Fixed Strike Asian Cap/Floor on CMS Rates with Lognormal Approach

Transcription

1 Fxed Strke Asan Cap/Floor on CMS Rates wth Lognormal Approach July 27, 2011 Issue 1.1 Prepared by Lng Luo and Anthony Vaz

2 Summary An analytc prcng methodology has been developed for Asan Cap/Floor wth fxed strke rates on CMS rates. The prce of an Asan Cap/Floor depends on deal parameters such as reset and payment structures, CMS rate Tenor and number of average CMS rates; and market data such as dscount curve, swapton volatlty curve, realzed CMS rates and correlatons between projected CMS rates. Ths prcng methodology can prce Vanlla Asan Cap/Floor (wth fxed averagng perods), as well as Plan Vanlla Cap/Floor. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page of 14

3 Contents 1 Introducton Scope Measure Change and Adjustment Future Extensons Background otaton Black Swapton Formula Forward Measure Modellng Swap Rates Under a Common Forward Measure Changng from A Swap Measure to T Forward Measure Changng from T Forward Measure to T P Forward Measure Analytc Prcng of Fxed Strke CMS Asan Cap/Floor 10 5 Parameter Determnaton n Analytc Prcng Formula 11 A Black Formula for Lognormal Random Varables 13 References 14 CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page of 14

4 1 Introducton 1.1 Scope It s desred to fnd a valuaton formula for an Asan CMS Cap/Floor wth a fxed strke rate. A CMS (Constant Maturty Swap) rate s equal to the par swap rate for a swap that starts mmedately for a specfed tenor. Denote R swap (t; T, M ) as the forward swap rate determned at tme t for a swap startng at tme T wth a tenor of M years, whch mples the swap matures at tme T + M. Then the CMS rate wth M -year tenor at T can be expressed as R CMS (T, M ) = R swap (T ; T, M ). An Asan CMS Cap s a seres of call optons or caplets on the average CMS rate observed every reset over a specfed tme perod. An Asan CMS Floor s a seres of put optons or floorlets on the average CMS rate observed every reset over a specfed tme perod. The average number of CMS rates could be fxed or varyng for each caplet/floorlet. A Vanlla Asan Cap/Floor s an Asan Cap/Floor wth fxed averagng perods for each caplet/floorlet. When the averagngperods are fxed at 1, the Vanlla Asan Cap/Floor becomes a Plan Vanlla Cap/Floor. The prce of a product at tme t s the present value of all future cash flows generated by the product. An Asan CMS Cap/Floor can be decomposed nto a seres of caplets/floorlets and ts prce at tme t equals to the sum of tme t prces of the caplets/floorlets whose payments occur after t. Denote T P (k), k = 1, 2,..., as the scheduled payment tmes of a Cap/Floor and V (t; T P (k)) s the tme t prce of the correpondng caplet/floorlet wth payment at T P (k), then the prce of the Cap/Floor at tme t s gven by P (t) = V (t; T P (k)). (1) k:t P ( k) >t The prces of the caplets/floorlets can be calculated ndependently. For the rest of the document, we focus on the analytc formula for the prce of one caplet/floorlet. Assume s the average number of CMS rates for a caplet/floorlet and T 1, T 2,..., T are the correspondng reset tmes, then the opton payo at tme T P (T P > T ) for $1 notonal s gven as follows V (T P ) = τ P ω 1 =1 R CMS (T, M ) K where we use the notaton [ ] + to mean max (, 0), K s the fxed strke rate, ω s the opton ID wth ω = 1 for caplets and ω = 1 for floorlets, T s the reset tme for CMS rate R CMS (T, M ), and τ P denotes the year fracton between payment dates. The value of the payo s unknown untl tme T and s pad on T P. To compute the value of the opton at tme t (t < T P ), we compute an expectaton n the T P forward measure; that s, +, V (t) = P (t, T P ) τ P E T P ω 1 =1 R CMS (T, M ) K + F t, t < T P (2) where P (t, T P ) s the dscount factor between t and T P, E T P denotes the expectaton under T P forward measure and F t denotes the nformaton fltraton at tme t. If t T, all the CMS rates CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 1 of 14

5 are known and we have [ ( + 1 V (t) = P (t, T P ) τ P ω R CMS (T, M) K)], T t < T P. =1 The dffculty n evaluatng the above expectaton s that we need to have dstrbutonal assumptons about the average CMS rate that are consstent wth market observatons on forward swap rates at tme t. It s a standard practce 1 to assume that forward swap rates follow lognormal dstrbuton. The lognormal forward swap model prces swaptons wth Black swapton formula, whch s the standard formula employed n the swapton market. As dscussed n the book by Haug [5], t s not possble to fnd a closed-form soluton for the valuaton of optons on an arthmetc average. The man reason s that the sum of lognormal varables wll not have a lognormal dstrbuton. Arthmetc average-rate optons can be prced by analytcal approxmatons or wth Monte Carlo smulaton. We adapt the approxmaton developed by Turnbull and Wakeman [11] by adjustng the mean and varance of a lognormal dstrbuton so that they are consstent wth the exact moments of the arthmetc average. The adjusted mean and varance are then used as nputs n the Black formula. The detals of the approxmaton s dscussed n Secton 4. A convexty adjustment and a tmng adjustment are derved to account for the effect of valung the CMS rates under a common measure. The adjustments are mplemented n terms of drft adjustments. Dscussons are gven below and n Secton 3 of ths report. 1.2 Measure Change and Adjustment The lognormal forward swap model assumes that a forward swap rate R swap (t; T, M) s modelled by followng SDE dr swap (t; T, M) = σ swap (T, M) R swap (t; T, M) dw A (t) (3) where 0 t T, σ swap (T, M) s the swapton volatlty correspondng to an opton expry T and an underlyng swap startng at T wth tenor of M years (maturty T + M), and the process W A s a Brownan moton under measure A. Ths assumpton mples that the CMS rate R CMS (T, M) = R swap (T ; T, M) s lognormally dstrbuted under measure A. Condtonal on the market nformaton at tme t, the lognormal varable R CMS (T, M) can be expressed as R CMS (T, M) = R swap (t; T, M) exp 1 2 [σ swap(t, M)] 2 (T t) + σ swap (T, M) [W A (T ) W A (t)], (4) where W A s a Brownan moton correspondng to the forward swap rate R swap (t; T, M) under measure A. To evaluate the Asan opton n equaton (2), all CMS rates need to be expressed under a common probablty measure. In partcular, they must all be expressed n terms of a common T P forward measure, where T P s the payoff tme. The measure converson s represented as a correcton to the expectaton of the forward swap rate at future tme s, expressed as the current swap rate R swap (t; T, M) plus a convexty adjustment and a tmng adjustment; that s, E T P R swap (s; T, M) F t R swap (t; T, M) + Cnvx (t, s) + T mng (t, s), t s T < T P. 1 See Brgo and Mercuro[1] and Hull[8]. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 2 of 14

6 Denote R CMS (t; T, M) as the expectaton for CMS rate wth s = T n the equaton above, then the CMS rate n equaton (4) have the followng expresson under the common T P forward measure and condtonal on the market nformaton at tme t. R CMS (T, M) R CMS (t; T, M) exp 1 2 [σ swap(t, M)] 2 (T t) + σ swap (T, M) [W T P (T ) W T P (t)], (5) where W T P s a Brownan moton correspondng to the forward swap rate R swap (t; T, M) under the common measure T P. Correlatons between W T P and W T P j can be ntroduced. The detals of these dervatons are explaned n Secton 3 of ths report. 1.3 Future Extensons In future, t s desred to develop a valuaton formula for an Asan CMS Cap/Floor wth a floatng strke rate. Although the valuaton analytcs are beyond the scope of ths report, ts prce can be approxmated by the smlar methodology developed n ths report for fxed strke Asan Cap/Floor. The floatng strke Asan CMS Caplet/Floorlet has a payoff at tme T P (T P > T ) gven by [ ( + 1 V (T P ) = τ P ω R CMS (T, M)) R CMS (T, M))]. =1 The value of the payoff s unknown untl tme T and s pad on T P. To compute the value of the opton at tme t, we compute an expectaton n the T P forward measure; that s, [ ( + V (t) = P (t, T P ) τ P E T 1 P ω R CMS (T, M) R CMS (T, M))] F t. =1 CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 3 of 14

7 2 Background 2.1 otaton Consder a swap that starts at tme T 0 0 and ends at tme T 0 + M wth a tenor of M years, t has n payments per year at T 1 <... < T nm, where T nm = T 0 + M. The frst cash flow exchange takes place at T 1. A plan vanlla swap contract conssts of two legs. One leg pays a floatng rate (such as, LIBOR or CDOR) plus a margn spread and the other leg pays a fxed rate called the swap rate. The swap rate s determned pror to the start of the swap. Let P (t, T ) be the dscount factor between t and T, then the value of the floatng leg at tme t (t T 0 ), V float (t), s gven by V float (t) = P (t, T 0 ) P (t, T nm ). The value of the fxed leg at tme t, V fxed (t), s gven by V fxed (t) = R swap nm j=1 τ j P (t, T j ), where τ j s the year fracton between T j 1 and T j. The par swap rate s determned so that fxed and floatng legs have the same value. Thus the t-tme par swap rate s determned as follows: R swap (t; T 0, M) = P (t, T 0) P (t, T nm ) nm j=1 τ. jp (t, T j ) The notaton R swap (t; T 0, M) ndcates that the par swap rate s determned at tme t for a swap startng at tme T 0 wth a tenor of M years, whch mples the swap matures at tme T nm. A CMS (constant maturty swap) s a swap contract where one leg pays the M-year swap rate (and possbly plus some margn) whle the other leg usually pays a floatng rate (such as LIBOR or CDOR). The CMS rates are usually set n advance. In partcular, the payment on T +1 depends on the CMS rate R CMS (T, M) whch s calculated at T for the swap startng at T and endng at T + M, that s, R CMS (T, M) = R swap (T ; T, M). 2.2 Black Swapton Formula A swapton s an opton grantng ts owner the rght, but not the oblgaton to enter nto an underlyng nterest rate swap. There are two types of swapton contracts. 1. A payer swapton gves the owner of the swapton the rght to enter nto a swap, n whch he would pay the fxed leg at the strke rate and receve the floatng leg. 2. A recever swapton gves the owner of the swapton the rght to enter nto a swap, n whch he would receve the fxed leg at the strke rate and pay the floatng leg. A swapton s characterzed by the followng: 1. strke rate (equal to the fxed rate of the underlyng swap), CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 4 of 14

8 2. length of the opton perod, 3. term of the underlyng swap, 4. notonal amount, and 5. frequency of payments on the underlyng swap. Fsher Black developed a smple formula for prcng swaptons based on the assumpton that swap rates are lognormally dstrbuted. The Black formula for swapton prcng ([1], [8]) can be formally derved by usng an annuty factor as a numerare to defne a probablty measure n whch the swap rate has a lognormal dstrbuton. Defne the annuty factor A(t; T, M) as follows. A(t; T, M) = +nm j=+1 τ j P (t, T j ). As a shorthand notaton, we use A to denote the probablty measure nduced by the numerare A(t; T, M). The swap rate R swap (t; T 0, M) can be expressed as follows R swap (t; T 0, M) = P (t, T 0) P (t, T nm ). A(t; T 0, M) Under probablty measure A 0, the swap rate s a martngale; that s, R swap (t; T 0, M) = E A 0 R swap (T ; T 0, M) F t, t T T 0. Ths mples that the stochastc dfferental equaton (SDE) of the swap rate can be wrtten as follows dr swap (t; T 0, M) = σ swap (T 0, M) R swap (t; T 0, M) dw A 0 0 (t), 0 t T 0. where σ swap (T 0, M) s the swapton volatlty correspondng to an opton expry T 0 and an underlyng swap startng at T 0 wth tenor of M years, and the process W A 0 0 s a Brownan moton under measure A 0. The payoff from a payer swapton at the opton expry T 0, whch corresponds to the start tme of the underlyng swap, s gven by nm V payer swapton (T 0) = P (T 0, T j ) τ j [R swap (T 0 ; T 0, M) K] + j=1 = A(T 0 ; T 0, M) [R swap (T 0 ; T 0, M) K] +. From the Fundamental Theorem of Arbtrage Free Prcng, t follows that V payer V payer swapton (t) = A(t; T 0, M) E A 0 swapton (T 0) A(T 0 ; T 0, M) F t = A(t; T 0, M) E A A(T 0 0 ; T 0, M) [R swap (T 0 ; T 0, M) K] + A(T 0 ; T 0, M) F t = A(t; T 0, M) E A 0 [R swap (T 0 ; T 0, M) K] + Ft CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 5 of 14

9 At tme t < T 0, the swap rate R swap (T 0 ; T 0, M) s has not be realzed. Under the lognormal assumpton of R swap (T 0 ; T 0, M), the tme t swapton value s computed by ( V payer swapton (t) = A(t; T 0, M) Black K, R swap (t; T 0, M), σ swap (T 0, M) ) T 0 t, +1, where the Black() formula s defned n the Secton A n Appendx. Smlarly, the payoff from a recever swapton at the opton expry T 0, whch corresponds to the start tme of the underlyng swap, s gven by nm Vswapton(T recever 0 ) = P (T 0, T j ) τ j [K R swap (T 0 ; T 0, M)] + j=1 = A(T 0 ; T 0, M) [K R swap (T 0 ; T 0, M)] +. Under the lognormal assumpton of R swap (T 0 ; T 0, M), the tme t swapton value s computed by ( Vswapton(t) recever = A(t; T 0, M) Black K, R swap (t; T 0, M), σ swap (T 0, M) ) T 0 t, Forward Measure The probablty measure that results from usng a zero coupon bond P (t, T ) as a numerare s called the T forward measure [1][8]. Consder a tradable securty S, under T forward measure; the followng quotent s a martngale S(t) P (t, T ) ; that s, S(t) P (t, T ) = ET S(s) P (s, T ) F t, t s T. Consder the forward swap rate R swap (t; T 0, M), the followng martngale property holds under T 0 forward measure snce a forward swap rate s derved from a tradeable securty and ts value s known at T 0. R swap (t; T 0, M) P (t, T 0 ) Wth s = T 0, we have = E T Rswap (s; T 0 0, M) P (s, T 0 ) F t, t s T 0. R swap (t; T 0, M) = P (t, T 0 ) E T 0 R swap (T 0 ; T 0, M) F t. It mples that the expected future swap rate s related to ts t-tme value by a smple dscount factor under T 0 forward measure. We shall use ths result n the next secton. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 6 of 14

10 3 Modellng Swap Rates Under a Common Forward Measure From our dscusson n Secton 2.2, t s standard practce to assume that a swap rate R swap (t; T, M), R swap (t; T, M) = P (t, T ) P (t, T +nm ) A(t; T, M), s modelled by (3), that s dr swap (t; T, M) = σ swap (T, M) R swap (t; T, M) dw A,t, where 0 t T, and σ swap (T, M) s the swapton volatlty correspondng to an an opton expry T and an underlyng swap startng at T wth tenor of M years, and the process W A s Brownan moton under measure A. Under probablty measure A, the swap rate s a martngale; that s, R swap (t; T, M) = E A R swap (s; T, M) F t, for t s T. A CMS rate wth M-year tenor at T can be expressed as R CMS (T, M) = R swap (T ; T, M). In the case of a fxed strke call opton, we wsh to compute the expectaton of the followng [ + 1 R CMS (T, M) K], =1 and n the case of a fxed strke put opton, we compute the expectaton of the followng [ K 1 + R CMS (T, M)]. =1 The dffculty s that all the CMS rates are modelled usng dfferent probablty measures. Hence to compute the expectaton, we must model all the CMS rates under a common probablty measure. The CMS rates R CMS (T, M) s the swap rate observed at tme T for = 1, 2,...,, whle the related payment s made at a later tme T P. In the followng subsectons, the swap rates wll be frst modelled under T forward measure and then modelled usng a common T P forward measure. An mpled convexty adjustment (from A swap measure to T forward measure) and an mpled tmng adjustment (from T forward measure to T P forward measure) wll be derved. These adjustments wll be used to adjust the expected value of CMS rate n (5). For smplcty, we assume that the accrual perods for swap rates are constant and equal to 1/n, where n s the number of coupon payments per year. 3.1 Changng from A Swap Measure to T Forward Measure Suppose that G (y ) s the prce of a future M-year bond at tme T that pays c/n coupon at the end of each payment perod, n s the number of coupon payments per year and y s ts yeld wth 2 G (y) = +nm j=+1 c (1 n + y ) n(tj T ). n 2 The prce formula neglects the fnal payment of the notonal amount under the consderaton that there s generally no exchange of prncpal n the CMS swap. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 7 of 14

11 It can be approxmated by ts second order Taylor seres expanded about y t as follows, where y t s the forward bond yeld at tme t (t < T ). G (y) G (y t ) + G (y t )(y y t ) G (y t )(y y t ) 2, (6) where G (x) and G (x) are the frst and second dervatves of G wth respect to x. Under the T forward measure, the expected future bond prce equals the forward bond prce, that s E T G (y) F t = G (y t ). (7) Takng the expectaton of (6) under the T forward measure wth the dentty (7) yelds G (y t ) E T y y t F t G (y t ) E T (y y t ) 2 F t 0. The expresson E T (y yt ) 2 Ft s approxmately y 2 t σ 2 y(t t), where y s assumed to follow a lognormal dstrbuton wth volatlty σ y. Hence t s approxmately true that E T y F t y t 1 2 wth G (y t) G (y t) y2 t σ 2 y (T t), E T y F t = 0 f y t = 0. The CMS rates R CMS (T, M) can be consdered as the yeld at tme T on a M-year bond wth a coupon equal to today s forward swap rate. Let Rt = R swap (t; T, M) and σr = σ swap(t, M) for = 1, 2,...,, and then the expected swap rate RT under T forward measure equals to E T R T Ft = R t + Cnvx (t, T ), where Cnvx (t, T ) = 1 2 ote that Cnvx (t, T ) = 0 f R t = 0. G (R t) G (R t ) (R t) 2 (σ R) 2 (T t) 3.2 Changng from T Forward Measure to T P Forward Measure To change the measure from T to T P, the Radon-kodym dervatve s used as follows. E T P RT F t = E T RT dqt P dq T T F t = E T RT P (T, T P )/P (t, T P ) P (T, T )/P (t, T ) F t = E T R T P (T, T P ) F t P (t, T ) P (t, T P ). (8) Suppose that ft := f fwd (t; T, T P ) s the forward rate (wth compoundng frequency n) durng future tme perod from T to T P at tme t wth 0 t T < T P, and H (ft ) s the forward prce of a future zero-coupon bond at tme T that pays 1 at tme T P, then ( H (ft ) = 1 + f t ) n (TP T ) = P (t, T P ) n P (t, T ). CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 8 of 14

12 At tme T, the zero coupon bond has value H (f T ) = P (T, T P ) P (T, T ) = P (T, T P ). The expected future zero-coupon bond prce equals the forward bond prce under the T forward measure, that s E T H (f T ) Ft = H (f t ) = P (t, T P ) P (t, T ). (9) Apply the martngale property n equaton (9) to obtan the followng. E T (R T R t) [H (f T ) H (f t )] Ft = E T R T H (f T ) R T H (f t ) R t H (f T ) + R t H (f t ) Ft = E T R T H (f T ) Ft E T R T Ft H (f t ). On the other hand, H (f T ) can be approxmated by ts frst order Taylor seres expanded about f t. Hence, E T (R T R t) [H (f T ) H (f t )] Ft E T (R T R t) [H (f t ) (f T f t )] Ft H (f t ) R t σ R f t σ f ρ R,f (T t), under the assumpton that f t follows a lognormal dstrbuton wth volatlty σ f. ote that H (x) s the frst dervatve of H wth respect to x and ρ R,f measures the correlaton between R and f. By comparng the two equatons above, we obtan E T R T H (f T ) F t E T R T F t H (f t ) + H (f t ) R t σ R f t σ f ρ R,f (T t). Thus equaton (8) becomes E T P R T Ft = E T R T H (f T ) Ft 1 H (f t ) wth Cnvx (t, T ) = 1 2 E T R T F t + H (f t ) H (f t ) R t σ R f t σ f ρ R,f (T t) R t + Cnvx (t, T ) + T mng (t, T ) G (R t) G (R t ) (R t) 2 (σ R) 2 (T t) (10) T mng (t, T ) = T P T 1 + f t /n R t σ R f t σ f ρ R,f (T t) (11) The convexty and the tmng adjustments on CMS rate are smlar to those gven n Hull[8], Equaton (32.2). CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 9 of 14

13 4 Analytc Prcng of Fxed Strke CMS Asan Cap/Floor Ths secton dscusses the analytc formula for prcng fxed strke CMS Asan caplet/floorlet based on the lognormal assumpton of CMS rates. Let T k 1 t < T k, then the CMS rates reset before t have been realzed and the remanng unknown CMS rates are R CMS (T, M) for k. The prce of the caplet/floorlet n Equaton (2) can be wrtten as V (t) = P (t, T P ) τ P E T P [ω (X ˆK)] + Ft, where X = 1 R CMS (T, M), =k ˆK = K 1 k 1 R CMS (T, M). (12) =1 ˆK s called the adjusted strke. We adapt the approxmaton developed by Turnbull and Wakeman [11] on equty Asan opton, that s, the sum of correlated lognormal varables, X, can be approxmated by a sngle lognormal varable by matchng ther frst and second moments. To smplfy the notaton, we rewrte the followng µ (t) = R CMS (t; T, M) = E T P R swap (T ; T, M) F t as the CMS rate mean condtonal on market nformaton on tme t. Then Equaton (5) becomes to R CMS (T, M) µ (t) exp 1 2 (σ R) 2 (T t) + σr [W T P (T ) W T P (t)], where µ (t) = R swap (t; T, M) + Cnvx (t, T ) + T mng (t, T ). (13) The convexty adjustment and the tmng adjustment of the expected forward swap rate are defned n Equaton (10) and (11) respectvely. Denote µ x as the expected value of X, σ x as the standard devaton of ln(x) and ρ j as the correlaton between W T P and W T P j. Then condtonal on market nformaton at tme t, the frst and the second moments of X can be derved as E[X F t ] = µ x = 1 µ (t), (14) =k E[X 2 Ft ] = (µ x ) 2 exp(σ x ) 2 = 1 2 =k j=k µ (t) µ j (t) expρ j σ R σ j R [mn(t, T j ) t] (15) In ths report, the nstantaneous correlaton between projected CMS rates s modelled by a twoparameter functon as follows, ρ,j = β 1 + [1 β 1 ] exp β 2 j, wth 0 β 1 1 and β 2 0. (16) It starts at 1 when = j and decreases to β 1 as j. The parameter β 1 can be nterpreted as the lmt of correlaton that s approached for rates far separated n tme, whle the parameter β 2 measures the speed of decay whch descrbes how fast the correlaton decreases and approaches β 1. β 1 could have negatve value, however we restrct t to be nonnegatve here. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 10 of 14

14 By applyng the assumptons above, the fxed strke CMS Asan opton prce can be calculated by the Black formula descrbed n Appendx A. P (t, T P ) τ P Black( ˆK, µ x, σ x, ω), f ˆK > 0 and µ x > 0; V (t) (17) P (t, T P ) τ P [ω (µ x ˆK)] +, f ˆK 0 or µ x = 0. ote that V (t) = 0 f = 0. Once the prces of all caplets/floolets are known, the prce of an Asan CMS Cap/Floor can be calculated as the sum of the caplets/floorlets prces, as shown n Equaton (1). 5 Parameter Determnaton n Analytc Prcng Formula An Asan CMS Cap/Floor can be decomposed nto a seres of caplets/floorlets. The prce of each caplet/floorlet wth $1 notonal s calculated based on the analytc prcng formula (17). The parameters n the formula are defned as follows, where the current tme (tme 0) corresponds to the curve date. t s the valuaton tme, t 0. T P s the payment tme of the caplet/floorlet, T P > t. P (t, T P ) s the dscount factor between t and T P. τ P s the accrual perod between payments wth the frst τ P starts from the effectve date. ω = 1 for caps and ω = 1 for floors. ˆK s the adjusted strke n Equaton (12), where K s the fxed strke rate. s the average number of CMS rate n the caplet/floorlet. T 1,..., T are the reset tmes of the caplet/floorlet wth T P = T +1. M s the tenor of the underlyng CMS rate. R CMS (T, M), = 1,..., k 1, are realzed CMS rates, T k 1 t. µ x and σ x are derved from Equaton (14) and (15), where µ (t) s defned n (13), whch s the expected value of R CMS (T, M) at tme t. ρ,j s defned n (16), whch s the correlaton between projected CMS rates. σ R s the volatlty of forward swap rate R swap(t; T, M). The valuaton tme, the reset tme and the payment tme are determned by the fracton of year between ther correspondng dates and the curve date based on the nput day-count bass. For parameters n Equaton (10) and (11), Rt and ft are derved from the nput dscount curve based on the followng formulas, where n s the number of coupon payments per year for the CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 11 of 14

15 fxed leg of the swap. R0 = R swap (0; T, M) = n P (0, T ) P (0, T + M) nm k=1 P (0, T + k/n) [ (P (0, f0 TP ) = f fwd (0; T, T P ) = n P (0, T ) ) 1/[n(Tp T )] 1 ], T P > T Here T s the reset tme for R swap (0; T, M). Dscount factors are log-lnearly nterpolated from the nput dscount curve wth flat extenson of contnuous compounded zero rates at both ends of the curve. The volatltes σr and σ f are lnearly nterpolated from the correspondng nput volatlty curve wth flat extenson of volatltes at both ends of the curve. The correlatons ρ R,f are lnearly nterpolated from the nput correlaton curve wth flat extenson of volatltes at both ends of the curve. CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 12 of 14

16 A Black Formula for Lognormal Random Varables Proposton A.1 Suppose V s a lognormally dstrbuted random varable such that E[V ] = m and the standard devaton of ln(v ) s s. Then Emax(V K, 0) = m Φ (d 1 (K, m, s)) KΦ (d 2 (K, m, s)) and Emax(K V, 0) = m Φ ( d 1 (K, m, s)) + KΦ ( d 2 (K, m, s)) where Φ() s the standard Gaussan cummulatve dstrbuton functon Φ(x) = 1 2π x e 1 2 λ2 dλ, d 1 (K, m, s) = ln(m/k) + s2 /2, s and d 2 (K, m, s) = ln(m/k) s2 /2. s Proof See [8] pages From the structure of the formulae n Proposton A.1, the Black formula [1] can be defned as follows. Black(K, m, s, a) = a m Φ (ad 1 (K, m, s)) a K Φ (ad 2 (K, m, s)) (18) Hence, Emax(V K, 0) = Black(K, m, s, +1) and Emax(K V, 0) = Black(K, m, s, 1). CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 13 of 14

17 References [1] Damano Brgo and Fabo Mercuro, Interest Rate Models - Theory and Practce, Sprnger Verlag, ew York, [2] H. Geman,. El Karou and J. C. Rochet, Change of umerare, Changes of Probablty Measures and Prcng of Optons, Journal of Appled Probablty, Vol. 32, , [3] Patrck Hagan, Convexty conundrums: Prcng CMS Swaps, Caps and Floors, Wlmott Magazne, March 2003, p [4] Mchael Harrson and Stanley Plska, Martngales and Stochastc ntegrals n the theory of contnuous tradng, Stochastc Processes and ther Applcatons, Volume 11, o.3, pp , [5] Espen Gaarder Haug, The Complete Gude to Opton Prcng Formulas, 2nd edton, McGraw- Hll, ew York, [6] Vcky Henderson and Rafal Wojakowsk, On the Equvalence of Floatng and Fxed Strke Asan Optons, Journal of Fnance, Vol. 52, o. 3, pp , [7] Vcky Henderson, Davd Hobson, Wllam Shaw, and Rafal Wojakowsk, Bounds for nprogress floatng-strke Asan optons usng symmetry, Annals of Operatons Research, Vol. 151, o. 1, pp , [8] John Hull, Optons, Futures, and Other Dervatve Securtes, 7th edton, Prentce Hall, ew Jersey, [9] F. A. Longstaff and E. S. Schwartz, Valung Amercan Optons by Smulaton: A Smple Least-Squares Approach, Revew of Fnancal Studes, Vol. 14, o. 1, pp , [10] Marek Musela and Marek Rutkowsk, Martngale Methods n Fnancal Modellng, Sprnger Verlag, ew York, [11] Stuart Turnbull and Lee Wakeman, A Quck Algorthm for Prcng European Average Optons, Journal of Fnancal and Quanttatve Analyss, Vol. 26, o. 3, September 1981, pp CMSOptonL v1p1.pdf (Lng Luo and Anthony Vaz) Page 14 of 14