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# 686 Class Note for MATH M0070 at Purdue

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Date Created: 02/06/15

Interest rate models and derivatives The LIBOR Market model Jos FigueroaL pez Purdue University Math 516 Stat 541 Fall 2008 75A mtmnar H L w 3 563 A032 w c 233 a 3m m3cvmm mnmom gmsmie m 33me in 3012 6 u 83562 2 Em oc5 mlt 2 Emmi 33 afgmgmf nm oam EH 3 cm E 3139 3 3 3279 1me ow 5351 335025 in 532 Em 22 53 5 m ltn 5 lt3 En Eire n4 nns arm rutlyl lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices 9 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices 9 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates 0 Moreover we would like to determine an arbitragefree model that can be easily calibrated to the market prices of traded capfloors lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices 9 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates 0 Moreover we would like to determine an arbitragefree model that can be easily calibrated to the market prices of traded capfloors e What will we gain lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices 9 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates 0 Moreover we would like to determine an arbitragefree model that can be easily calibrated to the market prices of traded capfloors e What will we gain The model will be easier to calibrate and implement lnlrodudion Motivations o The market quotes several important interest rate derivative specifically capfloors as if the spot LIBOR forward rates Lt S T were lognormal 9 Empirical evidence showed that parsimonious shortrate models does not account for the stylized features of the capfloor and swaption market prices 9 The inconsistencies inherent in shortrate models raise the problem of devising an arbitragefree model that is consistent with the lognormality of LIBOR rates 0 Moreover we would like to determine an arbitragefree model that can be easily calibrated to the market prices of traded capfloors e What will we gain The model will be easier to calibrate and implement We will be able to price any contract whose cashflow can be decomposed into functions of the LIBOR forward rates Review of Caps Mama pn ngwsn oesiomaps Market pacing practices tor caps Review of Caps 0 Let t be the settlement time of the cap Market pricing practices tor caps Review of Caps 0 Let t be the settlement time of the cap 9 Considera set of increasing maturities To lt T1 lt lt TN T with tg To Denote 6 T 7 TH called the tenor arm m cm Market pricing practices tor caps Review of Caps 0 Let t be the settlement time of the cap 9 Considera set of increasing maturities To lt T1 lt lt TN T with tg To Denote 6 T 7 TH called the tenor 9 A cap with cap rate R and resettlements dates Top t t TAM is a contract which at time T pays the amount Xi SIU Tiih Ti 7 9 where LT1 T is the spot LIBOR on 717177 1 7 pTie17 Ti 6ipTl 17Ti this payoff is reset by the market at time TM but paid until time T LTie1Ti arm m cm Market pricing practices lor caps Review of Caps 0 Let t be the settlement time of the cap 9 Considera set of increasing maturities To lt T1 lt lt TN T with tg To Denote 6 T 7 TH called the tenor 9 A cap with cap rate R and resettlements dates Top t t TAM is a contract which at time T pays the amount Xi 6il Ti71777 9 where LT1 T is the spot LIBOR on 717177 1 7 pTie17 Ti 6ipTi717 Ti this payoff is reset by the market at time TM but paid until time T o The claim contracted at time t that pays the amount Xi 6iLTie1 Ti 7 RM LTie1Ti to the holder is called the Tcaplet We can interpret this as a call option on the LIBOR rate LT1 T with strike R UBDF Maw m cm Marke pricing prac ces tor caps The Black model for Cap Pricing w our Markel pricing praclices lor caps The Black model for Cap Pricing o The Black formula is build on two principles whim14 Markel pricing pracliceslor caps The Black model for Cap Pricing o The Black formula is build on two principles The market is arbitragefree Markel pricing pracliceslor caps The Black model for Cap Pricing o The Black formula is build on two principles The market is arbitragefree The spot LIBOR rate Lt 774 m is log normal with constant volatility 7 underthe T forward measure 0 Concretely dLf777177 L1777177de7 1 lg TH arm m cm Market pricing practices lor caps The Black model for Cap Pricing o The Black formula is build on two principles The market is arbitragefree The spot LIBOR rate Lt 771 77 is log normal with constant volatility 7 underthe T forward measure 0 Concretely dLT Tgt177 L1777177de7 1 1 7771 9 Thus the quotBlack price of the T caplet is defined by the formula Cap150W 61PM Ti Mt T1217 TiNd1 7 RNd27 1 LtTT 1 d1 mltgt a nio d2d1iaT7t where a is a given constant called the Black volatility of the caplet aw niliyi Market pricing practiceslor caps The Black model for Cap Pricing o The Black formula is build on two principles The market is arbitragefree The spot LIBOR rate Lt 774 m is log normal with constant volatility 7 underthe T forward measure 0 Concretely dLU TH 77 LU T171 77deT f 1 T171 9 Thus the quotBlack price of the T caplet is defined by the formula Cap150W 61PM Ti Mt T1217 TiNd1 7 RNd27 1 than 1 2 d Ta lnlt R 2aiTI t 7 d27d1iauTiit where a is a given constant called the Black volatility of the caplet 9 Example pp 191 C amp S Flat term structure at 5 with continuous compounding Cap rate if 45 Principal K 1 m Tenor 6 3 months Black volatility a 10 The price of the caplet from 9 to 10 months is Cap5 963205 M12599 7 045 N11733025 0013 UBDF Maw m cm Marke pricing prac ces tor caps Market quotes and Black s volatilities at arm m cm Market pricing practices lor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities arm m cm Market pricing practices lor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Market pricing practiceslor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Suppose that Cap0 is the timet market price of a cap contract with resettlements dates T0 lt lt 774 Market pricing practiceslor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Suppose that Cap0 is the timet market price of a cap contract with resettlements dates T0 lt lt 774 Capfa Cap0 7 Cap211 is the stripped price of the 777 caplet Market pricing practiceslor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Suppose that Cap0 is the timet market price of a cap contract with resettlements dates T0 lt lt 774 Capfa Cap0 7 Cap211 is the stripped price of the 777 caplet The spot or forward Black volatilities 51 in are defined such that Capm 211Cap5taki1lv Market pricing practiceslor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Suppose that Cap0 is the timet market price of a cap contract with resettlements dates T0 lt lt 774 Capfa Cap0 7 Cap211 is the stripped price of the 777 caplet The spot or forward Black volatilities 51 in are defined such that Capm 211Cap5taki1lv The flat volatilities 51 an are defined such that Capra 2L1Capl t ii1N Market pricing practiceslor caps Market quotes and Black s volatilities 0 Traditionally the market quotes cap and floor prices in terms of Black volatilities similar to the market practice of quoting vanilla options in BlackScholes volatilities 9 There are two types of implied volatilities Suppose that Cap0 is the timet market price of a cap contract with resettlements dates T0 lt lt 774 Capfa Cap0 7 Cap211 is the stripped price of the 777 caplet The spot or forward Black volatilities 51 in are defined such that Capm 211Cap5taki1lv The flat volatilities 51 an are defined such that Capra 2L1Capl t ii1N 9 Example pp 212 C amp 8 Suppose that the we have two US Dollar cap volatility quotes on To Jan2195 say 61 1525 and 32 1725 with corresponding maturities T1 21 Mar95 and T2 21Jun95 Cap rate is 7 and principal is K 1 What are the cash prices of the caps LIE F kel model Market prlclrlg practices for caps Cap data and flat Black volatilltles Data obtained from L Hernandez Thesis Gatech Table 5 11m uple shew the usou memlela cf Caps Flows quoted in up Us hlnrket A154 Shawn in em table are he umber of Caplets Flea 1 par Cap IFbDr la the em most user tenors Avmjnbk hlnulrities a month tenor 6 much tenor Cap Mm of Caplets Cap Liam pr Caplets yam 3 enplecs 1 tablet 7 mplels Plats 1 caplm a Plats 13 mums 7 caplm 19 plats 9 clt1plgtgtrs rapists 13 Caplets a Eaplebx m Caplets Fur me e n p deer that a an yam matunt Cap conser uf weep caplels mm quot95 m the al a and 9 month mama and a mu vent Cap should Antlnde seven Caplas vliLh an alcheS 12 5 18V and 21 monthtinles 2005 Table l Flat Cnlp vnlmilih39 as repelml nn Blnmnhm on Max mm 003 Thia lablv elm rhe hids m 50pm nf hm dm39 Flat Vela 39ll39tlup hiatllrity l Volalllily 0 l yea 14 zuu 2 cnl UU 443w gtJlUl7l 371w 31700 10 wan 25700 At me mnney Caps That ls the cap rat swap rate with me same payment Schedule as me cap Mama pno mg praohces yor capsr Typical flat Black vols behavior Data obtalned from L Hernandez Thesls Galech 2005 MW Figure 23 This plot shows quotHmquot mm 39 dam m US Dollar CapsFloors quoted for May 12m 2003 see Table 6 The plot show Line rst order cubic and cubic brspline interpolations for that data In the us markets at Volatilitics are quoted for Capsmoms 7 and 10 yams 0t maturities l 2 3 4 new Maw may The ua on mama mode Generalities of the LIBOR model e qu new Maw may The L Ed imarket mode Generalities of the LIBOR model Setup Lg q 3 The Li d markei modei Generalities of the LIBOR model Setup o A set of discrete resettlement dates To lt T1 lt lt TAM The Liaea markei modei Generalities of the LIBOR model Setup o A set of discrete resettlement dates To lt T1 lt lt TAM o A set of zerobonds p p T with maturities To lt T1 lt lt TN mm mm The Liecn markei modei Generalities of the LIBOR model Setup o A set of discrete resettlement dates To lt T1 lt lt TAM o A set of zerobonds p p T with maturities To lt T1 lt lt TN 0 Let Lt Lt Ti1T denote the LIBOR fonNard rate for T1 Ti contracted at time t lP 7Ti71Ptv m 06 prT 6quot TiiTier mm mm The Lia onmarkei model Generalities of the LIBOR model Setup o A set of discrete resettlement dates To lt T1 lt lt TAM o A set of zerobonds p p T with maturities To lt T1 lt lt TN 0 Let Lt Lt Ti1T denote the LIBOR fonNard rate for T1 Ti contracted at time t lP 7Ti71Ptv m 06 prT 6quot TiiTier Notation The LIBOR model will be defined with respect to the forward neutral measures 0 07 i 1MN recall that Q is the martingale measure induced by the numeraire p LiBDFi Maw yinjyi The Li dn markei modei Definitions and assumptions mi our aw inujyi The Liaon markei modei Definitions and assumptions Definition We say that we have a discrete tenor LIBOR market model with volatilities a1t t taN if the ith fonNard LIBOR rate has dynamics 0 Lt Ltat dW t under 0 where W denotes a standard Wiener process under 0 aw mm The Lia on markei modei Definitions and assumptions Definition We say that we have a discrete tenor LIBOR market model with volatilities a1t t taN if the ith fonNard LIBOR rate has dynamics 0 Lt Ltat dW t under 0 where W denotes a standard Wiener process under 0 In order to construct an arbitragefree LIBOR market model we assume that aw met The Lia on markei modei Definitions and assumptions Definition We say that we have a discrete tenor LIBOR market model with volatilities a1t t taN if the ith fonNard LIBOR rate has dynamics d Lt Ltat dW it under 0 where W denotes a standard Wiener process under 0 In order to construct an arbitragefree LIBOR market model we assume that Key Assumption The LIBOR rates L have riskneutral deterministic volatilities ai Concretely under the fonNard measure ON dLt Lt mitdt aitdWNt where a is deterministic and W is a QN Wiener process didimensional UBDF Mum mujy Important consequences The an dn fmarka mode e our The Linen markei modei Important consequences 0 Under 0 the distribution of L is Iognormal Concretely t t IogLit 0 airedw39rsrgo iiaisii2ds mm mm The Li on markei model Important consequences 0 Under 0 the distribution of L is Iognormal Concretely t t IogLit 0 airedw39rsrgo llaisll2ds e The resulting riskneutral prices of the caplets induced by ON coincide with a Black s model with implied spot volatilities 52 1 T71llasll2ds 1mNt I 7 7 7 mm mm The Lia on markei model Important consequences 0 Under 0 the distribution of L is Iognormal Concretely t t IogLit 0 airedw39rsrgo llaisll2ds e The resulting riskneutral prices of the caplets induced by ON coincide with a Black s model with implied spot volatilities 1 Tlil 52Tiitt llasll2ds i1mNt 9 Then under some structural assumptions about the shape of the volatility one can calibrate the LIBOR model to recover quoted Black volatilities E in the market Popular specifications 39 70 7 0 S t S 7771 WU we TH E l S T ma WM 7 oer tic aw moi The Lia on markei model Important consequences 0 Under 0 the distribution of L is Iognormal Concretely t t IogLit 0 airedw39rsrgo llaisll2ds e The resulting riskneutral prices of the caplets induced by ON coincide with a Black s model with implied spot volatilities 2 1 Tlil 2 I 0Ti7tt llasllds 11MNt 9 Then under some structural assumptions about the shape of the volatility one can calibrate the LIBOR model to recover quoted Black volatilities E in the market Popular specifications 39 70 7 0 S t S 7771 WU we TH E l S T ma WM 7 oer tic 9 Once a model for the fonNard rates has been calibrated Monte Carlo methods can be used to compute the prices of exotic interestrate options see below aw man The Lia on markei model Construction of the LIBOR market model 0 Given the Assumption we know that the change of martingale measure from Q to 0 will only change the drift of L from m to say M 9 Thus the problem of constructing the LIBOR model is that of determining the QNdrifts m that will produce the desired driftless u 0 requirement of the LIBOR model when changing from ON to 0 Uaom Mum may The L aom mam mode Construction of the LIBOR market model Theorem Brace Gatarek Musiela q iv The Lia on markei modei Construction of the LIBOR market model Theorem Brace Gatarek Musiela Let 01 t t t aN be given deterministic functions tentative volatility structure for the fanvard rates mm mm The Lia on markei modei Construction of the LIBOR market model Theorem Brace Gatarek Musiela Let 01 t t t aN be given deterministic functions tentative volatility structure for the fanvard rates Define the processes N Mk0 7 7 3 N dLt 7 Ltki1 16kLk0aktatdt Ltat d W t i 1m N where W is a ddimensiona Wiener process relative ON mm mm The Lia on markei modei Construction of the LIBOR market model Theorem Brace Gatarek Musiela Let 01 t t t aN be given deterministic functions tentative volatility structure for the fonvard rates Define the processes N Skik0 N dLt 7Lit Z ma ai dt Litait d W t7 ki1 i 1m N where W is a ddimensiona Wiener process relative 0 Then the 0 dynamics of L is given by d LiU MUMf d W UL where W is a standard Wiener process under 0 mm mm The Lia on markei modei Construction of the LIBOR market model Theorem Brace Gatarek Musiela Let 01 t t t aN be given deterministic functions tentative volatility structure for the fonvard rates Define the processes N Skik0 N dLt 7Lit Z ma ai dt Litait d W t7 ki1 i 1m N where W is a ddimensiona Wiener process relative 0 Then the 0 dynamics of L is given by d LiU MUMf d W UL where W is a standard Wiener process under 0 Thus we have a LIBOR market model with volatilities 01 t t t aN exists The Lr ee marm moder Pricing under the LIBOR model 0 Suppose we wish to price an option whose payoff at time Tp depends on the future LIBOR forward rates Xp LpTp r r LNTp The Lia dh markei modei Pricing under the LIBOR model 0 Suppose we wish to price an option whose payoff at time Tp depends on the future LIBOR forward rates Xp LpTp t t t LNTpt 9 The price will be then 39o7 XI E Q 9 fun r d 2p p07 TN E Q Xp The Lia on markel model Pricing under the LIBOR model 0 Suppose we wish to price an option whose payoff at time Tp depends on the future LIBOR forward rates Xp LpTp t t t LNTPt 9 The price will be then 39o7 XI E Q 9 fun r d 2p p07 TN E Q Xp 9 The standard approach to evaluate E Q 29 is via Monte Carlo simulating the forward rates L backwards starting at i N as follows In Ln1h In Lnh 0nh WNn1h7 WNnh N h 15050717 21 1 aknhanhgt ki quotquotV39IIW V ngmn a QC aw may The an oa marka mode Examples 0 European put option on a TNbond The option s expiration date is To and strike is K Payoff X K7 pT0 TN aw may The Lia oa markei modei Examples 0 European put option on a TNbond The options expiration date is To and strike is K Payoff X K7 pT0 TNt Key observation N poo TN H 1 SILinequot i1 aw may The Lia oa markei modei Examples 0 European put option on a TNbond The options expiration date is To and strike is K Payoff X K7 pT0 TNt Key observation N poo TN H 1 SILinequot i1 9 European put option on a TNbond with coupon rate 0 The options exercise date is To and strike is K Payoff X K 7 pTo TN c where N71 PT07 TN 0 0 ZPT07TI1 0PTm TN 1 aw may The Lia oamarkei modei Examples 0 European put option on a TNbond The options expiration date is To and strike is K Payoff X K7 pT0 TNt Key observation N poo TN H 1 SILinequot i1 9 European put option on a TNbond with coupon rate 0 The options exercise date is To and strike is K Payoff X K 7 pTo TN c where N71 pTo TN 0 0 2pm Ti 1 cpTo TN i1 9 A European swaption with exercise date To gives the holder the right to enter into a swap arrangement at T0 with exchange times T1MTN 1 N X K6pToTiPTovTN1gt V

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