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# Advanced Probability And Options With Numerical Methods MA 51600

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This 39 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 51600 at Purdue University taught by Jose Figueroa-Lopez in Fall. Since its upload, it has received 43 views. For similar materials see /class/208108/ma-51600-purdue-university in Mathematics (M) at Purdue University.

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Interest rate models and derivatives Jos Figueroa Lopez Math 516 Stat 541 Fall 2008 Contents N 03 4 II The bond market and interestrate derivatives 1 Introduction The bonds market Interest rates Coupon bonds Forward contracts on bonds and forward rates Interest rate swaps gt gt gt gt gt GUIgtme Short rate models 21 Introduction 22 Money market account and short rate 23 The term structure equation Riskneutral modeling and inversion of the yield curve 3 Key ideas Calibration and the inversion of the yield curve Inversion of the yield curve in the HullWhite model Implementation of the HullWhite model Final remarks 21me Forward rate models 41 The HeathJ arrowMorton Framework 42 The HJM drift condition 43 The HJM program for valuation of derivatives Change of numeraire and option pricing 51 Introduction 52 Change of numeraire 53 Valuation of exchange options 54 Forward neutral measure 55 Applications 28 6 LIBOR Market model 34 61 Introduction 34 62 Market pricing practices for caps 34 63 The LIBOR market model 36 1 The bond market and interestrate derivatives 11 Introduction Introduction 0 Objective Develop a reasonable model that allows an arbitragefree pricing theory for the bond market 0 An arbitragefree pricing theory identi es a primary set of traded assets Which are used to price a set of secondary traded assets The basic pricing operation consists of constructing a portfolio of the primary assets dynamically rebalanced across time such that the cash ow and value of the portfolio replicate the cash ow and value of the secondary asset The primary set of traded assets compromises zerocoupon bonds and the money market account 0 The secondary set of traded assets compromises coupon bonds or interestrate options The later kind of assets includes all contingent claims Whose payoff depend on the evolution of the zerocoupon bond price curve or the socalled the term structure of interest rates 12 The bonds market Bonds 0 What is a bond A contract Where for a suitable price now the holder gets a xed pay ment called the principal value or face value at a prespeci ed xed time T called the maturity In addition to the principal the bond could entail the payment of a xed sum called coupon Which is paid periodically If no coupon is paid the bond is called a zerocoupon or pure discount bond As these assets provide the owner With xed cash ow they are also called xed income securities 0 Notation pt7 T Will represent the price at time t of a zerocoupon bond With face value 1 dollar and maturity T gt t For a xed it the graph of the functions T A ptT for the available maturities TS is called the term structure or the bond price curve at t Bond price behavior 0 pi7 T pT7 T 1 ast A T Why 0 t A pi7 T exhibit a very irregular trajectory Brownianlike o For each maturity T t A pi7 T can be considered the price process of a sepa rate asset 0 Of course all assets are correlated the smaller lTl 7T2 l the stronger correlation o It make sense to assume that if there were bonds of all maturities for a xed date it the graph of T A ptT the zerobond price curve would be smooth for T gt t Under the previous smoothness assumption it is possible to infer strippped the zerobond price curve from the purediscount and coupon bonds liquidly traded in the market The US bond market Most goveniments issue bonds for instance the US issues treasury securities of three types bills notes or bonds There are basically three markets to buy or sell a Treasury security 0 The spot market Most of the US treasuries are traded in the overthecounter market where dealers provide bidask quotes These quotes re ect not only today s interest rates determined for instance by the prices of the most recent issued bonds but also today s expectations of future interest rates Joshi Example 1 The following tables describes a typical overthecounter quote taken from the Wall Street Journal as of Dec 2nd 94 Rate Maturity Bid Asked Chg Ask yield 434 Dec 947i 99 z 29 99 31 1 497 Maturity Days to Mat Bid Asked Chg Ask yield Dec 15 94 10 4 70 460 lt13 The US bond market Cont o The forward market A forward contract is an OF C agreement made in the present between two parties to exchange a commodity say a bond at a predetermined future date but at a price agreed today called the forward price There is no exchange of money at the present for entering in the contract o The future market A future contract is an agreement to purchase a commodity at a prespeci ed date called the delivery date and for a given price called the future price The buyer and seller negotiate indirectly through an exchange say CME Chicago Mercantile Exchange The price is paid via a sequence of in stallments over the contract s life These installments are exercised at the end of each intermediate trading date and equal to the change in the future price on that day the future price is set according to an exchange 13 Interest rates The different interest rates What is an interest rate The amount of money that a borrower promises to pay the lender per dollar borrowed There are different ways to measure interest rates Exercise 1 A bank state that the interest rate on 1year deposits is 10 per year But what does this statement really mean If you deposit 100 dollars in the bank what will your terminal value ofthe investement be after time t l in years ifyour interest rate is a simple 7 twotimes I or c I m Solution 0 a If the interest rate r is simple A1 rt 1001i1 110 o b If the interest rate is compounded m times per year say m2 mt 1 2 Alt11gt 100lt17gt m 2 o c If the interest rate is continuously compounded Aequot 100e1i This is the limit as the compounding frequency m increases Why Rates of bonds 0 We can see a zerocoupon bond as a loan The writer of the bond borrows money from the holder of the bond 0 The simple annualized interest rate y associated with a purediscount bond is called the yield of the bond This is the number y such that 1 WT 1 M 7 t The yield is widely used in the US treasury market to quote treasury bills Exercise 2 The following quote of a treasury bill was obtained from the Wall Street Journal Maturity Days to Mat Bid Asked Chg Askyield Dec15 94 10 4 70 460 lt13 46 What is the asked price for the treasury bill Solution Need to solve 10 1 p 1 r0460gt r Then price 998722 0 In the case of a coupon bond the yield of the bond is the annualized interest rate compounded semiannually or continuously that When used to discount all cash ows gives a present price equal to the market bond price For instance for a coupon bond With maturity T in years par 1 and coupon rate c its market price p and yield y are related as follows When y is semiannually compounded 2T 7 c 2 1 c 2 p 1 on 1 1 y2gt2T Notice that the coupon rates is annualized and it is paid semiannually as a simple interest of the principal Exercise 3 Veri that the relation above holds for the ask price and ask yield in the quote below note 9929992932 and 9931993132 Coupon Rate Maturity Bid Asked Chg Ask yield 34 Dec 947 99 z 29 99 31 1 497 14 Coupon bonds Valuation of coupon bonds 0 Consider a coupon bond speci ed by the following data K is the principal T Tn is the maturity ci is the coupon paid at time T o What is the timet price pt of the bond in terms of zerocoupon bonds Bond Valuation Formula W KW Tn 611097 Ti7 o In most bond market there are only a few discount bonds traded Most bonds actively traded bear coupons An application of the relationship above is to infer strip implied zerocoupon prices consistent with the traded coupon bonds Exercise 4 Hull Section 54 Suppose the data below represents the prices of ve treasury bonds toda Principal Time to mat years Coupon rate Bond price 25 0 100 5 0 949 100 100 0 900 100 150 8 960 100 2 00 12 1016 Note The coupon rate quote is simple annual however it is paid semiannually 1 Determine the implied zerocoupon bond prices for maturities T15 and T20 2 Determine the continuously compounding treasury zero rates implied by the above quotes 3 Plot the term structure of bond prices and of zero rates zero curve Is it upward or downward slopping Remark 1 o The method outlined above is known as the Bootstrap Method cf Section 54 in Hull 0 In practice we might not have exactly the maturities we need say for computing the price ofa coupon bond 0 The approach often used is to interpolate between the closest available maturi ties 0 Say ifwe know that a 23year bond with coupon 6 sellsfor 98 and a 27year bond with coupon of65 sellsfor 99 it might be assumed that a 25year bond with a coupon of625 sells by 985 Hull 15 Forward contracts on bonds and forward rates Valuation of forward contracts 0 Consider a forward contract written now say at time t to buy a zerocoupon bond at time S with maturity T Let F t S T be the forward price contracted at time t Problem What is an arbitragefree value for Ft S T consistent with the bond market 0 Consider the following trading strategy i Sell an Sbond at time t ii Use the proceedings to buy pt7 539 dollars worth of Tbonds iii At time 539 pay the one dollar iv At time T get Describe the cash ow of the trading strategy and compare with the cash ow of the forward contract 0 We conclude that W7 T W 5 Ft S T Forward rates and LIBOR rates 0 It is more common to quote a forward contract in terms of its corresponding simple interest rate L that is L is de ned by the following equation 1 F1LTiS7 where F Ft S T is the forward price above Then immimm L TiSnTt Lt S T 2 L is called the simple forward rate for a 1 loan on 5 T con tracted at t or for simplicity the LIBOR forward rate 0 When S t the rate is called the spot rate or the LIBOR spot rate for L T and it is denoted by Lt7 T W T i 1 L tT i i aiwmnb This is the simple rate today of a loan of duration T and is just the yield of the discount bond with maturity T introduced before Remark 2 o The term LIBOR rate is widely used in industry LIBOR stands for London Interbank Offer Rate 0 A LIBOR quote by a particular bank is the rate of interest at which a bank is prepared to make a large wholesale deposit with other banks 0 Large banks typically quote 1month 3month 6month and 12month LIBOR in all currencies see Section 51 in Hull Continuouslycompounded forward rates 0 The continuously compounded forward rate R Rt S7 T for S7 T contracted at t is determined as follows 1 FeR Test where F F t S7 T is the timet forward price on a zerocoupon bond with maturity T and exchange time S Thus Rm Si 1 R flogp yTT 10675 0 When S i this is called the continuously compounded spot rate and it is denoted by Rt7 T log t T R t T I 777i lt gt T t This is just the continuouslycompounding rate today at time t of a loan of duration T Instantaneous forward rates 0 The forward rate contracted today at time t for a loan at time S of in nitesi mal maturity is de ned as ft S Thins Rt S T Thus f0 5 This is called the instantaneous forward rate at S contracted at t Blog t 5 7 as 0 When S i this is called the short rate at time t hereafter denoted by rt log LT Tt Zqll1f1 tiiT This is meant to refer to the log return today on a loan of in nitesimal matu rity Three approaches for bond market models 0 Notice that fort g s g T we have T ptT exp 71 fi7 sdsi o In view of the previous relation there are two ways to built a model for the bond market i By specifying the dynamics of all possible bonds pt7 T ii By specifying the dynamics of all possible forward rates f t T However historically the rst type of bond market models starts iii By specifying the dynamics of the shortrate 7 We shall see that by noarbitrage arguments all bonds can be computed as ex pectations of the discounted payoffs under a suitable riskneutral probability measure Q pi7 T EQ 6 ftT T5d5gti 16 Interest rate swaps What is an interest rate swap 0 An agreement to exchange a stream of xed interest rate payments on a notional amount for a stream of oating interest rate payments on the same notional o The oating interest rate is reset at the beginning of each period typically accord ing to a LIBOR rate oating on that period 0 The xed rate R called the swap rate is chosen such that the initialfair value of this contract is 0 o Concretely given some predetermined periods To T1 i i i Tn 1 Tn and swap rate R at each time Ti 239 17 i i i n one party 1 will received RTi 7 Ti1K and 2 will pay LTi17 Ti Ti 7 Ti1K to the counter party where ME 1 Ti stands for the LIBOR rate at time Ti 1 for a loan on Tl1 Ti Valuation using zerocoupon bonds 0 The fair value Ht of the contract at time t lt To should be n M K pa TH 1 6mm Ta i1 where we assumed a constant time span period equal to 6 called the tenor of the contract 0 The previous formula follows from the following facts i LTi1Ti ii The cost at time t of getting the amount at time T is pt TF1 0 Dumb question Ifthe cost now at time t ofobtaining the amount 6 at time T is cipt Ti how come the cost ofobtaining W at time T is not Computation of the swap rate 0 Recall that by de nition the swap rate R is chosen so that the fair value of the contract now is 0 Then R pm To 7 pa Ti 7 n 6 2151 PO Ti 0 In particular if To 0 meaning that one enters at the beginning of the rst payment period 1 7 1007 Tn R n i 6 211 1007 Ti Application to compute zerorates 0 Notice that 17 R6 2711 p0 T 0 T Z1 i p l 1 6R 0 Equivalently n71 R6 Zplt0Tigtlt1 6Rgtplt0Tngt 1 i1 which in the literature is interpreted as the following Swaprate Par yield namely the coupon rate at which the value of the bond is the same as the par or principal o The formulas above allow to compute zerorates that are consistent with swap rates via a bootstraplike method as we did with coupon bonds Example 2 Hull Section 63 The following table shows a marketmaker s quotes of a bid and an offer for the x rate they will exchange for oating Maturity years Bid Offer Swap rate 2 603 606 6045 3 62 624 6225 4 635 639 63 70 5 647 6 51 6490 7 6 65 6 68 6 665 10 687 6850 The zero rates for 6month 1year and 15year are respectively 55 575 and 59 per annum with continuous compounding Compute 2year and 25year zerorates that are consistent with the above swap rates 10 Why are interest rate swaps important Interest rate swaps are the most liquid of interestrate derivatives They are also important to strip the current zerorates Typically the short end of the LIBOR curve zerorate curve or term structure of yields is constructed from market LIBOR rates typically 1 3 6 and 12 month maturities The longer term zerorates are inferred from the market prices of other instru ments such as swaps Swaps are liquidly traded their rates observed and the resultant discount rates inferred Yoshi The curve inferred from swaps is the most relevant for pricing other exotic interestrate options as it is the curve at which the bank can directly trade swaps and hence hedge the option 2 Short rate models 21 Introduction Arbitragefree short rate model 0 Recall that our ultimate goal is to built a reasonable model for the bond market that is arbitrage free 0 The model must specify the stochastic evolution of the primary traded assets 0 In this part the starting point Will be to postulate a stochastic model for the spot short rate Tt the interest prevailing at time t for a loan of in nitesimal maturity o What conditions should be satis ed for the market to be arbitragefree We Will nd that under absence of arbitrage the prices of bonds With different maturities have to satisfy certain internal consistency relations Given a particular benchmark bond the price of any other bond with maturity prior to the benchmark Will be uniquely determined by the price of the benchmark and the money market account via a replication argument 22 Money market account and short rate The money market account 0 In addition to the bond market we assume the existence of another asset called the money market account The value of this asset at time t is given by dBt 7 tBt dt Bo 7 gt BE e13 uldur for a certain nonnegative process Mt called the shortrate 0 Interpretation The shortrate is meant to represent the log return today of a loan of in nitesimal maturity The value of B represents the continuoustime limit of a self nancing riskfree trading strategy that during a smalltime period t t dt gives the interest rate Ttdt and roll over the proceeding to the next period 0 In practice one takes Tt to be the yield of a liquid shortterm bond say of one month Wilmott LIBOR rates is also a typical proxy specially by derivatives traders active in the overthecounter market Hull Short interest rate models 0 In these models the short rate Tt is assumed to follow the dynamics dTt M05 Ttdt 0t Ttd Wt Where Mt T and 0t7 T are deterministic functions and W is a onedimensional Wiener process in which case we say that the model has onefactor o Intuition Tt At 7 Mt m mom at TtE a 5 N N01i 0 Simulation Ttn Ttn1ttn1 Ttn1 initn71atn1 Ttn1 Min 7 tn718n7 Where the en s are independent standard normal 23 The term structure equation Some important questions 1 Is the market consisting of all discount bonds and the money market account complete 2 Do you think that the market is arbitrage free 3 Is the price of a given zerocoupon bond P t T determined uniquely by the requirement that the market is arbitragefree and the dynamics of the short rate T 4 Is the price of a given zerocoupon bond P t T determined uniquely by the requirement that the market is arbitragefree the dynamics of the short rate T and the dynamics of other discount bond General description of main results In a frictionless arbitrage free bond market With a money market account 0 A11 bonds regardless of maturity have the same excess return With respect to the short rate per unit of risk or in other words the same risk premium per unit of volatility 0 Given a particular benchmark bond the price of any other bond or interestrate derivative With maturity prior to the benchmark can be replicated perfectly by a portfolio of the benchmark bond and the money market account dynamically rebalanced across time Assumptions 1 There exists a money market account With short rate Tt given by dTt p05 Ttdt 0t Ttd W 0 There exists a market for zero coupon bonds of all maturities T L For each maturity T the timet price of the Tbond is a smooth function of t and of the spot short rate Tt That is ptT FT WW for a smooth function FT t T 4 The market is arbitragefree and frictionless Notation o 011t represents the timet meanrate of return of the Tbond 0 0T t represents the timet volatility of the Tbond Thus dpt T pa T MM aTt awn Bond pricing theorem Theorem 1 The Bond price equation or term structure equation 1 There exists a random process Mt such that Act aT t 7 7107 O39T it regardless of T Such a process is called the market price of risk 2 The pricing function FT i7 T for the Tbond satis es the PDE 6 FT 6t 8F 1 82F 7 A0 a 502 BT27quot 7 TFT 07 for t lt T and T gt 0 with boundary condition FT T7 T l 3 Consider a European contingent claim with payoff X TT and maturity T fits timet value Ht is given by Ft7 Ttfor a smoothfunction F then F solves the above PDE with boundary condition FT7 T T N otational remarks 1 Notice that T has two meanings all over this part the random short rate Tt and the variable T of the function FT t7 T The meaning should be clear depending on the context 2 Also A has two meanings the market price of risk process Mt and the function Mt T appearing in the PDE of the bond pricing formula 3 The function Mt T is de ned in terms of the price function FS t7 T of a bench mark bond as follows 399th ptT TFS 02t 7 a27FS 7 T F5 A677 039tT67 FS with the natural notation th M5372 etc 4 The relation between the process Mt and the function Mt T is W WWW Riskneutral representation Theorem 2 FeynmanKac or riskneutral representation 1 The bond price func tion FT satis es FTtT 13 f WWW T Mihere the probability measure Q called the martingale measure is such that Wt Wt f Asds is a Wiener process under Q Thus under Q the dynamics ofT is A dTs p 7 Aads adWsi 2 Under the conditions of 3 in the quotBond pricing theorem the price Ht FtTt ofa contingent claim with payoff X TT and expiration T admits the representation Ft T W a ftTTltugtdultrgtTTTt T Additional remarks 1 The existence of a measure Q such that A t Wt I Wt Ads 0 is a Wiener process under Q is guaranteed under certain conditions by the Gir sanov s Theorem see Section 113 in Bjork 15 2 Concretely Q is de ned by QA E1A ifoTtdtiOTAEdti 3 The FeymannKac representation is proved using the following three steps see Section 55 in Bjork and in particular Exercise 512 i Find the dynamics of F 37 7 S using Ito ii Find the dynamics of e f T d Fs 7 S with respect to s using Ito iii Integrate from s tto s T and take conditional expectations give ftW Some additional consequences 1 Notice that the FeynmanKac representation can be written as follows an EQ HWWW Ft T W a ftTrltugtdu TTlTti Q The Markov s nature of the Tt will imply that 5 f3 100112907 T E Q 6 foT T000lu 3W e f ruduFt7T EQ 57 0T ruduqgtTT ftW L Hence under the riskneutral probability measure Q the discounted prices pro cesses of all zero bonds and related simple European interestrate derivatives are martingales gt Due to this property Q is also called a martingale measure of the bond market model 3 Riskneutral modeling and inversion of the yield curve 31 Key ideas Key ideas of the riskneutral modeling 1 For each riskneutral drift MQ a 7 A0 of the shortrate dynamics Tt there is an associated arbitragefree price model for the bond mar et E0 In other words if we postulate that the dynamics of the shortrate T under the martingale measure Q is as follows dTt Qwawt 0i To dWt for suitable functions a and a then the prices of all bond and interest rate derivatives are completely determined by the following procedure Riskneutral or martingale pricing procedure The arbitragefree price of European option with payoff X TT at time T is given by Ft Tt where the function Ft T satis es 8F 8F 1 282F 7 7 EMQE Ea WirFio FT7 7 39Tr 3 The riskneutral modeling is also called the martingale modeling 32 Calibration and the inversion of the yield curve Calibration and the inversion of the yield curve How do you determine the riskneutral martingale dynamics of the short rate Q A typical approach is the following i First proposed an explicit structure for the riskneutral dynamics of T which means that we shall specify the structure of the functions MQ and 0 ii Second determine values of the parameters of the model that closely match the market prices of certain interestrate derivatives 3 In practice market prices of liquid interestrate derivatives are used to calibrate the riskneutral dynamics Examples Bond prices and swaps gt The procedure is called calibration inversion of the yield curve or yieldcurve tting U1 Once a suitable dynamics is chosen one can derive the implied prices of other exotic interestrate derivatives using the pricing procedure Af ne models 1 A typical structure for the martingale dynamics drt MQ t Ttdt 0t Tt d Wt7 is of the form M t T WW W 0t 7 x7t39r 6t 2 Important examples 0 Vasicek d7quot b 7 ardt adW o HullWhite d7quot t 7 atrdt atdW o CoxIngersollRoss CIR d7quot ab 7 Tdt a dW Af ne Term Structure Models 1 If the term structure pt7 T T 2 07 0 g t g T has the form 106711 eAtT7BtTrt7 we say that the model possess an af ne term structure ATS 2 It follows that if drt 2th a d with MQ and a of the form M t T aw W 0t 7 x7t39r 6t then the model possess an af ne term structure 3 Moreover AG7 T and B t7 T satisfy the system 1 2 BtaB 7 EyB 717 1 At 7 BB 6B2 07 with AT T BT T 0 Two approaches for curveyield tting 1 A pragmatic seemly easy approach is to choose the values of the parameters 6 of the model such that the market prices of certain interestrate derivatives are replicated A k Equot e45 Tltugtdult1gtiltTltTgtgt 7 M 9 I 39 f g l 0 T We canthink of d as market implied parameters making an analogy With implied volatility 2 A second approach consists of introducing functional parameters eg i in the HullWhite model so that the resulting model can match perfectly the market bond prices 10 07 T for all maturities T 3 In practice one combines both approaches For instance in the model dr i 7 ardi adW one can use derivatives to estimate a and a and used implied zero prices to calibrate i 33 Inversion of the yield curve in the HullWhite model Inversion of the yield curve in the HullWhite model Theorem 3 i Consider a HullWhite modelfor the martingale dynamics ofr drt t 7 a Ttdi adWi7 with given known a and 0 ii Fixed a marketstripped term structure ofbondprices 10 07 TTgt0 subject only to the condition of being twice differentiable Then there exists a tnction 9 such that 1707 T 1707 Tfor all T gt 0 Con cretely 7 91 07T 02 2 T7Taf0T lie where 810g 10 0T f QT 7A Term structure of nward rate 34 Implementation of the HullWhite model A typical implementation of HullWhite 1 Get data of swap rates spot LIB OR rates coupon bond prices and other interest rate derivatives such as caps N Bootstrap zero LIBOR rates from swap rates and available spot LIBOR rates 9 Smoothly interpolate the term structure eg using cubic spline 4 Calibrate the volatility a and rate of mean reversion a to market option prices e g Caps or using historical short rates eg to estimate a 5quot Obtain the mean reversion parameter 9 using the previous theorem In the remainder of this part we illustrate some points from the above plant The data and graphs are obtainedfrom the Doctoral Thesis of L Hernandez at Georgia Tech 2005 Swap rates Obtained from Bloomberg using the page SWPM lhhll I l n A 39 prim d m prim Mmunv Bid 1 J 31b8 1 131m 2 mm 7 39nm Table 1 Wm swap mtw luuLUJ ILit 11391uh1u 1n the US Market AV qu Maturitin Sh rl Luu 1 v 5 mule 2 yours 7 yvnrx 12 wars 1 mum 11 U mnmhs 5 w H 2 lunlllhn i luumlb 1 V 3 un r11th 1 WM 5 W 391 1mmth 39 quot 1x 4 MUD 4 MW 4 tom 4 39JHD Swap rates data and bootstrapped zerobonds US Dollar Sup Cums for May 12 2003 l x I x L Eid rates 7 Ask39 rates 7 1 x v x 2n 25 an o 15 Mummy in yams Figure Swap ram curves as un May 12 2003 Shawn an the swap rate bid and ask quota corresponding to me 24 different maturities amnable in the US market 20 mmzca WNW as Dr My mm ms nu woe o 5 2m 25 30 rm mums a 5 Figure 2 Implied ml 1 ice cuive auxmed from Swap mm data as on May 12 2mm Some market conventions for swap rates oTh e swap rate with maturities of 1 week 1 2 and 3 months involve the exchange of only one payment at maturity The four month and ve month swaps involve exchange of payments at the third month and at maturity The six month swap rate involves the exchange of two quarterly LIBOR rate payments by one xed rate payment at maturity A nine maturity swap will exchange three quarterly LIBOR rate payments for two xed rate payment at the 6th and 9th maturity It can be proved that for maturities up to 6 months the implied spot LIBOR rate should match the quoted swap rate For 1 year and above maturities there are semiannual exchanges Missing spot LIBOR rates are interpolated linearly from the closest available rates The xed rate is on 30360 day counting convention while the oating is on an actual360 day counting convention 21 Mann rwmmu pammzm 5 w 3 2c 25 a 1mm V2313 Figurr 13 in prmm hem the pm of mean 139 en s7n parami tar function found min u 25 may the data m 1 mm dam a smooth enough ch pxice cum x rm mm a 0013 Um am we mu qumLit spLLmISI 39 u omm a pmm m w 1mm dam usmg the modal 3431 Mmquot reversion parameler o a 5 1 1 5 2 2 5 m vimemsquot mm 15 llnx plum Shows a dcmil ol Figun lb lax mummies shormr hurl Lhmo yum 35 Final remarks Does yieldcurve tting make sense 0 For One of the key problems in nance is that of option hedging Hedging is done by buying and selling some primary instruments at the market prices preferably liquid instruments So the option pricing model should at least match the market prices of the primary assets which are used to hedge 0 Against Yield curve tting is usually time inconsistent and hard The calibrated parameters could dramatic ally change in time t Very few models can accommodate the empirical features of the yield curve such as large slope at the shortend o Willmat A wrong model can still be useful and pro table We might loose money on the contract we are pricing but we should make that money back on the hedging assets 22 4 Forward rate models 41 The HeathJarrowMorton Framework Basic ideas of the model 1 The model postulates that for each maturity T gt 0 the instantaneous forward rate it7 T t g T satis es the SDE d ft7 T 0477 Tdt at7 T dWt7 NIT f07T7 Where 0 W is a didimensional Wiener process 0 at7 T and at7 T are random functions a is necessarily 1 X d dimen sional o f 07 T is a smooth function interpreted as the initial term structure of forward rates 2 Recall that f t7 T is aimed as the instantaneous rate at time T for a loan of in nitesimal duration contracted at time t 61 S m gti Applications and usage 1 The model serves more as a framework for analyzing interest rate models It provides tools to perform the following tasks in a simple manner i To replicate perfectly the initial term structure of bond prices which in needs to be generated from bootstrap and interpolation methods ii To handle multiple sources of randomness iii To price bond and other interestrate derivatives 2 Disadvantages 0 Calibration determination of the volatility and correlations structures is hard 0 Interestrate option pricing is computationally demanding since it typically relies heavily on Monte Carlo methods trees and PDErelated numerical methods are often not available or treatable Bond prices and the short rate Theorem 4 1 06 T e 1 mm 2 W ftt 3 If the market is arbitragefree then an arbitragefree price process Ht X for a European option with payo X TT is E Q 6 LT f5gtsd jigV where Q is an equivalent martingale measure for the bond market The bond and short rate dynamics Theorem 5 Suppose that d fi7 T ai7 Tdt ai7 T d Wt Then 1 dpt T 1005 T aTtdt aTtdWt with T T mt awds aTlttgtrlttgti altnsgtds uaTlttgtH2i 2 drt and btdWt with bt aw alttgt altt7tgtfTltotgt taTltu7tgtdu taTltu gtdWltugt aw Hm 42 The HJM drift condition Absence of arbitrage Necessity Fundamental question Does the system of bond prices MT NEW 0 s t lt T 1 precludes arbitrage opportunities Theorem 6 HJM driftcondition A necessary condition for the system of bond prices 1 to be arbitragefree is that there exists a d X 1 vector A A1 i i i My ofprocesses such that T atT aiT aTtsds 7 0i Ti7 z forallT 2 Oandt g T Absence of arbitrage Suf ciency and riskneutral dynamics Theorem 7 0 Suppose that the necessity conditions for the absence of arbitrage hold 0 Suppose that Z12 aft Slam5 ifot MSW is such that EZT 1i eg this is true if A is bounded or deterministic Then the following holds 1 The bond market pi T T gt 07 t g T is arbitragefree and QA 1E1AZTi is a martingale measure for the bonds and the moneymarket account 2 Under a martingale measure Q the dynamics of the forward rate is dft T atTdt 0t T dWQt with at7 T 0t T ET 0T t sds where WQ is a QWiener process 43 The HJM program for valuation of derivatives The HJM program 1 Specify the volatilities ai7 T 2 The riskneutral drift is now xed to be T at T m T aTt Susi t 3 Obtain the dynamics of the forward rates 1 t ft T f 0 T asTds as TdWQsi 0 0 4 Compute bond prices via pg T 5 f ftsds 5 Price interestrate derivatives using EQ e ftTf5gt5dX 3W An example 1 E0 Equot gt The HoLee model for the shortrate model has the general form drt trtdt atdWtr Being a particular case of the HullWhite model it can be shown that When t 1 07 t 0 the model match the initial term structure 1007 T 10 07 T for all T The HoLee model has an equivalent HJM formulation of the form dft7 T 01t7 Tdt adWt NIT f07TA This can be shown by inferring the dynamics of the shortrate via the H M pro gram 5 Change of numeraire and option pricing 51 Introduction General ideas 1 A numeraire is the unit of account in Which other assets are denominated 2 The message here will be that the employment of certain numeraires can facili tate computations of option prices 3 This Will be so in the case of interestrate derivatives Where we will nd out that by taking as numeraire a Tbond the computation of certain interestrate Toptions Will be simpli ed 52 Change of numeraire The general setup 1 The market consists of n1 tradable assets With respective prices So i i i Sn B SO is strictly positive prospective numeraire and B is the moneymarket account B0 1 Which may or may not exist 2 The dynamics of the prices are in terms of certain nontradable factors X1 i i i 7 X n r for instance volatility exchange rates etc 3 We assume the market is arbitragefree 4 We already select a quotsuitablequot riskneutral probability measure Q eg via a calibration method based on market prices under Which the dynamics of the factors and assets are determined as follows d Xit hitdt oitdWt d Sit Sid rtdt aitXdWti Where W is a standard Wiener process maybe multidimensional 5 The price process of a Ticlaim With payoff X is computed by HAM EQ e ftTTWduX ft mam EQ eiffwdux Change of numeraire and martingale measures Theorem 8 The tndamental theorem of nance Consider the normalized market where allprices are denominated in the numeraire SO also called de ator 5390 17 5391 3150 Hi 539 nSU 1 If the original market is arbitragefree then the normalized market is also arbitrage free 2 There exists a martingale measure Q0 2 Q so such that any normalize process 539 is a martingale under Q0 3 Q0 can be chosen such that 110 X 2 HQ t 2050 t is a Qomartingale E9 a ftTT d X ft sou EQO 50T 12c ft 4 Thus the Qodynamics 0ffit 110 X has 0 meanrate ofreturn d 11t a h dW0t where W0 is a standard Wiener process under Q0 A wellknown example 1 Before in the contest of shortrate models we consider a riskneutral martin gale measure Q for the bond market 2 Q is such that the discounted bond price process 6 lot Muldup 11th is a martingale under 3 Also the price process mo 1 E Q a f WWX is an arbitragefree price process for the Tclaim with payoff X 4 Q is a martingale measure associated with the numeraire Bt efot fwd the money market account 53 Valuation of exchange options Another example valuation of an exchange option 1 Consider two stocks with price processes dSlt a151tdt 0151tdWt ngt a2 Sgidt 0252tdWt where W is assumed to be a two dimensional Wiener process 2 Consider a Tclaim that gives the holder the right to swap one share of stock 1 for one share of stock 2 3 The payoff of the option is y 52T 51T 4 Using a change of numeraire write down the value of the option in terms of normal distributions N 54 Forward neutral measure TForward neutral measure 1 The martingale measure associated with the numeraire ptT t g T is pi7 T 2 This means that under the Forward Neutral Measure denoted QT any market price denominated in the zerobond currency is a martingale 1W WT E QT HTl 110 1007T EQT HT In particular the price Ht X 2 E Q e If 10 1qu ft of a TclaiIn is such that HOW METWQT lez H0X 1007TEQTX 3 What do we gain 1007 T is actually observable in the market and 1E QT X is easier to compute than E Q e foT T d 2 55 Applications Why is called forward neutral measurequot 1 Consider a forward contract on an asset S contracted at time t with exchange time T 0 At time t the parties agreed upon an amount 30 called the Tforward price of the asset SO is chosen so that the fair value of the contract at time t is 0 0 At time T the holder of the contract receives a unit of the asset worth ST 0 At time T the writer of the contract receives the money 539 t which was agreed upon at time t 51 ptgtT39 2 It turn out that Ed 3 Since under QT 3t is a martingale we can say that QT is the measure that makes all Tforward prices risk neutral The forward expectation hypothesis 1 Recall that the forward rate f t7 T is the riskfree rate of return on a loan during T7 T dT contracted or locked at time t 2 Also 7 T is the riskfree rate of return on a loan over T7 T dT that is in force at time called the Tforward neutral measure In the previous Theorem s notation So i r 3 It is a common market practice to View ft7 T as a prediction or estimate at time t ofthe unknown future rate 7 T at time T 4 Is this prediction unbiased m T E m 5 The answer is af rmative as far as we adjust our probability measure to the forward neutral measure 0 The forward rate ft7 T t g T is a martingale under QT c In particular ft7 T EQT 7 T Valuation of a call option Via change of numeraires 0 Let S be an asset and consider a European call option with strike K X PaYOff 5T 10 5T K1sTgtKA o By changing numeraires 110795 50Qs 5T 2 KKQT 5T 2K 2 where Q5 is the martingale measure associated with the numeraire S and QT is the Tforward neutral measure 0 Can we nd a more explicitformula Answer Yes if STt so deterministic volatility p 71 has dSt Sm 771mm 6Tt dWt V Deterministic A general call option pricing formula Theorem 9 GemanEl KarouiRochet Under an arbitragefree stochastic interest rate model and under the deterministic forward volatility assumption above the price of the call option on the asset S with strike K and maturity T is given by M 50Nd1 7 KP TNd27 where N is the distribution function of a standard normal and 1n50Kp07T7loT Hmwwt T 7 7 Vfo HUTWW T d1 d2 l waw 0 Note The option value depends only on the volatility 6 ofthe Tforward price ofthe asset and the current market price of the asset and the Tbond d2 30 Price of a call option on a zerobond 1 Under the HullWhite model d r i 7 ardi ad Wt the zerobond With maturity T2 gt T1 admits a deterministic Tlforward volatility condition That is 53913 t is such that 110 5T1tstuffdt 6T1tth7 With 6T1 t e awz tl 7 e T1quoti 2 Hence the price of a call option on a Tgbond with strike K and exercise date T1 lt T2 is given by H0 p0T2Nd1 7 Kp0T1Nd27 3 Where d2 lnp0 T2K O T1K z d1 d2 a and 2 2 A 7 a 72aT1 7aT27T1 a 7 2amp3 l e l e i 3 Note The formula is independent from 9 Caps and Floors 1 Idea A cap is a very popular nancial insurance contract that protect the bor rower from paying more than a prespeci ed interest rate the cap rate R Floor guarantee that the interest rate paid to lender Will never go below some predeter mined oor rate 2 A cap is equivalent to a portfolio of basic contracts called caplets Say the duration of the contract is QT and let To 0 lt T1 lt lt Tn T be equally spaced reset times 6 denotes the inter span Ti 7 TF1 Say K is the principal 3 The W caplet is a contingent claim that pays at time Ti Xi K6 LTi17 Ti 7 14r7 Where LTi17 Ti is the spot market set LIBOR rate on Tl17 Ti 1 PTi717Ti39 LTi717Ti 61411117171111 Question How could this claim help to cap the interest that a borrower pays How does it work Valuation of caps 1 One can price the Ti caplet in terms of a put option on a zerobond maturing Ti and exercise date TF1 E0 The trick is to write the payoff of the Ti caplet as follows Rquot T Xi if TF3 i 1001121713 13 p 1 gt where R 1 SR we x K 1 This is equivalent to the payoff 1 Xi R pTi717Tigt at time TF1 L Hence the value of the ith caplet is the value of Pi units of put options on a Tibond with exercise date TF1 and strike 1R 4 The price of the put can be obtained from the price of the call option via the putcall parity Example of caplet pricing 1 Using putcall parity the value of a put option on a Tgbond with strike K and exercise date T1 lt T2 is H0 KP07T1N 42 P07T2N 417 where d1 and d2 as in the case of the call option equation 3 2 Clewlow and Strickland Section 734 Take T1 16 T2 41 p0T1 9898 p0T2 0733 PTincipal 1 R Cap rate 7 a 0116 and a 10 The value of the caplet is 000227 Example Calibration of models using the caplets 1 Being a very liquid product caps are typically used for calibration purposes 2 For instance consider the HullWhite model d39r t 7 ardt ad VVti We want to calibrate t the parameters a and a to the market prices of caps L A standard approach will be to collect market prices of caps say F and deter mine the values of the a and a that minimize the below problem i 2 Iggy Fltgj z39 where E a7 a is the theoretical value of the caps which depend on a and a but not GUI see Clewlow and Strickland Section 734 32 4 Once a and a are calibrated one can determine 9 to match the observed stripped terms structure of bond prices 6 LIBOR Market model 61 Introduction Motivations l The market quotes several important interest rate derivative speci cally cap oors as if the spot LIBOR forward rates Lt S T were lognormal 2 Empirical evidence showed that parsimonious shortrate models does not ac count for the stylized features of the cap oor and swaption market prices 3 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates 4 Moreover we would like to determine an arbitragefree model that can be easily calibrated to the market prices of traded cap oors 5 What will we gain 0 The model will be easier to calibrate and implement 0 We will be able to price any contract whose cash ow can be decomposed into functions of the LIBOR forward rates 62 Market pricing practices for caps Review of Caps 1 Let tbe the settlement time of the cap 2 Consider a set of increasing maturities To lt T1 lt lt TN T witht g To Denote 6i Ti 7 TF1 called the tenor 3 A cap with cap rate R and resettlements dates To r r r TN1 is a contract which at time Ti pays the amount Xi 5i M11121 Ti RM where LTi17 Ti is the spot LIBOR on Tl17 Ti 1 i P Tiily Ti LTi717Ti this payoff is reset by the market at time TF1 but paid until time Ti 4 The claim contracted at time t that pays the amount Xi 5i M11121 Ti R7 to the holder is called the Ticaplet We can interpret this as a call option on the LIBOR rate LTi17 Ti with strike R 34 The Black model for Cap Pricing 1 The Black formula is build on two principles 0 The market is arbitragefree o The spot LIBOR rate Lt Ti1TZ 6LW is log normal with constant volatility ai under the Tforward measure QT Concretely dLt Ti1TZ Lt Ti1TiaidWTzt tg THi 2 Thus the Black price of the Ti caplet is de ned by the formula Capz t 0i 6iptTi LtTZ1TiNd1 7 RNd2 1 LtTi1TZ 1 2 d7l 7T7t dd7T7t 1 aimnlt R 201z 7 2 1 01 z 7 where 01 is a given constant called the Black volatility of the caplet 3 Example pp 191 C amp S Flat term structure at 5 with continuous com pounding Cap rate if 45 Principal K 1m Tenor 6 3 months Black volatility ai 10 The price of the caplet from 9 to 10 months is Capzf 963205 N1i2599 71045 N1117330125 10013 Market quotes and Black s volatilities 1 Traditionally the market quotes cap and oor prices in terms of Black volatil ities similar to the market practice of quoting vanilla options in BlackScholes volatilities 2 There are two types of implied volatilities 0 Suppose that Cap t is the timet market price of a cap contract with re settlements dates To lt lt TF1 o Capl t 2 Cap t 7 Camila is the stripped price of the Tr caplet o The spot or forward Black volatilities 61 i i i ampn are de ned such that Cap t 221 Capl t6ki 11 N o The at volatilities 61 i i i ampn are de ned such that Cap t 221 Capl tampii 11 N 3 Example pp 212 C amp S Suppose that the we have two US Dollar cap volatil ity quotes on To Jan2195 say 61 1525 and 62 1725 with corresponding maturities T1 21MaT95 and T2 21Jun95 Cap rate is 7 and principal is K 1 What are the cash prices of the caps Cap data and at Black Volatilities Data ech 2 T ble 5 rm mus 110 the man xunLunua 0 cm Floors qur A 310w ux m5 Lahlr are 119 nunher zi Caplets Fluorleb pzr must used tenors obtained from L Hernandez Thesis Gat d m the us blurkel ap Flour m the NC Table 6 quotde I slmrx the hub r vumilily a rupunml nu Blwumhum on May W ZULU Tim mhlw thm w Um wluy 7 m mm t or swim y 1 mph 3 I 1 WM 5 mm 2 mm 5 up EL mm 27 was 13 mp x Venn 39 mm 1 my 5 mm 7 mm m mmph n L 1 km W gm mimic on mum 0 31 up 111 94 0 Wm scum36me 7 39 m n k TI n m the 3 c a 2 1a 15 and m inunth tunes 39I ypical at Black VOlS behavior Data obtained from L Hernandez Thesis Gatech 2005 Figure 23 This 11101 Show quot atquot volatility data m L e Table 0 The plot shows line rst 01 115 for mm dam Ln l hP US markets m man a lnaru ti 1 4 and 1yea1 s Dalia 1 J 1 CapsFloors quoted rm mum and cubic Ir 39 me are quoted for CapsFL 9 J 015 63 The LIBOR market model Generalities of the LIBOR model Setup o A set of discrete resettlement dates To lt T1 lt lt TN71 0 A set of zerorbonds p With maturities To lt T1 lt lt TN 0 Let Li t Lt 1171 gt denote the LIBOR forward rate for 1171 cone acted at tiIne t Li plttT1 7 man 1 6T pm m 5quot iTislV 36 Notation The LIBOR model will be de ned with respect to the forward neutral measures Qi 2 QTI 239 17 i i i N recall that QTI is the martingale measure induced by the numeraire MO De nitions and assumptions De nition 10 We say that we have a discrete tenor LIBOR market model with volatil ities 01 i i i aN if the ith forward LIBOR rate has dynamics dLZt Litait dWit7 under Q where Wi denotes a standard Wiener process under Q In order to construct an arbitragefree LIBOR market model we assume that Key Assumption Q The LIBOR rates Li have riskneutral deterministic volatilities oi Concretely under the forward measure Q N dLZt Lit mitdt aiidWNi where al is deterministic and WN is a Q N Wiener process didimensional Important consequences 1 Under Qi the distribution of Li is lognormal Concretely t I 1 1 log Lit aisdWZs 7 E Hals H2ds 0 0 2 The resulting riskneutral prices of the caplets induced by Q N coincide with a Black s model with implied spot volatilities if 1 T171 2 at Ti 7 t HaltsgtH d8 Then under some structural assumptions about the shape of the volatility one can calibrate the LIBOR model to recover quoted Black volatilities 6 in the market Popular speci cations N 1MN L o O39it oi 0 S t S TFl o O39it a Tj1 S t S Tj o O39it mall1 7 73434714quot 4 Once a model for the forward rates has been calibrated Monte Carlo methods can be used to compute the prices of exotic interestrate options see below 37 Construction of the LIBOR market model 1 Given the Assumption we know that the change of martingale measure from QN to Qi will only change the drift ofLi from mi to say M 2 Thus the problem of constructing the LIBOR model is that of determining the QNdrifts mi that will produce the desired driftless pi 0 requirement ofthe LIBOR model when changingfrom QN to Qi Theorem 11 Brace Gatarek Musiela Let 01 i i i aN be given deterministic tnctions tentative volatility structure for the forward rates De ne the processes N 7 5kLkt N dLZt 7 iLla mama t dt Llama d W t 239 17 i i i N where WN is a ddimensional Wiener process relative QN Then the Qidynamics ofLi is given by d Lit LZ mlt d W t where Wi is a standard Wiener process under Qi Thus we have a LIBOR market model with volatilities 01 i i i O39N exists Pricing under the LIBOR model 1 Suppose we wish to price an option whose payoff at time T1 depends on the future LIBOR forward rates Xp LpTpmLNTpi 2 The price will be then T mo 2a EQ 10 P Wm pm TN 1W 2617 3 The standard approach to evaluate EQN Xp is via Monte Carlo simulating the forward rates Li backwards starting at 239 N as follows Inn7 1h lnLinh O39inh WNn 1h 7 WNnh 1 N Mam h faith g mammal nh Examples 1 European put option on a TNbond The options expiration date is To and strike is Payoff X K 7 10To7 TNr Key observation N PltT07TN H 1 5iLiTo71 E0 European put option on a TNbond With coupon rate 5 The options exercise date is To and strike is K Payoff X K 7 pTo7 TN 54r7 Where N71 pltToTN c e Z pToTi 1 cpTo TN 3 A European swaption With exercise date To gives the holder the right to enter into a swap arrangement at T0 With exchange times Tl7 r r r TN N X K621001071 PltT07TN 1 171

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