Forwards and Futures

Transcription

1 Handou #6 for Spring 2002 (lecure dae: 4/7/2002) orward Conrac orward and uure A ime (where 0 < ): ener a forward conrac, in which you agree o pay O (called "forward price") for one hare of he ecuriy a delivery ime. A ime : exchange O for one hare of he ecuriy. Remark: () I co nohing o ener a forward conrac. (2) O i no he price for he forward conrac. (3) O i no a coningen claim. ind O uch ha he price of he forward conrac i zero. * he "fair" forward price a ime i: Proof: O = [ / ] S = S * Mehod : ime price S O = = 0 Mehod 2: Wha aumpion are required for hi o be he fair price?

2 * When he inere rae are deerminiic O = S (+ r + ) (+ r ) (if conan rae r hen, O = S (+ r) - * Relaion o "Pu-Call Pariy": for a pair of uropean call and pu wih he ame expiraion ime and rike price e c - p = S - e [ / ], where = 0,,, If he rike price i he ame a he forward price a ime, hen he call and pu have he ame price a ime, ha i e = O c = p for any = 0,,,. Sock Pay Dividend Holder of he forward conrac doe no ge any dividend before ime * +. If he ock pay dividend hen he dicouned ock price S i no a maringale. Recall ha ock pay dividend a ime given by D whoe D dicouned value i equal o. he cumulaive dividend paymen: D = = D he cumulaive dicouned dividend paymen: D * = = D. hu, he um, * * D S + i a -MG. 2

3 ind he forward price O a ime uch ha he ime price of he forward conrac i zero. Aume ha he marke i complee. Claim: Proof: * D O = S = + Special cae: Inere rae i r and dividend i δ. Wha happen if he underlying i an ae ha can no be hored? xample: 4.4 Conider a wo period model wih Ω ={ω, ω 2, ω 3, ω 4 } and an ae wih ime 0 price A 0 = 00. a) If he bank accoun proce i deerminiic according o = (.05) for = 0,,2, and if he ae can be old hor hen wha i he ime 0 forward price O 0 of he ae for delivery a ime 2? b) If he bank accoun proce i random wih =.05, 2 (ω )= 2 (ω 2 ) =.2, 2 (ω 3 )= 2 (ω 4 )=.0, hen wha i he ime 0 forward price O 0 of he ae for delivery a ime 2? ind an expreion in erm of ({ω,ω 2 }). (Or eaier aume ha ({ω,ω 2 }) = 0.. c) Same a par (b) bu he ae can no be old hor. Wha can you ay abou he forward price a ime 0? d) Same a par (c), only now he ae incur a carrying charge of $5 per period. 3

4 uure Conrac uure conrac are ame a forward conrac excep: () raded on organized exchange (e.g., CO, NYMX) (2) a fuure conrac can be cloed ou by aking an oppoie poiion of he ame conrac in fuure marke. (3) uure price are adjued a he end of each rading day (marking o marke). Raionale Model You are given, and γ he price of he underlying ae a ime. 0 = iniial ime U = fuure price a ime If U > U - buyer receive he amoun U - U from he eller If U - > U eller receive he amoun U - - U from he buyer (for laer reference, wlog, buyer alway receive he amoun U - U from he eller) A ime, afer paying he U - - U o he eller and he underlying price γ receive he ae. Noice ha ideally U equal γ oherwie here i arbirage. How o compue U? 4

5 hink of fuure a a ecuriy ha ha oday price = 0 and pay a dividend of U - U - a ime. hu, wha we need i: = U U i a -MG. hi require: U U = 0 for = 0,,2,...,. hi i a difficul condiion o work wih. Suppoe inead ha i - meaurable -- ha i i known a ime (-). In uch a cae, we ay ha i predicable. Wha i he implicaion? hu, we obain ha he fuure price i a MG. hi i he fir iuaion in he coure where we ee ha an undicouned proce i a MG. Summary he fuure conrac i e up o ha i ha no value. We can cloe he accoun a any ime. If he underlying i he ock price and i predicable (r i known a ime -): U = ( S ). * he fuure price of a ecuriy a he delivery ime i he ame a he po price of he ecuriy, i.e., U = S. 5

6 6 * If inere rae are predicable hen he fuure price a ime (where 0 < ) i U = [U + ] * When he inere rae are deerminiic S U = (+ r + ) (+ r ) Remark: in uch cae, forward price and fuure price are he ame. * In general () U O when r and S are negaively correlaed ( ) U S S S O = = = (2) U O when r and S are poiively correlaed xample: ω 8 6 ω ω 3 3 ω 4 Suppoe =, 2 (ω,ω 2 ) = 7/6, 2 (ω 3,ω 4 ) = 9/8.

7 Opion on uure Price y reaing a fuure conrac of he underlying ecuriy a anoher (regular) ecuriy, uropean or American opion on he fuure price can be creaed and raded. he ame rik neural valuaion can be ued for pricing uch an opion, and a replicaing porfolio can be conruced uing he bank accoun and he underlying ecuriy (no he fuure conrac). ix ome ime : 0 < <=. ime- price: X, for all <=. Wha i he -meaure? Suppoe ha he fuure price follow a binomial model wih parameer, u, d, r. Le he CC be: (U - e) + hen: q = (-d)/(u-d), q ˆ = qu /( + r) (ame a before) ( + r) ( U e) + ~ e = U ~ 0 Pr{ X n } Pr{ Y n }; X ~ (, qˆ}, Y ~ (, q). ( + r) xample. N =, = 2, S 0 =, u =.2, d = 0.8, r = 0.04, r 2 = 0.02 if S =.2, and r 2 = 0.06 if S = 0.8. (a) Wha i he forward price proce wih delivery ime 2? (b) Wha i he fuure price proce wih delivery ime 2? (c) Wha i he ime 0 price of a call opion on he fuure price a in (b) wih expiraion ime and rike e =? 7

8 Soluion o xample: 4.4 Conider a wo period model wih Ω ={ω, ω 2, ω 3, ω 4 } and an ae wih ime 0 price A 0 = 00. a) If he bank accoun proce i deerminiic according o = (.05) for = 0,,2, and if he ae can be old hor hen wha i he ime 0 forward price O 0 of he ae for delivery a ime 2? O 0 = A 0 (+r) 2 = 00 (.05) 2 = 0.25 b) If he bank accoun proce i random wih =.05, 2 (ω )= 2 (ω 2 ) =.2, 2 (ω 3 )= 2 (ω 4 )=.0, hen wha i he ime 0 forward price O 0 of he ae for delivery a ime 2? ind an expreion in erm of ({ω,ω 2 }). (Or eaier aume ha ({ω,ω 2 }) = 0.. O 0 = A 0 / [/ ] = 00 0.x + 0.9x.2.0 = 0.97 c) Same a par (b) bu he ae can no be old hor. Wha can you ay abou he forward price a ime 0? d) Same a par (c), only now he ae incur a carrying charge of $5 per period. Carrying co i like he oppoie of dividend. So he forward price change o: * D O = S = = x price hould be? x = hu he forward 8

9 xample on page 6: ω 8 6 ω ω 3 3 ω 4 Suppoe =, 2 (ω,ω 2 ) = 7/6, 2 (ω 3,ω 4 ) = 9/8. 5 = (w,w 2 ) 8 + (w 3,w 4 ) 4! (w,w 2 ) = ¼, (w 3,w 4 ) = ¾ 8 = [((w )/(w,w 2 )) 9 + ((w 2 )/((w,w 2 )) 6) 6/7! ((w )/(w,w 2 )) = 5/6! (w ) = 5/24 and (w 2 ) = /24 Similarly, 8 ( w ) 9 ( w3, w4 ) ( w ) ( w, w ) ( w ) ( w, w ) = ==> = 0. 5! (w 3 ) = (w 4 ) = 3/ uure Price /24 x 9 + /24x6 + 3/8x6 + 3/8x3 = 5.5 5/6 x 9 + /6x6 = x x 3 = ω 6 ω 2 6 ω 3 3 ω 4 9

10 orward Price 0 2 5/(/4 x 6/7 + ¾ x 8/9) = 5/(4/7 + 2/3) = 5/(46/5) = 255/46 = /(6/7) = ω 6 ω 2 4/(8/9) = ω 3 3 ω 4 0

11 Soluion o opion of fuure (page 7) Price proce for ock q = q = q = Where, 0.60 = ( )/(.2-0.8) 0.55 = ( )/(.2-0.8) 0.65 = ( )/(.2-0.8) a) orward price proce Where:.224 =.2 x.02; = 0.8 x = /(0.6 x (/.02x.04) x (/.06x.04))

12 b) uure price proce = 0.55 x x = 0.65 x x =.44 x 0.6 x x 0.6 x x 0.4 x x 0.4 x 0.35 =.0736 (alo equal o:.224 x x 0.4 ) c) Price proce of call opion, expiraion ime =, rike e = = (0.224 x x 0.4)/.04 he call opion i aainable. here are wo poible S. One uing bank accoun and ock. he oher uing bank accoun and fuure. S () H = ( , 0.56) co = = S(2) H = ( 0.292, ) co = (a fuure co nohing) 2