Pricing and Valuation of Forward and Futures

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Julius Simpson
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1 Prcng and Valuaon of orward and uures. Cash-and-carry arbrage he prce of he forward conrac s relaed o he spo prce of he underlyng asse, he rsk-free rae, he dae of expraon, and any expeced cash dsrbuons of he underlyng asse before expraon. he heorecal or far prce s derved from he cash-and-carry arbrage. able. Cash-and-carry arbrage raegy Value oday Value a fuure me raegy I Purchase he secury C raegy II Purchase a forward conrac for prce - - Inves equvalen $ amoun for selemen a me a a rskfree neres rae r r 36 oal value for sraegy II r - 36 ource: Clarke R.G., Opons and uures: A uoral. he Research oundaon of he Insue of Charered nancal Analyss 99, s. 7. he purchase of he secury s a subsue for a rsk-free nvesmen. Boh sraeges begn wh he same value oday and resul n he nvesor ownng he asse a me, he endng values should be equal. ha s, r () C - 36 olvng for he forward prce gves r () - C 36 or a marke forward prce, we may nfer he mpled repo rae: C 36 (3) r - : marke forward prce.

2 ervaves on nancal arke able. Arbrage wh low forward prces raegy Value oday Value a fuure me ell he secury C Purchase a forward conrac for prce - - Inves equvalen $ amoun for selemen a me a a r rsk-free neres rae r 36 oal r ( ) - C 36 - > An nvesor may sell he underlyng asse and buy reasury blls or bonds earnng r. he frs sraegy (purchase of he underlyng asse) s beer han he second (purchase of a forward conrac) when r (4) > - C 36 or C 36 (5) r < r - able 3. Arbrage wh hgh forward prces raegy Value oday Value a fuure me Purchase he secury C ell a forward conrac for prce - - Borrow equvalen $ amoun for selemen a me a - r a rsk-free neres rae r - 36 oal r ( ) C 36 > he forward prce wh connuously compounded neres rae s: (r q) 36 (6) e q dvdend yeld.

3 ervaves on nancal arke Equy Index uures Prcng he far prce of he equy ndex fuures conrac s: r (7) - 36 equy ndex, r annualzed rsk-free rae, days o expraon, value of dvdends. (8) he mpled repo rae s: r Currency forward prcng (9) he far prce of a foregn exchange forward conrac s: ( d) ( ) he spo exchange rae, - nomnal domesc neres rae, d f f - nomnal foregn neres rae. me o expraon. () he mpled repo rae s: d ( ) f.4 Eurodollar uures Prcng Eurodollar fuures are quoed as an ndex formed by subracng from he annualzed hree-monh LIBOR (London Inerbank Offerred Rae) forward rae. he prcng formula for a Eurodollar fuures conrac s () (- f ) f s he annualzed hree-monh LIBOR forward rae begnnng a me. 3

4 ervaves on nancal arke reasury Bll uures Prcng he prce for fuures on reasury blls s calculaed as follows () (- d ) d he annualzed hree-monh forward dscoun rae on a reasury bll begnnng a me..6 Bond uures Prcng he bond fuures conrac requres he purchase or sale of he acual reasury bonds f s held o maury. he acual bond seleced for delvery by he shor seller s adjused n prce by a delvery facor o reflec a sandardzed 6 percen coupon rae. he fuures prce for any bond s calculaed usng he formula r a P ( ) - Bc (3) f P marke prce of bond wh accrued neres, B par value of he bond, r neres rae, c annualzed coupon rae, days o expraon, a days of accrued neres, f delvery facor of a bond. 4

5 ervaves on nancal arke Problem. Prcng and Valuaon of a forward conrac You own a sock currenly worh $ oday. You plan o sell n 6 days. o hedge agans a possble declne n prce, you ener no he forward conrac o sell he secury. he annualzed rsk-free rae s 3,5 percen. Calculae he forward prce on hs conrac. (b) uppose a marke dealer offers o ener no a forward conrac a $. How you could earn an arbrage prof. (c) uppose ha a make dealer offers an off-marke conrac. Calculae an up-fron fee and ndcae wheher paymen wheher paymen s made by he long o he shor or vce versa. (d Afer 3 days, he secury sells for $. Calculae he gan or loss o your poson. (e) A expraon he prce of he asse s $. Calculae he value of he forward conrac. oluon he far value s ( ) 5,75 (b) 5,75-3,75 ell he sock Inves money Buy forward (c) 3,73 Because he value s posve, he paymen s made by he long. (d) W V (, ) - (, ) ( ) - r ( ) ( ) ( ) ( ) 97,3 Gan o he long poson, loss o he shor. (e) Gan on asse -,, Value of forward - 5,75 94,5 for long he overall gan 5,75 5

6 ervaves on nancal arke Problem. Prcng and Valuaon of Equy orward Conracs he sock s sellng for $, and wll pay a $, dvdend n 9 days and anoher $, dvdend n 7 days. A forward conrac s exprng afer 3 days. he effecve annual rsk-free rae s,%. he acual prce of equy fuure s 5,. Calculae he smple neres raes for hree maures. (b) Calculae he presen value and fuure value od dvdends usng smple neres rae and effecve annual rae (c) Calculae he far value of he equy forward usng smple raes, effecve annual rae and connuously compounded rae. (d) Calculae he mpled repo rae. Wha arbrage posons would creae hs rae? (e) eermne he forward prce and value days laer. he sock prce s $8,. (f) eermne he value of he forward prce a expraon dae. he sock prce s $,. oluon he smple raes 9 9,58% 7 9,7398% r ( ) 3 9,7785% 36 9,8565% 365 9,863% he connuously compounded raes ,53% hese raes are dfferen r c ln 7 9,53% ,53% 36 9,53% 365 9,53% he las wo raes are he same. c ln( ) 9,53% (b) he presen value of dvdends s equal o n r 36 n ( ) 3,8 3,8 he V of dvdends 4,3 δ 365 ln 4,7355% 6

7 ervaves on nancal arke (c) he far forward prce can be calculaed as: r 36-4, or [ ]( ) 4, ( ) 4, ( ) c δ or e 4, (d) he mpled repo rae s r - 36,954% r ,36% or -,36% or ln c,683% 365 r c ln 36,683% e,36% he arbrage poson would requre buyng a sock and sellng a forward conrac. (e) days laer. W - ( ) ( ) ( ) ( ) he sock prce 8, he presen value of dvdends s equal o,9 he presen value of forward prce 99,4 -,7 or ( ) ( ) - - W - ( ) ( ) ( ) ( ) ( ) ( ) (f) 8,83 W -,7 - -, 7

8 ervaves on nancal arke Problem 3. Prcng of Equy Index uures he equy ndex fuures prce s 74,. he ndex s currenly a 66,. he fuures conrac expres n 73 days. he rsk-free neres rae s 8,%. Calculae he "far" fuures prce on hs conrac. (b) Calculae he mpled repo rae. Wha arbrage posons would creae hs rae. oluon r he far prce s - 7,87 36 (b) 36 he mpled repo rae s r - 3,68% he nvesor should buy socks and sell fuures conrac. 8

9 ervaves on nancal arke Problem 4. Prcng of Currency orward he forward exchange rae s 4,64 zł/u. Lfe of he conrac: 78 days. Curren spo exchange rae s: 4,579zł/U. omesc rsk-free neres rae s 8,%, foregn neres rae s 6,%. Calculae he "far" fuures prce on hs conrac and he mpled repo rae. Wha arbrage posons would creae hs rae. (b) I s now 3 days laer. he spo exchange rae s 4,56. Calculae he forward prce and he value of he conrac. (c) Wha s he value of he forward conrac a expraon. he spo exchange rae s 4,5. oluon he "far" forward exchange rae s he mpled repo rae s d ( d) ( ) f ( ) f 4, ,688 he annualzed cos rae s 3,7% plus ransacon coss. An nvesor should. borrow foregn currency a 6,%,. sell currency a he curren spo rae. 3. nves local currency a 8,%, 4. buy cheap currency forward. 3,7% Ad. Ad 3. W f ( ) ( ) r r 4,559-4,5496 -,49 4,5-4,64 -,4 9

10 ervaves on nancal arke Problem 5. Prcng of Eurodollar uures Eurodollar fuures conrac wh 3 days maury s equal o 98,845. LIBOR for 3 day deposs s equal o,%, and for days deposs s,6%. Wha s he mpled LIBOR forward rae? (b) Comparng hs rae wh fuures rae, should nvesor buy or sell fuures conrac? (c) Wha s nvesor's rae of reurn for days f he borrows money for 3 days? and sells fuures conrac? oluon he relaon beween spo raes and forward rae s: Impled forward rae s : r f r 36,7% (b) He should sell, because fuures rae,55% s lower han,7%. (c) Reurn s: r 36 f ( ) 36 ( ) r r f ((,% * 3 / 36 ) (,6% * 9 / 36 ) - )( 36/ ) 36,47%

11 ervaves on nancal arke Problem 6. Bond uures Prcng he par value of a bond s %, coupon rae s,%, Y 8,7%. aury s 7,5 years. Ineres are pad semannually. he nex coupon paymen occurs n 4 days. he las coupon paymen was 4 days ago. he fuures conrac expres n 8 days. elvery facor s equal o,. he shor-erm annualzed rsk-free rae s 3,8%. Calculae he marke prce of a bond. (b) Calculae he far fuures prce and he mpled repo rae when he marke fuures prce s 96. oluon Ineres paymen Ineres paymen he bond prce wh accrued neres oday 8 aury of he fuures conrac P y B c y y y Bca /3 8/3 3,894,883 (b) he "far" fuures prce r Bc Bc P ( ) f Bc r 36 he mpled repo rae Bc Bc f r 365 P Bc r - P f 6,65 convered prce 5, nerm coupon paymen,5 accrued neres receved, neres from renvesng coupon paymen,83 proceeds receved P,88 cos of he nvesmen 7,7% mpled repo rae

Selected Financial Formulae. Basic Time Value Formulae PV A FV A. FV Ad

Selected Financial Formulae. Basic Time Value Formulae PV A FV A. FV Ad Basc Tme Value e Fuure Value of a Sngle Sum PV( + Presen Value of a Sngle Sum PV ------------------ ( + Solve for for a Sngle Sum ln ------ PV -------------------- ln( + Solve for for a Sngle Sum ------