University of Rome Tor Vergata Department of Economics and Finance (DEF) Derivatives (Gianni Nicolini) Exercises book

Transcription

1 Univesity of Rome To Vegata Depatment of Economics and Finance (DEF Deivatives (Gianni Nicolini Execises book

2 Execises Pemise In this teaching notes thee ae some execises on deivatives. The stuctue and the topics ae consistent with the final test (exam of the couse Deivatives. It does NOT mean that the exam will be composed only by execises epoted in the following pages. Hence the likelihood that execises that ae not epoted in the pesent pape will be included in the final exam is not null. The following execises have to be used in ode to self-assess knowledge and skills in managing deivatives. Execises ae oganized by topics.

3 Fowad Execise # You ae a tade of an investment bank and eading the epot on the Euopean stock maket that you eceived fom the analysts, you think thee ae some inteesting tading oppotunities. The epot suggests a bea tend of the maket that will be paticulaly stong afte 3 o 4 months fom today. You think that, afte this downtun, a new bull maket tend will come pobably afte seveal weeks. Today the shae of you investment potfolio in Euopean stocks eplicates exactly the Euostoxx 50 index composition. Today the Euostoxx 50 value is points and the value of you stock potfolio is equal to 50,000,000. The analysis of the maket suggest that will not be paid any dividends o othe cash flows to the shaeholdes fo all the companies included in the index. The inteest ates you ae allowed to invest o to boow ae listed in the following table: Matuity Rates month months months months months months months (inteest ates ae compounded and epoted on an annual base Within you investment stategies thee ae even one that is based on the use of fowad contacts to be aanged with anothe investment bank that agee to be you countepat in the fowad. In this scenaio you ae invited to: Estimate the fowad pice fo contacts with 5. 6 and 7 months of duation. Descibe the maket stategy that is coheent with you maket expectations. nd even eply to the following question if you change idea and you think that the maket epot is wong, in case you would like to hedge you position, do you think that a fowad could be still an options? In case, what should be diffeent in the fowad contact? 3

4 nswes Pat The fowad pices fo 5, 6 and 7 months of duation ae estimated in this manne Due the fact that the inteest ates ae compounded. you have to convet them as exponential Using the following fomula R ln( c m These ae the exponential inteest ates Matuity R compound (R m R exponential (R c R month % months % 3 months % 4 months % 5 months % 6 months % 7 months % then the fowad pice fomula can be used F * We know that: Se * T 5 months 6 months 7 months S = 5,000 5/ = 4.679% T = 5/ F=5.000 e 4.679% 5/ = = 5,95,95 S = 5,000 6/ = 4.677% T= 6/ F=5.000 e 4.677% 6/ = 5,354,90 S = 5,000 7/ = 4.663% T=7/ F=5.000 e 4.663% 7/ = 5,43,575 Pat Due the fact that you plan you stategy on a bea maket expectation, this is a tading (speculative stategy. If you potfolio eplicate the index and its value is 50,000,000, you know that: You have to sell stocks (if you think thei pices will go down you should sell it know Ente in a fowad as a selle (shot position, using contacts with 5, 6 and 7 months of matuity, whose dimension is equal to 50,000,000 Pat 3 In case of hedging (to minimize o set to zeo the potfolio isk a shot position on fowad (sell obligation should be chosen, holding the stocks in the potfolio. In this case the gains/losses on stocks will be pefectly balanced by the losses/gains due to the fowad. 4

5 Execise # You ae financial manage of a fashion holding company and the US ae you main maket. You woy about the /US$ exchange ate, because the gowing euo value affects the evenues of you companies. The macoeconomic scenaio makes you be even moe woy because you expect that the euo will become even stonge in the next months. So you decide to use a cuency fowad and you asked fo it to you bank. In 8 weeks fom now you will eceive a 300,000,000US$ payment fom the selling of the new collections and you decided to completely cove the cuency exchange isk. You know that the bid-ask quotes fo spot tansaction offeed by you bank ae /S.40 and /$.40 and you bank let you invest and boow money at the same inteest ates. These inteest ates ae the following Matuity Rates in Rates in US$ week 4.306% 4.806% weeks 4.300% 4.80% 3 weeks 4.39% 4.893% month 4.474% 5.533% months 4.66% 5.60% 3 months 4.79% 5.70% 6 months 4.788% 5.70% You task is to estimate the pice of the fowad you need fo the hedging. 5

6 nswe You need is to sell US dollas (the payment fom you customes to buy euo. So the fowad position is the one that make you sell dollas and buy euo. The exchange ate /$ is efeed to the amount of dollas elated to. It means that the isk (of losses fo you companies will be ealised if the exchange ates (the amount of dollas you need fo each euo will gow. To hedge you isk you position in the fowad must let you gain when the exchange ate will gow: this will happen in case of long position. You need to buy euo fowad. The exchange ate is the pice (in dollas of. It means that the exchange ate that you bank will apply to you is an ask quotation:.40. You know that the elation between exchange ates and inteest ates is the following: T F S e e F D T Whee F = inteest ate fo foeign cuency (dolla D = inteest ate fo domestic cuency (Euo You have to: Find the inteest ate with the same duation of fowad 8 weeks = months 4.66 and 5.60 Switch the inteest ates (fo both euo and dolla fom compound inteest ate to exponential inteest ate R / ( = ln( = / ($ = ln( = c ln( 3 Fill the spot exchange ate (S, the two months inteest ate in euo ( and the two months dolla inteest ate ($ in the fomula and each the esults S=.40 / ( = / ($ = T= / F=.46 R m 6

7 Execise #3 Conside a stock whose maket pice is 400 and includes the payment of a dividend of 5 in exactly six months. Estimate the fai fowad pice (equilibium pice of a tade in one yea, knowing that the spot inteest ates (isk-fee ae as indicated in the following table. Matuity Rates month months months months months months months months months months months 4.80 months 4.85 (Inteest ates expessed on an annual basis unde the compound inteest ssume now that the undelying asset does not pay dividends and, always efeing at the initial date, a boke is willing to ente as a countepaty to a fowad contact with a matuity of one yea, quoting a fowad pice of 40. Descibe (in details and with numbes the abitage stuctue that leads to a iskless gain. C What is the gain (amount in fom the abitage? D pat fom the numeical example, can you summaize the elements that make hedging by futues uncetain? 7

8 nswe to question It s known that F t ; t 0; T ( S I e 0 e T nswe S= 400 I = 5 / = 4.85 in exponential inteest = ln ( = / = in exponential inteest = ln ( = t = 6/ T=/ F = (400-5 e / e / = nswe to question In this case the fai fowad pice is povided by the following fomula: T F S e S= 400 / = 4.85 in exponential inteest = ln ( = T=/ F= 400 e / = Due to the fact that the maket pice is equal to 40, it s convenient to sell fowad. So the abitage to apply is buy spot + sell fowad The abitage steps ae listed below: t initial date To boow a cash amount equal to the sum of money needed to buy the lfa stock; To buy the lfa stock on the spot maket (paying its pice; 3 To sell fowad the lfa stock (shot position in a fowad contact. t the settlement day 4 To settle the fowad (to sell the lfa stock, eceiving the ageed pice; 5 Repayment of the loan (capital and inteests. Tade Cash flows in t 0 Cash flows in t Loan e / = Stock puchase (to buy spot Stock sell (to sell fowad 0 +F * 40 Total =

9 Fowad Rate geement (FR Execise #4 It is assumed to be at the Decembe st, 007. On this day the maket zeo-coupon tem stuctue is following (system of simple capitalization. Matuity Rates 3 months 3.00% 6 months 3.50% 9 months 3.90% yea 4.40%.5 yeas 4.90% yeas 5.40% 3 yeas 5.90% tade, expecting an incease in the spot inteest ate (with yea matuity in the next yea bigge than the maket expectation implied by the tem stuctue, decides to ente into an FR with a notional of 5,000,000. List the contact tems/featues of the FR and estimate the FR ate unde the assumption of a no abitage oppotunity maket. ssume that on June st 008, the same tade decides to close his/he position. On this date, the maket zeo-coupon yield cuve (linea capitalization of inteest is the following: Matuity Rates 3 months.50% 6 months 3.00% 9 months 3.0% yea 3.40%.5 yea 3.90% yea 4.40% 3 yea 4.90% Descibe the two opeations that (at June st the tade can do in ode to close his/he position befoe the contact deadline (settlement day. Moeove explain the following statement "The two ealy closue tades should be consideed absolutely equal". C Identifies the gain / loss eaned by the investo in case of ealy closue. 9

10 nswe (0; = 4.40% (0; = 5.40% T = yea T = yeas (; = % T, T nswes ( ( 0, T 0, T T T T T nswes -C (0; 6/ = 3.00% ( 0; 8/ = 3.90% T = 0.5 yeas T =.5 yeas (0.5;.5 = % The two tades ae the following: To sell a FR 6 8 (Tying to balance the oiginal position with an opposite one To close (ealy the oiginal FR poposing an ealy closue to the countepat, knowing that V buye FR( k; T ; T FR(0; T ; T ( k; k; T The value of the contact on the evalutation date is equal to FRmaket = 4.86% FRoiginal = 6.3% (k;k;t = 3.90% (spot ate in t k fo matuity to T = 8 months Capital = 5,000,000 T -T = yea T -k =.5 V buye = -87,5,63 Capital ( T ( T k T 0

11 Execise #5 ssume the the intebank maket ates ae the following: Libo ates (matuity id quotes sk quotes month 3.60% 3.65% 3 months 3.77% 3.8% 6 months 4.0% 4.07% 9 months 4.46% 4.53% months 5.% 5.8% t the same time assume that a financial institution (e.g. bank poposes the following maket ates fo FRs: FR id sk quotes quotes FR % 4.35% FR % 6.95% Check the maket ates looking fo abitage oppotunities. In case an abitage oppotunity exists, descibe the abitage steps to be done in ode to gain money without isk. C Estimate the abitage gain unde the assumption that the notional value is equal to 0,000,000.

12 nswes Checking the maket ates we can ealize that an abitage window exist fo the FR 9. Looking at FR 3 6 ates, the limit fo the bid and ask quotations ae the following: T, T ( ( ( T id quote 0, T 0, T T T T T, T sk quote ( 0, T T ( 0, T T ( T T 6 ( 0,6 ( 6 0,6 ( 3 0,3 ( 3 0,3 3;6 ( 6 3;6 3 ( 3 6 ( 0, ( 0,040 6 ( 0, ( 0, ; % ( 6 3; % 3 ( 3 6 Due the fact that both the ate limits needed fo an abitage ae not eached ( 4.5% 3;6( maket 4,33% is false, as 4.35% 3;6( maket 4.6% is false too thee is no abitage windows on the maket. Looking at FR 9 ates, the limit fo the bid and ask quotations ae the following: T, T 9; ( ( ( T id quote 0, T 0, T T T T ( 0; ( 9 0;9 ( 9 ( 0,058 ( 0, ( 9 9; 7.49% T, T 9; sk quote ( 0, T T ( 0, T T ( T T ( 0; ( 9 0;9 ( 9 ( 0,05 ( 0, ( 9 9; 7.05% Hee we have an abitage window. If the bid-quote limit is not eached ( 6.85% 9; 7.49% if false the ask-quote limit has been ovepassed ( 6.95% 9; 7.05% is tue opening an abitage window.

13 The stuctue of the abitage is the following: t the beginning 9 months loan at ask-quote ate of 4.53%; months investment at bid-quote ate of 5.%; 3 Puchase of a FR 9 at ask-quote ate of 6.95%. 9 months late 4 Refinancing the initial loan fo 3 moe months; 5 Moneys fom the new loan ae used to pay back the initial loan (fom 0 to 9 months. months late (fom the beginning 6 The long-tem (0 to months investment is ove: both capital and inteest ae available; 7 The (newloan is ove (both capital and inteest have to be paid. Summaizing Tades Cash flows in T 0 Cash flows in T Cash flows in T Shot tem loan (fom 0 to 9 months + (+ 0,T T ( Long tem investment (fom 0 to months + (+ 0,TT + ( Long position on FR T T 0 + (+ 0,TT + ( {(+ 0,TT [+ T,T (T -T ]} Total 0 0 {( [ (- 0.75]} + (+ 0,TT ({(+ 0,TT [+ T,T (T -T ]} + (+0.05 {( [ (-0.75]} + (+0.05 {( [ (-0.75]} = ssuming that the notional value of the FR is 0,000,000, the abitage gain will be equal to: Gain = 0,000, =,600 3

14 Swap Execise #6 n investment bank issued a floating-ate bond with a seven yeas matuity. Its annual coupons ae linked to the months LIOR and the nominal value of the issues is 500,000. Thinking that the inteest ates will ise, the bank wants to ente in a liability swap that is consistent with its expectations. Questions List the contact tems of the Inteest Rate Swap that should be taded by the bank, specifying its position (long position on fix ate o on the floating ate. With the above Inteest Rate Swap in mind and using the following zeo-coupon ates Matuity Rates 5.00% 5.50% % % % 6 7.4% % % % % Estimate the IRS fix ate that should be used in the contact, epoting all the steps needed to estimate this value. 4

15 nswes nswe The swap featues ae the following: Matuity: 7 yeas Notional value: 500,000 3 Fix ate paye position (Floating ate eceive position 4 nnual fequency nswe Knowing that R fix _ swap ( 0; n ( n i 0; i T T n i R fix_swap = % 5

16 Execise #7 ssume an IRS with a notional value of 00 millions with semi-annual payments at a 8% fix annual ate (linea inteest, balanced by a 6-months LIOR payments. ssuming that ( the swap will end afte yea and 3 months, ( that Libo ates ae the following Matuity Rates 3 months 0.00% 9 months 0.50% 5 months.00% and (3 that the 6-months LIOR ate at the last settlement day was equal to 0.0%, estimate the IRS value fom the floating ate paye point of view. nswe We know that the value of the IRS fom the floating ate paye pespective is equal to V fix va So we need to estimate the bond values. 4,000,000 4,000,000 04,000,000 fix 0,5 0,75, ( 0.0 ( 0.05 ( ,906,000 3,7,000 9,8 98,898, ,000,000 00,000,000 va 0. 5 ( ,000,000 5,00, ( ,00, ( ,65,000 fix =98,898,000 va =0,65,000 Hence V = 98,898,000 0,65,000 = 3,77,000 6

17 Options Execise #8 It is Januay st, 008. The maket povides the following options: Option type Stike pice Matuity Pemium Call 0 Mach Call Mach Call 4 Mach Put 0 Mach Put Mach Put 4 Mach ll the options ae efeed to the same undelying asset. ssume that the inteest ate stuctue on the maket is flat and equal to 4.50% (compound inteest. If the maket pice of the undelying asset is equal to.50 and options ae taded on a egulated maket (exchange that follows the cycle of Mach : calculate the limits (uppe and lowe limits fo the pemiums fo both call and put options Check the consistency of maket pemiums with thei limits. C Check the pemium consistency between option types (consistency between calls and consistency between puts Use civil calenda (365 days pe yea (TT. Remembe that 008 was a leap yea Mach 008 MON TUE WEN THU FRY ST SUN

18 nswes nswe Estimate the exponential inteest ate fom the compound inteest of 4.50% R R exponential = ln ( = % c ln( R Option matuity (days until the expiation day is 80 days (3Jan + 9 Feb + 0 Ma = 80 days 80/365 = 0.9 m We know that c S p X e X e ( T t S c S p ( T t X So Type Stike pice Matuity Pemium Pmin Pmax Call 0 ma Call ma Call 4 ma Put 0 ma Put ma Put 4 ma Due the fact that a pice (pemium can not be negative Type Stike pice Matuity Pemium Pmin Pmax Call 0 ma Call ma Call 4 ma Put 0 ma Put ma Put 4 ma nswe The Call0 pemium fo Mach 008 is not consistent with the pemium limits. The lowe limit should be.69, while the maket pemium is.50. Even the Put4 pemium is not consistent with the option picing pemium limits. 8

19 nswe C The pemium fo the Put4 (Mach 008 is.0. It is not consistent with the values Put and Put0. In fact, given that a put with a stike pice of 4 is moe likely to be used, this option should be paid moe than a put with a stike pice of o 0 that have a lowe pobability to be in the money. The same aguments can be used compaing the Put with the Put0: even this time the two pemiums ae not consistent. 9

20 Execise #9 Point out which of the following stategies ae consistent with expectations of high volatility fo the undelying asset pice, if thee is no expectation about the diection of the maket (bull o bea maket: Naked call (long position Staddle 3 Stangle 4 Revese buttefly spead (o shot buttefly spead 5 ull-spead Fo each of them descibe the stuctue of the stategy and why it is o not consistent with the above mentioned expectations. nswe The stategies # (staddle and #3 (stangle ae the only ones consistent with the high-volatility expectations. Staddle should be pefeed when the tade has a stong belief on his/he expectations and when he/she own enough financial esouces (money. Stangle need less money but will povide (ceteis paibus lowe esults (lowe gains o highe losses. Stategy # (long call naked is not the ight option. If it will make the tade gain moneys when the maket ise, it will lead to a loss in case of downfall of the maket. Stagegy #4 (evese buttefly spead is not appopiate because in an high volatility scenaio will povide a loss. Stategy #5 is not appopiate. If the tade will benefit fom this stategy when the maket ise (even if the gain value will be capped, he/she will povide a loss when the maket will fall. bout the stuctue of the stategies: Naked call (long position: a. call (long position Staddle: a. call (long, stike pice S b. put (long, stike pice S 3 Stangle a. call (long, stike pice S b. put (long, stike pice S... whee S <S 4 Revese buttefly spead (o shot buttefly spead a. call (long, stike pice S b. calls (shot, stike pice S c. call (long, stike pice S 3... whee S <S <S 3 0

21 5 ull-spead (we assume that the bull spead is stuctued by call options a. call (long, stike pice S b. call (shot, stike pice S... whee S <S