QUANT FINANCIAL APPL
QUANT FINANCIAL APPL FIN 5883
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This 0 page Class Notes was uploaded by Domingo Parker on Sunday November 1, 2015. The Class Notes belongs to FIN 5883 at Oklahoma State University taught by Timothy Krehbiel in Fall. Since its upload, it has received 27 views. For similar materials see /class/232777/fin-5883-oklahoma-state-university in Finance at Oklahoma State University.
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Date Created: 11/01/15
Constructing LIBOR zero and forward curves from spot market LIBOR rates and CME eurodollar futures Presented below is an alternative method to bootstrapping from Treasury securities to construct the term structure of interest rates The term structure constructed from LIBOR CME futures and swap rates is of lower credit quality than the term structure constructed from Treasury securities A eurodollar is a dollar deposited in a US bank outside the US The rate earned on these deposits are benchmarked to the London Interbank Offer Rate LIBOR LIBOR refers to the rate at which banks are willing to lend LIBOR or borrow LIBID these funds The British Banker s Association posts a daily LIBOR value The British Banker s Association daily posting is usually the rate referred to when considering LIBOR Similar rates are available for all major currencies LIBOR rates from this market are referred to as spot market rates and quoted assuming addon interest based on a 360day year BridgeTelerate US GLUS3M US GLUS6M USGLUS1Y LIBOR USGE3M USGE6M USGE1Y EURIBOR Futures contracts offered at the Chicago Mercantile Exchange CME on the 3MO eurodollar rate have delivery dates of near months plus March June September December cycle extending up to lOyears in the future Settlement prices for the CME eurodollar contract can be obtained on a daily basis from the exchange httpwwwcmecom The delivery date for these contracts is the third Wednesday of the delivery month A futures contract price of 96 implies a futures rate of 4 This futures rate is expressed with quarterly compounding and an actual360 day count convention Conversion of the futures rate to a forward rate requires three sequential adjustments 360 gt365 quarterly gtcontinuous a convexity adjustment Adjustment to futures contract price necessary to account for first two conversions Adjusted futures 3659llog19l360 001 lOOSettlement Price The futures rate f is biased upwards relative to the forward rate due to nonlinearity of the bond pricing relationship CME Eurodollar futures prices 100 7 f re ect a present value bond price Convexity adjustment HoLee model TS Ho S Lee 1986 Term Structure Movements and Pricing Interest Rate Contingent Claims Journal of Finance vol 41 F f 05a 2t1t2 where F forward rate f adjusted futures rate 039 volatility of annual changes in L t1 time in years till futures contract delivery date t2 time in years till maturity of the rate underlying the futures contract F Adjusted futures 7 ConveXity adjustment Estimating volatility of annual changes in 3MO LIBOR httpwwwf 39 39 vegov releases The LIBOR zero curve term structure is constructed from observations of spot LIBOR and forward rates calculated from CME eurodollar futures contract prices Conversion to continuous compounding simplifies this process however may not be in accordance with day count conventions I to quotquotquotquot quot Zl quotquotquotquotquotquotquotquotquot quott1 I t t F1 t t 22 t Definition of forward rate The forward rate for a specified forward period is the rate earned during the forward period making total return of a sequential investment equivalent to the total return of a long term investment tit tit tit 210 J F21 J 220 J e1 365 gtxlt e 12 355 e z 355 Given 21 spot market rate and F11 forward rate derived from futures contract Fl2t239 t1 210139 to 2 0239 t0 The set ofLIBOR zero rates is 39 39 J by I this r quot to periods corresponding to the March June September December eurodollar futures cycle After constructing the LIBOR zero curve from spot rates and forward rates linear interpolation between zero rates can be used to find zero rates corresponding to the payment dates of financial contracts interest rate swap Utilizing zero rates for a swap contract s payment dates the relationship above can be inverted to determine the relevant forward rates for determining the equilibrium swap rate For example if t1 is a reset date and t2 the corresponding payment date the forward rate F1 determines the variable rate payment for purposes of calculating the equilibrium swap rate is 2202 39 toquot 210139 to Fm I241 182 108quot Fixed for oating interest rate swap valuation The timing of cash ows for a plainvanilla swap the realization at time ti reset time of the spot rate L spanning the period t tm determines a oating payment per unit of notional principal at time tm payment date of magnitude Li l7i per dollar of notional principal The distance 17 is given by the number of days in the period t tm divided by 360 or 365 as dictated by the appropriate conventions For a plainvanilla swap the fixed payment per unit of notional principal X E also occurs at time tm Li Ti l l X l l to t 1H1 L is the spot 3MO or 6MO LIBOR reference rate for swap contract rate prevailing at time t Absence of arbitrage swap pricing Given both LIBOR forward and spot rate curves the unknown cash ows in the oating leg of the swap must be set equal to the forward rates to prevent arbitrage opportunities Defme P0t the price of a discount bond maturing at time t Using continuous compounding the P0t are functions of the LIBOR zero rates zt P0t exp 21t365 Defme F the LIBOR forward rate for the period t t 1 The equilibrium swap rate coupon rate is defined as the xed rate X such that today s present value of the fixed and oating rate payments over the swap s tenor legs are equal n n 2NPXrP0ri1 2NPFzriP0rz1 i1 i1 n n NPquotFzHP0Ji1 ZFiquotP0Jz1 X i1 2 i1 n n ZNPliP0til ZP0til 39 i1 1 1 X is the equilibrium swap rate that makes today s value of the swap equal for both counterparties X is a weighted average of the projected forward rates analogous to the calculation of Macaulay s duration After an equilibrium swap has been entered the swap will in general no longer have zero value since interest rates will in general not have followed the implied forward curve The swap will maintain zero value only if the realized values L are the projected forward rates on the date the equilibrium zero value swap was struck Deviations of realized L from time 0 projected forward rates will cause the swap value to deviate from zero Replacement value of a swap after the initial date can be found from the initial equilibrium swap rate and the forward rate curve on the valuation date The fixed rate payor s replacement value at date t where n is the number of remaining cash ows at t n Vt ElEPtati139X0Ptati1lNPli i1 n VI X139 X0 2 Ptati1NPTi i1 Dividing Matrices Matrix division is useful primarily for solving equations and especially for solving simultaneous linear equations AX B Matlab provides two matrix division processes left and right and In general given appropriate matrix dimensions X AB solves for X in AX B X BA solves for X in XA B In general matrix A must be a nonsingular square matrix ie it must be invertible Left division Right division A 2 l 3 BX A l9 7 5 23 31 29 B eigA 378139 00930 49956i BA 00930 49956i 67 48 48 ans x3 2 30000 20000 10000 1 BAX B l l 76 160 AB ans 30000 20000 10000 Bootstrapping the term structure of interest rates gzero rate curve from market data Given n bonds each with maturity T l n B1 value of bond with maturity t cit cash ow of bond with maturity t at time i ctt par value of bond with maturity t plus nal coupon payment b0t l1z mt price of one dollar received at time t ie discount factor We have a system of n equations in n unknowns Bl c11b01 32 c12b01 c b02 B3 c13b01 c b02 c b03 l l l l l B c1nb01 c b02 c 7b03 c b0n B Cb B vector ofbond prices C diagonal matrix of bond cash ows b unknown vector of discount factors bCB The zero rates are obtained from the relationship Z 1 11 1 0quot b0t C110 0 0 10 110 0 10 10 110 B99 975 96 Find zero rate curve for this set of market data The plot of the zero curve against time to maturity is commonly referred to as the yield curve The yield curve is useful for pricing default free assets or if pricing default risky assets the zero rates can be augmented by an appropriate risk premium The discount function present value factor as a function of time to cash ow can be found easily from zero rates df 1 zt t f compounding frequency peryear t time to cash ow in units of compounding frequency The yield curve is the basic input for deriving implied forward rates 71 fnilJz 1Znn1Zniln 1 fut the annualized implied forward rate from the end of year nl to the end of year n The definition above ignore for the most part issues regarding compounding frequency Constructing matrices C and B from market data is confounded by different security types market reporting conventions liquidity and taX issues Ontherun Treasury securities are the most recently auctioned Treasury issues The secondary dealer markets for the ontherun issues will be most liquid for these issues At auction the coupon rate of Treasury notes and bonds will be set such that the bonds are issued at par value After issue these bonds will trade in the secondary market at premiumdiscount prices as the level of interest rate changes There are taX consequences 39 A with r 39 39 39 bonds which can materially affect demand 1 Treasury bonds and notes are quoted on an actualactual and bond equivalent yield basis with prices in 32nds 64ths 128ths of a percentage of par value Prices for Treasury notes and bonds are reported on a clean basis For a transaction the invoice amount equals the clean price accrued interest Treasury bills are quoted on an actual360 and bank discount yield basis Some Treasury notes and bonds are callable The Financial Toolbox in MatLab contains functions helpful in creating zero discount and forward curves The market data must be provided in a specific format to utilize these functions Treasury bills Maturity Daysto Maturity Bid Ask AskYield Treasury notes and bonds ICoupori rate IMaturity Bid Price Ask Price Ask Yield All rates Coupon Bid Ask and Ask Yield stated in decimals All prices Bid Price Ask Price stated in decimals Term structure functions format compact a XLSREAD hclassesspring04fin5883s04treasury7022304bXls39 bills b XLSREAD hclassesspring04fin5883s04treasury7022304bXls 39notes39 c XLSREAD hclassesspring04fin5883s04treasury7022304bXls39 bonds settlement date settle datenum 0223200439 convert Excel datenum to Matlab datenum to check dates use datestr command datestra l2 a lX2mdatea 1 b2X2mdateb2 c2X2mdatec2 create appropriate data format a35001a35 bl 5001b1 5 b3f1Xb3l0032b3 Xb3 b4f1Xb4l0032b4 Xb4 cl 5001c1 5 c3f1Xc310032c3 fiXc3 c4f1Xc410032c4 fiXc4 Parse notes and bonds Eliminate data rows with maturity before last bill maturity date from notes and eliminate bond rows with maturity before last note maturity endbillaend l endnotebend2 notedatesb2 bonddatesc2 bbnotedates gt endbill ccbonddates gtendnote convert bill to bond comparable information a bleonda combine data allabc convert data into form ready for bootstrapping bonds prices yields r2bondsall bootstrap zero curve from bond data and prices zerorates curvedateszbtpricebonds prices settle The bootstrapped zero curve is not useful for identifying over or under priced securities from the set of securities used as inputs The zero curve will fit their prices exactly There is a vast literature on the best methods to fit a function to the yield curve C Nelson and A Siegel parsimonious Modeling of Yield Curve Journal ofBusz39ness 60 no 4 1987 473489 M Fisher D Nychka and D Zervos Fitting the Term Structure of Interest Rates with Smoothing Splines Working Paper No 951 1995 Finance and Economics Discussion Series Federal Reserve board 1995 R Bliss Testing Term Structure Estimation Methods Advances in Futures and Options Research 9 1997 197231
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