Markov Functional interest rate models with stochastic volatility

Transcription

1 Markov Functonal nterest rate models wth stochastc volatlty New College Unversty of Oxford A thess submtted n partal fulfllment of the MSc n Mathematcal Fnance December 9, 2009

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3 To Rahel

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5 Acknowledgements I would lke to thank my supervsor Dr Jochen Thes for advsng me throughout the project and proof readng of ths dssertaton. Furthermore I want to extend my grattude to d fne GmbH for gvng me the opportunty to attend the MSc n Mathematcal Fnance programme. But above all I am ndebted to my famly, especally to my wfe Rahel, for ther great support and patence.

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7 Abstract Wth respect to modellng of the forward nterest rate term structure under consderaton of the market observed skew, stochastc volatlty Lbor Market Models LMMs have become predomnant n recent years. A powerful representatve of ths class of models s Pterbarg s forward rate term structure of skew LMM FL TSS LMM. However, by constructon market models are hgh dmensonal whch s an mpedment to ther effcent mplementaton. The class of Markov functonal models MFMs attempts to overcome ths nconvenence by combnng the strong ponts of market and short rate models, namely the exact replcaton of prces of calbraton nstruments and tractablty. Ths s acheved by modellng the numerare and termnal dscount bond and hence the entre term structure as functons of a low dmensonal Markov process whose probablty densty s known. Ths study deals wth the ncorporaton of stochastc volatlty nto a MFM framework. For ths sake an approxmaton of Pterbarg s FL TSS LMM s devsed and used as pre model whch serves as drver of the numerare dscount bond process. As a result the term structure s expressed as functonal of ths pre model. The pre model tself s modelled as functon of a two dmensonal Markov process whch s chosen to be a tme changed brownan moton. Ths approach ensures that the correlaton structure of Pterbarg s FL TSS s mposed onto the MFM, especally the stochastc volatlty component s nherted. As part of ths thess an algorthm for the calbraton of Pterbarg s FL TSS LMM to the swapton market and the calbraton of a two dmensonal Lbor MFM to the dgtal caplet market was mplemented. Results of the obtaned skew and volatlty term structure Pterbarg parameters and numerare dscount bond functonal forms are presented.

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9 Contents 1 Introducton 1 2 A revew of Lbor Market and Markov Functonal Models The Lbor Market Model Non log normal forward Lbor dynamcs Incorporaton of stochastc volatlty Markov Functonal Models Defnton and examples of Markov Functonal Models A Lbor Markov Functonal Model Mult dmensonal Markov Functonal Models A Lbor Market Model as pre model for a Markov Functonal Model Pterbarg s term structure of skew forward Lbor model The forward Lbor dynamcs Swap rate dynamcs under the FL-TSS model Dervaton of the forward swap volatlty level Dervaton of the forward swap skew The effectve skew and volatlty formulaton The effectve forward swap skew The effectve forward swap volatlty Calbraton of the FL-TSS model Forward rate volatlty calbraton Forward rate skew calbraton Calbraton results A Markov functonal model wth stochastc volatlty Pterbarg s FL TSS Lbor Market Model as pre model The pre model wth two Brownan drvers A smplfcaton of the pre model process

10 4.4 Constructon of a two dmensonal Lbor Markov functonal model A two dmensonal Lbor Markov functonal model n the termnal measure Calbraton results Concluson 53 A Mathematcal detals 55 A.1 The drft term n the Lbor Market Model A.2 The dervatve of the forward swap rate w.r.t the forward Lbor rates A.3 Dervaton of the coeffcent c mn A.4 Proof of corollary A.5 A recurson scheme for a system of tme dependent Rccat equatons A.5.1 An analytc soluton for D t, T A The case g A The case g = A.5.2 An analytc soluton for A t, T A The case g A The case g = A.5.3 Summary of the soluton A.6 Dervaton of relaton A.7 2d Markov functonal ntegraton B The Heston Model 75 B.1 Specfcaton of the model dynamcs B.2 The characterstc functon B.3 The soluton of the Heston ODE B.3.1 Boundary condtons B.3.2 A system of Rccat ODEs B.4 Opton prcng by transformaton technques B.5 Calbraton of the Heston Model C Tables and fgures 87 Bblography 93

11 Lst of Fgures 3.1 Volatlty level λ 10 t and skew β 10 t of forward rate F 10 t for tmes = 0 y t < T 10 = 10 y Proxy forward rate F 10 T as functon of z t = 0, z 2 at reset tme T 10 = 10y. Ths corresponds to the zero correlaton case, Γs The numerare dscount bond as functonal of F 10 T 10,z T10 : DT 10, T 11 ; F 10 T 10,z T C.1 The numerare dscount bond as functonal of F 1 T 1,z T1 : DT 1, T 11 ; F 1 T 1,z T1. 89 C.2 The numerare dscount bond as functonal of F 3 T 3,z T3 : DT 3, T 11 ; F 3 T 3,z T3. 89 C.3 The numerare dscount bond as functonal of F 5 T 5,z T5 : DT 5, T 11 ; F 5 T 5,z T5. 90 C.4 The numerare dscount bond as functonal of F 10 T 7,z T7 : DT 7, T 11 ; F 7 T 7,z T7. 90

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13 Chapter 1 Introducton Ths study s dedcated to the ncorporaton of stochastc volatlty nto a Markov functonal framework. The class of Markov functonal models MFM was ntroduced by Hunt, Kennedy and Pelsser n [15. A major motvaton whch lead to ther development was the desre to have models that can exactly replcate prces of lqud calbraton nstruments n a smlar fashon to market models whle mantanng the effcency of short rate models n calculatng dervatve prces [13, [18. Latter are formulated n terms of the short rate or nstantaneous forward rate whch cannot be traded n the market. As a consequence the prces of dervatves n these models are qute nvolved functons of the underlyng process whch s beng modelled. Ths fact makes t dffcult to capture the most characterstc features of a dervatve product wth models of ths knd. However, ther strong pont s that the short rate process s easy to follow and hence mplementon s straghtforward [13. Unlke short rate models the class of market models s formulated n terms of market rates whch are drectly related to tradable assets. Thus they exhbt better calbraton propertes than short rate models. However, as these models capture the jont dstrbuton of market rates, they are hgh dmensonal by constructon and tedous to mplement. The frst formulaton of a market model was provded by Brace, Gatarek and Musela n the context of forward Lbors LMM [4. A forward swap market model was developed by Jamshdan n 1997 [16. In these approaches the underlyng rates are modelled as log normal martngales under ther own probablty measure. However, the presence of a volatlty skew n the caplet and swapton markets ndcate that a pure log normal forward dynamcs s not approprate. In ths respect modfed forward rate dynamcs were ntroduced, e.g n the context of constant elastcty of varance CEV and dsplaced dffuson models n whch mxtures of pure normal and log normal dynamcs are consdered [5, [23. Amng at a proper modellng of the skew term structure, stochastc volatlty extensons of Lbor and Swap Market models were ntroduced. Ths was 1

14 acheved by modellng the forward rate varance as CEV process. Approaches of ths knd are the ones by Andersen and Andreasen [2 and Pterbarg [19, [20. Especally Pterbarg s stochastc volatlty extenson accounts for a Lbor specfc term structure of forward rate skews and volatltes. In the study at hand ths forward Lbor term structure of skew model FL TSS s used n the constructon of a Markov functonal model wth stochastc volatlty. The MFM framework s based on formulatng the numerare and the termnal dscount bond as functonals of a low dmensonal Markov process whose dynamcs can be followed easly. The functonal forms n turn are obtaned by calbraton to prces of lqud dervatves at partcular dates whch are relevant to the product beng prced. As the dscount bonds at earler tmes are obtaned by applyng the martngale property of numerare rebased assets the resultng model s arbtrage free by constructon. Thus, MFMs combne the strong ponts of market and short rate models. The ncorporaton of stochastc volatlty nto a MFM s based on the concept of a pre model whch depends on a low dmensonal Markov process [14,[17. By regardng the forward Lbors and hence the dscount bond processes as functons of the pre model process, the calbraton can be formulated n terms of the latter. Thus the correlaton structure of the pre model s ncorporated nto the MFM. In ths study a pre model s selected whch s an approxmaton of a calbrated Pterbarg FL TSS model and depends on a two dmensonal brownan moton. Ths proxy s then employed n constructng a Lbor MFM whch nherts the stochastc volatlty structure of Pterbarg s FL TSS. Tractablty s mantaned snce calbraton nvolves the ntegraton of the known probablty dstrbuton of the two dmensonal Markov process. The thess s structured as follows: In chapter 2 the concepts of LMMs and MFMs are revewed. Workng under the termnal forward measure drft terms for a stochastc volatlty LMM are derved. Furthermore mult dmensonal extensons of MFMs are dscussed. Chapter 3 s dedcated to the detaled study of Pterbargs FL TSS LMM. Ths model s calbrated to the swapton market, and the resultng term structure of skews and volatltes s presented. The ncorporaton of stochastc volatlty nto a MFM s the topc of chapter 4. Here a two dmensonal Lbor MFM s constructed whch uses an approxmaton of Pterbarg s FL TSS as pre model. The model s calbrated to the dgtal caplet market, and resultng numerare dscount bond funtonals are presented. The thess concludes wth chapter 5. Besdes result tables and fgures, the appendces contan mathematcal detals and a thorough presentaton of the Heston Model. 2

15 Chapter 2 A revew of Lbor Market and Markov Functonal Models In ths chapter we revew the class of Lbor Market and Markov functonal models whch have become prevalent n the last ten to ffteen years. In secton 2.1 we dscuss the LMM under consderaton of non log normal forward rate dynamcs and stochastc volatlty. In partcular, workng under the termnal forward measure forward rate drft terms are derved. The Markov functonal framework s ntroduced n secton 2.2. Theren mult dmensonal extensons are dscussed as well. In secton 2.3 the dea of constructng MFMs n terms of a pre model s ntroduced. 2.1 The Lbor Market Model As already mentoned n the ntroducton the Lbor Market Model focusses on modellng the dynamcs of forward Lbor rates F t; T, T 1 whch reset at tmes T, = 1,...,N. In a determnstc volatlty settng employng K ndependent Brownan drvers these are modelled as log normal varables wth respect to ther martngale measure Q 1 whch s nduced by takng the dscount bond D t, T 1 as numerare, df t; T, T 1 = λ tf t; T, T 1 K k=1 σ,k tdw 1 k t = λ tf t; T, T 1 σ t T dw 1 t, for tmes t < T T N where dwt 1 s a K dmensonal vector of orthogonal Brownan motons under Q 1 and λ t are postve contnuous, real valued functons. In partcular the relaton dw k, dw l = δkl dt holds. The K dmensonal vector σ t contans the load factors of the orthogonal brownan motons onto forward rate F whch satsfy K k=1 σ,kt 2 = 1. Thus they defne a correlaton matrx through the relaton ρ j t = σ tσ j t T = {ρ j } =1,...,K. 1 Indeed, n ths formulaton the covarance of forj=1,...,k 1 We assume that K = N,.e., that each forward rate has ts own drvng brownan moton. 3

16 ward Lbor yelds s gven by df t F t, df jt = λ tλ j t F j t K σ,k tσ j,l t dw 1 k t, dw 1 l t }{{} δ kl dt σ,k tσ j,k t dt = λ tλ j tρ j tdt, k,l=1 K = λ tλ j t k=1 } {{ } =ρ j t from where t becomes apparent that the brownan correlatons ρ j t as well as the volatltes λ t contrbute to the forward rate correlaton. Snce the forward Lbors follow a log normal process n ther own martngale measure caplet prces are gven by the Black76 formula, comp. [3. Therefore the volatltes λ t can be obtaned by calbraton to quoted caplet prces. For ths a wdespread approach s to assume a parametrc shape for the volatltes as functon of tme to expry whch correctly captures ther dynamcs observed n the market. The parameters are then determned n the course of the calbraton process, comp. [21. Wth respect to the correlaton functon the most reasonable approach s to model t n parametrc form as well. The reason for ths s that t s not easy at all to extract nformaton on the nstantaneous correlaton ρ j t out of quoted dervatve prces, e.g., swapton volatltes, because the latter depend on the hstory of λ tλ j tρ j t on the tme nterval whch starts at and ends on swapton expry. The parametrc form proposed by Rebonato n [21 and [22 s gven by ρ j t = ρ 1 ρ exp δ T t ǫ T j t ǫ, 2.1 wth constants 1 ρ 1, δ, ǫ > 0, and wll be adopted n what follows. Of course, to use the LMM n practce the dynamcs of all forward Lbor rates have to be formulated n a sngle measure. In ths respect a convenent choce s the termnal measure Q N1 whch s nduced by takng the termnal dscount bond D t, T N1 as numerare. As a consequence only the forward Lbor F N s a martngale, and accordng to Grsanov s theorem all other forward Lbors wll be modfed by addtonal drft terms, comp. [13. Indeed, when changng from the termnal measure Q N1 to the martngale measure of forward rate F the process dw N1 k t µ N1 tdt also s a brownan moton under Q 1. Hence n the termnal measure the forward rate processes become df t; T, T 1 = λ tf t; T, T 1 σ t T[ µ N1 tdt dwt N1, = F t; N1 T, T 1 λ t [ µ tdt σ t T dwt N1, 2.2 N µ N1 α l λ l tρ l tf l t =, µ N1 N t = 0, 1 α l F l t l=1, t < T T N, = 1,...,N 4

17 where the drfts µ N1 t are gven by A.10 whch s derved n appendx A Non log normal forward Lbor dynamcs The forward Lbor process 2.2 presented above models the forward Lbors F as a log normal processes wth respect to ther martngale measure. In the termnal measure a log normal behavour s mantaned for forward Lbor F N. However, the caplet/floorlet market dsplays a volatlty surface n terms of terms of mpled Black volatltes,.e., the volatlty vares as opton expres and moneyness changes. Specfcally, the observed volatltes are monotone decreasng functons of the forward Lbor level, a behavour whch s denoted as volatlty skew. Its presence ndcates that the forward Lbors do not follow a log normal process, for n that case the mpled Black volatltes should be constant. One proposal for an alternate forward rate dynamcs provded by Rubnsten [23 s a dsplaced dffuson whch combnes a log normal and a normal process. Another approach s to model the forward Lbors as constant elastcty of varance CEV processes whch was proposed by Cox and Ross [5. In both cases the forward Lbor change can be wrtten n terms of a volatlty functon ϕ F whch mposes a rate level dependence onto the forward rate volatltes,.e the forward Lbors are modelled as df t; T, T 1 = ϕ F λ tσ t T dw 1 t, = 1,...,N. For example a dsplaced dffuson model s establshed wth a functon of the form ϕ F t; T, T 1 = β F t; T, T 1 1 β F T0 ; T, T 1, where the dsplacement parameter β s a real valued constant. Obvously the case β = 1 corresponds to a log normal dynamcs. For β = 0 a normal process s recovered. A generalzaton of ths wth tme dependent paramter βt wll be consdered n the next chapter where Pterbarg s forward Lbor term structure of skew model FL TSS wll be presented. A CEV model s obtaned by defnng ϕ F t; T, T 1 = F t; T, T 1 β, 2.3 wth 0 β 1. As wth the dsplaced dffuson model the case β = 1 corresponds to log normal dynamcs whereas β = 0 results n a normal model. In ths model the yeld or percentage volatlty,.e., the volatlty of df F s gven by ϕf β 1 F λ t = λ tf t; T, T 1 whch for 0 < β < 1 s a monotone decreasng functon of forward Lbor F n accordance wth the observed market behavour. A model wth ths knd of local volatlty functon was frst ntroduced by Dupre n modellng equtes, comp. [7. 5

18 Based on these deas a generalsed forward Lbor process can be formulated n the termnal measure Q N1, df t; T, T 1 = ϕ F λ tσ t T[ µ N1 tdt dwt N1, = ϕ N1 F λ t [ µ tdt σ t T dwt N1, 2.4 N µ N1 α l λ l tρ l tϕ F l t =, µ N1 N t = 0, 1 α l F l t l=1 t < T T N, = 1,...,N whch was proposed by Andersen and Andreasen, comp. [1. The drft terms are gven by equaton A.9 whch s derved n appendx A Incorporaton of stochastc volatlty As the market observed volatlty skews cannot be solely captured by the ntroducton of a local volatlty functon stochastc volatlty extensons of the LMM were devsed. In ths respect one approach s to extend the forward Lbor process 2.4 wth a stochastc varable whch accounts for the volatlty level and as such modulates the local volatlty functon ϕf. A convenent choce s the square root of a varance process Σ t whch follows a one dmensonal CEV process, comp. [1. Assumng that the brownan drver of the varance process s correlated wth each forward rate drver,.e., dv N1 t, dw N1 k t = Γ k tdt k = 1,...,N, the process 2.4 can be generalsed to df t; T, T 1 = ϕ F λ t N1 Σ t [ µ tdt σ t T dwt N1 = ϕ F λ t N1 Σ t [ µ tdt σ t T Ωt T dz N1 t σ t T ΓtdVt N1 2.5a dσ t = Θ Σ 0 Σ t dt η Σt dv N1 t, 2.5b, wth dv N1 t, dw N1 k t = Γ k tdt, < dv N1 t, dz N1 k t, and t < T T N, = 1,...,N, k = 1,...,N, where the K dmensonal vector of brownan drvers dwt N1 was decomposed nto orthogonal components dz N1 t and dvt N1 accordng to A.3. The drft terms are gven by 6

19 equaton A.8 whch s derved n appendx A.1: µ N1 t = N α l λ l tϕ F l Σ t 1 α l F l t l=1 [ σ l t T T Ωt Ωtσ t σ l tγt σ tγt, 2.5c µ N1 N t = 0, wth matrx Ωt = { 1 Γk t 2 δ kj }k=1,...,k j=1,...,k and vector Γt = {Γ k t} k=1,...,k. Of course, the ntroducton of an addtonal Brownan drver ncreases the dmensonalty of the model, and the addtonal correlaton coeffcents Γ k t enlarge the parameter space. However, for the stochastc volatlty model we wll work wth n the followng chapters, namely Pterbarg s FL TSS LMM, the rate and varance processes are ndependent. Thus Γ k t = 0 and the drft reduces to µ N1 t = Σ t N l=1 α l λ l tρ l tϕ F l 1 α l F l t, µ N1 N t = Nevertheless the process 2.5 wll be referenced n secton 2.3 where the dea of a pre model n a Markov functonal context s dscussed. 2.2 Markov Functonal Models The class of Markov functonal nterest rate models was orgnally ntroduced by Hunt, Kennedy and Pelsser n [15. A major motvaton whch lead to ther development was the desre to have models that can ft observed prces of lqud nstruments n a smlar fashon to the market models whle mantanng the effcency of short rate models n calculatng dervatve prces, comp. [13, [18. Ths s acheved by specfyng a low dmensonal process whch s Markovan n some martngale measure and formulatng pure dscount bond prces as functons of ths process. Snce effcent algorthms to compute condtonal dstrbuton functons are known for ths set up, the valuaton of dervatves n a Markov functonal framework s much more effcent when compared to prcng usng market models. Although market models are Markovan as well, they are naturally of hgh dmenson. Moreover an essental feature of these types of models s the freedom to choose the functonal form of the dscount bond prces n such a way that market prces of calbraton nstruments are replcated. Ths dstngushes Markov functonal from short rate models n whch the functonal form of dscount bond prces wth respect to the Markovan short rate s fxed. Therefore Markov functonal models combne the strong ponts of market as well as short rate models, namely the fttng to observed prces of lqud nstruments and tractablty. 7

20 2.2.1 Defnton and examples of Markov Functonal Models In ths secton we want to gve a formal defnton of Markov functonal models and also present some examples. We begn by ctng the defnton gven by Hunt and Kennedy: Defnton Hunt and Kennedy [13. An nterest rate model s sad to be Markov functonal f there exsts some numerare par N, N and some process x sucht that: 1. the process x s a tme nhomogeneous Markov process under the measure N; 2. the pure dscount bond prces are of the form D ts = D ts x t, t S S, for some boundary curve S : [0, [0, and some constant ; 3. the numerare N, tself a prce process, s of the form N t = N t x t t. Obvously the boundary curve S s ntroduced so that the model does not need to be defned over the entre tme doman 0 t S. The most common choce for the boundary curve s for some constant T. S = { S, f S T T, f S > T, Thus the man ngredents of a Markov functonal model are the drvng Markov process x t whch descrbes the state of the economy and the functonal forms of 1. the dscount bond D S S x S D S, S; x S on the boundary curve S ; 2. the numerare N t x t for tmes t. The reason for ths s that the functonal forms of bonds at earler tmes t < S are determned by the functonal form of the dscount bond on the boundary curve by the martngale property of numerare rebased assets, comp. [10. D t, T; x t = Nt x t E N [ D S, T; x S N S x S F t, t S T, 2.7 One partcular choce of measure s the termnal forward measure N = Q N1 whch s nduced by takng the dscount bond D t, T N1 as numerare. Takng TN1 as boundary we thus have S = mn S, T N1, Nt x t = D t, T N1 ; x t, T0 t T N1, 8

21 and the prce of a dscount bond maturng at S T N1 becomes D [ t, S; x t = D t, TN1 ; x t E Q N1 DS, S; xs D S, T N1 ; x x t S = D [ t, T N1 ; x t E Q N1 1 D S, T N1 ; x x t, t S T N1, 2.8 S where the expectaton s condtoned on x t because of the Markov property of the underlyng process 2. If one consders nterest rate dervatves lke caplets/floorlets or swaptons exprng at tme T m wth strke K ther payoff functon V m T m, K depends on the dscount bonds DT m, T j ; x t, wth j = m 1 for caplets/floorlets and j > m n the case of swaptons, and by 2.8 s a functon of the numerare dscount bond DT m, T N1 ; x Tm at tme T m. Thus V m T m, K V m Tm, K, DT m, T N1 ; x Tm and by applcaton of the fundamental theorem of asset prcng the dervatve value at tme t s gven by V m t, K; xt = D t, TN1 ; x t E Q N1 [ Vm Tm, K, DT m, T N1 ; x Tm D T m, T N1 ; x Tm x t. 2.9 Hence, f the Markov process x t and thus the condtonal probablty dstrbuton px Tm x t s specfed, ths relaton provdes a means to extract the functonal form of the numerare dscount bond DT m, T N1 ; x Tm at tme T m from market observed dervatve prces snce the payoff functon V m Tm, K, DT m, T N1 ; x Tm s known. However, n order to proceed along these lnes one has to assume that the dscount bonds are monotone functons of the underlyng Markov process. It has to be emphaszed that due to the functonal dependence the specfed underlyng Markov process x t determnes the probablty dstrbuton of dscount bonds. As the prces of mult temporal nterest dervatves depend on the jont probablty dstrbuton of forward rates and thus on the jont dstrbuton of dscount bonds at those tmes relevant to the product at hand 3, the drvng Markov process encodes all nformaton on the correlaton structure. So n desgnng a Markov functonal model for a specfc product class the process x t has to be chosen n such a way as to capture the characterstc product features whle retanng low dmensonalty. Referrng to the process dmenson, the underlyng Markov process should not have more than two brownan drvers. An example of a smple one dmensonal underlyng process s consdered n the next secton where a Lbor Markov functonal model s explored. 2 For a Markov process x t the relaton E [ fx t F t = E [ fxt x t holds. 3 E.g., for a bermudan swapton the sngle exercse dates are the relevant tmes. Therefore the probablty dstrbuton of the process x t only needs to be known on these dates. 9

22 2.2.2 A Lbor Markov Functonal Model We know consder the set of forward Lbor rates F t; T, T 1 whch reset a tmes T, = 1,...,N, and specfy a Markov functonal model. For ths we choose the fnal tme S = T N1 and work n the measure N = Q N1 whch s nduced by takng the dscount bond D t, T N1 as numerare. The underlyng Markov process x t s chosen to be a tme changed brownan moton. Assumng that σt s a determnstc, postve real valued functon on [, T N1 we defne x t := t σsdw s, where dw s s a brownan moton under Q N1. Clearly, due to the Markovan character of the brownan moton x t s a Markov process whose condtonal probablty dstrbuton s normal, p x s 1 x t = 2π 1 2 s t σu2 2 du exp 1 2 x s x t 2 s, s t t σu2 du Havng specfed the underlyng process the functonal form of the numerare dscount bond D t, T N1 ; x t remans to be determned. Ths wll be done at dscrete tmes T = 1,...,N accordng to a recurson scheme n whch the functonal from wll be determned by calbraton to dgtal caplet prces. To start wth we observe that at tme T N we observe that the forward rate F N t; TN, T N1 ; x TN s a log normal martngale under Q N1. Thus ts dynamcs s governed by df N t; TN, T N1 = σtfn t; TN, T N1 dwt, t T N, where dw t s a brownan moton under Q N1, whch ntegrates to F N TN ; T N, T N1 ; x TN = FN T0 ; T N, T N1 exp 1 TN σs 2 ds 2 = F N T0 ; T N, T N1 exp 1 TN 2 TN σs 2 ds x TN σsdw s Wth ths relaton at hand the functonal form of forward rate F N at tme T N s known. Because forward rates and dscount bond prces are related by F m Tm ; T m, T m1 ; x Tm = 1 DT m, T m1 ; x Tm α m T m, T m1 DT m, T m1 ; x Tm, 1 m N, the functonal form of D T N, T N1 ; x TN unfolds tself as monotone decreasng functon of x TN, D T N, T N1 ; x TN = 1 1 α N T N, T N1 F N T0 ; T N, T N1 exp TN σs 2 ds x TN. 2.11

23 Ths result serves as the bass for the recursve calculaton of functonal forms at earler tmes T m < T N whch wll be extracted from market observed dgtal caplet prces. The payoff of a dgtal caplet exprng at tme T m wth strke K s gven by V m Tm, K; x Tm = D Tm, T m1 ; x Tm 1FmT m;t m,t m1 ;x Tm >K, and at tme < T m < T N the numerare rebased dervatve value s therefore [ V m T0, K; x T0 D Vm Tm, K, DT = E QN1 m, T N1 ; x Tm, T N1 ; x T0 D T m, T N1 ; x x Tm [ D Tm, T = E QN1 m1 ; x Tm D 1 FmT T m, T N1 ; x m;x Tm >K x Tm [ [ 1 = E QN1 E QN1 D T m1, T N1 ; x x Tm Tm1 1 FmT m;x Tm >K x, 2.12 where n the last lne the martngale property of numerare rebased dscount bonds 2.8 was used. Thus f the functonal form of the numerare bond D T m1, T N1 ; x Tm1 s known at tme T m1, the expected value on the rght hand sde can be calculated 2.12 because the condtonal probablty dstrbuton of x t s known. Indeed, snce the forward rate F m s a monotone functon of x Tm through ts dependence on the dscount bond D T m, T m1 ; x Tm, there exsts a unque value x for whch the forward rate matches the dgtal caplet strke Tm value K, F m t; Tm, T m1 ; x Tm = K 4. Thus 2.12 s equvalent to V m T0, K; x [ T0 D =, T N1 ; x T0 x Tm 1 D T m1, T N1 ; x Tm1 p x Tm1 x Tm dxtm1 p x Tm x T0 dxt0, 2.13 whch provdes a relaton between x and market derved dervatve values. Indeed, because D Tm, T N1 ; x T and Vm T0, K; x T can be observed n the market at tme T0, the left hand sde of 2.13 s known. Wth the known functonal form D T m1, T N1 ; x Tm1 the ntergrals on the rght hand sde can be calculated numercally for varyng values of x. That value of Tm x for whch the left and rght hand sde of 2.13 are equal s the Tm desred target value whch satsfes the relaton F m Tm ; T m, T m1 ; x Tm = K. 5 Conductng ths matchng procedure for a seres of optons wth dfferent strke values K j j = 1,...M and values V m T0, K j ; x T0 results n a set { x Tm,j F m Tm ; T m, T m1 ; x } Tm,j = Kj { [ = x Tm,j D T m, T N1 ; x 1 D T m, T m1 ; x 1 } Tm,j = Tm,j αn T N, T N1 K j D T m, T N1 ; x, Tm,j As mentoned above we assume that the dscount bonds are monotone functons of x t. Due to ther relaton ths behavour transfers to the forward Lbor rate. 5 In practce the soluton for x s found by a numercal root fndng method, e.g., the Brent algorthm. Tm 11

24 where n the second lne the relaton K j = F m Tm ; T m, T m1 ; x Tm,j = 1 DT m, T m1 ; x Tm,j = α m T m, T m1 DT m, T m1 ; x Tm,j 1 D T m,t N1 ;x Tm,j DTm,T m1;x Tm,j D T m,t N1 ;x Tm,j α m T m, T m1 DTm,T m1;x Tm,j D T m,t N1 ;x Tm,j, 1 m N, was used. Because DTm,T m1;x Tm,j corresponds to the nner ntegral the bracket term D T m,t N1 ;x Tm,j of equaton 2.13 ts value has already been determned n the course of fndng x Tm,j. Therefore the set dentty 2.14 defnes the numerare dscount bond at tme T m as functon of x Tm,j. Obvously, because only a fnte number of optons wth dfferent strkes can be observed n the market the sets n 2.14 are dscrete. Therefore contnuous functonal forms have to be obtaned by nterpolaton between the sngle set elements. Followng the above reasonng the recurson scheme starts at tme T N 1. Performng the calbraton accordng to 2.13 the expected value of the nverse of the already known functonal D T N, T N1 ; x TN gven by 2.11 s calculated. As a result the set 2.14 and thus the functonal forms D T N 1, T N1 ; x TN at tme TN 1 are obtaned. These n turn serve as nput for the calbraton at tme T N 2 where they enter the nner ntegral of Pursueng the recurson along these lnes untl tme T 1 the functonal forms of the numerare dscount bond D T, T N1 ; x T are establshed for tmes TN, T N 1,...,T Mult dmensonal Markov Functonal Models In the prevous secton we presented the Lbor Markov functonal model n whch a one dmensonal Markov process was used as underlyng for the dscount bond term structure. However t s also possble to consder hgher dmensonal underlyng processes. As mentoned n the prevous secton the drvng process should be selected n such a way that the essental features of the dervatve product for whch the model s desgned are captured. As an example for whch a two dmensonal drvng process s requred the class of spread optons can be consdered. The payout structure of ths knd of dervatves can depend on the level of two dfferent rate types whch follow ndvdual dynamcs. Therefore a two dmensonal process s requred n order to model the separate rate components. Another example s the ncorporaton of stochastc volatlty whch we focus on n ths thess. In ths context a two dmensonal Markov process would encompass a rate and a volatlty component. As dscussed above the specfcaton of the Markov process s only one part n the specfcaton of a Markov functonal model. The second s the determnaton of the functonal 12

25 forms for the numerare and the dscount bond on the boundary curve. These are obtaned by a calbraton procedure for whch t s essental that the functonals are monotone functons of the drvng Markov process. But when hgher dmensonal Markov processes z t are consdered the monotoncty of functonals can no longer be mantaned because e.g., n a two dmensonal extenson of the matchng equaton 2.13 more than one tuple z t = z t,1, z t,2 would be obtaned as target value. For an n dmensonal Markov process z t ths problem can be overcome by the ntroducton of a functon π : R R n R, t,z t πt,z t =: x t, whch serves as projector to the one dmensonal real axs, comp. [13. In general, the process x t = πt,z t thereby defned wll not be Markovan. However ths fact poses no mpedment snce the functon π merely serves as a means to facltate the calbraton to dervatve prces observed n the market. Followng ths approach the numerare dscount bond at tme T m becomes a functonal of the mult dmensonal Markov process, D T m, T N1 ;z Tm, and n the calbraton procedure the expected values are calulated wth respect to the condtonal dstrbuton of z t. Therefore the only modfcaton whch needs to be appled to the calbraton equaton 2.12 s the change from one to mult dmensonal ntegrals: V m T0, K; x T0 = π,z T0 D, T N1 ; x T0 = π,z T0 [ Vm Tm, K, DT = E QN1 m, T N1 ; x Tm D T m, T N1 ; x x Tm [ [ 1 = E QN1 E QN1 D T m1, T N1 ; x x Tm Tm1 [ [ 1 = E QN1 E QN1 D T m1, T N1 ; x x Tm Tm1 [ [ 1 = E QN1 E QN1 D T m1, T N1 ;z z Tm Tm1 [ = x Tm =πtm,z Tm 1 FmT m;x Tm >K 1 xtm >x Tm 1 πtm,ztm >x Tm x x z 1 D T m1, T N1 ;z Tm1 p z Tm1 z Tm dztm1 p z Tm z T0 dzt0, 2.15 wth x T m = Fm 1 T m ; T m, T m1 ; K. The varable z descrbes a curve n the n dmensonal Tm state space for whch x = πt Tm m,z at tme T Tm m. Thnkng about possble choces for the functon πt,z t the concept of a pre model was devsed [14. It s based on the dea that π can be defned as approxmaton to a model whch has already been calbrated to market prces, e.g., a Lbor Market Model where the drft terms have been frozen to ther ntal values. Snce the so defned pre model s an 13

26 approxmaton only t s not arbtrage free. However, snce the no arbtrage requrement s nherent n equaton 2.15 a calbraton to market quotes va the pre model wll result n an arbtrage free model. 2.3 A Lbor Market Model as pre model for a Markov Functonal Model In ths secton an approxmaton of a forward Lbor Market Model LMM wll be consdered whch wll then be used as pre model for a Markov functonal model. Workng n the termnal measure Q N1 the forward rate F t; T, T 1 s modelled accordng to 2.4, F t; T, T 1 = F T0 ; T, T 1 t [ exp µ N1 s 1 t 2 λ s 2 σ s 2 ds λ sσ sdw s, µ N1 t = K l=1 α l T l, T l1 λ l tρ l tf l t; Tl, T l1 2.16a 1 α l T l, T l1 F l t; Tl, T l1, 2.16b t T, = 1,...,N, 1 K N, where dw s s a K dmensonal Brownan moton under Q N1 and σ s the load vector whch encodes the effect of the ndvdual orthogonal Brownan drvers on forward rate F. From ths dynamcs t s obvous that the change of forward rate F depends on the state of all forward rates F l l > at tme t whch s why the ndvdual processes F t are not Markovan 6. Therefore one usually resorts to Monte Carlo methods n numercal evaluatons whch gets qute expensve as the number of factors ncreases. However, computatons can be allevated by referrng to an approxmaton proposed by Rebonato due to whch the forward rates F l t; Tl, T l1 n the drft term 2.16b are replaced by ther tme T0 values F l T0 ; T l, T l1, a process whch s also denoted as partal freezng [22. A further smplfcaton can be acheved by replacng the Brownan moton terms wth normally dstrbuted varables whch exhbt the same mean and varance. Employng these deas the forward rate vector can be approxmated as Ft Ft = F exp µ N1 0 t Mt z t, 6 However, the process for the entre forward rate vector Ft s Markovan snce ts change at tme t only depends on the state of the forward rate vector at tme t. 14

27 wth vectors and the matrx where µ 0 N1 { t [ µ N1 0 t = t = K l=1 N1 µ 0 s 1 2 λ s 2 σ s 2 } ds, =1,...,N α l T l, T l1 λ l tρ l tf l T0 ; T l, T l1 1 α l T l, T l1 F l T0 ; T l, T l1, z t = { Z t, }=1,...,K, where Z t, N 0, t, {[ 1 Mt = t Indeed, the terms M k tz t,k satsfy and [ K E M k tz t,k = k=1 t λ s 2 σ k s 2 ds 1 K M k te [ [ K Z t,k = 0 = E k=1 [ t = E λ sσ sdw s t } 2. =1,...,N k=1,...,k k=1 λ sσ k sdw k s, because λ sσ k sdw k s are Ito ntegrals, [ K [ K K var M k tz t,k = E M k tm l tz t,k Z t,l = M k tm l te [ Z t,k Z t,l k=1 = = = = K k,l=1 K k=1 k,l=1 M k tm l tcov [ Z t,k Z t,l }{{} =δ kl t t t k,l=1 λ s 2 σ k s 2 ds K [ t E λ s 2 σ k s 2 ds K [ t 2 E λ sσ k sdw k s k=1 k=1 t by Ito s sometry K = λ s 2 σ k sσ l s E [ dw k sdw l s k,l=1 }{{} =cov[dw k s,dw l s [ K t = var λ sσ k sdw k s k=1 [ t = var λ sσ sdw s, = 1,...,N Above proxy processes can be related to the orgnal forward rates F t by ntroducng monotone functons g whch act as perturbatons on F t,z t. Thus F t = g F t,z t, 15

28 and one can defne projector functons π := g F by wth π : R R K R, t,z t π t,z t := g F t,z t, K F t,z t = F exp µ N1 t M k tz t,k, for = 1,...,N, k=1 whch were ntroduced n the prevous secton. Hence the numerare dscount bond becomes a functonal of the Markov process z Tm through ts dependence on F m T m = π m T m,z Tm = g m F m T m,z Tm. Based on ths approach the ncorporaton of stochastc volatlty nto a Markov functonal framework wll be devsed n chapter 4. 16

29 Chapter 3 Pterbarg s term structure of skew forward Lbor model Ths chapter s dedcated to a survey of Pterbarg s term structure of skew forward Lbor model FL TSS whch was ntroduced n [19. As wth other forward Lbor Market Models the man motvaton s to capture the dynamcs of the jont dstrbuton of forward Lbors throughout tme. To facltate ths the forward Lbor dynamcs has to be flexble enough to capture nformaton on the margnal dstrbutons whch s encoded n caplet and/or swapton prces. Snce for these products the market mpled volatltes exhbt a skew,.e., the Black mpled volatltes appear to be functons of the opton strkes, Pterbarg consders a weghted sum of log normal and normal dynamcs for the forward Lbors. Mathematcally ths s expressed by the ntroducton of a skew parameter. Reference to the swapton market necesstates ths parameter to be tme dependent n order to reproduce swapton skews across expres and underlyng swap maturtes. It s ths tme dependent skew parameter whch renders the model more flexble over earler constant skew models, e.g., Andersen and Andreasen [2. Furthermore, to account for the market observed varablty of volatlty levels, smlar to the formulaton of [2 these are modelled as stochastc CEV processes whose Brownan components are assumed to be uncorrelated wth the stochastc drvers of the forward rate processes. Hence the FL-TSS model belongs to the class of stochastc volatlty models. 3.1 The forward Lbor dynamcs Followng the qualtatve descrpton of Pterbarg s model we know set out to specfy the dynamcs of forward Lbors F t; T, T 1 resettng at tmes T, = 1,...,N. Workng n the measure Q N1 nduced by choosng the termnal dscount bond D t, T N1 as numerare the forward rates are modelled as 17

30 df t = β tf t 1 β T0 F T0 λ t [ Σ t µ N1 tdt K l=1 σ,l tdw N1 l t, 3.1a dσ t = Θ Σ 0 Σ t dt η Σt dv N1 t, 3.1b for tmes t < T T N = 1,...,N. All forward rates are drven by K ndependent Brownan motons dw N1 l t whch are assumed to be uncorrelated wth the stochastc volatlty drver dvt N1 and therefore dvt N1, dw N1 l t = 0. Ther nfluence on the forward Lbors F s medated by the load factors σ,l t whch contan nformaton on forward Lbor correlaton snce the relaton K l=1 σ,ltσ j,l t = ρ j t holds. The drft terms µ N1 t = 1,...,N result from workng n the termnal measure and are gven by expresson 2.6. It vanshes for the last forward rate F N t; TN, T N1 snce t s a log normal martngale under Q N1. Hence µ N1 N t = 0. The model parameters are hence the tme dependent forward skews β t and volatlty levels λ t as well as the tme ndependent volatlty of varance η and mean reverson speed Θ. Pterbarg chooses the latter to be constant whch reduces the degrees of freedom to the set of tuples { β t, λ t } =1,...,N assocated wth forward Lbors F t; T, T 1 for tmes t T T N and = 1,...,N. These parameters characterse the dstrbuton of each forward Lbor as they are not affected by a change of measure. Reference to 3.1a clearly reveals the role of the skews β t as parameters mxng a purely log normal wth a normal forward Lbor dynamcs. From there t also s apparent that the parameters λ t determne the level of the stochastc volatlty Σ t whch s governed by 3.1b. Together wth correlatons ρ j t among forward rates F and F j these parameters determne the jont dstrbuton for tmes t T N. The term structure of forward Lbor skews and volatlty levels have to be obtaned by calbraton to market prces of caplets/floorlets or swaptons. As outlned above the FL- TSS model s calbrated to the swapton market. Snce forward swap rates are the natural swapton underlyngs, t s therefore necessary to formulate a consstent forward swap rate dynamcs and relate the resultng forward swap skews and volatlty levels to ther forward Lbor counterparts. Ths s done by requrng that the process governng the forward swap rates should have the same structure as the forward Lbor dynamcs. The detalng of ths dea wll be presented n the followng secton. 18

31 3.2 Swap rate dynamcs under the FL-TSS model To derve a consstent forward swap rate dynamcs we frst observe that the par rate S mn t of a forward swap startng at tme T m > t and maturng at T n > T m can be expressed as a weghted sum of ts consttuent forward rates F m,...f n 1. : S mn t = Dt, T m Dt, T n n 1 l=m α lt l, T l1 Dt, T l1 n 1 = l=m α l T l, T l1 Dt, T l1 n 1 l=m α lt l, T l1 Dt, T l1 } {{ } =:w l t F l t; Tl, T l1 n 1 = w l tf l t; Tl, T l1, 3.2 l=m whch follows from the forward rate defnton F l t; Tl, T l1 = Dt, T l Dt, T l1 α l T l, T l1 Dt, T l1, where α l T l, T l1 denotes the year fracton of the perod [T l, T l1 and Dt, T j stands for the dscount factor correspondng to tme T j. Thus the forward swap rate s a functon of ts consttuent forward rates, S mn F m,...,f n 1, and a stochastc dfferental equaton s arrved at by applcaton of Ito s lemma: ds mn t = = = n 1 l=m n 1 l=m 1 2 n 1 l=m S mn t F l t df lt 1 2 S mn t F l t n 1 l,k=m n 1 l,k=0 2 S mn t F l t F k t df ltdf k t β l tf l t 1 β l T0 Fl T0 λ l t Σ t }{{} =:ϕf l t [ σ T l t µ m,n 2 S mn t F l t F k t ϕf ltϕf k tρ lk tdt l dt dw m,n t S mn t F l t ϕf ltλ l t Σ t σ T l tdwm,n t drft terms. 3.3 Above expresson s formulated n the swap measure Q m,n whch s nduced by usng the present value of a bass pont P mn t = n 1 l=m α lt l, T l1 Dt, T l1 as numerare 1. The K dmensonal drft vectors µ m,n l account for the change from the termnal measure Q N1 to Q m,n, under whch dw m,n t s a K dmensonal Brownan moton. Addtonal 1 Strctly speakng the quantty P mnt refers to a notonal of 1 and therefore represents the present value of 10,000 bass ponts. 19

32 drft terms arse from the non zero correlatons ρ lk t = σ T l tσ kt = K j=1 σ l,jtσ k,j t between forward rates F l and F k. But snce covarances and therefore volatltes and correlatons reman nvarant under a change of measure, one can consder above swap dynamcs under a new measure Q wth assocated K dmensonal Brownan moton d Wt n whch the drft terms vansh. Under Q 3.3 transforms nto ds mn t = n 1 l=m S mn t F l t ϕf ltλ l t Σ t σ T l td Wt. 3.4 As was already mentoned n the ntroducton we are lookng for a forward swap rate dynamcs whch has the same structure as the forward rate process. Ths s equvalent to requrng a dynamcs of the form ds mn t = βmn ts mn t 1 β mn T0 Smn T0 λ mn t Σ t } {{ } =:ϕs mnt K l=1 σ mn,l td W l t, 3.5 wth forward swap rate skews β mn t, volatlty levels λ mn t, and Brownan loadngs σ mn,l t l = 1,...,K. In order to ensure consstency between the two formulatons we set 3.4 equal to 3.5 and n addton match the slopes of both expresson wth respect to all forward rates F l. Ths reasonng results n drect relatons between swap and forward rate volatlty levels and skews, respectvely, whch wll be presented n the followng subsectons Dervaton of the forward swap volatlty level Matchng of both expressons for the forward swap rates results n the requrement n 1 l=m S mn t F l t ϕf ltλ l t Σ t σ T l td Wt = ϕs mn t λ mn t Σ t σ T mntd Wt from whch the forward swap volatlty level can be derved: λ mn t σ T mnt = n 1 l=m n 1 = λ mn t = l=m S mn t F l t S mn t F l t ϕf l t ϕs mn t λ ltσ T l t 3.6 ϕf l t ϕs mn t σt l t σ mntλ l t, 3.7 snce σ T mnt σ mn t = 1. Whereas the load factors of the orthogonal forward Lbor drvers can be extracted from the correlaton matrx 2 ths does not apply to the Brownan motons 2 We assume that the correlatons between forward Lbors F and F j are gven n the parametrc form 2.1 propsed by Rebonato [21: ρ jt = ρ 1 ρ exp δ T t ǫ T j t ǫ 20

33 drvng the swap rate. In above case of K > 1 Brownan swap rate drvers we therefore have to retreve the swap rate volatlty by referrng to 3.6 nstead of the volatlty level λ mn t tself. It s obvous that forward skew and volatlty parameters as well as ther swap counterparts are ntertwned n above expressons, whch s due to the presence of the skew functons ϕf l t and ϕs mn t. Ths dependency s resolved when 3.6 s consdered at the money,.e., at the tme forward Lbor and swap rate ponts. In ths case ϕf l t and ϕs mn t become skew ndependent, and by further freezng Smnt F l t by A.11 at ther ntal values 3.6 smplfes to λ mn t σ T mnt = n 1 l=m S mn t F l t whch s the expresson on whch calbraton wll be based Dervaton of the forward swap skew whch are gven F l t=t0 S mn λ ltσ T l t, 3.8 The forward swap and Lbor skew can be nterpreted as slopes of the respectve skew functons ϕs mn t and ϕf l t whch are lnear n the underlyng rate varable. Hence we have β mn t = ϕs mnt S mn t β l t = ϕf lt F l t βmn ts mn t 1 β mn T0 Smn T0 =, ϕs mn t β l tf l t 1 β l T0 Fl T0 = for l = m,...,n 1, F l t and a relaton between both quanttes can be establshed by matchng the slope of ds mn t n formulatons 3.4 and 3.5 wth respect to the forward rates F k t. Referrng to 3.4 and assumng that the dervatves Smnt F l t F l t vares wth tme, we thus obtan ds mn t n 1 = F k t l=m n 1 l=m = S mnt F k t S mn t ϕf l t F l t F k t l = m,...,n 1 do not vary sgnfcantly as 2 S mn t ϕf l t λ l t Σ t σ T l F k t F l t }{{} td Wt 0 S mn t F l t β k tδ lk λ l t Σ t σ T l td Wt t=t0 β k tλ k t Σ t σ T k td Wt, 3.9 t=t0 where our assumptons took effect n the second lne by freezng Smnt F l t at ther ntal values and neglectng second dervatves of the swap rate wth respect to the forward Lbors. 21

34 Focussng on the formulaton n terms of swap volatlty levels and skews 3.5, a smlar analyss yelds ds mn t F k t = ϕs mnt S mn t S mn t F k t λ mn t Σ t σ T mntd Wt β mn t S mnt F k t t Σ t σ T mntd Wt t=t0 λmn Matchng of the forward swap slopes wth respect to Lbors F m,...,f n 1 n equatons 3.10 and 3.9 results n a system of n m equatons between forward swap and Lbor skews: β mn t λ mn t σ mn t = β k tλ k tσ k t k = m,...,n 1, to whch no unque soluton exsts. Therefore one has to revert to a least squares optmsaton to fnd an approxmate soluton. For ths sake we consder the functonal J βmn t n 1 2, = βmn t λ mn t σ mn t β k tλ k tσ k t k=m and obtan the optmal soluton for the forward swap skew by requrng the dervatve of J wth respect to β mn t to vansh: 0 = dj d β mn t = 2 λ mn t σ T mnt = β mn t = 1 n 1 n m k=m n 1 βmn t λ mn t σ mn t β k tλ k tσ k t k=m λmn t σ T mnt λ k tσ k t λmn t σ T mnt λmn t σ mn t β kt As the central result of ths secton we hence have establshed the relaton between forward swap skews and volatltes to ther forward Lbor counterparts for each swap rate S mn t over tme nterval t T m. It s ths tme dependence of the forward Lbor parameters whch defnes the central dea of a skew term structure n Pterbarg s FL TSS model. As mentoned earler, n calbratng the FL TSS model to the swapton market Pterbarg uses skew and volatlty parameters of a Heston model as market nput. By constructon Hestons s model parameters are tme ndependent whch necesstates the ntroducton of a tme-averagng method for the swap rate parameters wthn the FL TSS. Ths requrement results n the formulaton of effectve skew and volatlty parameters whch are central n pavng the way to calbraton. 22

35 3.3 The effectve skew and volatlty formulaton Ths secton s dedcated to the formulaton of swap rate effectve skews and volatltes wthn Pterbarg s FL TSS model. In summary, these facltate the transton from a swap rate dynamcs wth tme dependent parameters 3.5, ds mn t = βmn ts mn t 1 β mn T0 Smn T0 λmn t Σ t σ T mntd Wt, to a formulaton based on tme ndependent ones, ds mn t = β mn S mn t 1 β mn Smn T0 λ mn Σt σ T mntd Wt, where for each swap rate S mn t the effectve skews β mn and volatltes λ mn are n prncple gven as wegthed tme averages of the tme dependent quanttes 3.8 and The detalng of ths central concept wll be provded n the followng, where at frst attenton wll be pad to the effectve skew n subsecton after whch effectve volatlty s covered n The effectve forward swap skew Pterbarg arrves at the effectve swap rate skew by consderng two dffuson processes where one has a tme dependent local volatlty functon and the other a tme ndependent one. The latter s defned as a weghted average of ts tme dependent counterpart. Focussng on the weght functon wt on tme nterval [0, T, Pterbarg derves an explct expresson such that the average of dfferences between european swapton prces across an nfnte range of strkes, calculated wth respect to the respectve processes, tends to zero as valuaton tme approaches. In detal, the followng theorem holds: Theorem Pterbarg [19. For T > 0, let f C 1 [0, T R, R be a local volatlty functon satsfyng the usual growth requrements. Let σt, t [0, T be a functon of tme only. Fx x 0 R. For any ǫ > 0 defne a re scaled local volatlty functon and assume wthout loss of generalty whch mples f ǫ t, x = f tǫ 2, x 0 x x 0 ǫ, ft, x 0 1, t [0, T f ǫ t, x 0 1, t [0, T. 23

36 Let wt, t [0, T be a weght functon satsfyng T 0 and defne an averaged local volatlty functon f ǫ x 2 = wtdt = 1, T 0 Further defne two famles of dffusons ndexed by ǫ, f ǫ t, x 2 wtdt dx ǫ t = f ǫ t, Xǫ t λt ΣtdWt, dy ǫ t = f ǫ Yǫ t λt ΣtdWt, X ǫ 0 = x 0, Y ǫ 0 = x 0, 3.13a 3.13b 3.13c 3.13d for t [0, T wth dσ t = Θ Σ 0 Σ t dt η Σt dv t, dv t, dwt = 0. If the weghts wt are gven by the expresson then wt = vt 2 λt 2 T 0 vt2 λt 2 dt, wth vt2 = E [Σt 2 X 0 t x 0, 3.14 E [ Yǫ T K E [ Xǫ T K dk = O ǫ for ǫ 0. 3 By applyng above theorem to the swap rate S mn t wth local volatlty functon f t, S mn t = ϕsmnt S mn and replacng tme zero wth, a formulaton wth tme ndependent effectve skew parameters s obtan accordng to ds mn t = βmn ts mn t 1 β mn T0 Smn T0 λmn t Σ t σ T mntd Wt, = S mn f 1 t, Smn t λmn t Σ t σ T mntd Wt, S mn f 1 Smn t λmn t Σ t σ T mntd Wt, = β mn S mn t λmn 1 β mn Smn T0 t Σ t σ T mntd Wt, 3 Whle above theorem provdes a relaton between the two dffusons, t has to be observed that 3.15 s not formulated n terms of absolute values. Hence the fact that Y ǫt comples wth relaton 3.15 n the lmt ǫ 0 does not generally ensure convergence to X ǫt n probablty. 24