Reading 31: Valuation of Contingent Claims

University:
California State University, Northridge
Course:
ACCT 230 | Introduction to Managerial Accounting
Academic year:

2023
Views:

252

Pages:

31
Author:

SophGr

Page No. 89Date/ /Reading 31: Valuation ofContingent ClaimsFoundationCall Option : The right , but not the obligation to buy anasset at a fixed price at some point in the futurePut Option : The right f but not the obligation, to sell an asset ata fixed price at some point in the futureEuropean Option : Can only be exercised at expirationAmerican Option : Can be exercised at any time prior to expirationThe Binomial ModeloXVEIContinues Calcu=edie XVEIn the binomial model, the underlying (eg stock price) willincrease or decreases in value based on one of two possibleValues : u (up factor) or D (down factor) 1 uA binomial tree can be used to model stock price movements,and hence the value of call of put options on that stockThe size of, & spread bet" u & D will impact the optionvalue Page No 90Date / /As+the change in option price toA= = C-C-ASthe change in stack priceS+_ 5tcAS-1toind6PayoffsLoan PMTC hereu:Ast - c+- -45+ +cFirst determined: A5-C- -AS + cc+ & C5 & 5no arbitrage price for C:then find A that makestc - AS - PVC-DS + +c+) =0Payoff = PayoffC = AS + PVC-As+ + c+)then solvePayofftorPayoffExample :-The binomial tree is constructed using the following data:U= Size of up p-move - = It %up = 1.33D = Size of down-move - = 1- % down = 0.75I1.33TTu = Probability of up-move =LtRf. - D -=0.556U-PRisk neutralTD = Prob of down-move - = 1-TTu = 0.45 time value lostBino mial tree : At each node use the intrinsic value orCalculated value - whichever is higher(Relpace the calculated with interesceExample : Early Exercise $ it is higher)A stock is currently trading at $50 & the periodicallyCompounded int rate is 3% Using an up factor of 1.3& down of 0.8 Calculate the value of a two -periodEuropean-style put option with an excersis price of $50TTu = Prob of up-move = 1.03 -0.8= 0.461.3 -0.8TD 0.46 +D-0.545++ = $84.500.465t=$65p++ = $00.46pt0.5456-$ 50St = 52C=4.956.46pt- = $00.545= $40P- 9.440.545- = $32.p- = $18Todaylyr2 yr Eurpoean option Page No.96Date//P = (0x0.46) t ( 18 x o. 54) - 9.441-03p+ = $0P. (0x0.46) + (944 x0.54) = $4.95103Que: Useing Same que Calculate the value of a two-periodAmerican - style put option with an excersi price of $50AmericanOptionpt=00.465th = 84.5 > 50ptt = O046St=650.53=50i+ 52 >500Pneo = 10pt- = 00.545=$40.00.46exersise8/7/945=0.5432,50-32r (IV) Intensic 10 (50-40)ValueP-- = 18P.= (0x0.46) + ($10x 0.54)1.03= $5.24NotesCallsearly excersise results in loss of time valuebetter to sell than exercise.no change in valuing C bet Eu & AmercianPutearly exercise results in a premium over European style optionwitgo Page No. 97Date / /*The Binomial ModelContinuing our original exRb 7 , 7.Callix5+ 30 x 1.33 = 40 Delta call = A Call plise0.35C+ = max (o, 40-30) = $10 option D stock priceSo=$30=6006= 5-14S = 30 x0.75 = 22.5a -170 40-225C = max Co, 22.5-30)=50=10/5 to 0.571TodaylyrOption Valuation Using the Hedge Ratio22.506 = hs. + (-hs - c+) = hs. + (-h5+c)CIt R6)(l+Rf)h is the hedge ratio : h = [C+ c][s+ s-]In our ex : h = 10 -0 = 0.571440 -22.5hS.hst at(o = 0.5714 X30 + (-0.5714 X40 +10) = $5.14(1.07)NOTEsIf the value of option > Value of the replicated folio arbitrage can becarned by selling the option & buying the replicated folio Page No98Date/of Can the value of option < Value of the replicated folio, arbitragethe optionbe earned by selling the replicated folio & buyingExampleif the call is trading at $6.4 (too high) write 100 calls & buy57.14 share Net a Cash outflowborrow l100# of call XH = 571At t=0 : burrow (57.14 X 30) - (100 x 6.5) = 1064 @7%At t=1 : stock price is either 40 or 22.50If S, = 40 value = (57.14 x 40) - (100 x 10) = 1,286If 5, =22.5 value = (57.14x225) - (100 X 0) = 1,286Repay loan = 1064 (1-07) = 1138Arbitrage profit = 1286 - 1138=148PV of arbitrage = 148 1.07 = 138The Binomial Int Rate ModelThe binomial int rate model is used to generate anint rate Hee .Int rate trees :Based on assumptions about rate volatilityYieldbnomial lattice of +ve int rate pathYou will not be generating int late trees OA the exam Page No 99Date / /An int rate tree looks like thisi2 ,44TTu = ITo = 0.5iloiz, ou11.01200Today14r2yrInt Rate OptionsTo Calculate the value of an European int rate option :Step 1 : Calculate the payoff at option expiry :- Call payoff = notional principal X [Max Co, reference rate - excercise rate)- Put payoff = notional principal X [Max Co, excersi rate -reference rate)long FRA but Right not obligationShort FRAStep 2 : Calculate expected value of payoff in each terminal nodeStep 3: Disc expected values back through treeExampleGiven the two-period int rate tree on the following slideWhat is the value of a two -period European int ratecalloption with an exercise rate of 5.5% & a national principalIof $1 m ( Assume that option cettle at time t = 2)C5.5 Pay c Fix(2)E Rec Reffoo Page No.100Date / /Call X = 5.5%NP=$1mto disc Hom toto t10.7383%R for dis= 5.2383% x Im = $52,38257883%D.Sc+32,7847.1981%3%C+ = 1.6981 % x $lm = $16,98119882388%C-8173to4.8250%ti( 0.00% Xlm = $0tz2Sol" C+= = C 52,383 x (0.5) + (16,981X0.5) = 32,784Long CallShortPut1.05788Pay FixPay-FixC = (16,981 X 0.5) + (0 x o.5) = 8173Rec RefRec - Ref1.0388RE> FixRefX) a Risk neutral probWhere:T= Time to maturity Cas % of 365d yr)So = asset priceGod is to replicate option payoffeX = exercise priceDiscwith stocks & Zero-coupon bondis = Continuously Compounded RFR div yieldSb = Volatility of underlying r stock returnRemoving CFNC.) = Cumulativenormall prob. N(d)) equivent to ZComponet Interpretationdi2StockComponentS.N(di)Bond Component : xe. N(d2) or XeCalls = leveragedinvest" in NCd.) units of Stock of stock usinge-r XN (d2) of borrowed founds Call = Stock Compunet- bond com poneRV(x)Pats = Investin NC-dw units of bond using NEdi) shortproceedsfromstock Stock Component : S.NC-di)Bond Component Xert NG-2)Arbitrage similartobinomial model using leveragedinvest analogieslemongs Call cost : ns.s + NB BN (dd) =ns>0- N(d2) = nekoPage No 103Buy stocksell bondDate / /Replicate Payoff of long callas S rises , v(d1) fises,di = LN (x) t E T x [co2 12) RFR I } as s drops NCdr) dropstaXVTMust hedge by buying more stock ind2 =sising markets & sell stock in falling marketsBond = e-rtCamulative losses of buying high & selling low =Cioby 'Law of One price total cost of stock t bondtExampleblioShort of XZ, Inc is currently trading at $50 So Suppose that thereturn volatility is 25% & the continuously compounded RFRis3%Calls & Puts with a Strike price of $45 & expiringin Six months are trading at $ 7 &$ P. respectivelyT=05yrCoifNCd, = 0.779 & N(d2) -0723 Calculate the value of replicatingfolio & any arbitrage s profits an both options No continuesSor Replicating Portfolio : CallN(d)50Long 0.779 X share: Cost CO-779 x $ 50)= 38.950.03(0.5)Borrow $45XeTX(0.723)= 32.05N(dz)Net Cost: 38.95 - 32.05= 6.90Market price of the call is $7 (too high)sell (ShortWritea call at $7 buy the replecating folio for $6.9 toyield an arbitrage profit of $ 010 per CallReplicating Portfolio : PutN (-di) = 1-N(d) = 1-0-779 = 0.221N(-d2) if 1-0.723 = 0.277N(d7) Page NolohDate/X 0.03(0.5) Grate, NEds)Long bond $45 X e X (0.277) = $12.28Short 0.221 share : proceeds (50 x0.221) = 11.05SoNet cost: 12.28-11.05M-d1)= 1.23Market price of the put option is $1 (too low)Sell the replicating folio for $1.23 buying put for $ for anarbitrage profit of $0.23 per put*BSM Model for Currency OptionsCall [s.e-" N(d))] - [Xe" (P)T N(d2) ]=Put = [Xe C PPT N (-d2)] - [.S.e" N(-d2)r(B), ((P) = ContinuouslyCompoundedbaseCurrencyandpriceint rate respectively.currencyCarry benefit = base Currency int rated, LN (x) 5-9E + {TxICoz /2) + RFR] }=OXITq= Continuously Compounded div yield*Black Model for Options on FuturesSo = PVCfuture Price) PVCa$strick)Co = eFT N(di) - XN (d2))NOJust Sub PV of FT for So in the BSM STPo is e[XN (-d2) - F.(T) N (d) )Page No 105Date//Valuation of European option on currencies using BBM modelCall = { S. X e Int late of base currency XT XN (di) I3 - Bond x N(d2) ]Bond = e IntrateofpricecurrencyXTCall = [Fxe X N(di)] - [Bond x N(Cdz) ]hedge egaistPut = [ Bond XN (-d2)] - [Fx e TT x NC-dis]hedge aginstintRatedi = LN (FX e/x it ) + { I x [co2 (2) + RFR }6xIT* Swaptions(bengfit when nt Rate1)ReceiverSwaption :- Right to receive fixed rate & pay flacting ratePayer swaption :- Right to pay fixed & receive floating rate(benefit when Int Rel)AP x PVA {CSFR x N(d)) - x x N(d2) ]} x Notional amtValue of payer swaption:Ex: Fix >RefPay RofPay RefIMP Equivalencies short Rec-FixbayRec-FixEx Ref FixPayer Swap:- beaj receiver swaption +shortpayer swaptionReceiver Swop:- Longreceiver swaption + Short payer swaption Ex Fix RefRec - Rif Pay -Fix (because of short get flif Swap rate increase before expiration buye. r enters into a pay fx@XF>lan enter an offsetting receive fixed pay floating swapfleating leg offset - anxuity = fixed lute OXR Page No 10621" Feb 23 ThueDate/UsingBlackBSM for value EquityCall=SpotXN(di)-Bond x N(d2)]Pat =I-[-itBond XN (-d2)Spotx N (-di)54+di = LN(X)(2)tRFRb XVT*Using Black Model to Value European Int rate options &SwaptionCall =AP xe365 Actual )X[FRA X N(di).]-X-NCd2)]XNotional antAP(Accured period) [(N-M)30.360]Equivalencies (int rate Option) VIMPPet-FixLong FRA : Long Int rate call + short int rate putPay-FlotingShort FRA : Short Int rate call + Long Int rate put1st rate Cap: A series of int rate Call optionsUsed by a fleating rate borrower to hedgethe risk ofin int ratesInt late floor : A Series of int rate put optionsUsed by a flooting rate lender to hedge theriskofin int ratePayer Swap: Long int rate cap + Short int rate floorReceiver swap : Short int rate Cap + Long int are floor Cap on flouting Rde Note = stries of Kant short putQPage No 107Date//"Greek Risk" & The BSM ModelCallPutBSM femula has five inputs1SpotNol115 the asset price (Delta) Ain pliceTime20 Volatility (Vega)ARFR3] Rf the int rate (Rho)a] T passage of time (Theta)5x, the excersie priceChanging an input will change the value of the option(Call or put)Each sensitivity (except X ) is a " Geek "Option GreekOTHATMITMDelta of0.5Change in the value of option for a given change in thevalue of underlying stockIt is a slope of the prior - to - expiration curveCall option : Delta = +vePut option : Delta = -VeIn-the- money option : Delta is close to -itAt - the -money option: Delta is b/w o & eiltOut-of-the - money option : Delta is close toODelta, = e-It X N (di)AC=ASX DelthteadfillDeltap = -eit X NC-d, Page No. 108Date/21VegaChange in the value of option ($term) for a givenchange in volatilityIt is a based on future volatilityHv.is highest when the option is -the money andto expiryCall option : Vega = +vePut option : Vega - +veRho Ca Same as delta in direction )Highest when away from expireChange in the value of options for a given change in RFRCall option : Rho atvePut option : Rho = -VeSellaShort Call veBuyerLong callShort put Gamna Gamma+ve GammaLong PutChange in delta for a given change in the value ofunderlying StockMeasure of Carvature of the option value in relation tothe value of StockGamma It is is same for both Call & put optionto exipisy highest when the option is at - the - money and closeAC = AS x Delta + 1/2 Gamma X A52Seller option using time Itve theta Theta Buyer Kill Ve The tutimeH is rate at which time value of option declines as theChange the in value of option for given change in calender timeoption approaches expiry.Highest when Option ATM & close to expiry Page No 109Date / /Relationship bet" option value & time to maturity is +veRelationship bet" option value & time IS -Veto PersonalSensitivity FactorInputCallsPuts"GreekDeltaAsset price (s)Positively relatedNegatively relatedDelta>oDelta <0Vegagive option valueVolatility (o)Positively relatedPositively relatedVega >0Vega > 0RhoRFR (r)Positively relatedNegatively related Proceeds AKX)PayPv(R)Rho >0Rho <0ThetaTime to expirationValue$0 asValue usually$0(T)callmaturityas putmaturityTheta <0Theta ZOExercise Price (x)NegativelyPositively felatedtelatedDelta and Dyanamic HedgeDelta :- change in the price of an option for a 1-unit changein the price of the underlying stockFrom BSM moldstIf not div payingDelta call = e -ST NCd,) Co = SoeNCdilremove e-st the D N(d)Detta put = -ST NCdV$1Detta hedging refers to managing folio detta by entering intoadditionalpositionsDelta -neutral positions : Folio detta is set &maintained at zerd Page No 110Date//2Detta-neutral folio needs to be rebalanced continually to maintainthe hedge (aka dynamic hedging)No. of options = No. of shareNo of options will be higher than theDeltano. of share.*AGraphical Depiction of Deltadelta = the slope of the finto- expirationCall payoff priorexpirationstock price (1)out of moneyin the moneyExercisePriceDetta SummaryCall deltas range from 0 to e-stFar out-of the money : Delta approches 0Far in -the -money : Delta approaches e-STPut dettes range from -edt to 0Far out of the money : Delta approaches Leist 0.5TFar in the - money : Delta approaches -Example : Delta : Contineous -Timee -ST N (d) from the BSM model is 0.58 calculate the approxchange in the price of a call option on the Stock ifthe Stock priceby $0.75 Page No 111Date//sofDC 2 0.58 X 0.75 = $0.435Dynamic Hedging HeDelta HedgingDelta - neutral hedge :- Combination of long stock & short callsso the folio value doesn't change as stock price changes# of Short Calls needed = #Share hedge12delta callNote: Helpful hint : Delta <1 , So always need more callsthan shares S=00.5Example#shure hedoSuppose you own 60,000 Share of Arthurall Company commonStockthat is currently selling for $50 A call option onArthwall with a Strick price of $50 is selling for $4 & hasa detta of 0.60 Determine the no of call gotion necessarytocreate a delta - neutral hedge.Calculate the effect on postfolio value of a $1 increase in theprice of Artharall stocklong on the cakNo. of option needed to detta hedgeshortBecause we are long the stock market , we need toshorttheoptions# Options - 60,000 = 100,000 options0.60 Page No 12Date/2fthe price of Arthurall Stock increased instantly of $1.00value of the Short call option position would demasres by$06 Therefore the net impact of the price change 6 thevalue of the hedge felio would be zelo.Total change in value of stock = 60,000 X $1 = $60,000Total valuechange in of option = 100,000 x -$0.60 = -$60,000Total Change in felia value = $60,000 - $60,000 =0Role of gamma risk in options tradingfor Stocks -0Gamma = rate of charge in delta as stock price changestve for both calls & putsLargest when option is at-the-money 2 close to expirationSmall for deep in-the-meney 2 deep out of -the- moneyoptions not close to expirationGamma Risk : When Stock price jumps abruptly (a violation of BOM-Risk of leaving a delta- hedge folio unhedged because ofa studen jump in stock prices leale BSMGamma Captures non-linearity risk of the risk that remains oncethe folio is delta neutral.Measuring Volatility of Stock ReturnsImplied velatility of Continuous returns on underlying stockis o' from Bsk model that makes model value = market price Page No. 113Date / /The volatility (o) "implied" by the option priceIf is the voladility in BSM model that yield thatmarket price of the option=>Implied volatility is often used by traders to gauge marketperceptions or to quote pricesif expected volatility > implied volatity t take long position on optionsif expected volutility < implied volatility, take hag short position on optionExampleXSuppose market price of 3-month 45 Vanilla Inc call is $725Volatility InputValue from BSMCalls on blue stock are currently trading25%6.11at an implied volatility of 22% A30 %6.44trader est that the future volatility35%6.79will actually be closer to 257. To40 %717Capitalize on her beliefs, the trader45%7.59should Implied Vol < Expected (25%) V4Buy call(22)VolT PliceA. Change her modelBased on the trader beliefs CallB. buys calls on Blue stock options on Blue stock are underprisedC. write calls on Blue stock in the market Accordingly she shouldby the calls Page No. 1kDate//2OptionsCallPutSShort ForwardLSBullBearBearBull416 Delta-Ve Delta+ve DeltaLong ForwardOTMATMITMDelta No0.5Forward/FurtureStockLongshortLongShort Delta HDelta -1Delta NHDeltar -1Long CaltShort callDeltaLong putshort pall+--+Gamma+-+Vegat+-Theta-+11-tRho+-+ Page No IISDate//Delta HedgeEx:- Long10,000Calloption;Delta = 0.7DeltaNeutralhedge?Intrument PositioncityDeltapriceTotal deltaLong10,000 x to.7=all option+ 7000}StockShort7000x-1=-7000Sign de satEx2 :- Short put 50,000 aty S= 1000 X 1 1 200 Delta 0.65matterDelta nutral+Hedgingintrumentcall optionS= 1000 & Delta =0.57 ,=900Qty?aba xbIntrumentdirectQtyDelta priceTotal DeltaPutShort50,000+0.65+32,500CallShort57018--0.57-32,500Ex3 :- Current PositionCityDelta priceBuid a detLong Call10,0000.3Neutral folioShort Put20,000-0.7Long Put20,000-0.85Hedge Instrument:-PutDeta --0.35IntrumentDirectionQtyDeta Pricetotal deltaCallLong10,000+0.33000Putbishort20,000to.714,000PutLong20,000-0.85-17,0000Already0 original delodeltaAimplice x GarnPage Noifdepend on price changeNew deltDate/2Dynamic HedgeBuild initialdue to GammaRe-calibrate thehedgespotChargesDelha will changehedgeEx :- Short Put 100,000 QtyDelta = -0.72 ; Gamma = 0.07a) Build a delta netial hedge (stock)b) If stockpricegoesup$2 Rebalance thehedge ?501" :-a] Invest"DirectionQtyDelta PuTotal deltaPutShort100,000+0.7272,000Stockshurt72,000-72,000bOTM0.5currentITMabsolute value delta willt0.72go towards zero 0.72stock plice$2ifgainy otm12x007(+ve) if going TM0.58InstrumentDirectionQtyDelta PUtotal dettaPutShort100,000+0.5858,000Stockshort72,000-1- - 72,000-14,000StockLong14,000+1+ 14,000ArbitrageEx:- Share price = 100 if goes up150RFR = 10% Pagoes up70expiry = lys0 Cull option X=90;Value = 27.27 Page No 117Date / /P= H+107. 0.7 = 0.4 = 50%50%150 601.5 -0.7 0-8100SOT 70 Obif Actual price option is 35 Us e 10,000options Qty Determine the step to exploit arbitraysL Hint Delta)Delta Call = A Call Price=60-0 = 60 =0.75optionStock price150 -70 80Today : Step 1: Price 35 VS 27.27ValueOption is expSELLShort 10,000 Call options @ 3510,000 x -0.75Receive = 35,000= - -7500 DeltaStep 2 : We need to Keep folio Delta Neutral7500 x 1Buy 7500 stockCt 7500 )7500 x100 = 750,000 (need)= +7500 DeltaStep 3: Borrow 400,000 @ 10% Arbitiage :- INo price Risk (delta=0)2No investAfter lyr (expiry)Down ScenarioUp ScenarioValue Stock (asset) = 7500Value of stock = 7500 x 70X150= 525,0001,12,500Pay" made due 10000 X-60Liabilit repay" = -440,000Short call optionPay" made short call =0=-600,000Liability Repay" loan = -440,000Net +85000Net +85,000 Page No 118Date/2Profit per option 85000 = 8.5Pigit per = 8.510,000optionPV profit per persoption I 8.5 - 7.73PV profit per optive = 7.73(Itca)(35-27.27)IAlternative WaySell 10,000 options = 350,0002Buy 7500 stock = 750,0003350000True ValueMispricing19000 x 27.27= 272,70077,300 (take homeuse to buy Stockborrow - 750,000 - 272,700= 477,300After lyrUp ScenarioDown ScenarioStock value 7500 X150 = 11250007500 x 70 = 525,000Loan repay" 477,300 x 1.1 = 525030-5 2500Option Liab 10,000 x 60 = -600.000NO6aoEx:- Stock 500u= 1.25Xcall400lyr expiryRFR = 5% P.a.Value => 119Price of theoption( Hirt : Delta Neutral)70 today Use 100 options & plait aihing Page No. 119Date / /Profit today per Option = 119 - 70 = 49Today'2Sell 100 stock @ 500 = +50,000 [Delta = 100 x -1 = -100Buy 100 option @ 70 = -7000 [Delta =100 x +1 = +1003 Invest 43,000 @ 57.Up ScenarioDown Scenari1100 Option X225 = +225,001 Option value 100 X 0 =0100 share stock x -625 = -62500t2shard pt stock 100 x -400 =400.003Invest Tedempt value = 145,1502Invest value = 4515043,000 x 1.055150Net Benefit=5150SISO /100- 51.5per option= 51.51.05= 44.04= 49,04