Pricing Interest Rate and currency Swaps. Up-front fee. Valuation (MTM)

Transcription

1 Pricing Ineres Rae an currency Swas. U-ron ee. Valuaion (MM) A lain vanilla swa ricing is he rocess o seing he ixe rae, so ha he iniial value o he swa is zero or boh couneraries. hereaer i is osiive or one counerary making i an asse an negaive or anoher counerary making i a liabiliy. I is oun by comaring he agree ixe rae o he curren ixe rae on a swa having he same mauriy as he original agreemen. he rocess o valuaion is calle mark-o-marke. A lain vanilla ineres swa s rae is is ixe rae. Alhough he swa ixe rae is quoe o he reasury yiel curve, i is rice o an aroriae orwar curve corresoning o he loaing reerence rae on he swa. he swa s ixe rae shoul be esablishe a he level so ha he resen value o he ixe cash lows evenually ajuse by an u-ron ee equals he resen value o he loaing cash lows imlie by he orwar raes. hese orwar raes may be observe, calculae or esimae. he aroriae so raes are use as iscouning raes. Swa wih a LIBOR/WIBOR as a reerence rae he FRA ineres raes In he Unie Saes he FRA rices are erive rom he observe LIBOR orwar curve (Euroollar uures) because i inicaes irecly he levels o he loaing ineres rae ha can be locke in by arbirage ransacions. In Polan insea WIBOR uures o no exis an he FRA rices may be erive only using imlie orwar raes, which are inerre rom he WIBOR an WIBID so ineres raes an evenually ineres raes observe in he FX swa marke. Swa wih a enor u o years In he Unie Saes an in Polan he ixe rae o he ineres rae swa wih a enor o u o wo years is esablishe using he observe FRA ineres raes. Swa wih a enor beween an 0 years In he Unie Saes he ixe rae o he ineres rae swa wih a enor o beween an 0 years is esablishe using he observe LIBOR orwar curve (Euroollar uures). In Polan WIBOR uures o no exis. he orwar curve or WIBOR mus be esimae o he orwar curve or reasury securiies. he ollowing roceure or a U.S. swa exceeing 0 years alies. Swa wih a enor exceeing 0 years In he Unie Saes he ixe rae o he ineres rae swa wih a enor exceeing 0 years is esablishe using esimae orwar raes. hese esimae LIBOR orwar raes are reice using he orwar raes or reasury securiies an ineenenly orecase ED (reasury Euroollar Dierence). he orwar raes or reasury securiies are imlie orwar raes calculae using so reasury curve. he so reasury curve is someimes irecly observe bu i is usually erive using he boosraing roceure.

2 Derivaives on Financial Marke Swa wih a reasury yiel as a reerence rae Assuming ha loaing-rae aymens are mae on he basis o a/ an ixe-rae aymens are mae on he basis o a/ (oher assumions will be inrouce subsequenly) - s + z + z where - - orwar rae, number o ays in erio noional rincial in erio, s swa ixe rae, z so rae (iscouning rae, PP 0 u-ron ee, erio (,,...,), s Swa ixe rae is equal o: - + z + z + PP 0 Swa wih a WIBOR (LIBOR) rae - s () + z + z Swa ixe rae is equal o: s z - + z

3 Derivaives on Financial Marke Problem. Pricing FRA raes using Euroollar uures Suose oay is: hree-monh LIBOR is 5,5% he hree-monh orwar raes or Euroollar uures conracs are ollowing: Mauriy Rae -98 5,9% ,86% ,86% ,89% -99 5,6% ,0% ,09% ,5% Require (a) Calculae he rices or a series 3 x 6, 6 x 9, 9 x, x 5, 5 x 8, 8 x, x 4 o FRA ransacions base on Euroollar uures. (b) Calculae he rices or a series 6 x, x 8, 8 x 4 o FRA using raes calculae in ar (a). Soluion (a) A 3 x 6 FRA rae is calculae as a weighe average o December an March Euroollar uures raes. he number o ays beore an aer March uures conrac ivie by he number o ays or a FRA ransacion are use as weighs x 6 FRA 3 ays 66 ays Euroollar uures conracs ,9% 4,86% he FRA raes or a series o seven conracs: FRA Meho o calculaion 3 x 6 (5,9% * 3 ays + 4,86% * 66 ays) : 89 ays 4,95% 6 x 9 (4,86% * 5 ays + 4,86% * 67 ays) : 9 ays 4,86% 9 x (4,86% * 4 ays + 4,89% * 68 ays) : 9 ays 4,88% x 5 (4,89% * 3 ays + 5,6% * 69 ays) : 9 ays 5,09% 5 x 8 (5,6% * ays + 5,0% * 68 ays) : 90 ays 5,05% 8 x (5,0% * 30 ays + 5,09% * 6 ays) : 9 ays 5,07% x 4 (5,09% * 9 ays + 5,5% * 63 ays) : 9 ays 5,3% 3

4 Derivaives on Financial Marke Calculaions are mae using he ollowing able Fuures Fuures FRA Fuures FRA Forwar Days ( mauriy rae s ae mauriy n ae ) Days ( ) rae -98 5,9% ,5% ,86% ,95% ,86% ,86% ,89% ,88% -99 5,6% ,09% ,0% ,05% ,09% ,07% ,5% ,3% (b) FRA Meho o calculaion 6 x ( + 4,86% * 9/ ) ( + 4,88% * 9/ ) ( + FRA 6x * 84/) hus FRA 6x 4,90%. x 8 ( + 5,09% * 9/ ) ( + 5,05% * 90/ ) ( + FRA x8 * 8/) hus FRA x8 5,0%. 8 x 4 ( + 5,07% * 9/ ) ( + 5,3% * 9/ ) ( + FRA8 x4 * 84/) hus FRA 8x4 5,3%. 4

5 Derivaives on Financial Marke Problem. Pricing Ineres Rae Swa O he FRA Curve he curren FRA erm srucure is FRA Rae Days ( ) 0 x 6 5,33% 8 6 x 4,904% 84 x 8 5,036% 8 8 x 4 5,34% 84 he noional rincial o he swa is $00 million. (a) Deermine he ixe rae on he wo-year ineres swa using FRA raes an he ollowing ay-coun convenions: "30/", "acual/" an "acual/". (b) Deermine he ixe rae on he wo-year ineres swa uner he assumion ha he ixe-rae receiver will ay u-ron ee equal o % o he noional rincial? Soluion (a) k a - - a 0 x ,3% 0,503 0,58% 0,58% 97,48%,5 6 x ,90% 0,5 0,5% 05,5% 95,0% 4,9 x ,0% 0,506 0,58% 07,86% 9,7% 7,3 8 x ,3% 0,5 0,6% 0,69% 90,34% 9,7 he ixe swa raes or semiannual selemens are no equal o ixe swa raes or quarerly selemens. a s 30/ a s a/ a s a/ 0 x 6 48,7 5,66% 48,3 5,044% 49,0 5,33% 6 x 96,3 5,0869% 96,3 5,0874% 97,6 5,077% x 8 4,6 5,08% 4,5 5,57% 44,5 5,0456% 8 x 4 87,8 5,434% 88, 5,370% 90,7 5,0666% he ixe swa rae on a wo-year ineres swa is 5,434% (30/ basis), 5,370% (a/ basis), an 5,0666% (a/ basis). (b) As he ixe-rae receiver ays an u-ron ee, he swa ixe raes are lower. a s 30/ a s a/ a s a/ 0 x 6 48,7,0584% 48,3,067% 49,0,055% 6 x 96,3 3,0099% 96,3 3,00% 97,6,9690% x 8 4,6 3,7087% 4,5 3,73% 44,5 3,664% 8 x 4 87,8 4,0785% 88, 4,0734% 90,7 4,076% 5

6 Derivaives on Financial Marke Problem 3. Pricing an Valuaion o FRA he curren erm srucure o WIBOR is 9 -ay WIBOR 5,00% 8 -ay WIBOR 5,0% he noional rincial is $ (a) Calculae FRA. (b) I is 6 ays laer an he relevan erm srucure is 3 -ay WIBOR 4,99% 0 -ay WIBOR 4,84% Deermine he marke value o he FRA. (c) On he exiraion ay, 89-ay WIBOR is 4,00%. Deermine he aymen. Soluion (a) he FRA rae is: z + z + 9 5,00% 8 5,0% 5,34% (b) ,00% ,0% 6 3 4,99% 0 4,84% 6

7 Derivaives on Financial Marke he value o FRA will be V or + ' ' ' ' z z + + ' ' z + -37,7 4,77% V (c) A exiraion, he ayo is -37,7 V or + '' ' z + '' z '' ' z + V -33,45-33,45 Problem 4. Pricing an Valuaion o IRS Consier a one-year ineres swa wih semiannual aymens. (a) Deermine he ixe rae on he swa. he curren srucure o WIBOR so raes is given as ollows. Days 84 WIBOR 4,50% 4,60% (b) 53 ays laer, he erm srucure is as ollows: Days 3 WIBOR 4,99% 4,77% Deermine he marke value o he swa rom he ersecive o he ary aying he ixe rae an receiving he loanig rae. Assume he noional rincial o $ million. 7

8 Derivaives on Financial Marke Soluion (a) s + z + z 4,55% 84 z 4,50% 4,60% /(+z /) 0,9778 0, ( - - )/ 0,504 0,4959 Σ ( - - )//(+z /) 0,499 0,474 0,9670 (b) - + s MM + z + z + z he resen value o loaing aymens Number o ays 84 Floaing rae (leas reerence ae) 4,50% Cash lows,07 Number o ays 3 So rae 4,99% Presen value acor 0,996 Discoune cash low,084 he resen value o remaining ixe aymens: Number o ays 84 8 Swa ixe rae 4,55% 4,55% Cash lows 0,09,06 Number o ays 3 So rae 4,99% 4,77% Presen value acor 0,9958 0,9730 Σ Discoune cash low 0,08 0,9950,078 Dierence 0,0006 x