. This approach is necessary when dealing wih a large number of quaniies and/or when algebraic manipulaion based on he index is needed for example we could add 1 differen cash flows by wriing: 1 i=1 CF i;. This would no have been possible by eiher of he oher wo approaches. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 4 of 29

5 Exoic FX Swap Version: ver 1. 1 A word on Symbols c By appending some label as subscrip. Le s assume our reasury bond is called xyz and our sock is called is called XYZ. A naural expression would be: CF xyz; for he firs cash flow and CF XYZ; for he second. In he cases where an index or a label is needed, his is added always on he boom righ of he quaniy symbol. This choice is inspired from he convenional usage in he case of ime. Time is denoed by. Differen ime insans are ypically denoed by i. 4. Mos of he quaniies of ineres in Finance are a funcion of one or more ime variables. For example he forward swap rae R crv;, 1,..., N is a funcion of he 2 + N ime variables,, 1,..., N. Neverheless a single ime variable is more imporan because i serves as he ime variable for he associaed sochasic process. In he case of he forward swap rae his is. We follow he convenion o place his ime variable a he boom righ as a subscrip. So only a couning index/label or a ime variable will be ever placed a he boom righ. Everyhing else goes o he op. The following able liss all he symbols used in his aricle. In he lef column several versions of each symbol are presened, according o he discussion above. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 5 of 29

6 Exoic FX Swap Version: ver 1. 1 A word on Symbols Symbol CP or CP i or CP R i N or N i or N R i m or m i or m R i s or s i or s R i I or I i or I R i τ or τ i or τ R i T or T i or T R i F or F Q or F Q i C or C Q or C Q i Inerpreaion Table 1: Table of symbols i h coupon received on a paricular sream of cash flows associaed wih R Noional applied in he calculaion of he i h coupon associaed wih R Muliplier applied in he calculaion of he i h coupon associaed wih R Spread applied in he calculaion of he i h coupon associaed wih R Index applied in he calculaion of he i h coupon associaed wih R Accrual inerval applied in he calculaion of he i h coupon associaed wih R Paymen ime of he i h coupon associaed wih R Floor applied on he sochasic quaniy Q. i is couning index or idenifying label Cap applied on he sochasic quaniy Q. i is couning index or idenifying label r or r or r or Forward libor rae a for a libor saring a, maured a 1 r, 1 or r crv;, 1 and being associaed wih he curve crv R or R or R or Forward swap rae a for a swap saring a, having coupon daes 1,..., N R, 1,..., N or and being associaed wih he curve crv R crv;, 1,..., N S or S or S ccy1 ccy2 S ; N or N R or i D or D or D α or R or R; Sar or Forward FX rae a wih mauriy a associaed wih he FX Number of coupons associaed wih R Time. i is couning index or idenifying label ccy1 ccy2 Day-coun fracion of he ime inerval. α indicaes he day-coun convenion used Se ime of he forward swap rae R ime of he forward swap rae R i or R;i, i 1 Paymen ime of he i h coupon associaed wih he forward swap rae R T or T S T or T S; Mauriy ρ or ρ or ρ α,β Se ime of he forward FX rae S of he forward FX rae S Correlaion a beween α and β Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 6 of 29

7 Exoic FX Swap Version: ver 1. 1 A word on Symbols CF or CF or CF i; V or V or V α Cash Flow occurring a ime. i is couning index or idenifying label Value a ime of some raded asse. α indicaes he raded asse V or V 1 CF 2 Value a ime 1 of he cash flow CF 2 wih 1 2 P or P or or P crv;t Value a ime of a riskless bond wih noional = 1 mauring a ime T wih T. crv is he indicaor of he curve associaed wih he bond P or P or 1 or 1,T 2 or P crv1;t 1,crv2;T 2 N or N or N i; Q or Q N Value a ime of he raio beween wo riskless bonds T 1, crv1 and T 2, crv2 Numeraire a ime. i is economy indicaor Measure w.r.. numeraire indicaor N E or E Q X Expecaion of he random variable X condiional on he maximal available informaion as of ime w.r.. measure Q PmCcy i RaCcy i RepCcy i ccy1 ccy2 F or F or F α Paymen currency. i is couning index or idenifying label Rae currency. i is couning index or idenifying label Repor currency. i is couning index or idenifying label Defines he ype of he exchange rae beween wo currencies. The ccy1 ccy2 associaed spo FX rae S ; will equal he number of unis of currency ccy1 needed o buy 1 uni of currency ccy2 a ime Informaion i.e. σ algebra available by he processes indicaed by α a ime C or C α MAXX, Y MINX, Y N or Nx σ or σ or σ X Correcion facor applied on he random variable indicaed by α Maximum of X and Y Minimum of X and Y Sandard normal cumulaive disribuion Percenage volailiy of process X a ime µ or µ or µ X Percenage drif of process X a ime w or w or w i Value of he i h orhogonal componen of a sandard vecor Wiener process a ime ξ or ξ or ξ X M or M or M i; F i; or G i; Condiional expecaion of Radon-Nikodym derivaive indicaed by X a ime Maringale process. i is couning index or idenifying label Io process. i is couning index or idenifying label Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 7 of 29

8 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. 2 Inroducion This documen is concerned wih he analyical pricing of a swap consised of wo legs. Each leg describes a series of coupon paymens CP i, i = 1,..., n. Each CP i is paid a ime T i in some currency referred o as Paymen Currency, and is given by: where CP i = N i m i I i + s i τ i 2.1 N i = The noional a he sar of he i h accrual period in unis of he Paymen Currency. m i = A consan possibly ime-dependen called muliplier. I i = The index applicable for he i h coupon. s i = A consan possibly ime-dependen called spread. τ i = The lengh of he i h accrual period in years, according o he respecive daycoun convenion. Addiionally he index I i can be resriced o vary in he inerval [ F I i, C I i ], where F I i C I i is some period dependen Index Floor level is some period dependen Index Cap level We se = for oday s ime o simplify he noaion. Le he value oday of a cash flow CF occurring a any ime be V CF. Then due o he lineariy of he value operaor V and since V s i = i s i we conclude: and V SwapLeg = V CP i = N i m i V I i + i s i τ i 2.2 N SwapLeg i=1 V CP i = N SwapLeg i=1 N i m i V I i + i s i τ i 2.3 where N SwapLeg is he number of coupons in he swap leg SwapLeg and i is he price oday of a riskless bond on a 1 currency uni noional having mauriy T i. The challenge in he above formulas is o calculae V I i. The purpose of his documen is o derive closed form formulas for his quaniy, for differen definiions of I i. In he following secion we sar wih he simples case where he index is jus some forward swap rae on he same currency as he paymen currency and we derive he formula for he respecive convexiy correcion. Each addiional secion will exend his basic seup by adding complexiy o he definiion of he index. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. This simples case covers he so called CMS swaps. In a ypical CMS swap he coupon of he floaing leg is given by 2.1 where he index I i is jus a generally forward swap rae ha ses a R wih accrual sar a and remaining coupon paymen imes given by 1,..., N, where N is he number of he coupons. For example, N = 2 for a 1-year semi-annual CMS rae. Using full noaion, we could have wrien R, 1,..., N for his rae, bu we prefer he simpler noaion since he conex here is no R ambiguous. Replacing for he index I R i in 2.1 and 2.2 we ge: Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 8 of 29

9 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. where we dropped he coupon index i for simpliciy. The amoun CP is paid ou a ime T as shown in figure 1. CP = N + s τ 3.1 mr V CP = N mv + s τ 3.2 R T 1 N Valuaion Dae CMS Se Paymen CMS Sar CMS Firs Coupon CMS Mauriy Fig. 1: Relevan daes for a coupon when he index is a CMS rae In order o calculae V we choose he bond P R T maured a T as numeraire. Le Q P he associaed measure. We know from he general heory[3, 4] ha he assumpion of no arbirage opporuniies implies he value of each raded asse divided by his numeraire is a maringale w.r.. Q P. We also assume he marke is complee here is some asse ha replicaes any given final cash flow here is some asse whose value a T equals. V R is nohing else bu he value of his asse as of. Wriing V R for he value of his asse a any ime we herefore have: V = V R and V T = R Since V is a maringale in Q P we ge: V = E QP VT T where E Q X is he expecaion w.r.. Q of he random variable X condiional on he informaion available a ime. Using he wo previous equaions and he fac T = 1 we ge: V = P R T E QP R 3.3 The placemen order for he paymen ime shown here represens jus one of he possibiliies. In general i may occur a any ime afer. The res of he imes should be ordered as shown. he concep of asse encompasses self-financing rading sraegies as well Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 9 of 29

10 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. I seems we only need o calculae E QP. Wih a bi of foresigh hough, we would raher find he R whole disribuion funcion of noe ha is a random variable adoped o he informaion. This R R F is because laer we will deal wih caps and floors on he rae = we will have o calculae expressions R like E QP MAX, K, where K can be any number. Wih some more foresigh, we would beer op for R he whole hing, i.e. he diffusion process of R,. This ulimae knowledge will enable us laer o calculae more exoic payoffs based on R, like knock-ou barriers. Also noe ha knowing he disribuion of R for each is no enough. There are many differen diffusions which lead o he same uncondiional disribuions. Our aim is herefore o derive he Sochasic Differenial Equaion SDE of R, in he measure Q P. Here is he resul: Resul 3.1 [ SDE of F orward Swap Rae R in Q P ] 1 R dr d = µr d + σ R dw w.r.. Q P for 3.4 where µ R = σ R σ R + σ P σ r, N 3.5 σ R, σ r, N and σ P are all deerminisic funcions of ime and σ P is approximaed by he expression 1 P N σ rt, N if T N σ P 1 σ r N,T if N T P N R is a brief noaion of R, 1,..., N which is he forward swap rae as seen a ime when he underlying swap sars a ime and pays coupons a imes 1,..., N. The order holds: < 1 <... < N. σ R is he percenage volailiy of he forward swap rae process R. I can be inferred from he swapion marke. a σ r 1, 2 is he percenage volailiy of he forward libor rae process r 1, 2. I can be inferred from he cap marke. b σ P is he percenage volailiy of he bond raio process P N Q P is he equivalen maringale measure associaed wih he numeraire being he bond price mauring a he coupon paymen ime T. Also noe he assumpions 3.2, 3.3, 3.4 and 3.5 apply. a Secion 3.1 b Secion As a firs applicaion of he resul 3.1 we can calculae explicily he expecaion in 3.3: Group Risk Conrol ExoicFXSwapAnalyics.ex V R = R e Exoics Pricing Dae: January 24, 22 Page 1 of 29 µr u du 3.7

11 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. This in urn can be se in 3.2 o ge he price oday of he whole coupon. By comparing 3.7 wih he well known resul V = r in he special case when is an amoun r equal o he spo a libor rae from o T paid a T i.e. when R,T r we mean he forward rae r,t, we derive he following resul: r r,t,and where by r Resul 3.2 [ CMS Convexiy Correcion for Cash F lows linear in R ] When he coupon paymen CP a ime T is linear on R, 1,... N, R T and < 1 <... < N, i.e. if CP = ar + b, wih a, b consans, hen is value oday is: V CP = ar C R + b where C R is some correcion facor, ypically called he convexiy correcion facor, given by: 3.8 C R = e µr u du 3.9 where µ R is given by 3.5. We remind is oday s price of he riskless bond wih mauriy T and R R, 1,... N is he forward swap rae as seen from oday. Finally we may wan o value some generic possibly non-linear in R cash-flow CF T paid ou a ime T. The following resul holds: Resul 3.3 [ V aluaion Expression for Cash F lows which are funcions of R ] Le CF T some cash-flow paid a ime T, which is a possibly non-linear funcion of R R, 1,... N, T and < 1 <... < N, i.e. CF T =. Then is value oday is given def fr by: V CFT = P T [ E Q P fr ] 3.1 where is he price a of he riskless bond wih mauriy T and Q P is he equivalen maringale measure associaed wih. The expecaion can be in principle calculaed by making use of he SDE of R, w.r.. Q P according o resul 3.1. Proof of resul 3.1 By he definiion of R i follows ha R can be wrien in erms of bond prices as: R = P P N i=1 τ i P i where i, i =,..., N, are as shown in figure 1. and τ i refers o he daycoun fracion associaed wih he accrual inerval i 1, i. he symbol r is specially reserved for libor raes. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 11 of 29

12 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. Consider he measure Q R associaed wih he numeraire i=1 τ i P i. The expression above implies R is maringale w.r.. Q R. Furher on, we need o make he following assumpion: Assumpion 3.1 [ P osiive Raes M odel ] The bond prices are posiive and monoonically decreasing wih mauriy, i.e. P 1 > P 2 for all 1 < 2 This resuls in R > which in urn implies R is an exponenial maringale[6, Page 73] and herefore obeys he following SDE in Q R : Here w is he sandard Wiener process 1 R dr d = σr dw w.r.. Q R for 3.11 and σ R is some sochasic process sufficienly consrained so ha R is indeed a maringale. A sufficien condiion is he so-called Novikov condiion:[5, Theorem 6.1] E QR e 1 2 σ R 2 d < Below we will resric our sudy o he so called lognormal model, according o which he swap rae R for fixed is lognormally disribued. This is equivalen o imposing he addiional assumpion below: Assumpion 3.2 [ Lognormal Swap Rae Model ] The percenage volailiy σ R of he forward swap rae R is a deerminisic bounded funcion of ime. Obviously he Novikov condiion is hen saisfied = he rae R is well defined as an exponenial maringale. The nex sep is o use Girsanov s heorem[2] which in his case implies ha in he measure Q P he process R will sill keep he same percenage volailiy σ R bu will acquire some drif µ R iself an Io process. So we may wrie for he SDE of R in Q P : 1 R dr d = µr d + σ R dw w.r.. Q P for 3.12 Laer we will show ha he deerminisic funcion of ime σ R can be inferred from he marke prices of european swapions. So we only need o calculae he drif µ R. One way o proceed is by calculaing he condiional expecaion of R in Q P by means of he change of measure formula: E QP R = 1 ξ E QR R ξ for 3.13 where ξ = E QR dq P and dqp is he Radon-Nikodym derivaive of Q P w.r.. Q R. dq R dq R I is a well known fac for example [1, Page 191] ha ξ equals he raio of he respecive normalized numeraires. The normalized numeraire a ime associaed wih Q P is and he one associaed wih Q R is i=1 τ i P i i=1 τ i P i. So we have: as shown in [6] he Wiener process and he associaed volailiy are generally k-dimensional = he expression σ R dw is acually a shorhand of k i=1 σi dw i. Bu we may replace k i=1 σi dw i wih σ eff dw eff where σ eff and w eff are boh 1-dimensional. To achieve compleeness we need of course o resric he associaed filraion o he one generaed by w eff which is smaller han he original produc filraion generaed by w i, i = 1,..., k Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 12 of 29

13 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. ξ = i=1 τ i P i i=1 τ i P i = i=1 τ i P i i=1 τ i P i When we replace ξ in 3.13 wih he expression above, he facor E QP i=1 R = τ i P i E QR R i=1 τ i P i i=1 τ i P i = 1 M E QR drops ou and we ge: R M 3.14 where in he las sep we replaced he fracion ouside and inside he expecaion by M and M respecively, whose process is defined by: Now, because of 3.12 we have: M = def T P i=1 τ i P i On he oher hand, he posiiviy of R M implies: E QP R = R e µ u du 3.15 E QR R M = R M e µrm u du 3.16 where µ RM is he percenage volailiy of R M. The assumpion 3.3 implies: µ RM = σ R σ M 3.17 where σ M is he percenage volailiy of M which since i is a posiive maringale w.r.. Q R, i will saisfy: 1 dm = σ M d w.r.. Q R for M d 3.18 Now we replace he expecaions in 3.14 from 3.15 and 3.16 o ge: R e µ u du = R e µrm u du µ R u du = and since is arbirary we conclude µ R = µ RM and by using 3.17 : µ RM u du In order o find σ M we rewrie M as: µ R = σ R σ M 3.19 Group Risk Conrol ExoicFXSwapAnalyics.ex M = i=1 τ i P i = i=1 τ i P i P P N P N P Exoics Pricing Dae: January 24, 22 Page 13 of 29 = R P P N = R F 3.2

14 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. where F = def P T P P N is some posiive Io process. In order o simplify he subsequen formulas we will make he following assumpion: Assumpion 3.3 [ One F acor Model ] All bond prices are driven by he same facor. In oher words all yield curve quaniies including swap raes and bond prices are perfecly correlaed. Then we can wrie he SDE of F as follows: 1 F df d = µf d + σ F dw w.r.. Q R for 3.21 where dw is he same as he one appearing in he SDE for R in 3.11, because of he assumpion 3.3. Now from 3.11,3.18,3.2 and 3.21 we conclude: σ M = σ R + σ F 3.22 I remains o calculae σ F. We rewrie firs F in erms of he forward libor rae r, N, which is he libor rae associaed wih he period [, N ]. For noaional simpliciy we se r = r, N below. By he definiion of he simply compounded forward libor rae we have: Plugging his in he definiion of F we ge: r r, N = def P P N N P N F = 1 1 N r P N 3.23 Now we define he posiive Io process, N or simply P by: P, N = T P def P N 3.24 Noe ha P is jus a forward discoun facor if T > T M. Oherwise i equals he inverse of a discoun facor. We can now wrie 3.23 in erms of P : F = 1 1 N P r 3.25 Based on our assumpions so far, r and P are perfecly correlaed posiive Io processes. Le σ r, N σ P he corresponding percenage volailiies. Then 3.25 implies: and Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 14 of 29

15 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. σ F = σ P σ r, N 3.26 I remains o calculae σ P. We consider 3 cases: Case 1 T < N Then he forward libor rae r T, N can be defined as: r T, N = 1 N T P N 1 = = P = 1 + N T r T, N 1 N T P 1 We know he percenage vol of N T r T, N is σ rt, N, i.e. he percenage volailiy of he forward libor rae r T, N. Wih some simple algebra i urns ou ha he percenage volailiy σ P of P is given by: Case 2 T = N Then obviously σ P =. Case 3 T > N σ P = N T r T, N 1 + N T r T, N σ rt, N = Then we wrie P in erms of he forward libor rae r N,T : P = [ 1 + T N r N,T 1 P N ] 1, N rt σ The percenage volailiy of r N,T is σ r N T. Wih a bi of algebra we find: σ P = 1 + T N r N,T T N r N,T σ r N,T = 1 P N σ r N,T Recapiulaing, we have: 1 P N σ rt, N if T N σ P = 1 σ r N,T if N T P N 3.27 By combining 3.19,3.22,3.26,3.27 we ge for he drif of he forward swap rae in Q P : We use he fac ha for a posiive process F : df d1+f F = µd + σdw =...d + F 1+F 1+F σdw We use he fac ha for a posiive process F : df F = µd + σdw d[1+f 1 ] 1+F 1 =...d F 1+F σdw Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 15 of 29

16 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. µ R = σ R σ R + σ P σ r, N The equaion above is exac bu no very pracical because boh σ P and σ r, N are generally non-deerminisic sochasic processes. To proceed we need o make he following wo simplifying assumpions: Assumpion 3.4 [ Lognormal Libor Raes Model ] The percenage volailiies σ r, N and σ rmint, N,MAXT, N of he forward libor raes r, N and r MINT, N,MAXT, N respecively, are deerminisic bounded funcions of ime. Assumpion 3.5 [ Lognormal Bond Raio Model ] The percenage volailiy σ P of he bond raio process, N is deerminisic bounded funcion of ime. One migh hink ha we canno have boh swap raes and libor raes lognormally disribued. Noe hough ha he libor raes involved in assumpion 3.4 are no he same wih hose used in he definiion of he swap rae. Therefore he assumpion 3.4 does no necessarily violae he no arbirage condiion. Clearly hough, assumpion 3.5 is no compaible wih assumpion 3.4, due o he relaion 3.27 In order o saisfy assumpion 3.5, we could in some approximaing sense replace he fracion of bond prices in 3.27 wih is iniial value. We do no need o change he libor rae volailiies since hese are already deerminisic bu possibly ime dependen from assumpion 3.4: 1 P N σ rt, N if T N σ P 1 σ r N,T if N T P N 3.1 Special Case: European Swapion Assume he special case when he fixing ime, he sar ime and he paymen ime T all coincide, i.e. = = T. Le furher he amoun paid a N be given by i=1 τ i P i MAX R K, where K some consan. Noe his is no of he form 3.1 and paricularly is no linear in. Insead, i can be verified his R is he payou a he expiry ime of a uni noional european swapion wih srike a K, when he underlying swap is he one associaed wih he swap rae R being se a ime. Ineresingly enough, moving o measure Q P and making use of resul 3.3, is no helpful here because he payou canno be wrien as a funcion of R. The way o value he european swapion is by choosing he posiive process i=1 τ i P i, as numeraire. Le Q R he associaed equivalen maringale measure. Le V he value of his swapion a any earlier ime. Then V divided by he numeraire is maringale w.r.. Q R V = i=1 τ i P i E QR MAX R K, The SDE of R in Q R given in 3.11 leads o he immediae calculaion of he expecaion: where E QR MAX R K, = R Nd 1 KNd Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 16 of 29

17 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. d 1 = R ln K σr σ R d 2 = d 1 σ R N = Sandard normal cumulaive disribuion funcion σ R = σr u du 3.29 and R sands for he forward swap rae R, 1,..., N as seen a ime, i.e. oday. We ge herefore he final resul: V = The expression i=1 τ i P i is also called he spo dv1 of he swap. N i=1 τ i P i [R Nd 1 KNd 2 ] can be solved for σ R when he price V is known from he marke. In his conex, he σ R is called he implied Black swapion volailiy. For he purpose of exoics pricing we would raher know σu R for all imes u no jus is ime average up o some fixed ime. We will see in secion 3.3 ha a european opion wih expiry on a forward swap saring a he laer ime, provides informaion on he ime average volailiy up o ime. If we knew he σ R for all possible expiries such ha for fixed, we could inver 3.29 and solve for σu R, u, as a funcion of ime. Alernaively, if we canno find liquid opions on forward swaps, we may simply use he volailiy σ R of he vanilla european swapion i.e. opion on a spo swap expiring a and assume σu R is consan, i.e. σu R = σ R for all u such ha u. 3.2 Special Case: Caple We assume he special case when he fixing ime and he sar ime coincide. Le he rae R fixed a ime, consised from he single period [, 1 ], < 1, i.e. N = 1. Then we are dealing wih a spo w.r.. fixing ime libor rae, so le denoe i by r r, 1 Le also he cash-flow. τ MAX r K, paid a 1 i.e. we assume he paymen ime T and firs coupon ime 1 coincide as well. This is he payou of a uni noional caple on wih srike a K. Since his r can be wrien as a funcion of, we can value i easily in he QP measure by making use of he resul r 3.3. In his special case Q R = Q P, since he respecive numeraires are he same apar from a consan muliplier he SDE 3.11 applies in Q P as well. Now direc applicaion of 4.8 gives: V V τ MAX r K, = P 1 [ E Q P ] τ MAX r K, = P 1 τ [ E Q P ] MAX r K, As discussed, he SDE 3.11 applies, which leads o he final resul: Assuming he swapion formula 3.3 known, we could simply apply i here, since a caple is nohing bu a swapion wih he underlying swap consised of a single period. Neverheless we provide here a derivaion independen of he swapion resul. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 17 of 29

18 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. V = P 1 τ [r Nd 1 KNd 2 ] 3.31 where d 1 = ln r K σr σ r d 2 = d 1 σ r σ r = σr udu 3.32 and r sands for he forward libor rae r, 1 as seen a ime, i.e. oday can be solved for σ r when he price V is known from he marke. In his conex, he σ r is called he implied Black caple volailiy. Like in he swapion case discussed above, for he purpose of exoics pricing we would raher know σ r u for all imes u no jus is ime average up o some fixed ime. We will see in secion 3.4 ha a european opion wih expiry on a forward libor saring a he laer ime, provides informaion on he ime average volailiy up o ime. If we knew he σ r for all possible expiries such ha for fixed, we could inver 3.32 and solve for σ r u, u, as a funcion of ime. Alernaively, if we canno find liquid opions on forward libor, we may simply use he volailiy σ r of he vanilla caple i.e. opion on a spo libor expiring a and assume σ r u is consan, i.e. σ r u = σ r for all u such ha u. 3.3 Special Case: European Opion on a Forward Swap This is an exension of he european swapion of secion 3.1. There he underlying a opion expiry was a spo swap. Here he underlying is a forward swap. Mahemaically we express his fac by allowing he fixing ime being earlier han he sar ime, i.e.. In he limi when =, we recover he european swapion case. Like in secion 3.1, we assume he payou is received a fixing ime, i.e. T =. One can verify he N appropriae amoun is i=1 τ i P i MAX K,, where is he forward swap rae fixed a ime R R, i.e. R, 1,..., N. R We value his insrumen by he same procedure followed in secion 3.1. We choose he numeraire i=1 τ i P i, wih he associaed measure Q R. Then he maringale propery implies: V = MAX K, R i=1 τ i P i E QR The SDE of R in Q R given in 3.11 leads o he immediae calculaion of he expecaion: where E QR MAX K, = R R Nd 1 KNd Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 18 of 29

19 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. d 1 = R ln K σr σ R d 2 = d 1 σ R σ R = σr u du 3.34 and R sands for he forward swap rae R, 1,..., N as seen a ime, i.e. oday. We ge herefore he final resul: V = N i=1 τ i P i [R Nd 1 KNd 2 ] 3.35 This is acually he same formula wih 3.3 excep from in he hree equaions 3.29 being replaced wih in he hree equaions The pracical resul is ha for a fixed underlying swap saring a, he opion on his swap expiring a i.e. he vanilla swapion requires knowledge of he ime average volailiy of σu R up o ime, whereas he opion on he forward of his swap expiring a requires knowledge of he ime average volailiy of σu R up o ime. This jusifies he claim we made in secion 3.1 regarding he recovering of he ime dependen funcion σu R from he opion on forward swap marke daa. 3.4 Special Case: European Opion on a Forward Libor This can be considered as an exension of he vanilla caple, where he underlying fixed a expiry is some forward libor rae insead of he spo libor rae. Like before we denoe by he expiry of he opion, which is he same wih he fixing ime of he forward libor running from o 1. The payou occurs a ime 1 i.e. T = 1 and concerns he amoun τ MAX K,. We may value his insrumen like in secion 3.2, r by choosing as numeraire he bond P 1 and applying he maringale relaionship. Alernaively, we may observe his is a special case of he european opion on a forward swap in he limi when he swap reduces o a single period. Then we can apply 3.35 direcly o ge: where V = P 1 τ [R Nd 1 KNd 2 ] 3.36 d 1 = ln r K σr σ r d 2 = d 1 σ r σ r = σr udu 3.37 and r sands for he forward libor rae r, 1 as seen a ime, i.e. oday. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 19 of 29

20 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. This is acually he same formula wih 3.31 excep from in he hree equaions 3.32 being replaced wih in he hree equaions The pracical resul is ha for a fixed underlying libor saring a, he opion on his libor expiring a i.e. he caple requires knowledge of he ime average volailiy of σ r u up o ime, whereas he opion on he forward of his libor expiring a requires knowledge of he ime average volailiy of σ r u up o ime. This jusifies he claim we made in secion 3.2 regarding he recovering of he ime dependen funcion σ r u from he opion on forward libor marke daa. 3.5 Special Case: Libor in Advance Floaing Coupon We consider here he valuaion of a coupon on he floaing leg of a uni noional vanilla swap, where he index is he spo libor r r, 1, < 1 where, 1 is he defining period of he libor rae. We also assume he paymen follows exacly a he end of his period, i.e. T = 1. Noe we do no impose any requiremen on he coupon accrual period which deermines τ. Also since he rae is jus a libor rae, N = 1. Finally he amoun paid a 1 is τ r. We value his coupon by using resul 3.3 direcly. Le V is value oday. Then 4.8 implies: V = P 1 [ E Q P ] τ r = P 1 τ [ E Q P r ] bu since r, is a maringale w.r.. Q P : V = P 1 τ r 3.38 where of course r r, 1 is he forward libor rae for he inerval, 1 as seen from oday implies we can price his simple case of he vanilla coupon where he index is he spo libor and he coupon is paid a he end of he libor period by replacing he floaing amoun r wih he consan r and hen rea i as a fixed paymen paid a 1, i.e. muliply i wih P 1. We will see in secion 3.6 we can sill apply he same prescripion even when we replace he spo libor wih he forward libor in he definiion of he index, o he exen he coupon is sill paid a he end of he libor period 1. Unforunaely his logic breaks down when he coupon is no paid exacly a he end of he libor period or when he index is no a forward on some libor rae. The laer case has been already reaed in resul 3.2, where we saw we may sill apply he prescripion menioned here bu wih he precauion ha he forward swap rae R needs o be muliplied wih he convexiy correcion facor C R. The former case is reaed in secion. 3.6 Special Case: Forward Libor Floaing Coupon We consider here he valuaion of a coupon on he floaing leg of a uni noional swap, where he index is a forward libor r, 1, r < 1 where, 1 is he defining period of he underlying libor rae. We also assume he paymen follows exacly a he end of his period, i.e. T = 1. Noe we do no impose any requiremen on he coupon accrual period which deermines τ. Since he rae is jus a libor rae, N = 1. The amoun paid a 1 is τ. r We value his coupon by using resul 3.3 direcly. Le V is value oday. Then 4.8 implies: spo wih regard o he fixing ime Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 2 of 29

21 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. V = P 1 [ E Q P bu since r, is a maringale w.r.. Q P : ] τ = P 1 r τ [ E Q P r ] V = P 1 τ r 3.39 where of course r r, 1 is he forward libor rae for he inerval, 1 as seen from oday implies we can price his variaion of he vanilla coupon where he index is he forward libor and he coupon is paid a he end of he libor period by replacing he floaing amoun r wih he consan r and hen rea i as a fixed paymen paid a 1, i.e. muliply i wih P 1. As we have already menioned in he commen of secion 3.5, his is clearly he same prescripion wih he one we used in ha secion for he vanilla case. 3.7 Special Case: Exoic Forward Libor Floaing Coupon This is an exension of he insrumen reaed in secion 3.6. The aribue exoic refers o allowing he coupon paymen imet occur a any ime on or afer he fixing ime. In secions 3.5 and 3.6 insead, he resricion T = 1 was imposed. The only resricion here compared o he mos general linear payou case reaed in resul 3.2 is ha he index consiss of a single period, i.e. N = 1. So le r, 1 be he forward libor rae fixed a ime r and le τ be he amoun ha is paid a T wih r < 1 and T. Le V he value of his amoun oday. We may apply resul 3.2 o ge a = τ and b = : where he convexiy correcion facor C r is given by: V = τ r C r 3.4 and µ r is given by 3.5, ha is in our case: C r = e µr udu µ r = σ r σ r + σ P σ r, 1 = σ r σ P since r sands for r, 1 in he case we examine here. So our final resul for C r is: C r = e σr σ P du 3.41 Noe ha σ P is approximaed according o 3.6 by he expression: Group Risk Conrol ExoicFXSwapAnalyics.ex 1 P 1 σ rt, 1 if T 1 σ P 1 σ r 1,T if 1 T P 1 Exoics Pricing Dae: January 24, 22 Page 21 of

22 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. Noe ha σ P = for T = 1, which implies C r = 1 in 3.41, as expeced from our discussion in secion is only useful if we know he exac ime dependence of σ r and σ rmin T, 1,MAX T, 1. More ofen han no, we only know he ime averages σ r and σ rmint, 1,MAXT, 1 of hese wo quaniies from he cap marke. Then we could approximae 3.41 by: C r e σr σ P 3.43 where 1 P 1 σ σ P rt, 1 if T 1 1 σ r 1,T if 1 T P Special Case: Libor in Arrears Floaing Coupon Here he index is he spo libor rae r r, 1 fixed a he same ime when he coupon is paid, i.e. = = T and < 1. This represens a subcase of he insrumen sudied in secion 3.7 we may apply 3.4 o ge: V = P τ r C r 3.45 where r r, 1 is he forward libor rae for he period, 1 as seen from oday. The convexiy correcion facor is approximaed as C r e σr σ P from Furher on, σ P is given by 3.44 where we noe ha T = and herefore he upper expression applies. We also observe ha σ rt, 1 = σ r, 1 = σ r. Therefore we have: which leads o he final resul: σ P 1 P 1 P σ r C r e 1 P 1 P σ r 2 = e τ r 1+ τ r σ r where in he las sep we replaced P 1 Also τ D 1 P by using P 1 P = 1 1+D 1 r is he day-coun fracion of he inerval, 1 according o he day-coun convenion used in he paricular definiion of he single compounded rae r. As usually, r r, 1 is he forward libor rae for he period, 1 as seen from oday. The same approximaing resul for libor in arrears swaps 3.46 can be found in he finance lieraure, for example in [7, Page 126], where hough, he derivaion appears o be quie complex hrough muliple applicaions of Io s lemma. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 22 of 29

23 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. 3.9 Special Case: Forward Libor in Arrears Floaing Coupon Here he index is he forward libor rae r, 1 fixed a some ime r. We demand ha he libor period sar a he same ime when he coupon is paid, i.e. = T and < 1. This represens an exension of he libor in arrears case sudied in secion 3.8. Basically he libor period, 1 says he same, bu he coupon is now based on he fixing of he forward libor rae a he earlier ime. I s ineresing o see wha effec will his variaion have in he convexiy facor. We noe his sill represens a subcase of he more general insrumen sudied in secion 3.7 we may apply 3.4 o ge: V = P τ r C r 3.47 where r r, 1 is he forward libor rae for he period, 1 as seen from oday. The convexiy correcion facor is approximaed as C r e σr σ P from Furher on, σ P is given by 3.44 where we noe ha T = and herefore he upper expression applies. We also observe ha σ rt, 1 = σ r, 1 = σ r. Therefore we have: which leads o he final resul: σ P 1 P 1 P σ r C r e 1 P 1 P σ r 2 = e τ r 1+ τ r σ r where in he las sep we replaced P 1 P by using P 1 P = 1 1+D 1 r Again, r r, 1 is he forward libor rae for he period, 1 as seen from oday. We observe ha he only difference beween 3.46 and 3.48 is he in 3.46 is replaced by in I s ineresing o check he limi of 3.48 when we le. We ge C r 1 which is wha we would expec, since in his case he index becomes deerminisic from oday s scope. In he limi where we recover 3.46 as we should. 3.1 Special Case: Forward CMS Swap Floaing Coupon wih Cap and Floor Here he index is some forward swap rae R, 1,..., N fixed a some ime R and resriced o lie in he inerval [F, C], where F and C are consans called Floor and Cap respecively. The coupon is paid a ime T wih he only consrain being T. Formally, he index I R in his case is no jus bu R raher is given by he expression I R = MAX MIN, C, F. The amoun paid ou is τ I R R. We assume for simpliciy ha he muliplier m = 1 and he spread s =. Oherwise he full coupon paid on a uni noional would be: τ mi R + s. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 23 of 29

24 Exoic FX Swap Version: ver 1. 3 Vanilla CMS: Domesic Index Paid in Domesic currency. We may rewrie he index in he form F + MAX F, MAX C,, which implies we are acually R R dealing wih an exoic caple spread porfolio of long, srike F, exoic caple and a shor, srike C, exoic caple plus a fixed paymen of F. By exoic caple we mean he caple-like insrumen where he index is a forward swap rae insead of a spo libor or where he index is a forward libor bu he paymen may occur a a ime oher han he end of he rae period. Le V he value of his cash-flow oday. We may calculae V by applying resul 3.3: V = [ E Q P ] [ τ I R = τ F + E QP MAX F, R E QP ] MAX C, R We can easily evaluae he wo expecaions by making use of he SDE 3.4, which implies a lognormal disribuion for he random variable R. The resul is: E QP E QP MAX MAX F, R C, R = R e µr N = R e µr N d F 1 d C 1 FN CN d F 2 d C 2 where H below is any posiive consan d H 1 = R ln H + µ R σr σ R 3.49 d H 2 = d H 1 σ R 3.5 σ R = µ R = σr u du µr u du As usually R sands for he forward swap rae R,,..., N as seen a ime, i.e. oday. µ R is given by 3.5 of resul 3.1. We presen below he final resul: V = {F [ ] } τ + R e µr N d F 1 N d C 1 FN d F 2 + CN d C where d F 1, df 2, dc 1, dc 2 are given by 3.49 and 3.5 and µ R is given by Due o he imporance and he wide applicabiliy of he above resul, we presen below he corresponding expression when he coupon involves a muliplier m and a spread s as well. More precisely, V m,s below is oday s price of he cash-flow τ mi R + s a ime T : V m,s = {F [ ] } τ m + R e µr N d F 1 N d C 1 FN d F 2 + CN d C 2 + τ s 3.54 Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 24 of 29

25 Exoic FX Swap Version: ver 1. 4 Quano CMS: Foreign Index Paid in Domesic currency. The valuaion formula 3.53 has he desired convergence behavior: Namely, F C V τ F, as expeced since in his case he index will end o equal F, i.e. in he limi he coupon paymen a T will be he fixed amoun τ F. Also, if we remove he barriers, i.e. we le F and C, we ge V τ R e µr, which clearly agrees wih he expression 3.8 of resul 3.2, which applies on cash-flows linear in R. 4 Quano CMS: Foreign Index Paid in Domesic currency. We urn now our aenion o he more complex case where he index I is a forward swap rae bu wih respec o he risk free yield curve of some currency RaCcy referred o as Rae Currency possibly differen from he currency PmCcy referred o as Paymen Currency in which he paymen akes place. The ime diagram of figure 1 applies here as i is. We will use he ilde symbol o refer o quaniies relaed wih he rae currency economy. So we denoe he rae currency forward swap rae as R, 1,..., N, or briefly when he conex is clear as R. Like in secion 3 we reserve a special noaion for rae currency libor raes: r, 1 or briefly r. Noe also he same noaional discipline applies for he symbols,, 1,..., N, since hey are also quaniies relaed wih he rae currency. In conras we wrie T wihou a when i comes o he paymen ime, since his is a quaniy relaed o he paymen currency. The fundamenally new sochasic process of his secion is he FX rae. Similarly o our handling of he ineres rae, we will reserve he symbol S primarily for he forward FX rae, since he spo FX rae is nohing bu a forward whose mauriy equals he observaion ime. Formally we wrie S PmCcy RaCcy ; T T, where T is he se ime or observaion ime, T is he mauriy of he forward and PmCcy RaCcy defines he involved currencies. PmCcy RaCcy In paricular S ; T is he spo FX rae a ime T which equals he number of unis of paymen currency T PmCcy needed o buy 1 uni of rae currency RaCcy. In oher words i represens he value a ime T of one uni RaCcy in erms of PmCcy. In wha follows we will assume he values of all asses are w.r.. PmCcy and we will wrie S T T or briefly o mean he forward FX rae associaed wih he FX S T PmCcy RaCcy. We may hink of he rae currency RaCcy as he foreign currency and of he paymen currency PmCcy as he domesic currency. Replacing R for he index I i in 2.1 and 2.2 we ge: where we dropped he coupon index i for simpliciy. CP = N m R + s τ 4.1 V CP = N mv R + s τ 4.2 is he price oday of a riskless bond w.r.. PmCcy wih mauriy T. The challenge is o calculae V R. Like in secion 3, le Q P be he equivalen maringale measure associaed wih he bond price, T chosen as numeraire. Le V = V def R. Then V is a maringale w.r.. Q P and obviously V T = R. We use a o refer o imes relaed wih he FX rae Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 25 of 29

26 Exoic FX Swap Version: ver 1. 4 Quano CMS: Foreign Index Paid in Domesic currency. Therefore he maringale propery leads o he resul: V = E QP VT T V = E QP R = E QP R 1 Alhough he equaion 4.3 only requires he evaluaion of he expecaion E QP R, we would prefer o deermine he SDE of R, w.r.. Q P. Here is he resul: Resul 4.1 [ SDE of F oreign F orward Swap Rae R in Q P ] R d R d = µ R ρ R,S σ Rσ S d + σ Rdw w.r.. Q P for 4.4 where R is a brief noaion of R, 1,..., N which is he foreign i.e. w.r.. RaCcy forward swap rae as seen a ime when he underlying swap sars a ime and pays coupons a imes 1,..., N. The order holds: < 1 <... < N. µ R is he drif of R in he foreign economy w.r.. Q P and is given by 3.5, where all quaniies should be undersood as he corresponding foreign ones. Here Q P is he equivalen maringale measure in he foreign economy w.r.. he riskless foreign bond price. σ R is he percenage volailiy of R. I can be inferred from he swapion marke in he foreign economy. Q P is he equivalen maringale measure associaed wih he numeraire being he domesic i.e. w.r.. PmCcy riskless bond price mauring a he coupon paymen ime T. We may now apply resul 4.1 o calculae he expecaion in 4.3: V = R e µ R u du e ρ R,S u σ R u σ S u du 4.5 This in urn can be se in 4.2 o ge he price oday of he whole coupon. By comparing 4.5 wih he well known resul V = r in he special case when is an amoun paid r a T and equal o he paymen currency spo a libor rae from o T i.e. when R,T r and where by r we mean he forward rae r,t, we derive he following resul: See he relevan discussion in secion 3. r r,t, Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 26 of 29

27 Exoic FX Swap Version: ver 1. 4 Quano CMS: Foreign Index Paid in Domesic currency. Resul 4.2 [ Quano/CMS Convexiy Correcion for Cash F lows linear in R ] When he coupon paymen CP a ime T is linear on he foreign forward swap rae R R, 1,... N, T and < 1 <... < N, i.e. if CP = a R + b, wih a, b consans, hen is value oday is: V CP = a R C RC S ; R + b Here C R is he same convexiy correcion facor given by 3.9 and is due o he index differing from he sandard libor. C S ; R is a new correcion facor, ypically called he quano correcion facor, given by: 4.6 C S ; R = e ρ R,S u σ R u σ S u du 4.7 where ρ R,S, σ R, σ S are as in resul 4.1. We remind is oday s price of he domesic i.e. w.r.. PmCcy riskless bond wih mauriy T and R R, 1,... N is he forward swap rae w.r.. RaCcy as seen from oday. Finally we may wan o value some generic possibly non-linear in R cash-flow CF T paid ou a ime T. The following resul holds: Resul 4.3 [ V aluaion Expression for Cash F lows which are funcions of R ] Le CF T some cash-flow paid a ime T in currency PmCcy, which is a possibly non-linear funcion of R R, 1,... N, T and < 1 <... < N, i.e. CF T = f R. Then is def value oday is given by: V CFT = P T [ E Q P f R ] 4.8 where is he price a of he domesic w.r.. PmCcy riskless bond wih mauriy T and Q P is he equivalen maringale measure associaed wih. The expecaion can be in principle calculaed by making use of he SDE of R, w.r.. Q P according o resul 4.1. Proof of resul 4.1 Wihou repeaing he argumenaion used a he same sage of he proof of resul 3.1, we will deermine he SDE of R hrough is condiional expecaions in Q P. We may wrie R in erms of he foreign i.e. w.r.. RaCcy riskless bond prices P as: R = N P P i=1 τ i i P where i, i =,..., N, are as shown in figure 1. and τ i refers o he daycoun fracion associaed wih he inerval i 1, i, ha is τ i D i i 1. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 27 of 29

28 Exoic FX Swap Version: ver 1. 4 Quano CMS: Foreign Index Paid in Domesic currency. We consider now he measure Q P, defined as he equivalen maringale measure in he RaCcy economy associaed wih he RaCcy-numeraire. As we found in secion 3, he rae R is diffused w.r.. Q P according o 3.4, which we rewrie here wih he new -noaion: 1 R d R d = µ Rd + σ Rdw w.r.. Q P for 4.9 An imporan fac for he res of he proof, is ha Q P is also an equivalen maringale measure w.r.. he PmCcy economy wih S being he corresponding numeraire. Noe ha S is he spo FX rae a and herefore S is he value of he RaCcy-bond in PmCcy erms i represens he price of a PmCcyeconomy raded asse. So we prove he following more general resul: Proposiion 4.1 Le economies A and B associaed respecively wih he class A and B of raded asses, when he economies do no inerac. Assume nex ha he agens in economy A are allowed o rade a any ime on he asses of he economy B afer hey conver he B-prices by muliplying hem wih he FX rae S. We assume his is also rue for he agens of B, wih he only difference ha hey need o conver he A-prices by dividing hem wih he FX rae S. This addiional rading possibiliy will expand he class of A-raded asses from A o A SB, where SB = {S B, B B} Assume nex he exisence of an equivalen maringale measure Q B in B associaed wih he numeraire N B. Then Q B is also an equivalen maringale measure in A bu associaed wih he numeraire S N B. Proof By he definiion of equivalen maringale measure we have ha he raio B B. B NB is maringale w.r.. Q B, Now he asses available o he agens in A are represened by he class A SB. I is enough o prove ha boh saemens below hold: 1. A A A S NB is a maringale w.r.. Q B. 2. B B S B S NB is a maringale w.r.. Q B. The second saemen is obvious due o he definiion of Q B as he equivalen maringale measure in B. For he firs saemen, observe ha A S is a raded asse in he economy B. Q.E.D. This follows from he definiion of he exchange rae as a 1-dim diffusion S wih he propery: A = asse price a in RaCcyeconomy S A = asse price a in PmCcy-economy. Group Risk Conrol ExoicFXSwapAnalyics.ex Exoics Pricing Dae: January 24, 22 Page 28 of 29