INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on he fron page of answer bookle and insrucions o examinees sen along wih hall icke carefully and follow wihou excepion 2. * ou have 15 minues a he sar of he examinaion in which o read he quesions. ou are srongly encouraged o use his ime for reading only, bu noes may be made. ou hen have hree hours o complee he paper. 3. ou mus no sar wriing your answers in he answer shee unil insruced o do so by he supervisor 4. he answers are no expeced o be any counry or jurisdicion specific. However, if Examples/illusraions are required for any answer, he counry or jurisdicion from which hey are drawn should be menioned. 5. Aemp all quesions, beginning your answer o each quesion on a separae shee. 6. Mark allocaions are shown in brackes. A HE END OF HE EAMINAION Please reurn your answer book and his quesion paper o he supervisor separaely.
Q. 1) A deposi insrumen offered by a bank guaranees ha invesors will receive a reurn during a one-year period ha is greaer of zero and 30% of he reurn provided by NSE Nify index over he one-year period. An invesor is planning o pu Rs. 5,000,000 in he insrumen. (a) Describe he payoff as an opion on NSE Nify index. (1) he risk-free rae of ineres is 6% per annum (wih coninuous compounding), dividend yield on NSE Nify is 2% per annum (wih coninuous compounding) and he volailiy of NSE Nify is 20% annum (wih coninuous compounding). NSE uses Black-Scholes formula o price he opions on NSE Nify index. (b) Wha is he value of he insrumen? (5) (c) Explain wheher or no he insrumen is a good deal for he invesors when he/she compares he minimum reurn from he produc wih a risk-free invesmen? (2) Q. 2) Consider he following wo porfolios: Porfolio A: One long European call on NSE Nify wih exercise price 100 One shor European call on NSE Nify wih exercise price 150 One long European call on NSE Nify wih exercise price 200 One long European pu on NSE Nify wih exercise price 100 One shor European pu on NSE Nify wih exercise price 150 One long European pu on NSE Nify wih exercise price 200 Porfolio B: One long European call on NSE Nify wih exercise price 150 One long European pu on NSE Nify wih exercise price 150 a risk-free deb of face value 50 mauring a ime. he deb is redeemed a face value on mauriy. All he opions maure a ime. (a) Make a able showing he payoff of each porfolio as a funcion of Nify value a ime ( S ). (4) (b) Make a able showing he payoff of each porfolio for he following values of NSE Nify a ime : S = 0,100,150, 200. (2) (c) Which porfolio requires greaer iniial oulay o esablish? (2) [8] r (d) For an arbirage free marke, prove ha c1 2c2 c3 25e. Where c 1, c 2 and c 3 are he prices of he European call opions on NSE Nify a ime 0 wih exercise prices 100, 150 and 200 respecively and mauriy ime. r is he risk-free rae of ineres wih coninuous compounding. (3) Page 2 of 6
Q. 3) Consider a risk-free asse B and wo socks, and, where db rb d, B 0 1 ds S d S dw, S 0 s ds S d S dw, S 0 s Where W is a P-Wiener process. Assume ha he wo socks pay coninuous dividend where he dividend processes are given by: dd dd S S d d (a) Using Girsanov s heorem, derive he sochasic differenial equaions of S and S under Q-Wiener process (risk-neural measure). (5) (b) Consider a conrac wih mauriy ime where an invesor a ime receives he amoun D D, where 0. If D D is posiive, he invesor receives D D and if D D is negaive, he invesor pays D D. Deermine such ha he price of he conrac is 0 a ime 0. (5) Q. 4) Consider a sandard Heah-Jarrow-Moron (HJM) forward rae model under he risk neural maringale measure Q of he form: df(, ) m(, ) d s(, ) dz( ) z () is assumed o be a Q-Wiener process. (a) Explain he HJM drif condiion. Wha does i say? Why is i needed? How is i derived (only qualiaive answer is needed)? (3) (b) Given he above for ward rae dynamics, derive he dynamics of shor rae under Q. Does he HJM drif condiions imply resricions on he parameers of he shor rae under Q? (8) Q. 5) Consider a sandard Black-Scholes marke, ha is, a marke consising of a risk-free asse, B, wih P-dynamics given by db( ) rb( ) d B(0) 1 [10] Page 3 of 6
and a sock S, wih P-dynamics given by ds( ) S( ) d S( ) dw ( ) S(0) s(0) Here W denoes a P-Wiener process and r, and are assumed o be consans (a) Verify wheher he porfolio wih holding unis in sock S a ime defined by h ) B ( unis in risk free asse B and h ( ) S h( ) B S S( ) B( ) h( ), h( ), B( ) S( ) is self financing or no. (4) (b) Deermine wheher he following process represens a radable asse or no 2r ( ) S( ) where 2 (4) [8] Q. 6) A sock price is currenly Rs.1000. Over each of he nex wo 1-year periods i is expeced o go up by 10% or down by 10%. he probabiliy of an up move is 0.55 under he original P-measure. he risk-free ineres rae is zero. (a) Compue he price of an up-and-ou pu opion (barrier opion) wih barrier H = 1050 and exercise price K = 1050. (2) (b) Find he replicaing porfolio for he opion in (a) and verify ha he porfolio is selffinancing. (4) Q. 7) Suppose ha he prices of zero-coupon bonds wih various mauriies are given below in he able. he face value of each bond is Rs. 1000. Mauriy (ears) Price (Rs.) 1 934.5794 2 869.3715 3 804.9606 4 741.8753 5 680.5831 [6] (a) Suppose ha an invesor buys oday (a ime = 0) hree-year mauriy zero coupon bonds wih face value of Rs. 10 million. How many five year mauriy zeros would you have o sell o make your iniial cash flow equal o zero a ime = 0? (2) (b) Wha are he cash flows on his sraegy in each year? (2) (c) Wha is he effecive wo-year ineres rae on he effecive hree-year-ahead forward loan of principal value of Rs. 10 million? (2) Page 4 of 6
(d) Confirm ha he effecive wo-year ineres rae equals 1 f )(1 f ) 1. ( 4 5 Where f 4 is he forward rae per annum (wih annual compounding) beween he years 3 and 4 and f 5 is he forward rae per annum (wih annual compounding) beween he years 4 and 5. (3) [9] Q. 8) (a) Prices of long-erm bonds are more volaile han prices of shor-erm bonds. ield o mauriy of shor-erm flucuaes more han yields of long-erm bonds. How do you reconcile hese wo observaions? (2) (b) A fixed income porfolio manager is unwilling o realise a rae of reurn of less han 5% per annum (wih annual compounding) over a six-year invesmen period currenly valued a Rs. 10 million. Four years laer, wo-year zero rae is 7% per annum (wih annual compounding). Wha mus be he value of he porfolio a his ime such ha he porfolio manager is assured of achieving he minimum possible reurn? (2) (c) In wha way is owning a corporae bond similar o wriing a pu opion? A call opion? (2) (d) ou will receive bonus nex monh ha you hope o inves in long-erm corporae bonds. ou believe ha bonds oday are selling a quie aracive yields, and you are concerned ha bond prices will rise over he nex few weeks. How migh you use financial fuures o hedge your risk? (2) (e) Suppose ha he spo price of one US dollar is Rs. 45. he one year fuures price is Rs. 46. Is he ineres rae higher in India or in Unied Saes? Explain. (1) (f) wo bonds have idenical imes o mauriy and coupon raes. One is callable a 104, he oher a 108 on he same call dae. Given all he oher facors idenical, which should have he higher yield o mauriy? Why? (2) Q. 9) Alpha has jus purchased NSE Nify fund currenly selling a Rs. 5900 per share. o proec agains losses, Alpha also purchased an a-he-money European pu opion on he fund for Rs. 110, wih hree-monh ime o expiraion. Gamma, Alpha s financial adviser poins ou ha Alpha is spending a lo of money on he pu. He noes ha 3-monh pus wih exercise price of 5800 cos only Rs. 70, and sugges ha Alpha use he cheaper pu. (a) Analyse Alpha s and Gamma s sraegies by making ables showing he profi for sock-plus-pu posiions for various value of sock funds in hree monhs. (3) (b) When does Gamma sraegy do beer? When does i do worse? (2) (c) Which sraegy enails greaer sysemaic risk? (1) [6] Page 5 of 6
Q. 10) (a) Should researchers use real-world or risk-neural defaul probabiliies for (i) calculaing credi value a risk and (ii) adjusing he price of derivaive for defauls? (2) (b) A long forward conrac subjec o credi risk is a combinaion of a shor posiion in a no-defaul pu and long posiion in a call subjec o credi risk. Explain his saemen. Assume ha defauls happen only a he end of he life of he forward conrac. (4) (c) he posiion of a buyer of a credi defaul swap is similar o he posiion of some one who is long a risk-free bond and shor a corporae bond? Explain his saemen. (3) (d) Explain he srucure of a CDO. (2) Q. 11) (a) Suppose ha European call opion on Infosys sock wih ime o mauriy hree-monhs and srike price Rs. 3000 are selling a an implied volailiy of 28% per annum. Infosys sock currenly is Rs. 3000 per share, and he risk-free rae per annum (wih coninuous compounding) is 6%. If you believe he rue volailiy of he sock is 30% per annum, how can you rade on your belief wihou aking on exposure o he performance of Infosys. How many shares of sock you will hold for each opion conrac bough or sold such ha i will give you profi when he opion prices come ino alignmen? (3) (b) Using he daa in problem 11(a), suppose ha hree-monh pu opions wih a srike price of Rs. 3000 are selling a an implied volailiy of 32%. Calculae a dela neural porfolio comprising in calls and pus ha will profi when he opion prices come back ino alignmen. (3) (c) he curren risk-free rae of ineres is 6% per annum (wih coninuous compounding). he 3-monh fuures price for Infosys sock is Rs. 3045, where as he 6-monh fuures price is Rs. 3100. Is here an arbirage opporuniy here? If so, how would you exploi i? (3) ************************* [9] Page 6 of 6