Reading 30: Pricing and Valuation of Forward Commitment

Page No 5519th Feb 2023 SundayDate//Reading 30 : Pricing And ValuationOf Forward CommitmentFoundation ConceptsNo. - arbitrage principleThere should be no riskless profit from combining forwardCor future) Contract with other instrument.Zero Value PrincingForward contracts are priced such that the Contract hasZero value at initiation (preventing Rf arbitrage)Pricing a Forward Contract- Settlement price for the underlying assetThe most basic cost of carry is (periodically compounded ) Rf rateCosts of corry increase the forward priceForward Price = So X C +Rf)Spot pliceTwoValue of forward / future at initiation is zeroInvestment = 0fundamental rules for arbitrage is ZeeExposure to market risk =0 Page No.56Date//Example :Three - month forward on a Zelo- coupon bondCurrently selling for $ 500 ( $1000 fare)Rf rato is 6xSpotWhat is forward priceAny a then priceFP = $500 X (1.06)Ht = $ 507.34 arbitiage opportunity& there will beExample 2 :FVof130135.6short110- LongSpot price - 100RfR = 10%Fr of 100130.00Maturity = 1yr134as25.05S= 100105.88is 10588S=130t2524.11long = 11029.1100.01PVis134.051Price of the Contract (0.6) = 130 X (1+10x) 0.4 = $134.05Value of the contract at expiration (1) = 135-110 = $ 25.05Value of the Contract today (0.6) = 25 = $24.11CHOOPY*Cash&CarryArbitrageThis Suppose the quotes forward price is $ 510 (tooTo exploit would this allow liskless arbitrageSellhigh)mispricinguseaCash- -Buy in the Cash market & Carry to the and forward Corry urbitrage marketExampleToday :- Borrow $ 500 bond at 6% for 90 days (FV = $ 507.34)Buy the for $ 500Short the forward (contract to sell for $ S10) Page No 57Date//In 90 days:- Deliver the bond & receive $510Repay the loan with $ 507.34Arbitrage profit = $2.66Reverse Cash & Carry ArbitrageWhat if the price is too low?Instead of a forward price of $ 510 Suppose it is $502This would also allow riskless arbitrage.BuyExampleToday :- Short the bond (receive $ 500)Invest proceeds at 6% for 90 days (FV = $ 507.34)In 90 days:- Receive $ 507.34 from invest"Buy the bond for $ 502 ; Cover the shortArbitrage profit $ 5.34Arbitrage Opportunity4 FP > Sox (1+R()If FP < So X (1+R6)Forward is overplicedForward is under pricedCash & Carry arbitiageReverse Cash & Curry arbitrageBorrow So moneyBorrow assetLong assetShort assetShort forwardlend moneyaRfProfit = FP So X (1+R/)TLong forwardProfit So X (1+Rf) - FP alongforwardContractduringthelifeoftheThe no -arbitrage value ofContractPVof forward pricePage No 58Vt during of long life position of Contiact) = St - [ (HR6)T-t FP]DateValue of the Short position =Ve(Because its a zero sum game, byofthePositionValue aforwardCorliactlwin 4 you loos)Value of long position at initiation = 0- Because Contract is priced to pevent arbitrageValue of long positions at maturity = ST - FPValue of long position prior to maturity = PV (ST) -PVCFP)Contract price underlying pricePricingEquityForwardContractWe can determinethe value of forwardInterim CF (eg div € Coupons) are ConsidContract during itslife time.Benefit Essential of + they coiry offset the costs forward of price carry.As the underlyingCollyContract Price chang will the forwedvaleaccuratePricing equity forwards - formula :Long(Buy) : Profit of price?10 PV of DivFP (on equity Security ) = ( S. - PVO)Short : Profit Phielx(1FP (on equity Sec) = [ So- (1+R6)-FVDBothversionsaremath ematical-bothequivalentreduceforwardpricebyvalueUse # daysofdiv365ExampleCalculatethenoforward on a to stock that is currently priced forarbitrageforwardpricea100-day& is expectedaat $ 30payadiva$0.9in85 days&$0.5in175daysis 15day , $60 $0.4The annual Rf is 5% , & the yield curve is flat Page No 59Date//ool Note- The find div (175 days) occurs after maturityand hence is not relevant.PVD= 0.4 + 0.40 = $0.79461.05 15/3651.05 85/365FP= $30 -$0.7946] X 1.05.00 = $29.6What about the value of an equity forwardafter initiation?Formula :-Vt (long Position) = (St - PVD6)-FP(1+Rf) (T-t)*Volving Equity Forward -ExAfter 60 days the value of the stock in the previous exis $36 Calculate the value of the equity forward contractassuming the risk-free rate is still 5% & the yiddCurve is flatNote : There is only one div remaining (25 days)15d85dPVD60 = 0.4 = 0.3987tt/1.0525/36525GodSoln V60 (long) = (S60 - PVD6).-FP(1+1) (40/366)V60 (long) = ( 36 -0.3987) -Los 4013652966 Page No SoDate/*Pricing Index Forward ContractsMain point : Viewed as paying a continous divEquity index = Basket of many StockIn this Case, the continuous div is an ffset to the lostof Carry-Think: Borrowing cost takes money away continousdiv bring some of it backFormula: :-FP Con equity index) = So X e(Rf-5')xs= ( 5.XTRf'xTXeXRf- Continously Compounded RfReS' - Contineous diu yield.ExampleSoCalculate the RfR is 4.6% & the Contineals div yield is 2.1%The Commpounded value of the S & P 500 index is 1,140 The continuouslycontract on the no-arbitrage Index price of a 140 day forward ScTFP= 5.x e (Rfc s') xTFP 1140 X e(0.046 x (160)365= 1151 Page No 61Date//Forward Contracts on Fixed Income SecFixed - Income sec typically pay interim CFs to the holderAs with div-paying stocks, these CFs offset the cost ofCarry & therefore + reduce the future prices.Key paiat: Princing is analogous to equity : replace PVDCPV of div ) with PVG CPV of coupon)Formal:FP (on Fi sec) = ( S. PVC) X (1+Rf) T= S. x (1+R) - FVCThe value after initiation isVt Clong position) = C SE - PVCE) -FP(1+Rf (T-t)ExampleanotherwiseCalculate semi annual the entil price of a 250-day forward contract on a 7%us. Treasury bond with a spot price of $1500 S. (includingCouponaccused int ) that has just paid a Coupon & will make anotherCoupon pay" in 182 days The annual RfR is 6%Note: Treasureies make semiannual Coupon pay"C= $1000 X 7% x 0.5182 d= $35250 dPVC = $35 = $391.06192/365250FP = ( S. PVC) x (1 +Rf) 365FP = [$1050 -$34 Tx 1.06250/365=$ 1,057.37 Page No. 62Date/After 100 days the value of the bond in the previousexample is $1090 Calculate the value of the forwardContract cn the bond to the long position assumptionthe annual R/R is 6%100d182dNote ; There is only one coupon pay" remaining beforematurity (82 days)PVC =35= $34.541.06 82/365Sol" Vellong) = ( 5100 - PVC100) -FP(ItRj)(150/365)Vellong) = C 1090 $34,54)-1057. 37]= $23.111-06 (LER)Bond Futures Contracts Value of Future reset to O after MtoMIgnoring Bond mark - to - market future are the same as forwardin which case, they are priced based on cheapest -to-futures may allow short a div delivery option-deliver (CTD) bond.The quotes (or clean) price does not 80 include accured intAccrued Int (RI) (( days since last Coupon pay"days bet" Coupon pay")Coupon amtDiets plice-Full price = clean price accrued int.Transaction price QFP to price of futureIf we want to ConverPage No 63ContractOFR xa CFDate//Adj the forward pricing formula to acc for theShort delivery optionwill be givenEach delivery bond is assigned a conversion factor (CF) toadj the settlement pay" for delivery of higher or lowerCoupon bondsUse the CF for the cheqpest - to - deliver bondQuatedutuleQFP = LFull price (1+Rf) - Alt -FVC] x CF tpriceBotAl.AccusedInt atExample quoted + Al.maturityLearthPliceCalculate the quoted futures price for a 1.2-yr T-bond futurecontract if the cheapeast-to-deliver bond is a 7% Fbord withexactly 10 yr remaining and a quated price nea of $ 1040+A10 Bowith a Conversion factor of 1.13 1fullbecause no A full price = quated priceThere is currently no accrued int as the bond has justpaid a coupon The annual RfR is 5% The accrued ioton the bond at maturity of the future Contract willhe$14ExampleThe full prices is $ 1040 as there is no accrued int :- Semiannual Coupon = $1000 X7% x0.5 x0.51N+= $35.000.51.2- Coupon 1.0 S are due 0.5 yr (0.74 to maturity of the Contract)& yr (0.2 yr to maturity of the contract)Sol"-FVC=(35X1050+($35X1.05=$71.56QFP =Full price X(1+Rf) - AlT - 71.56 xI14CF Page No. 64Date//1QFP = [C$1040 X $ 14 - $71.56]11.13= $900.1'Longflow much(900)3x1.137 1414Ineed topy1031 16*Forward Rate Agreement (FRA)Price of FuturelendFRA : Agreement to borrow (long) or land (short) at a fixedrate in the futureUssuallybasedonLIBORwith # 365 days day-count conventionLong FRA = Pay fixed & received floating rateShort FRA = Pay flouting & receive fixed rate Serie f short FRFixed rate = Forward late (price)Characteristics of us LIBORH is a rate at which one bank lends another bankFor Short termCurrency in usoIssued out of usH is an add-on rateDiff 360 LIBOR exist for different maturitiesdayconventionisusedIX 3FRAt30+602 x5 FRAt60H903x6 FRAtt90 Page No 65Date//2x6 FRA60tt120In a "2x3 FRA the long agress to borrow money startingtwo monthy from today at a fixed rate for aterm that ends three months from todayThat is, the underlying in a 2x3 FRA is a 30-daysloan 60 days from now.223 FRA Start in 2month & end in 3mothsStart End3 month2month30-day loanin 60 daysToday\ month2 months3 month30 days60 days90 daysExpiry ofEnd of nationalFRAborrowing / lendingPricing an FRAThe price of the 2x3 FRA is the implied 30-day forwardrate in 60 days calculate using the 60- & 98-day spotiate [Note: The Same process is used in the fixedIncome study session ]Step 1 : De- annualize relevant LIBOR Spot satesStep 2 : Calculate period rate as( + long)I(itshort)Step 3 : Annualize Page No66Date//ExamplesidCalculate the price of a 1x4 FRA given the following Spotlates30-day LIBOR is 4%1205120 - day LIBOR is 57.SorThe Do- annualized 30-day rate is:R30 = 0.04 X 30 = 0.00333360The De-annuolized120-day rate isR120 = -0-05X120=0.01667150ToT30Tee1.00333tHF101667IXG FRA = 1.016671= 01331.00 333Implied forward rate FRA price isstarted as an annualizedsimple rate: 1.33 x 360 = 5.32%90 Page No 67Date//FintreeFRA : 4x5 Nominal amt : $10,000,000Position : shortLIBOR (today)LIBOR(after 90 day)30-day300 bps30 day370 bps60320603809036090450120390120520150Goor150580Sol"Formula:150It (0.04 X3603604.34 %x=1201+(0.039 X)30360150Logic4X360= 1.67%t$100150days$101.671203.9xt360= 1.3%5100120days$601.3Rate for To days: PV = -101.3 FV=10167 N=1 CPT0.362 Z0.36230day4.34% 360 days12021.3%Magic4 X 150/3603.9X3600.37%1.67//150days120 day150d0.3730days4.44%360 days fixed fate = Writing put & Buying callPage No 68Pay erDate/Price of new FRA (after 90 days)t120 days30dayst1t30days ,90days+ (0-083 X 60/360)360 = 3.89 1.It Co.037 X 30/360)30Value of the FRAx30 360 30 ) = $3,750I + (0.038) x (60/300)Valuing an FRAaThree Keys for valuing an FRA2. Value is determined by rate changes(eg You contracted to borrow at 5.3% and rates T to6%)2. The long wins when rates go up3. Paidrealized in arrears : The impact of rate changes wont beuntil the end of the borrowing [ leading periodExampleTaking our orginal example (1x4 FRA) priced at 5.32%assume the national principal was $1m10so201-:-To? floatinga series of short FRAsPage No. 69Rate=Date//Payer(Swap)10 days have now passed & LIBOR is now:- The 20-day rate is 5.7%- The 110-day rate is 5.9 %What is thevaluetothelongparty?4 - Step Process20R20 =5.7%x360=0.3171]De - annualize :Ruo =5.9%110X360=1.8031.2] Calculate what the fixed ratewouldnowbeonanewFRA(1 +long)t(H short)3Compare the fixed rate on the original FRA to the fixedrate on a new FRA to compute payoff.aDisc the poyoff back to todayT-10To voluationTzoTnodatett1.00317tIFFOriginalinitiation date1.018032 The fixed late on a new FRA10 days later for the same=1.0180390-day loan period :1.00317)X 360 = 5.926 %903 & 440ToT20Fro(0.05926-0.0532) -90xx360flm=$1515Original FRA @5.32%initiation date1515x 360 110= 1488PV at I Date//Note :FRA ( OTC)- underlying is a int rate on a depositlong = floating rate receiver (Pay fixed)Short = Fixed rate receiver (Pays flooting)- No exch of notional amt- FRA price is the fixed int rate that eliminates arbitage3x9 FRA- the rate on a deposit that begins in 3((90)months 4 lasts for 6 monthsiaL(270)3F(3,6) = FRA fixed rate30/360((90)iiiL(180) tF(3,3)FRA expieFRA payoffdisc backadvanced set advanced settled(FRAs)Receive bloatingNA ( bloating rate - fixed rate) 1 days /360)[I + floating rate (days /300)Receive fixed:- :-NA (fixed rate - blesting rate ) L days/360)I + floating rate (day((360)I Page No 71Date/Pricing & Value of currency forward ContractsSpot price: 66.505 ($ Int rates: India=7% USA = 27. Position : LongInvest" 1,000,000 maturity : 6month After 2 month: Spot = 68.151Int late : India = 6.1USA = 1.5%Price of the Contract : 66505 X (1:07 1.02=268.11554hzPrice of the new Contrat : 68.15 XC1.0691015)= Rs 69.3377Value of the Contract : 1,00,000 x 1.2233=Rs 1,195 342.941.0694112Fixed Income ForwardIFull price = Clean Price t Accrued Intquoted price2Fixed Income future Contracts are based on a generic boodi.e Treasury Bond Futures - one contract Involves the delivery of$100,000 FV / 6% Semi - annual coupon.i underlying # deliverable - short position decides what to deliverEg- T-Bond futuresany govt bond > 15 yrs to maturity as ofDayl of the delivery period.i. need a conversion factor - quoted price a bond would have onthe first day of the delivery period on the assumptionthat the int sate for all maturities = 6% Page No 7219th Feb 2023 MondayDate//(Quoted futures price X CF) + Aldefines the price to beT-bond quotepaid to the short sideof delivery3Cheapest - to - - delivery C CTD )Since the short side decidewhichbondtodeliverthey will choose the CTDOn day T : Cost to purchasedeliverable bond = STShort side receives(FT XCF)+ACheapest -to - deliverST --(F-XCF)ExamplesemiFull price : $1020 Maturity: 1.4yr Coupon: 12% Semi-annual RJR : 10%annyConversion factor : 1.17C=60C=605= 1020AIitO0.50.960x1.05046.5(62.38)60x105051020 x105 1.4%(65.5)11691041.120.4Accurued int (60 X0.5) :(48)Future Price :993.1Conversion factor : 117Quoted future price ( 993. 9931 1.7 )848 Page No 73Date / /Eg FT = 93.25BondPVCFCost to- delivery\99.501.038299.50 - (93.25 X 10382) = 2.692143.501.5188143.5 - (93.25 X 1.5188) = 1.873119.751.2615119.75 - (93.25 X 1.2615) = 212BorrowOITBoy bondS.Pay AIAlS. t Al - 1/1155) I (Ht)T Al Owe (So+A) 1/11rs (1tr) FSell ST = !GetAL (Known amt)Fo= [ (s.tal)-17 - (1+r)+ (1+r)T -AlFo = OF. X CFQF. = FoCFICTD60 days122 days t148 d35dr=++127.0TPV=120270 daysCF=14Fo= (5.+A1) - 1/(11r) ] (1+r)T - AI(120 t 6(60) (82)6(11)THS = 124.65 - 6 (193)(H) 121/365119.799QF. =Fo/CF =119.794= 85.5711.4 EPage No.75Date//*Currentry Forwards FuturesIf - Foreign RFRCurrency (an be sceen asr - domestic RFRhaving a Known yieldF6 =Sb(1+r)orFd=Sd(1+r)dd(1+r)bf(1+16)DR(8-1)TFbd Sfrd eFalf = Salf e(1-1) FEgUSD= 0.62(Aud = 5%(USA = 7%AUDT=2 yrFusa =Spotusp(1+ (w)FusaAuD=AuDSuse e(L+rAuo)2AUDAUD=0.62(1.07)4(0.87-0-05)2(1.05)2= 0.6200 ea= 0.6953= 0.6438Note : fid or d/ddoesnotmatterif ff=rthen Fo = 50if rb >rthenFFId>Sfid&FairAB$5MValuationUsing the DFs , + one can calculate PV as :- PV (CF) = E&DF; X CF }Each CF Collespounds to a settlement dateForValue of swap = )outflowsa currency swap Since CF are in diff currencies,you must adj one using the current exch rate Page No 83Date / /40100390360ttExamplebadContinuing with the uso GBP quarterly swap ex, after 300 dthe 60-day USD rate is 54% & the 60-days GBP rateis 66% The exch rate is now GBP 0.52USDCalculate the value of the $5,000,000 quarterly swap for theUSD fixed receiver CGBP fixed payer)Sol" After 300 days the remaining CF are the final int pay"& principal pay" in 60 days Relevant disc factor areUSD: DF60 =\= 0.99108 x $ PV $601+10.054 X 360)GBP : DF60 =\= 0.98912 x £ = PV £60It (0.066X360)Value (USD) = 0.99108 X $ 5,055,000 = $5,009,909coupon (5M + Coupon 5SK)Value (GBP) = 0.98412 X £2,542,500 = £ 2,512,83842.5K + 2.5mBefore the value can be neted they must be converted tothe same currencyValue in USD (GBP) = £ 2,514,838 = $ 4,836,2270.52Value (reeive $ >ide) = $ 5,009,909 - $ 4,836,227Based on que = $173,682 Page No.24Date / /Example : Price & Value of Currency swapsUso for Pricefor ValueUK LIBOR (today)UK LIBUR 30d)us LIBOR stoday)us LIBOR(30d)god110 bpsGod90bps90d180lpsGod190 bps180d130 bps150d100 bps180d200bpsIsod250 bps270d150 bps240d110bps270d220 bps240d300 lops360d170 bps330d120 bps360d250 bps330d320 bpsNaMaturity: lyr (quarterly pay)Notional amt£ ImSpot late :$1.5£Spot rate after 30d $1.75£Use forNotion (Price of the swap (UK) : 1-24= 0.42 x4=1.687PrincipalZitZztZztZnPrice of the Swop (us) :1-Z4= 0-6175x4 =247%ZztValue of USD bond2ClouponXZ,+Zz+Z3+Z4)]t (FV x 2W){[1.5mX247%X1/4XCo.9968 +0.9896+0.9803t0.9715)]+(115m(009715)3$ 1,493,733up no bazed Page No 85Date / /Value of GBP bond{[lm X 1.68% x 99/360 X (0.9985 +0.9958 + 0.9927 +0.9891) It (Am x 0.9891) 31,005,819 = $1,760,184 (1,00 5,819 x (.75)Value of Swap (USD receiver) : Value of USD bond - Value of GBP bond= 1,493,733 - 1,760,184= ($266,450)*Equity Swaps : Who Pays What ?Equity return payerPays any tve return on equityReceiver fixed rate pay" + any ve equity retuinFixed-rate payerPays fixed rate + -ve equity returnReceives tve equity returnEquity SwapsValue = PV C Cash inflow) - PVC Cash outflows) PricePV C equity side (F)=Current index /vlnationalindex Ivl at least settlement500$ x 1.035£ in1.45 $ GBP Page No 86Date//OTC ContractOne party pays a variable Series determined by an egorythe other pay1 Variable rate series determine byanother equity or rate2a fixed series.floatingfloatingareceive equity payfixedb] pay equity receivefixedEquityequitystockUnderlying for the equity legIndexCustom portfolioEquity leg CF can be with or without divReceive equity return Pay fixed NA (equity r - fixed r)Pay fleating NA (equity - floating r)Pay equity NA (equity r- equity )A $10mExample1.513% & principal value equity swap has fixed quarterly late ofindex the other side pays the quarterly return on an indexThe the index is currently trading at 985 After 30 days have passedbeen stands at 996 The value of the pay fixed side hascalculated as $0 993993 per $ of notional principal Calculatethe value of the swap to the fixed -rate payer on day 30.Sol": Value (Pay fixed) = 0.993993 X 10,000,000 = $ 9,939,930Value (pay equity) = $10,000,000996X985 = $ 10,111,675 Page No. 87Date / /Value (to fixed payer) = $10,111 675 - $9,939,930= $171,745Ex : Equity for Equity SwapsAn investor is the stock A return payer (& stock B return sezain a$1 m quarterly -pay swap After one month, stock A is up1.3% & stock B is down 0.8% Calculate the value of theswap to the investor.The investor pay the stock A return and as the return on B isnegative also pay that.Bis DownValue = (-1.3% -0.8%) X $ 1,000,000 = $ 21,000Pay A Receive BEx Fintree: M=14 National ant : Rs 1.5min Equity index today R527600Equity index after sed : 29,300LIBOR (today)LIBOR ( 30d)god180 bpsGod190 bpsPrice of the Avg: 1-21180d200 bpsisod250 bpsZi+ZztZz+Z4270220 bps240d300 bps= 0.6179 x4360d250 bps330d320 bps= 247Value of the bondS[1.5m x 247% X4 x Co.9968 + 0.9896 + 0.9803 + 0.9715)]+L1Smxo.975= Rs 1,493 733Value of the equity index : 1.5min X 29300= Rs 1,592, 39127600 Page No88Date//Value of swap = 1,592, 391 - 1,493,733= 98,658One period Binomial modelSigmaPeriod of the SigmaexpiryUup factor (u)10%Pa10 x 10.253me10%Pa6m107.XV0.5e210%Pa145e10%10%Per period10%One periodeA10%Per periodtwo periodloxxvosebutfor twoFor - two periodjusttusee2