Caibration of a Lib rate and the SABR-LMM

Transcription

1 Caibration of a Libor Market Mode with Stochastic Voatiity Master s Thesis by Hendrik Hüsbusch Submitted in Partia Fufiment for the Degree of Master of Science in Mathematics Supervisor: PD. Dr. Vokert Pausen Münster, August 27, 214

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3 Contents 1. Introduction 3 2. Preiminaries Forward Rates and Swaps Basic Definitions Basic Derivatives Bootstrapping Market Data The Bootstrapping of Forward Rates The Stripping of Capet Voatiities The SABR and SABR-LMM mode The SABR Mode The SABR-LMM Mode The SABR-LMM Dynamics under any Forward Measure P The SABR-LMM Dynamics under the Spot Measure P spot Swaps Rates in the SABR-LMM A SABR mode for Swap Rates Swap Rates Dynamics in the SABR-LMM Parametrization of the SABR-LMM Mode The Voatiity Structure The Correation Structure Approximation of P through a proper Correation Matrix Caibration of the SABR-LMM to Swaptions Preparation for the Caibration The Caibration of the Voatiity Structure The Caibration of the Correation to Swaps i

4 Contents 7. CMS Spread Options and Swaps Markovian Projections of CMS spreads Convexity Correction for CMS spreads Caibration to CMS Spread Options Impementation and Empirica Study Impementation Empirica Study The Data Caibration to Swaption Prices as of Caibration to Market Prices as of Critique on Caibration Soey to CMS Spread Options Concusion 113 A. Appendix 115 A.1. Parameters Obtained from the Caibration on Data as of A.2. Parameters Obtained from the Caibration on Data as of ii

5 List of Figures 2.1. Two Exampes a stripped forward curve The Capet Voatiity Surface from The Impact of the SABR Parameters on the Impied Voatiity Different Shapes of the Voatiity Functions g and h Simpest Correation Matrix Possibe Shapes of the (2SC) Parametrization Possibe Shapes of the (5L) Parametrization Invoved Entries of a Correation Matrix in the Swap Dynamics The Difference of (5L) and (2SC) as of The Difference of (5L) and (2SC) as of Invoved Entries of a Correation Matrix in the Swap and CMS Spread Dynamics A.1. Correation Matrices from Based on (2SC) A.2. Correation Matrices from Based on (5L) A.3. Error Surface for Capet Prices as of A.4. Error Surface for Swaption Prices as of , when (5L) is used A.5. Error Surface for Swaption Prices as of , when (2SC) is used. 119 A.6. Correation Matrices from Based on (5L) and Caibrated to Swapts A.7. Correation Matrices from Based on (2SC) and Caibrated to Swapts A.8. Correation Matrices from Based on (5L) and Caibrated to CMS Spreads A.9. Correation Matrices from Based on (2SC) and Caibrated to CMS Spreads iii

6 List of Figures A.1.Error Surface for Capet Prices as of A.11.Error Surface for swaption prices as of , if Caibrated to Swaps and (5L) is used A.12.Error Surface for swaption prices as of , if Caibrated to Swaps and (2SC) is used A.13.Error Surface for Swaption Prices as of , if Caibrated to CMS Spreads and (5L) is used A.14.Error Surface for Swaption Prices as of , if Caibrated to CMS Spreads and (2SC) is used iv

7 List of Tabes 5.1. Parameters for Figure Parameters for Figure Pricing Errors for Capets from Pricing Errors for Swaptions from Pricing Errors for Capets from Pricing Errors for Swaptions from , if Caibrated to Swaps Pricing Errors for Swaptions from , if Caibrated to CMS Spreads A.1. Parameters for g and h from A.2. Parameters k i from A.3. Parameters ζ i, from A.4. Parameters for P Based on (5L) and from A.5. Parameters for P Based on (2SC) and from A.6. Parameters for g and h from A.7. Parameters k i from A.8. Parameters ζ i from A.9. Parameters for P Based on (5L), Caibrated to Swaps and from A.1.Parameters for P Based on (2SC), Caibrated to Swaption Prices and from A.11.Parameters for P Based on (5L), Caibrated to CMS Spreads and from A.12.Parameters for P Based on (2SC), Caibrated to CMS Spreads and from v

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9 Eidesstattiche Erkärung Hiermit versichere ich, Hendrik Hüsbusch, dass ich die voriegende Arbeit sebstständig verfasst und keine anderen as die angegebenen Queen und Hifsmitte verwendet habe. Gedankich, inhatich oder wörtich übernommenes habe ich durch Angabe von Herkunft und Text oder Anmerkung beegt bzw. kenntich gemacht. Dies git in geicher Weise für Bider, Tabeen, Zeichnungen und Skizzen, die nicht von mir sebst erstet wurden. Hendrik Hüsbusch, Münster, August 27, 214 1

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11 1. Introduction The market for interest rate derivatives is one of the biggest financia markets in the word and is easiy bigger than the stock market. A huge part of the market for options on interest rates is the over-the-counter (OTC) market. Different from stock exchanges where prices of products are pubicy quoted on consent, in the OTC market the invoved parties negotiate over prices ony between themseves behind cosed doors without making them pubic. In the goba OTC derivatives market positions of amost 6 triion USD were outstanding in 213 [39] and a of those derivatives can be found as positions in the baance sheets of financia institutions. To evauate those positions and to set prices in OTC trades sophisticated modes are needed. The main underyings for products in the word of interest rate derivatives are forward rates. Roughy speaking, those rates give the interest rate as of today for some future time period. The most famous forward rates are the Libor/Euribor forward rates. Those rates dictate the conditions to which big and iquid financia institutions end money to each other. A majority of the derivatives that have to be priced depend on more than one forward rate. In 1997 the first Libor Market Modes (LMMs) to describe a set of forward rates consistent with each other were pubished in [2] and [11]. This was a rea breakthrough, because, firsty, the forward rates are modeed directy and, secondy, it enabed market participants to evauate whoe books of options depending on a range of forward rates in arbitrage free manners. One drawback in the eary simpe LMM is the incapaby to incorporate the observabe smie effect due to the deterministic voatiity structure. So the mode is ony capabe to evauate European options on those strikes which are used for the mode caibration and, even worse, it can ony be used to evauate a European options for exacty one strike. In most cases this strike is at the money. Obviousy a reasonabe mode shoud be abe to price options on any strike. A simpe mode for forwards rates that is capabe to incorporate the smie effect is the SABR mode and was introduced by Hagan [18]. It is popuar since, because it easy to understand and to caibrate at the same time. The SABR mode is a one asset 3

12 1. Introduction mode with stochastic voatiity and thus gives a way to incorporate the market smie. Therefore, it is possibe to evauate a book of European options on more than one strike at east for books depending on ony one underying. The mode is a step in the right direction, but for the sake of evauating whoe baance sheets consisting of options on more than one underying it is not enough. Rebonato proposed in [36] the SABR-LMM, which is a hybrid of the SABR mode and the LMM. The SABR-LMM is a market mode which can do both, it incorporates the market smie and it describes the dynamics of a set of forward rates. Simutaneousy, it tries to preserve the simpe SABR dynamics for the singe assets as cose as possibe. A main issue is to caibrate the SABR-LMM to the market to refect the dynamics of the rea word. The goa of this work is to tacke this probem by giving the right frameworks for an impicit caibration to current market prices. We wi focus on the caibration to cap and swaption prices and on the caibration to cap and constant maturity swap (CMS) spread option prices. The first was introduced in [36] and is revised thoroughy in this work. For the second caibration approach we extend the work of Kienitz & Wittke [22] to our SABR-LMM environment. A subprobem in the caibration task is to find an appropriate parametrization of the mode coefficients and structures. A styized parametrization is required to guarantee a stabe impementation. To describe the correation-structure of the SABR-LMM through a proper parametrization we wi research the two approaches coming from Lutz [29] and Schoenmakers & Coffey [25]. In addition, we wi test the caibration methods and the different parameterizations on two different data sets consisting of rea market prices from two different dates. The work is organized as foows. In chapter 2 we introduce basic products, derivatives and bootstrapping techniques. Then, in chapter 3 we expain the simpe SABR mode and the SABR-LMM. In addition we cacuate the invoved asset dynamics under common measures. In chapter 4 we approximate the induced swap rate dynamics for the SABR-LMM in a simpe SABR framework. The parametrization of the SABR-LMM is covered in chapter 5 were we expain how we styize the mode voatiities and correations. The impicit mode caibration to cap and swaption prices is expained in chapter 6. Afterwards, we introduce in chapter 7 the concept CMS spread options and describe their dynamics in the SABR-LMM mode. Further, we show how to caibrate the correation structure to CMS spread option prices. In chapter 8 we outine the out-carried impementations of a caibration procedures expained in the previous chapters and test the caibration methods by reprising the invoved products using 4

13 Monte Caro simuations. Here we use market prices from two different dates. Last, chapter 9 concudes the resuts of this works. 5

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15 Acknowedgments I woud ike to thank Christoph Mo for providing the opportunity to write my thesis in cooperation with zeb/information.technoogy GmbH & Co.KG. Further, my gratitude goes to Nies Linnemann for being a great motivator (or mentor) and for his power to have the right questions for every situation. I thank Josef Üre for his fu support throughout my study. Last but not east, I thank my ove Jana for her amazing patience and encouragement during the months of writing. 7

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17 2. Preiminaries This work discusses the caibration of a market mode for forward rates. To understand the mode we first need to understand the concept of forwards and to comprehend the caibration processes we first have to grasp the concept behind derivatives on forward rates. For this we give a proper mathematica environment and expain a the needed concepts. Amongst others we give definitions for basic financia instruments and show how to bootstrap the needed market data. This data wi be essentia in the caibration part. Further we give a method to evauate options market consistent Forward Rates and Swaps In this section we first introduce two basic instruments that wi be the basis of this work. We show how they are reated and give important formuas that wi accompany us throughout this thesis. After that, we estabish derivatives on these products and give evauations formuas for pricing. This section foows [7] and [8] Basic Definitions To tak about financia requires a concept of time and time steps. A tenor structure (T i ) i {,...,N} is a set of rea numbers consists of a time points of interest to the mode. Usuay the structure starts at T = which can be interpreted as the vauation date we are ooking at. At this point in time a market prices are known. Formay a tenor structure can be defined as 9

18 2. Preiminaries Definition 1 (Tenor Structure). A tenor structure (T i ) i {,...,N} is a finite, stricty monotonousy increasing sequence of non-negative rea numbers, hence T < T 1 < < T N <. Further we define δ i := T i+1 T i, i {,..., N 1} as the i-th time step. The T i don t have to be equay spaced in genera, so we do not force δ i to be constant. But in fact, when it comes to impementation it seems possibe (and in fact turns out to be possibe) to set δ i = δ constant without getting to infexibe. The most basic product in market is the zero coupon bond and can be defined as foows Definition 2 (Zero Coupon Bond). A zero coupon bond, pays at some tenor point T its notiona N and has no other payments in between. We write B(t, T ) for its price at time t T, where B(T, T ) = N and ca T maturity and T t time to maturity of the zero coupon bond. In most cases we just write bond instead of zero coupon bond. In this work it hods N = 1 throughout and the bond price is assumed to be positive a the time, which means B(t, T ) > for a t. In addition we assume the price process B(t, T ) is decreasing in T. Hence, it hods B(t, S) < B(t, T ), if and ony if S > T. That means the onger the time to the payment the ess worth is the bond. This agrees with our intuition about pricing. Further, we define bond prices B(t, T ) as the discount factors for the periods [, T ]. This makes sense, since a bond pays in T exacty one and therefore gives today s vaue of one unit of money in T. One of the most important concepts for this work is the forward rate. A forward rate F i t for the time interva [T i, T i+1 ] at time t gives the current interest rate for that time interva, which is consistent with the bond prices (B(t, T i )) i {Ti T i t}. Here F i t is normaized inear to one year, that means over the period [T i, T i+1 ] the occurring interest rate is (1 + δ i F i t ). Mathematicay the rate is defined as This strange definition comes from market convention. 1

19 2.1. Forward Rates and Swaps Definition 3 (Forward Rate). The forward rate at time t over time [T i, T i+1 ] is defined as Ft i := 1 B(t, T i ) B(t, T i+1 ). (2.1) δ i B(t, T i+1 ) Remark. In particuar, our assumptions about the bond prices impy Ft i for a t < T i and i,..., N 1. For carification we emphasis: The forward rate F i over the period [T i, T i+1 ] has the expiry date T i, which means that the rate is fixed to a certain vaue after this time, and that any payments are done a time step ater, at the settement date T i+1. To check the interpretation of a forward rate as an interest rate for a certain period define the vaue ( 1 ) 1/(T t) r t,t := 1 B(t, T ) This is the risk-ess interest rate the bond pays unti maturity. If we consider a bond with maturity T i, i > 1, the risk-ess rate r,ti is an interest rate over more than one time period. Foowing the mentioned intution about forwards it seems equay pausibe to foow one of the two foowing strategies of which one is equivaent to the other. The first is to buy a bond with maturity T i+1 and get for (T i+1 t) periods the interest rate r t,ti+1. The second is to buy a bond with maturity T i and a forward rate agreement for the period [T i, T i+1 ]. A forward rate agreement (FRA) is a contract that guarantees at time t an interest rate of exacty F i t over the period [T i, T i+1 ]. Both strategies shoud have the same payoff in the end. To verify this caim we cacuate both portfoio payoffs in T i+1 and obtain (1 + r t,ti ) (Ti t) (1 + δ i Ft i 1 ( ) = 1 + B(t, T i) B(t, T i+1 ) ) B(t, T i ) B(t, T i+1 ) 1 = B(t, T i+1 ) = (1 + r t,ti+1 ) (T i+1 t). Another important financia product is the swap. A swap over a time horizon [T m, T n ] is a contract between two parties the ong and short party which exchanges the forward rates F i, i {m,..., n 1}, in each period against a fixed rate K. In a payer 11

20 2. Preiminaries swap the ong party pays the fixed rate K and has to receives the foating rates F i. In a receiver swap the ong party receives the fixed rate K and pays the foating rates F i. A swap over the period [T m, T n ] expires in T m. On that date a rates are fixed to the vaue FT i m and don t change over the exchange time from T m to T n. The difference T n T m is the tenor of the swap and describes the ength of the exchange period. For the ong party a payer swap at time t T m has the vaue Swap m,n m 1 t := δ i B(t, T i+1 )(Ft i K) = i=n m 1 i=n B(t, T i ) B(t, T i+1 ) δ i B(t, T i+1 )K m 1 = B(t, T m ) B(t, T m ) δ i B(t, T i+1 )K, (2.2) since the B(t, T i+1 ) are the discount factors and the forward rate F i is paid in T i+1. In the market the vaue of K is chosen such that the expression in (2.2) is equa to i=n zero. In this case K is caed swap rate. Since it hods we define the foowing: m 1 = B(t, T m ) B(t, T m ) δ i B(t, T i+1 )K i=n K = B(t, T n) B(t, T m ) m 1 i=n δ ib(t, T i+1 ), (2.3) Definition 4 (Swap Rate). The swap rate at time t T m over the period [T m, T n ] is given as S m,n where we define the swap numéraire as t := B(t, T m) B(t, T n ), (2.4) i=m A m,n t A m,n n 1 t := δ i B(t, T i+1 ). 12

21 2.1. Forward Rates and Swaps Remark. A swap rate over a time interva can be interpreted, due to the reation in (2.3), as the average interest rate over this period. An important feature of a swap rate is that it can be written as a weighted sum of the invoved forward rates. To reaize this we write S m,n t = B(t, T m) B(t, T n ) = = =: where the weights are defined as A m,n t n 1 i=m B(t, T i) B(t, T i+1 ) A m,n n 1 i=m ω m,n n 1 i=m δ i B(t, T i+1 ) A m,n Ft i t t ω m,n i (t)f i t, (2.5) i (t) := δ ib(t, T i+1 ). (2.6) A m,n t The ast equations (2.5) and (2.6) are extremey important since they enabe us to see the direct ink between forward rates, which we are paning to describe in a market mode, and swaps, one of the most iquid products in the market. Further the sum structure shows that swaps depend on the interpay of the forwards. reevant in the caibration part of this thesis. This wi be Basic Derivatives Later we want to caibrate our mode to market prices. It is practice to use ca or put-ike derivatives for this purpose, since these simpe products are the most iquid ones. High iquidity favors the reiabiity of the observed prices since the associated products are more ikey traded on a census price. We wi start with derivatives one forwards and then come to options on swaps. The most simpe derivative on a forward rate is a capet, which is a simpe ca option. It enabes to hedge against rising interest rates for a period of ength δ i. 13

22 2. Preiminaries Definition 5 (Payoff of a Capet). A capet on a forward rate F i with strike K pays in T i+1 the foowing δ i (F i T i K) +. (2.7) So a capet payment at the settement date T i+1 is fixed one period earier at the expiry date T i. However, in the market amost no capets are quoted directy. They are quoted in whoe portfoios of capets which are caed caps. A cap over the period [T m, T n ] is a sum of capets with expiry dates T i, i {m,..., n 1}, where each capet has the same strike K. This impies the foowing proposition: Proposition 1 (Cap Price). The cap price of a cap ranging from T m to T n and strike K is given as C m,n (K) := n 1 i=m C i (K), (2.8) where C i (K) is defined as the vaue of the capet on F i with strike K. Remark. As for swaps the difference T n T m is the tenor and T m the expiry date of the cap. Apart from capets, foorets which form the counterparts of the capets and are puts on forward rates exists. Therefore we can define the payoff of a fooret as: Definition 6 (Payoff of a Fooret). A fooret on a forward rate F i with strike K pays in T i+1 the foowing δ i (K F i T i ) +. (2.9) Again those derivatives are not quoted directy in the market. There are ony foors a sum of foorets quoted. Foors can be seen as the counterpart of caps. The price of a fooret is given as: Proposition 2 (Foor Price). The foor price of a foor ranging from T m to T n and strike K is given as P m,n (K) := n 1 i=m P i (K), (2.1) where P i (K) is defined as the vaue of the fooret on F i with strike K. 14

23 2.2. Bootstrapping Market Data Remark. Simiar to caps the difference T n T m is the tenor and T m the expiry date of the foor. Up to now we have discussed derivatives on forward rates. Now we want introduce options on swap rates. Those options are often referred to as swaptions. A swaption with strike K gives the right to enter a payer swap or receiver swap, respectivey, with strike K. In our case, the swaption and swap have the same expiry dates a the time. Therefore, because of (2.2), the payoff of a payer swaption in T n is given as ( n 1 i=m δ i B(T m, T i+1 )(F i T m K)) +. With the resut in (2.5) we are abe to rewrite this payoff in the foowing proposition: Proposition 3 (Payoff of a Swaption). The payoff of a swaption on a payer swap S m,n at time T n is given as where A m,n is the swap numéraire from definition 4. A m,n T m (S m,n T m K) +, (2.11) We want to emphasize that, unike as in the case for caps it is not possibe to decompose the payment (2.11) nor the vaue of a swaption in more eementary payoffs or prices. This is a huge distinguish feature of caps/foors and swaptions Bootstrapping Market Data In the caibration part we wi rey on some fundamenta data which we wi assume as given. This incudes the prices of capets and foorets in any given tenor as we as prices for swaptions with any expiry date and any tenor. Further, we wi need the current forward rates at the vauation date. Unfortunatey, those cannot be obtained directy and have to be stripped as we The Bootstrapping of Forward Rates To cacuate the current forward rate which are consistent with the corresponding swap prices we are going to rey on the definition of forward rates as a quotient of bond prices (2.1) and on the definition of swap rates as in (2.4). Our pan is to 15

24 2. Preiminaries cacuate the forward rates based on a set of swap rates starting at the vauation date and having growing tenors up to the maxima tenor T N T. Those swaps are quoted for a very ong tenors up to 5 years. Therefore they provide the right environment to cacuate a the needed forward rates. We want to cacuate the forward prices basing on bond prices. To achieve this we now cacuate the needed bond prices iterativey. It is cear that the first forward rate F starting at the vauation date and setting in T 1 corresponds to the swap rate S,1. From this we get B(, T 1 ) = δ F. (2.12) This is our initia vaue. Next, et us consider definition 4, namey which is equivaent to S,n = B(, T ) B(, T n ) A,n = 1 B(, T n) A,n, S,n A,n = 1 B(, T n ) S,n ( A,n 1 + δ n 1 B(, T n )) = 1 B(, T n ) B(, T n ) = 1 S,n A,n δ n 1 S,n B(, T n ) = 1 S,n n 2 i= δ ib(, T i+1 ). (2.13) 1 + δ n 1 S,n On the eft hand side of (2.13) we find the n-th bond price and on the right hand side we find a function depending on the n-th swap rate and the first n 1 bond prices. Therefore, the formua gives us a way to cacuate the bond prices one by one by just knowing the swap rates S,n for each tenor point T n. If we cacuated a bond prices, we can cacuate the forward rates through the formua in (2.1) F i = 1 B(, T i ) B(, T i 1 ). δ i B(, T i 1 ) However, not a needed swap rates can be found in the market and have to be interpoated. We decided to interpoate inear. This method doesn t guarantee positive forwards, but in our case we did not get any. 16

25 2.2. Bootstrapping Market Data 3,% 3,% 2,5% 2,5% Forward Rate 2,% 1,5% Forward Rate 2,% 1,5% 1,% 1,%,5%,5% Expiry Expiry Figure 2.1.: The stripped forward rates foowing the approach above. On the eft we used inear interpoation and get shape that is in [14] reffered to as a saw tooth shape. On the right we used spine interpoation and got a smoother shape, but a smaer maxima forward rate. The data is from the and was obtained from Boomberg. The stripping was impemented in F# and the pot was done in Matab. Sti, there are other possibiities. In [14] the (C 1 /C 2 ) spine interpoation is suggested and expained, but this method doesn t guarantee positive forwards either. Another possibiity is the Forward Monoton Convex Spine introduced in [33]. This method incorporates the idea of occurred interest, meaning that a forward rate F i is paid over the interva [T i, T i+1 ] and not ony at T i+1. A the above methods work ony in an environment of greater certainty about the input data. If the vaidity of the data is questionabe one coud buid a swap curve by using a Neson-Siege or Svensson curve as described in [1] and [16]. Those curves have a parametrization that forces them in a range of ideaized swap curves. Since those interpoation methods are behind the scope of this work we stick to inear interpoation The Stripping of Capet Voatiities Capet voatiities wi be one of the corner stones of our caibration procedure ater on. As described in section capets are not directy quoted in the market, but indirecty as caps. In this section we wi describe a stabe approach to cacuate capet voatiities from cap voatiities. This procedure is caed capet stripping. For the genera framework we rey on [24]. 17

26 2. Preiminaries In the market we find for each set of strikes (K i ) i a set of cap prices, for caps ( C 1,j (K i ) ) i,{1<j N} with expiry date T 1 growing tenors up to T N T 1, given in Back voatiities σ cap (j, K i ). So in the market caps are quoted indirecty. The cap price can be obtained via j 1 C 1,j (K i ) = C k (K i ) k=1 j 1 = C k (F k, K i, σ cap (j, K i ), T k ), (2.14) k=1 where C k (F k, K i, σ(j, K i ), T k ) is the price of the k-th capet assuming that Ft k Back s mode [1]. Therefore, it hods due to (2.7) foows C k (F k, K i, σ(j, K i ), T k ) = δ k B(, T k+1 )(F k N (d 1 ) K i N (d 2 )), (2.15) where d 1/2 := n ( F k K i ) ± 1 2 (σ(j, K i )) 2 T k σ(j, K i ) T k. So a cap is priced by using an a in voatiity σ cap (j, K i ) for a capets. Knowing this we want to cacuate the capet prices for a capets ( C j (K i ) ) in Back voatiities ( j,i σ cp (j, K i ) ) j,i at any tenor point and strike K i. To achieve this, we first fix a strike K i and therefore ony consider the set of caps ( C 1,j (K i ) ). As in the case for forward rates the stripping of capet voatiities {1<j N} is done iterativey as foows: It is cear from (2.8) that the cap price C 1,2 (K i ) agrees with the capet price C 1 (K i ). This is our initia vaue. Then we sove iterativey the foowing equations for 1 < k < N k 1 C k (F k, K i, σ cp (j, K i ), T k ) = C 1,k+1 (K i ) C j (F j, K i, σ cp (j, K i ), ) (2.16) to obtain a capet voatiities σ cp (j, K i ). We do this for a strikes and get the whoe capet voatiity surface. Simiar as in the bootstrapping of forward rates not a cap voatiities for a tenors we are interested in may exist in the market. We gain the missing tenors by spine j=1 interpoation, since we want a smooth voatiity surface. 18

27 2.2. Bootstrapping Market Data Remark. It is cear from the definition of the payoff of a fooret (2.9) and from the definition of the price of a foor (2.1): It is possibe to strip fooret voatiities ( σ ft (j, K i ) ) in the same fashion as stripping capet voatiities. The fooret voatiities σ ft (j, K i ) then agree with the capet voatiities σ cp (j, K i ) for the same j,i strikes, underying prices and expiries, because of the ca-put parity. 7% 6% Voatiity 5% 4% 3% 2% 1 2 Expiries 3 4 ATM 1.% 3.% Strikes 6.% 1.% 14.% Figure 2.2.: The voatiity surface stripped from Euro caps as of the with a haf year tenor (δ i.5) and a time horizon of over 2 years. We obtained the data from Boomberg. The impemention was carried out in F# and the pot was done in Matab. 19

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29 3. The SABR and SABR-LMM mode The main goa of this work is to provide an environment to price options depending on a range of forward rates consistenty. To achieve that we set up a mode that can do both. It wi incorporate the observabe smie effect and provide a dependency structure for the modeed assets. We wi deveop the mode in two steps. In the first one we give a simpe mode that can ony hande one asset, but is capabe of incorporating the smie effect. Further, the mode provides a anaytic formua that transates the coefficients which are describing the mode into an impied mode smie. This feature wi come in handy ater on, since it wi enabe us to caibrate efficienty to given prices. Then we extend the mode to a fu market mode. By doing so we try to preserve the dynamics from the simper mode as we as possibe The SABR Mode The SABR (σ, α, β, ρ) mode is a mode of stochastic voatiity and can describe exacty one asset F. It was first pubished by Hagan [18] and it has been popuar since, because it easy to understand and to caibrate. The stochastic voatiity gives a way to incorporate the market smie. Further there exists a formua that gives depending of the mode parameters the smie generated by the mode, whereas a change of mode parameters can be directy interpreted in changes of the mode induced smie in a ogica way. In addition there exist very effective simuation schemes which reduce the pricing procedure through Monte Caro simuations drasticay [6]. For this reason it is a perfect too to manage a book of options on a singe asset. Theoreticay F can be any asset but in our context F wi be ony a forward rate or a swap rate. 21

30 3. The SABR and SABR-LMM mode Definition 7. (The SABR Mode) In the SABR mode the dynamics of an asset F maturing at time T is given by df t = σ t F β t dw t, F = F (), t T dσ t = ασ t dz t, σ = σ() (3.1) d W, Z t = ρdt, where σ, α R + und β [, 1], ρ [ 1, 1]. Further W and Z are one dimensiona Wiener processes. To be consistent we define df t for t > T. The process σ t is the voatiity of the mode and σ as the eve of the voatiity. The coefficient α is named the vo-vo, that s short for the voatiity of voatiity and ρ is the so caed correation or skew. The coefficient β is the CEV parameter. In the specia case β = we get a norma mode, since in this case F t is approximatey normay distributed. With this choice for β the process (F t ) t can get negative and in Asian markets practitioners often choose this parametrization to mode forwards, since they tend to be negative in this markets from time to time [18]. If we set β = 1, we obtain a og-norma mode. If β (, 1), we get a CEV mode. The choice of beta wi be important in the caibration part ater in this work. We do not favor the SABR mode ony because of the simpe structure above other ones ike Heston [26] or Bates [37], which aso coud be extended to a fu Libor- Market-Mode [12]. The main advantage of the SABR mode over the other ones is the anaytic function for impied Back voatiity depending on strike and underying price. If this voatiity is put in to Backs pricing formua, it yieds the mode price for a ca. To describe the impied Back voatiity coser, we first consider Back s Mode [1] in which an asset F foows the SDE df t = σ imp F t dw t, F = F (), where σ imp > is a rea number and W a Wiener process. It is we known [1] that the ca price for a ca with strike K, expiry date T ex and settement date T set can be cacuated as [ ] C(F, K) = B(, T set ) F N (d 1 ) KN (d 2 ) (3.2) 22

31 3.1. The SABR Mode and, respectivey the put price for a put with strike K, expiry date T ex and settement date T set can be cacuated as where [ ] P (F, K) = B(, T set ) KN ( d 2 ) F N ( d 1 ), d 1/2 = n ( F ) K ± (σ imp ) 2 T ex σ imp (3.3) T ex and B(, T set ) is the discount factor for the time interva [, T set ]. The voatiity σ imp which yieds in (3.2) and (3.3) the same price for a ca and put the SABR mode woud produce is caed impied (Back) voatiity. It is we known that in rea market situations the impied voatiity depends on the underying price and strike. Thus, a voatiity surface can be observed which reduces to a voatiity smie, if we fix the underying price F to some vaue. In the SABR mode exists a formua to cacuate the impied voatiity surface based on the mode parameters σ, α, β and ρ. The formua is given in [18] and was improved in [23]. The improved formua goes as foows where σ I (F, K, β, α, ν, ρ, T exp ) :=IH (F, K, β, α, ν, ρ) (1 + I 1 H (F, K, β, α, ν, ρ)t exp), (3.4) IH 1 (F, K, β, α, ν, ρ) := (β 1)2 α 2 24(F K) 1 β + ρναβ 2 3ρ2 + 4(F K) (1 β)/2 24 ν 2 and αk β 1, if F K = 1 n(f /K)α(1 β), if ν = IH F 1 β K (F, K, β, α, ν, ρ) := 1 β ( ν n( F K )/ 1 2ρz 1 +z1 n 2+z ), 1 ρ 1 ρ, if β = 1 ( ν n( F K )/ 1 2ρz 2 +z2 n 2+z ) 2 ρ 1 ρ, if β < 1 where z 1 := ν n(f /K) α 23

32 3. The SABR and SABR-LMM mode and z 2 := ν α F 1 β K 1 β. 1 β A the above expressions are purey anaytic and no numerica integrations or some σ I β=.5, ν=.5, ρ=.5,f =.35 α=.2 α=.35 α=.5 σ I α=.2, ν=.5, ρ=.5, F =.35 β=.4 β=.5 β= σ I.5.1 Strike α=.2, β=.5, ρ=.5, F =.35 ν=. ν=.2 ν=.4 σ I.5.1 Strike α=.2, β=.5, ν=.5, F =.35 ρ=.8 ρ=.5 ρ= Strike.5.1 Strike Figure 3.1.: The graphic shows how the impied voatiity σ I changes with changes of the SABR mode parameters. α has an impact on the eve the smie, whereas a higher ν produces a more pronounced shape. A change in β effects the eft end of the smie, that is in the area with sma strikes. The parameter ρ has a genera impact on the skew of the smie. The pots give the impression that changes in ν and ρ can substitute changes in β very we. The pot was done in Matab. 24

33 3.1. The SABR Mode simiar cumbersome procedures are needed. This makes this formua highy tractabe and efficient. However, we want to emphasis that there exist other approximation for the impied voatiity. For exampe, other formuations are given in [38], [3] and [27], whereas the formuation in the ast source is the most exact one according to market opinion. But, the impementation of those significant more compicate formuas is behind the scope of this work. Note however that it was shown in [18] and [23] that the above version in (3.4) works quite we. Ceary, the formua enabes us to cacuate prices, which our mode produces, for puts and cas with different strikes and underying prices without doing cumbersome Monte Caro simuations. Further, the formua enabes us to cacuate prices on portfoios of put and cas, ike straddes, butterfy spreads, covered cas, protective puts, etc. But, we can go the other way around as we. It is market practice to quote prices of cas and put in Back voatiities indirecty. Hence, if we observe impied voatiities of European options we can easiy caibrate the SABR mode to market prices by minimizing the difference of quoted voatiities and impied mode voatiities depending on the mode parameters α, β, ν and ρ. However, in this work we wi fix β to.5 or 1., depending on the assets we are ooking at. So we ony have to estimate the three parameters α, ν and ρ. We have two reasons to do so. First, the impact of β and ρ, in combination with ν, on the shape of the curve is very simiar as can be seen in figure 3.1. By fixing β we obtain a more unique soution. Second, we want to mode forward rates and to set β =.5 seems to be market conform, as argued in [36]. Discussion with traders showed that most of them indeed choose β =.5 in their CEV modes and β fixed at this vaue eads to a ower variation of the other parameters over time. So the mode caibration is onger approximatey vaid and a onger vaidity speaks in favor of a fixed β. To estimate the SABR mode parameters we simpy minimize the square of the sum over the squared errors between market prices and mode prices. That impies our estimated parameters α, ρ and ν are obtained by ( α, ρ, ν) = arg min α,ρ,ν [ σm (F, K i ) σ I (F, K i, β, α, ν, ρ, T i ) ] 2, (3.5) i where σ M (F, K i ) is the in the market quoted Back voatiity for a ca or put with strike K i, underying price F and expiry date T i. The minimization probem in (3.5) can be taiored to ones needs by mutipying a weights. This technique can be used to 25

34 3. The SABR and SABR-LMM mode weight uncertain data ower than certain one or to emphasis on a range of strikes. In this case the minimization probem becomes [ ( α, ρ, ν) = arg min ω i σm (F, K i ) σ I (F, K i, β, α, ν, ρ, T i ) ] 2. (3.6) α,ρ,ν i For exampe, by choosing (ω i ) i = (σ M (F, K i ) 1 ) i the reative differences wi be minimized. If not written differenty we wi use (3.5) The SABR-LMM Mode In this chapter we wi combine the simpe SABR mode with the cassic Libor market mode (LMM) under deterministic voatiity as deveoped in [11] and [2]. In a LMM a number of forward rates with a dependency structure are modeed. The dependency structure is given through the correation which describes the comovement of the assets in the mode. The probematic part in a simpe LMM is the ack of possibiity to mode smie effect which is observabe in the rea market. This means we are ony abe to evauate capets or swaptions on those strikes which are used for the mode caibration and, even worse, it is ony possibe to evauate capets or swaptions for exacty one strike. In most cases those strikes are at the money. Obviousy a reasonabe mode shoud be abe to price options on any strike. The SABR mode can reproduce the smie, but since it is a one-asset mode no dependency of two or more processes can be considered. It is definitey no good soution to mode a number of assets simpy by taking a number of uncorreated SABR processes. For exampe, this shows the vauation of swaptions based on forward rates. So the LMM and the SABR together have the needed features pus the SABR mode gives us the usefu formua for the impied voatiity. In the foowing we wi combine both modes and deveop two stabe caibration methods. The first method wi be a caibration on capets and swaptions and the second wi be a caibration on capets and CMS spread options. In both cases we wi heaviy depend on the formua for impied voatiity to hit quoted market prices. The overa goa in both approaches wi be to keep the SABR dynamics for the forward rates as cose as possibe, since that mode has so many good characteristics. The SABR-LMM is defined as foows: 26

35 3.2. The SABR-LMM Mode Definition 8. (The SABR-LMM Mode for Forward Rates) In a N-dimensiona SABR-LMM mode the N forward rates (F i ) i {1,...,N 1} have under their individua forward measure P i the foowing dynamics: dft i = σt(f i t i ) β dwt i, t < T i, F = F () (3.7) σt i = gtk i t i (3.8) dkt i = h i tktdz i t, i t < T i, k = k() (3.9) d W i, W j t = ρ ij dt, i, j {1,..., N 1} (3.1) d Z i, Z j t = r ij dt, i, j {1,..., N 1} (3.11) d W i, Z j t = R ij dt, i, j {1,..., N 1}, (3.12) where β [, 1], ρ i,j, r i,j, R i,j [ 1, 1] for a i, j {1,..., N 1} and the deterministic functions g, h : R + R fufi T g 2 i (s)ds < and T h 2 i (s)ds < for a i {1,..., N 1} and < T T i. Further, we set for competeness F i t = F i T i for a t > T i. We define the super correation matrix of the mode as ( ρ P := R T ) R. (3.13) r The Matrix (ρ ij ) i,j consists of a forward/forward correations, the entries of (r ij ) i,j are the voatiity/voatiity correations and (R ij ) i,j carries a the forward/voatiity correations. Notice, ony P, (ρ ij ) i,j and (r ij ) i,j are symmetric. The matrix (R ij ) i,j is asymmetric in genera. Remark. From time to time we wi use the forward rate F which is not contained in the SABR-LMM above. This forward rate is the interest rate for the period [T, T 1 ]. Since we assume that a prices in T are known F is not stochastic. Obviousy, its dynamics doesn t have to be modeed. The SABR-LMM incorporates not ony the SABR into the LMM mode it has even 27

36 3. The SABR and SABR-LMM mode time dependent parameters given through g i and h i. The stochastic voatiities σ i of the forward rates F i can be separated into a deterministic part g i and in a stochastic part k i. Therefore g i is often caed the deterministic voatiity of F i and k i the stochastic voatiity, respectivey. Further, the function h i describes the deterministic voatiity of voatiity. We woud ike to highight the foowing feature: If k i is constant for a i, e.g. h i for a i, we obtain an ordinary LMM. This is because the stochastic voatiity vanishes The SABR-LMM Dynamics under any Forward Measure P To simuate option prices in a Monte Caro setup or to examine the mode it is necessary to express a asset dynamics in the mode under one common measure. A possibe choice for such a measure is the forward measure P for some. In the specia case = N 1 we ca P N 1 the termina measure and in most cases we use this measure for our simuation. In the foowing, we aways assume that our modeed assets are forward rates with dynamics as given in definition 8. change of measures by change of numéraire techniques. Theorem 1 (SABR-LMM Dynamics under different P ). The idea is to cacuate the In the SABR-LMM mode, as in definition 8, the dynamics under a certain forward measure P of the forward rates F j and the stochastic voatiities k j are given as j+1 i df j t = σj t (F j t )β dw j t +1 i j ρ i,j δ i σt i(f t i)β dt + dw j 1+δ i Ft i t ρ i,j δ i σt i(f t i)β dt + dw j 1+δ i Ft i t, if j <, if j =, if j > (3.14) and j+1 i dk j t = hj t kj t dz j t +1 i j R j,i δ i σt i(f t i)β dt + dz j 1+δ i Ft i t R j,i δ i σt i(f t i)β dt + dz j 1+δ i Ft i t, if j <, if j =, if j > (3.15) where σ j t = hj t kj t stays the same. 28

37 3.3. The SABR-LMM Dynamics under any Forward Measure P Proof. We wi carry out the proof with by means of induction. First, we concentrate on the dynamics of the F i. It hods per definition, since F i is a forward rate: Ft i = 1 ( B(t, Ti ) B(t, T i+1 ) ) for a t T i, δ i B(t, T i+1 ) where the B(t, T i ) are stricty positive bond prices at time t for bonds which pay at maturity T i exacty one unit of money. Further, the F i are oca martingaes under P i. Therefore the probabiity measure P i can be seen as the measure under which every tradeabe asset reative to the numéraire B(t, T i+1 ) is a oca martingae. In the first step we cacuate the dynamics of F i t under Pi 1 and therefore reativey to B(t, T i ). For this we need the Bayes formua. We give the formua without proof. Proposition 4. (Bayes Formua) Let (Ω, A) be a measurabe space with probabiity measures P and Q. Further et B A be a sub-σ-agebra. Then it hods for an integrabe and A measurabe random variabe Y It foows that E Q[ Y B ] = E P[ dq dp Y B] ( E P[ dq dp B]) 1 P-a.s.. (3.16) dp i 1 Ft dp i = B(t, T i) B(, T i+1 ) B(t, T i+1 ) B(, T i ) P i -a.s., (3.17) since the expression is a probabiity measure, because (1 + δ i F i t ) B(,T i) is positive P i+1 -martingae with an expected vaue of 1. Further, et B(,T i+1 ) = dpi 1 Ft ( ) dp i Xt B(t,T i ) t T i be a P i 1 martingae. Then it hods with the Base formua (3.16) for t T i E Pi[( dp i 1 dp i ) X Ti B(T i, T i ) F t ] = E Pi[( B(T i, T i ) B(T i, T i+1 ) = E Pi 1[ X Ti B(T i, T i ) F t E Pi[ B(T i, T i ) B(T i, T i+1 ) = X t B(t, T i ) B(, T i+1 ) ) B(, T i ) ] B(, T i+1 ) B(, T i ) X Ti B(T i, T i ) F t F t ] This shows, that the measure impied by density in (3.17) agrees with the probabiity measure P i 1. Therefore the notation dpi 1 dp i Ft is justified. Now it foows with the ] 29

38 3. The SABR and SABR-LMM mode Ito-Formuas [9] and by considering the SABR-LMM mode dynamics from definition 8: [ ( dp i 1 )] d n Ft dp i = d n((1 + δ i Ft i ) B(, T i) B(, T i+1 ) ) = d n(1 + δ i F i t ) = = δ i δ 2 i 1 + δ i Ft i dft i δ i Ft i d F i t δ i δ 2 i 1 + δ i Ft i σt(f i t i ) β dwt i δ i Ft i (σt) i 2 (Ft i ) 2β dt. (3.18) According to Girsanow s Theorem [4] is X a oca P i -martingae if and ony if is a oca P i 1 -martingae. ( dp i 1 ) Y := X X, n F dp i (3.19) A change of measure produces a drift that maintains the martingae property. In the finance iterature this drift is often caed Convexity Correction. If we use (3.19) on F i, we get, with the hep of (3.18) and (3.1), the foowing dynamics under P i 1 d F ( dp i = df i d F i i 1 ), n F dp i = σ i (F i ) β dw i σ i (F i ) β δ i 1 + δ i F i σi (F i ) β. (3.2) Now we want to cacuate the dynamics of F i under P i 2. In anaogy (3.18) to foows [ ( dp i 2 )] d n Ft dp i 1 = d n(1 + δ i 1 Ft i 1 ) = 1 2 δ i δ i 1 F i 1 t δ 2 i δ i 1 F i 1 t σ i 1 t (Ft i 1 ) β dwt i 1 (σt i 1 ) 2 (Ft i 1 ) 2β dt. (3.21) 3

39 3.3. The SABR-LMM Dynamics under any Forward Measure P Together with (3.2) foows for F i under P i 2 by considering dpi 2 dp i d F ( dp i = df i d F i i 2 ), n F dp i = d F i d F ( dp i i 2 ), n F dp i 1 = σ i (F i ) β( ( dw i δ i 1 + δ i F i σi (F i ) β ρ i,i + δ i δ i 1 F i 1 t = dpi 2 dp i 1 dp i 1 dp i )) σt i 1 (Ft i 1 ) β ρ i 1,i Now foows the theorem for F j in the case of j >. The case < j foows in the same way and we ony note: It hods dp i ( Ft dp i 1 ) 1 ( dp i 1 = Ft B(t, Ti ) = dp i B(t, T i+1 ) B(, T i+1 ) ) 1 P i 1 -a.s.. B(, T i ) The dynamics of the stochastic voatiies k i under the forward measure P i is through (3.9) as dk i = h i k i dz i. In the same fashion as for the forward rates we first cacuate the stochastic differentia equation of k i under the measure P i 1 and P i 2. With Girsanow (3.19) foows, with the hep of (3.18), for the dynamics of k i under the measure P i 1 ( dp d k i = dk i d k i i 1 ), n F dp i = h i k i dz i h i k i δ i 1 + δ i F i σi (F i ) β R i,i. Therefore, we obtain by considering (3.21) for the dynamics of k i under P i 2 ( dp d k i = dk i d k i i 2 ), n F dp ( i = h i k i( dz i δ i 1 + δ i F i σi (F i ) β R i,i + δ )) i δ i 1 F i 1 σi 1 (F i 1 ) β R i,i 1. Again, per induction foows the theorem for k j in the case of j >. The case < j foows anaogousy and therefore is omitted. 31

40 3. The SABR and SABR-LMM mode Remark. Notice that the cacuated mode dynamics in Theorem 1 don t agree with the ones in [36]. There we find in the dynamics of k j instead of R i,j the Term R i,i ρ i,j and the function g j. However our version coincides with the dynamics in [17] The SABR-LMM Dynamics under the Spot Measure P spot Another measure under which we can cacuate the dynamics of the Forwardrates is the Spot Measure P spot. In this measure processes of the form ( X t G t ) t are oca martingaes, B(t,T where G t := γ(t) 1 ) 1 i γ(t) 1 B(T i 1,T i ), and γ(t) := inf { k N T + k 1 i= δ i > t } = inf { k N T k t }. Theorem 2 (SABR-LMM Dynamics under P spot ). Under P spot the SABR-LMM dynamics given in definition 8 are the foowing: df j t = σj t (F j t )β( γ(t) i j ρ i,j δ i σ i t (F i t )β 1 + δ i F i t + dw j t ), (3.22) and where σ j t = hj t kj t dk j t = hj t kj t stays the same. ( γ(t) i j r j,i δ i gt ihi t ki t (F t i)β ) 1 + δ i Ft i dt + dz j t, (3.23) Proof. A proof can be found in [7]. Aternativey one can carry out the proof in anaogy to Theorem 1. Since the numéraires of P spot and P are known one can cacuate the density for the change of measure, ike in (3.17). Then just the drifts comming from Grisanovs theorem have to be cacuated to obtain the dynamics under the spot measure. 32

41 3.4. The SABR-LMM Dynamics under the Spot Measure P spot To interpret P spot we write G t in a different way. It hods B(t, T γ(t) 1 ) G t = 1 i γ(t) 1 B(T i 1, T i ) = (1 + δ i 1 FT i 1 i 1 )B(t, T γ(t) 1 ). 1 i γ(t) 1 So G t can be seen as the time vaue process of a portfoio with the foowing strategy: The portfoio vaue in the beginning is exacty one. Then, from period to period, the portfoio reinvests its capita with the actua one period spot rate. To get the time vaue at time t the portfoio vaue is discounted by B(t, T γ(t) 1 ). The reason for considering different measures is the effort of cacuating the drifts terms in simuations. Amost haf of the simuation time comes from the drift cacuation the other haf comes from generating random numbers. In the spot measure the processes F j t and k j t have drifts consisting of (j γ(t) + 1) summands as shown in (3.22) and (3.23), respectivey. In the termina forward measure the processes F j t and k j t have drifts consisting of (N j) summands as shown in (3.14) and (3.15). It is natura to choose the measure with the minima cost of drift cacuation. We concude the foowing thumb rue: If ony forwards with short expiries have to be simuated, we choose the spot measure and if forwards with onger expiries are invoved, we choose the termina forward measure. 33

42

43 4. Swaps Rates in the SABR-LMM Swap rates depend directy on underying forward rates, since we can write them as sum of forwards as shown in Section in equation (2.5). This structure particuary yieds a direct dependence of the swap rate on the interpay of the forward rates. We want to anayze how the dependence of the interpay can be described in terms of the super correation matrix P which we defined in (3.13). To achieve this, we first give a way to mode the swap rate dynamics in a SABR environment. Then we approximate the swap s SABR coefficients by taking the structure as a sum of forward rates into account. Here we assume that the forward rate dynamics are governed by the SABR-LMM. The approximated SABR coefficients wi depend on P. By doing this we find a proper way to describe a swap rate dependent on P, which we wi ater use to estimate the matrix impicity by using market quotes of swaption prices. More on this can be found in chapter 6 which covers caibration to swaptions A SABR mode for Swap Rates A swap rate depends directy on forward rates, because of (2.5). Since we chose SABRike dynamics for a the forwards it makes sense to assume that a swap rate does not evove in a competey different stye and can be described by a SABR mode under the swap measure P m,n as we. For this section we depend on [36]. We define the swap rate dynamics as foows: 35

44 4. Swaps Rates in the SABR-LMM Definition 9 (The SABR mode for Swap Rates). The SABR dyanamic of a swap rate S m,n with expiry T m and tenor T n T m is under the swap measure P m,n defined as ds m,n t dσ m,n t = Σ m,n ( t S m,n) β m,n t dw m,n t, S m,n = S m,n () (4.1) = Σ m,n t V m,n dz m,n t, Σ m,n = Σ m,n () (4.2) d W m,n, Z m,n t = R m,n dt, (4.3) where V m,n, Σ m,n R + and R m,n [ 1, 1]. Further, W m,n and Z m,n are onedimensiona Wiener processes. Remark. Notice, we write for the swap SABR coefficients capita etters, whereas we write for the SABR-LMM coeffcients, except for the forward/voatiity correation matrix R, sma etters Swap Rates Dynamics in the SABR-LMM We want to estimate the swap rate dynamics in a SABR-LMM framework, where we assume that the swap rate evoves under the swap measure P m,n governed by a simpe SABR mode as in definition 9 above. There the swap process has the deterministic voatiity V m,n. Now, the chaenging part is the foowing: If in a LMM the forward rates have deterministic voatiity under the forward measures P i the swap rates have in genera stochastic ones under any P i. This simpy comes from the sum-weights ω m,n i (t) in the sum representation (2.5), because they are quotients of stochastic processes. This even happens in the case of a LMM with deterministic voatiity ony. To circumvent the probem we wi simpy freeze the weights to their initia vaues to make them deterministic again. The approximation wi be done step by step. First, we approximate the initia eve of the swap voatiity Σ m,n and the vo/vo V m,n. Then, the correation R m,n is approximated. In this section we rey on Rebonato [36], but have thoroughy revised the derivations. For a start, we describe the swap rate S m,n dynamics as ds m,n t = Φ m,n ( t S m,n ) β m,n dw m,n t, (4.4) 36

45 4.2. Swap Rates Dynamics in the SABR-LMM where Φ m,n t := Φ m,n t ({ Ft,..., Ft N 1 }, {σt,..., σt N 1 }, (ρ ij ) i,j ), is a stochastic voatiity depending on the SABR-LMM parameters. Now, our goa is to approximate Φ m,n and get an idea of its genera structure. If we cacuate the swap rate dynamics using Ito s formua [9] we get by keeping in mind the SABR-LMM dynamics for forwards (3.7) ds m,n n 1 t = S m,n t F =m t n 1 n 1 j=m ωm,n j F =m t n 1 = =m df t ()F j t ω m,n ()df t n 1 j,=m n 1 j,=m S m,n t F j t F d F j t, F t n 1 j,=m S m,n t F j t F σ j t t σ tf j t F t ρ j dt (4.5) S m,n t F j t F σ j t t σ tf j t F t ρ j dt. (4.6) Here we used in (4.5) the sum formua (2.5) and freezed the weights ω m,n j to their initia vaues. This is a common technique in financia mathematics and is a quite good approximation for fat underying yied curves. The [... ]dt term can be interpreted as the drift correction from Girsanow [4] due to the change of measures from the forward measures P j to the swap measure P m,n. We cacuate the quadratic covariation of (4.6) as n 1 t = d ω m,n ()df t d S m,n =m n 1 = d ω m,n ()σ ( ) F βdw t t (4.7) =m Now (4.7) is, because of (4.4), equivaent to ( S m,n t n 1 ) 2β m,n( Φ m,n) 2 = t j, ( Φ m,n n 1 ) 2 t = j, ω m,n j ()ω m,n () ( F j ) β ( ) t F βσ j t t σ tρ j ω m,n j () ) β m,n ( S m,n t ω m,n () ) β m,n ( S m,n t ( F j) β ( ) t F βσ j t t σ tρ j (4.8) 37

46 4. Swaps Rates in the SABR-LMM We define and rewrite (4.8) as ( Φ m,n t W m,n n 1 ) 2 = (t) := ωm,n j, n 1 j, W m,n j ( S m,n t () ( Ft ) β ) β m,n (4.9) (t)w m,n (t)σ j t σ tρ j W m,n j ()W m,n ()σ j t σ tρ j, (4.1) where we froze the ratios (4.9), which resuts ony in a sma ose of precision. This comes from the observation that the ratio ( F t ) β ( S m,n) β m,n t is ony sowy varying over time due to the high correation of swaps and forwards, as Hu and White argued in [21]. Now (4.1) eads to Φ m,n t n 1 j, W m,n j ()W m,n ()σ j t σ t ρ j. (4.11) Therefore, the swap rate dynamics in (4.4) can be approximated as ds m,n t n 1 j, W m,n j ()W m,n ()σ j t σ t ρ j ( S m,n) β m,n t dw m,n t. With the hep of this representation we pan to approximate the SABR coefficients Σ m,n () and V m,n. From (4.11) we obtain, by writing E m,n for the expected vaue under P m,n E m,n[ T m ( Φ m,n) 2dt ] t E m,n [ T m n 1 W m,n j ()W m,n ()σ j t σ tρ j dt ]. (4.12) j, 38

47 4.2. Swap Rates Dynamics in the SABR-LMM For the right hand side of this equation we obtain E m,n[ T m ( Φ m,n) 2dt ] t E m,n [ T m Now, from the definition of the quadratic variation X t = X 2 t + 2 t X t dx t ( Σ m,n) 2dt ] t. (4.13) we get, by knowing Σ m,n t has a bounded variation since V m,n R + and our time horizon is finite what impies that the expected vaue of t X tdx t vanishes, Hence, and therefore Tm d dt Em,n[( Σ m,n ) 2 ] d t = dt E m,n[( Σ m,n t E m,n[( Σ m,n ) 2 ] Tm s ds = ( V m,n ) 2 t = ( V m,n) 2 E m,n [( Σ m,n t E m,n[( Σ m,n ) 2 ] s ds ) 2 ]. ) 2 ] ( = Σ m,n) 2 ( exp V m,n t ) = ( Σ m,n) 2 ( exp V m,n t ) dt ( Σ m,n V m,n ) 2 ( exp ( (V m,n ) 2 T m ) 1 ), (4.14) which gives us the right hand side of (4.13). Now, we come back to (4.12) and use our ast equation (4.14) together with the definition of the σt in the SABR-LMM (3.8). This eads to ( Σ m,n V m,n ) 2 ( exp ( (V m,n ) 2 T m ) 1 ) n 1 j,=m Tm ρ j W m,n j ()W m,n () g j (t)g (t)e m,n[ k j t k t] dt. (4.15) Further, the definition of the quadratic covariation gives in the same fashion as above E m,n[ k j ] t k t k j k exp ( r j ĥ j t ), (4.16) 39

48 4. Swaps Rates in the SABR-LMM where we negected any drift terms for the k coming from the change of measures from P to P m,n and ĥ j t := 1 t t Overa we get with (4.16) together with (4.15) ( Σ m,n V m,n T m ) 2 ( exp ( (V m,n ) 2) ) 1 h j sh sds. n 1 j,=m Tm ρ j W m,n j ()W m,n ()k j k g j (t)g (t) exp ( r j ĥ j t ) dt. A Tayor approximation from second order of both sides and equating the terms of the same order gives Σ m,n 1 T m n 1 j,=m Tm ρ j W m,n j ()W m,n ()k j k g j t g tdt (4.17) and V m,n n Σ m,n T m j,=m Tm ρ j r j W m,n j ()W m,n ()k j k g j t g t (ĥj t ) 2tdt. (4.18) This gives a good approximation for two of the four SABR parameters in the swap mode in definition 9. Further, those equation wi be from major importance in the caibration part, when it comes to estimating the two correation ρ and r of the SABR- LMM mode. To describe the swap correation R mn in an environment of a SABR-LMM Rebonato approximates in [36] n 1 R m,n Ω j R j, (4.19) j, 4

49 4.2. Swap Rates Dynamics in the SABR-LMM where he defined the matrix (Ω j ) j as Ω j := 2ρ jr j W m,n j ()W m,n ()k j k TM g j ) 2tdt t g t(ĥj ( ) V m,n Σ m,n 2. T m Notice that, n 1 j,=m Ω j 1 due to equation (4.18). Further we define n 1 β m,n := =m ω m,n ()β. (4.2) This choice is reasonabe since is exact for β = and β = 1. In the case β (, 1) [4] impies that the error we produce is very sma and can be negected. In his book [36] Rebonato showed that the approximations in the equations (4.18) to (4.19) works with great precision. He tested his approach by evauating swaptions with different strikes. The accuracy gets better the onger the tenor of the swap is. In the case of swaptions on swaps with expiry 5 years and tenor 15 years or with expiry 1 years and tenor 1 years the approximations are working amost perfect for a strikes. If the expiry becomes shorter the derivations in terms of the voatiity smie grow sighty for higher strikes. 41

50

51 5. Parametrization of the SABR-LMM Mode The acceptance of a market mode stands and fas with its tractabiity, the quaity of its produced prices and the vauation time needed for pricing. One huge factor for the tractabiity is the parametrization of the mode, since it determines the amount of required parameters and, in terms of caibration, the caibration time and the caibration stabiity. A parametrization reduces the describing mode parameters drasticay by using ideaized functions to catch the reevant characteristics. In our case for the SABR-LMM, the underying functions that parametrize certain mode coefficients or structures, ike correation, beonging to a certain forward rate F i is the same for a forwards and is individuaized by using some dependency on the expiry or simiar. For exampe, this techniques aows us to reduce the needed parameters for the correation matrix ρ from (N 1)N 2 to 5, or even 2, numbers. Another exampe is the use of the same underying function g t to describe a the N 1 functions gt i by shifting the time parameter t depending on the expiry of F i. In the foowing chapters we wi first describe how to chose the parameterizations for the functions g i and h i, respectivey. Second, we give parameterizations for the correation structure consisting of ρ, r and R The Voatiity Structure If we want to parametrize the voatiity functions g i and h i in a reasonabe way, we have to take into account the basic properties and genera shapes voatiities empiricay have depending on time. According to [35] the parametrization for the voatiity shoud be time-homogeneous, because empiricay the voatiities of forward rates deveop a in the same way when expiry gets smaer. That means if we are currenty at time point T i and go one time step to T i+1 the k-th forward rate shoud have roughy the 43

52 5. Parametrization of the SABR-LMM Mode same voatiity as the (k 1)-th forward rate one step back at T i. Therefore, we want to parametrize the deterministic voatiity and the vo-vo of the SABR-LMM as g i t := g(t i t) (5.1) and h i t := ζ i h(t i t) (5.2) for some functions g and h and coefficients ζ i wi are cose to 1. Further, a typica voatiity structure over time either has its maximum in a range of 1.5 years to 4 years to expiry or fas monotonousy and concavey with rising times to expiry. In addition, it is observabe that the voatiities have a certain termina eve. To describe a this behavior we define as in [35] the underying functions g and h for g i and h i, respectivey, as g(t) := (a g + b g t) exp( c g t) + d g (5.3) and anaogousy h(t) := (a h + b h t) exp( c h t) + d h, (5.4) where a g + d g, a h + b h > and d g, d h > some rea numbers. Note that, the instantaneous voatiities are given as and im g(t) = t ag + d g (5.5) Further, the termina voatiities are given as im h(t) = t ah + d h. (5.6) and im g(t) = t dg (5.7) im h(t) = t dh. (5.8) 44

53 5.2. The Correation Structure The extrema of g and h are given at t = 1 c g ag b g (5.9) and respectivey. t = 1 c h ah b h, (5.1) Notice, that the g i and h i are square integrabe, as required in definition 8 for the SABR-LMM. Furthermore, cosed form soutions exists for those integras, which wi vauabe when it comes to caibration since we can sove the integras anayticay rather then by cumbersome numerica integration. We wi use the knowedge about instantaneous and termina voatiities to choose initia vaue for the caibration ater on. In addition, we wi incorporate the styized fact about extrema occurring in a range of 1.5 years to 4 years to expiry The Correation Structure The heart of the SABR-LMM is the super correation matrix P. The correations are describing the direct dependence of the forward rates on each other and the voatiities. A part of P describes the cross skew of the mode, that is the correation between forward rates and voatiities. Atogether, the super correation matrix is the main difference and biggest advantage over the standard LMM mode [2], [11]. The matrix does not carry the eve of the mode voatiity, but most of the other informations over the shape of the voatiity surface, ike its skew and how strongy it is pronounced. As in definition 8 of the SABR-LMM we write ( ) ρ R P = R T, r and notice that the Matrix consists of 4(N 1) 2 parameters, from which we ony have to estimate N(N 1) due to symmetry. Since this number is way to big we want give some styized parametrization depending on maxima 9 parameters and minima 6, for the whoe super correation matrix P. We we do this in the same fashion as for the voatiities in chapter

54 5. Parametrization of the SABR-LMM Mode a =.7,b =.2, c =.6, d =.75 a =.17, b =.37, c = 1.12, d =.3 a =.2, b = 2.6, c = 2.2, d =.7 a =.5, b =.7, c = 1.5, d =.1 a =.3, b = 1.5, c = 5, d = Voatiity Time to Expiry Figure 5.1.: Here a range of possibe shapes for the function g and h are shown. The parametrization can be cassified in two groups. In the first group are the ones that produce a rea humped shape and in the second group are those parametrization where the voatiity fas stricty. According to Rebonato in [36] and [35] the humped shaped voatiity functions are characteristic for norma market situations and the faing voatiity functions occur in excited market. The termina voatiity is ceary visibe and agrees with the parameter d. The parameters are from [36] and the pot was done in Matab. 46

55 5.2. The Correation Structure We wi give for each sub matrix of P a parametrization and gue them together in the end. For the guing we wi need an optimization agorithm that gives us the nearest correation matrix, since just sticking together three parameterizations for ρ, r and R do not give a we-defined correation matrix because the eigenvaues of the resuting matrix can be negative. The most important part of P is the sub matrix ρ, which directy governs the interpay between the forward rates. In this way ρ has the most impact on the pricing quaity of our mode. Therefore, we put specia emphasis in modeing the forward/forward correation. In the ater, we discuss correation matrices in genera and refer to a correation matrix using the symbo ρ. Obviousy, a the discussions wi hod for the voatiity/voatiity correation r as we. To give proper parameterizations we first have to give some criteria which properties a correation matrix must have and which it shoud have. According to Lutz [29] and [36] for the correation matrix (ρ ij ) ij has to hod (A1) (A2) (A3) ρ has to be rea and symmetric, ρ i,i = 1 for a i {1,..., N}, ρ has to be positve semi-definite. In addition, we demand two further properties, which describe empirica observations and whose vaidity is market consents. (B1) j ρ ij shoud fa strong monotonousy for j > i, (B2) i ρ i+p,i grows for fixed p {1,..., N 2}. The first property assures that two forward rates whose expiry is farther apart a ess correated then two rates who expire cosey together. The second property assures that, if we have two pairs of assets and in each pair the distance between the expiries is the same, then the pair of assets which overa expiry is further in the future is stronger correated then the other one. For exampe, ets consider two pairs of forwards. The first pair consists of forward rates expiring in 1 and 3 years and the second pair consists of forwards expiring in 2 and 22 years. Intuitivey, it is cear that the ast two forwards shoud be more strongy correated than the first two. 47

56 5. Parametrization of the SABR-LMM Mode The simpest matrix that fufis the properties (A1) (A3) is given through ρ ij = exp( β i j ), (5.11) where β > is the decorreation coefficient. It is obvious that this matrix does not obey (B1) and (B2), since ony the index distance of the tenor points (T i ) i matters. The simpe structure and the dependence on ony one parameter β is nevertheess attractive. This is especiay usefu in situations where we have to set up a matrix under high uncertainty or we beieve the correations behave uniformy. We want to further deveop the approach in (5.11) in a trivia way. For this, we first notice that, if (ρ ij ) i,j is a correation matrix then ( ρ ij ) i,j defined through Figure 5.2.: Some exampes for the correation matrix in (5.13). The simpe structure is obvious. On the eft the parameters are β (1) =.3 and ρ (1) =.8 on the right the parameters are β (2) =.1 and ρ (2) =.. The pot was done in Matab. ρ ij := ρ + (1 ρ )ρ ij (5.12) is a correation matrix as we, where ρ [, 1). The coefficient ρ describes the termina correation and it hods Now we enhance (5.11) to ρ ij j ρ. ρ ij = ρ + (1 ρ ) exp( β i j ), (5.13) 48

57 5.2. The Correation Structure where ρ [, 1). Indeed, we wi use the above parametrization to mode the correation between the voatiity processes, hence we wi use it for the submatrix r. However the parametrization in (5.13) is way to simpe and infexibe to mode the correation of forward rates. The reason for this is the uniform correation coefficient β. Empiricay, forward rates with shorter expiry are way ess correated with other forward rates then forwards with arger expiry. This means we need a way to mode the correations on the short end of the tenor structure more independenty from the ong end. The first step in this direction is the Doust parametrization [31]. This parametrization gives a genera framework for matrices that fufi (A1) (A3) and therefore for correation matrices. A matrix (ρ ij ) ij obeys the Doust parametrization if there exists a set {a k a k [ 1, 1], k {1,..., N 2}} such that ρ i,i = 1, for a i {1,..., N 2} ρ 1,j = and therefore ρ can be written as ρ = j 1 k=1 a k = ρ j,1 ρ i,j = ρ j 1 1,j = a k (5.14) ρ i,1 k=i 1 a 1 a 1 a 2 a 1... a N 2 a 1 1 a 2 a 2... a N a 1... a N 1 1 This impies ρ has the Choesky decomposition ρ = LL T, 49

58 5. Parametrization of the SABR-LMM Mode where 1 a 1 1 a L := a 1 a 2 a 2 1 a a a 1... a N 1 a 2 a 3... a N 2 1 a a 2 N 2 which yieds for ρ the correation matrix property. However, this representation cannot grantee that either (B1) or (B2) hods. Another probem is the dependence on N 2 coefficients, which is simpy too much since in practice N ies in the range of 2 to 4. Schoenmakers and Coffey further deveoped in [25] the approach from Doust. They gave a parametrization, which gives a correation matrix that fufis (B1) and (B2) as we. In its most common formuation it depends on N parameters, but the number can be efficienty reduced to two parameters. A correation matrix (ρ ij ) ij foows the Schoenmakers & Coffey parametrization if and ony if there exists a growing sequence 1 = b 1 < b 2 < < b N (5.15) such that and with b 1 b 2 < b 2 b 3 < < b N 2 b N 1 (5.16) ρ ij = b j b i, for a 1 j i N 1 (5.17) ρ ij = ρ ji. Here the definition of the entries in ρ via fractions (5.17) corresponds to the definition of the Doust parametrization (5.14). The two additiona requirements (5.15) and (5.16) yied the desired properties (B1) and (B2). Further, it was shown in [25] that a matrix ρ obeys the Schoenmakers & Coffey parametrization, if there exists a sequence 5

59 5.2. The Correation Structure (b i ) i {1,...,N 1} such that (5.17) hods and for a rea sequence 1,..., N. b i = exp ( N 1 j=1 min(j, i) j ), (5.18) With the hep of equation (5.18) it is possibe to generate correation matrices with properties (5.16) and (5.17) without non inear bounded parameters b i. This is an important fact for the impementation since it fastens up the computation time. Let s choose in (5.18) the foowing parameters ( N i 2 ) i := α, for 1 i < N 1 and N 1 := γ N 4 N 2 α (N 3) (5.19) 6 where α := 6η (N 2)(N 3), This gives the optima two parametric correation matrix (ρ ij ) ij from Schoenmakers & Coeffey [25] with where [ j i ( )] ρ ij = exp γ + ηh(i, j) for a i, j {1,..., N 1}, (5.2) N 2 ( i h(i, j) := 2 + j 2 + ij 3(N 1)i 3(N 1)j + 3i + 3j + 2(N 1) 2 N 5 ) (N 3)(N 4) and η, γ, γ η. (5.21) Here the parameter exp( γ) is the termina correation just ike in (5.12). From here on we wi refer to the above parametrization as the (2SC) parametrization. Schoenmakers & Coffey caim in [25] that the representation in (5.2) is fexibe enough to describe a wide range of different correation matrices. However, we found in the empirica work in chapter 8 that the parametrization works quit we, but may be too infexibe. This is due to the high dependency between the shape of the short end of the matrix the area for assets with shorter maturity 51

60 5. Parametrization of the SABR-LMM Mode Figure 5.3.: Here different (2SC) parametrizations are shown. From the eft to the right the coefficients are given in tabe 5.1. The imitation of the parametrization is ceary visibe, since the back of the matrix heaviy depends on the front. The pot was done in Matab. and the ong end of the matrix the area for assets with onger maturity. The strong dependency of those two areas comes from property (5.16), which says that the ast = b i b N row of the matrix ρ i,n is amost inverse proportiona to the first row of the matrix ρ i,1 = b 1 i. We beieve that the two parameters of the (2SC) representation are simpy not enough to give the fexibiity which we desire. Especiay not if, we try to describe the correation of assets whose prices highy depend on the correations between the underying assets, ike Constant Maturity Swaps Spreads (CMS spreads). Nevertheess, the (2SC) parametrization is very good since it works highy efficient with ony two parameters, which aows for a very fast caibration and, particuary, a very stabe one. Further, it reduces the risk of over-fitting, because it does not react to every minor disturbance in the data and rather keeps a genera textbook shape. 52

61 5.2. The Correation Structure Tabe 5.1.: Parameters for Figure 5.3 Pot No γ η To get a more fexibe parametrization we chose to incorporate more parameters. A good parametrization which can reproduce a wide range of styized shapes is given by Lutz in [29]. His correation matrix depends on 5 parameters and the representation is derived by its Choesky decomposition. For this, he first gave a genera framework for the decomposition: Theorem 3 (Lutz Choesky Decomposition for Correation Matrices). Let ρ be a N N correation matrix with fu rank. Further define I 2 := {2,..., N} and I 1 := {1,..., N}. Then there exists functions f : I 1 [ 1, 1] and g : I 2 I 2 R, such that ρ ij = f(i)f(j) + (1 f(i)2 )(1 f(j) 2 ) a i a j min(i,j) k=2 and the Choesky decomposition L of ρ can be written as with f(i), for j = 1 L ij = h(i, j) 1 f(i) 2 a i, for 1 < j i,, otherwise a 1 = 1, a i = i h(i, k) 2, i I. k=2 h(i, j)h(j, k) for a i, j I 1 Proof. The proof is simpe, but doesn t give any new insights. We refer to [29]. Remark. The correation matrix of Schoenmakers & Coffey can be retained by setting f(i) = 1 b i and h(i, j) = b 2 j b2 j 1, i, j I 2, 53

62 5. Parametrization of the SABR-LMM Mode where we choose b i according to (5.18) and with i given in (5.19). If we choose in Theorem 3 f(i) = exp( βi α ), for α, β >, i I 1 and [ ( i 1 h(i, j) = exp N 2 γ + N 1 i )( j )] N 2 δ i 1 and combine the resuting matrix with the termina correation extension as in (5.12) we get Lutz 5 parametric form (5L). The parametrization is the foowing with [ ] ρ ij = ρ + (1 ρ ) exp( β(i α + j α )) + H(i, j, α, β), ρ [, 1) (5.22) H(i, j, α, β) := θ ij θii θ jj (1 exp[ 2βi α ])(1 exp[ 2βj α ]) for α, β > and i, j I 1 and with 1, if min(i, j) = 1 θ ij := min(i, j) 1, if min(i, j) > 1, ξ i ξ j = 1,, if min(i, j) > 1, ξ i ξ j = 1 (ξ i ξ j ) min(i,j) /(ξ i ξ j ) where ( ξ i := exp 1 ( i 1 i N 2 γ + N i 1 )) N 2 δ, γ, δ R. Notice, the parametrization for ρ given above in (5.22) need not necessariy fufi the desired requirements (B1) and (B2) since Lutz Theorem ony affects the genera correation matrix properties (A1) (A3). This coud be a possibe probem, when it comes to caibration, because we ose some contro over the genera shape of the matrix. In [29] Lutz argues that, given good enough input data for the caibration procedure the resuting shape shoud fufi the properties (B1) and (B2). His opinion comes from empirica studies of correation matrices for forward rates from the years 54

63 5.2. The Correation Structure Figure 5.4.: Here are possibe shapes of the (5L) parametrization shown. We kept ρ fixed at.2. Observe the high independency of the back end of the matrix from the front end and fexibiity comparing with the (2SC) parametrization in figure 5.3. The parameters for this pot are from [29] and can be found in tabe 5.2. The pot was done in Matab. 24 to 28. There, without given any parametrization for the matrix, the average resuting shape fufied (B1) and (B2). In addition, in our empirica work we gain the same opinion, because a our estimated correation matrices for different dates and parametrized through (5L) obeyed a desired properties. Further in this work we wi use the parameterizations (2SC) (5.2) and (5L) (5.22) to mode the correation ρ of the forward rates in the SABR-LMM. We wi compare both approaches visuay and by the accuracy of produced prices. Further, we mode the correation of the voatiities with the simpe parametrization given in (5.13). The ast structure we have to define is the one of the cross skew R, which describes the correation between forward rates and voatiities. Here, we foow Rebonato [36] 55

64 5. Parametrization of the SABR-LMM Mode Tabe 5.2.: Parameters for Figure 5.4 Pot No α β γ η and choose R ij = sgn(r ii ) R ii R jj exp ( λ 1 (T i ) + + λ 2 ( T i ) +), λ 1, λ 2 >. (5.23) We wi see in the caibration part that, the R ii are the individua forward/voatiity correations for each forward rate F i. Overa, we have given a piecewise parametrization of whoe super correation matrix ( ρ P = R T ) R. r 5.3. Approximation of P through a proper Correation Matrix In chapter 5.2 we have shown how to parametrize the pieces of the super correation matrix P. But, by simpy sticking those pieces together it is not guaranteed that we reay obtain a correation matrix by definition, since the eigenvaues can be negative. This is extremey probematic, because we need to be abe to do a Choesky decomposition of P for a Monte Caro simuation. To fix this probem, we approximate the matrix P through the nearest correation matrix P. Hence, we sove the probem P = arg min A C ω ij (A ij P ij ) 2, (5.24) ij where C is the set of a correation matrices and the ω ij are weights. In most cases we won t retain P as a soution, but the weights enabe us to put specia emphasis on some areas of P to maintain the most crucia characteristics. Obviousy, in the SABR-LMM the matrix for the forward/forward correation ρ is of 56

65 5.3. Approximation of P through a proper Correation Matrix major importance because it describes the direct interpay of the rates. Another important part of P is the diagona of R, since it consists of the individua forward/voatiity correations, which have an huge impact on the individua mode smie of a the rate F i. Rebonato stressed in [36] the importance of the R ii and our empirica studies confirmed him. The impact of the changes in the skew were visuaized in figure 3.1. So we decided to use the ω ij to overweight ρ by 8 and (R ii ) i by 8. The minimization agorithm is behind the scope of this work. More on this topic can be found in the Cresnik s thesis [28], where he used the majorization technique, described in [34]. Further, the agorithm in [34] can sove the probem (5.24) extended to P = arg min A C k ω ij (A ij P ij ) 2, (5.25) where now C k is the set of a correation matrices of rang k 2(N 1). ij This enabes us to reduce the simuation time immensey since there is a one to one reation between rank and number processes we have to simuate for pricing. In this work we wi use k = 1. 57

66

67 6. Caibration of the SABR-LMM to Swaptions Now that the theoretica foundation for the SABR-LMM is given, we want to show how the mode can be caibrated to the market. We wi give a method to caibrate impicity to current market prices and which is not based on historica prices for some past period. Historic prices have a certain backwards character and do not carry more information than current market prices. These aready incorporate a reevant expectation of future price deveopments. Therefore, historic prices cannot hep to estimate the future any better. The basis for the caibration wi be the prices in Back voatiities of caps and swaptions. The goa is to caibrate the induced mode prices as cose as possibe to the observed market prices. To cacuate the induced mode prices fast and with satisfactory accuracy, we wi use the formua for the impied Back voatiity for the simpe SABR mode (3.4) together with the resuts in chapter 4. There we demonstrated how to transfer the swap dynamics from the SABR-LMM to the SABR mode, where we can use the impied voatiity formua. This chapter is spit into three parts. In the first part we expain what kind of data we need and how it can be stripped. The second describes the caibration of the voatiity structure, that is the estimation of the parametrization for the g i and h i, which is done soey based on capets. This part of the caibration wi be exacty the same when we caibrate the mode to CMS spread options ater in chapter 7.3. The third part of this chapter concerns the caibration of the correation structure to swaption prices. In the foowing, by using four parameters in our notation we wi ony sighty rey on the parameterizations of the g i and h i. The parametrization of the super correation matrix does not matter either, because we wi give a formuas ony depending on the entries of P. So the described procedures can easiy used with other parametrizations then the ones introduced in chapter 5. 59

68 6. Caibration of the SABR-LMM to Swaptions 6.1. Preparation for the Caibration For the caibration we need certain basic data. Here, I want expain in short how to obtain it. First, we need the initia vaues F i for the forward processes. Those can be obtained by a forward bootstrapping as expained in chapter Further, we have to get the bond prices. Those are necessary to cacuate the swap numéraires A m,n when it comes to caibration on swaps. We cacuate the bond prices by using the forward rates as i 1 B(, T i ) = (1 + δ j F j ) 1, j= where we obtained the formua by simpe iteration of the definition for forward rates (2.1) The Caibration of the Voatiity Structure One of the big objectives of the caibration of the SABR-LMM is to keep the dynamics of the forward rates as cose to the simpe SABR mode as possibe. If this works the SABR-LMM s acceptance wi be highy strengthened since the SABR mode has aready been approved in practice. In particuar, the foowing caibration procedure wi assure that we can use the formua for impied voatiity (3.4) of the SABR mode to approximate prices of European options on forward rates and swaps even in the SABR-LMM as we. The first step in the caibration of the voatiity functions is to bootstrap the impied capet voatiities from quoted cap voatiities as described in the preiminary chapter Then we caibrate for each forward rate F i a simpe SABR mode by using the formua from (3.5) (σ i, ρ i, ν i ) = arg min α,ρ,ν [ σm (F i, Ki j ) σ I(F i, Ki j, β j, α, ν, ρ, T i ) ] 2, (6.1) j where the σ M (F i, Ki j ) are the stripped capet voatiities, the (Ki j ) j the avaiabe strikes for the i-th capet and the β j are the betas of forward rates in the simpe SABR modes. In our case hods β j.5, but other choices are possibe, as expained in [36]. We 6

69 6.2. The Caibration of the Voatiity Structure woud ike to emphasize that, the set (σ i, ρ i, ν i ) i (6.2) wi pay a centra roe in a caibration procedures from here on. The functions g i can be interpreted as the deterministic voatiity of F i. Since we chose h i square integrabe the stochastic voatiity k i is on average k as can be seen in (3.8). Therefore, we foow [36] and caibrate the g i ike in the simpe LMM with deterministic voatiity [3]. We define 1 ĝ i := T i Ti ( g i t ) 2dt as the average squared voatiity and then use the minimization probem from (3.6) where C is defined as min (a g,b g,c g,d g ) C N ( ) 2, ω i σi ĝ i (6.3) i=1 C := { (a, b, c, d) R 4 a + b > ɛ 1, d > ɛ 2 }, (6.4) with ɛ 1 and ɛ 2 representing some beieves about instantaneous and termina voatiity, as expained in (5.5) and (5.7). The weights ω i are defined as ω i := (.2 +.5T i ) exp(.2t i ) +.2 and overweight the σ i for the range 1.5 years to 5 years. If the parametrization is abe to fit today s voatiity structure perfecty the weights shoudn t have any impact. With (6.3) we are abe to caibrate the deterministic voatiity averaged over a forward rates. The derivation for the vaidity of the formua is exacty as in the LMM with deterministic voatiity, if we approximate kt i ki 1. The ast approximation is vaid, because to achieve that the SABR voatiity σ i agrees with the SABR-LMM voatiity σt i on average for each forward rate, we cacuate the initia vaue ki of the stochastic voatiity as k i = σ i ĝ i. 61

70 6. Caibration of the SABR-LMM to Swaptions So the k i shoud be near to one if the fits of the gi are adequate. To estimate the parameters for h we foow [36] and define ĥ i t := 1 t t and set up the optimization probem min (a h,b h,c h,d h ) C [ (a h + b h (T i s)) exp( c h (T i s)) + d h] 2 ds N i=1 ν i ki [ Ti ( ) 2 g i 2 (ĥi ) ] 2tdt 1/2. σ i T t t (6.5) i In addition, we go a step further then in [36] and cacuate the correction factors ζ i, in order to hit the vo-vo for each forward rate exacty, as / ( k ζ i = i [ Ti ( ) ν i 2 g i 2 (ĥi ) ] 2tdt 1/2 ) σ i T t t. i Again, the ζ i shoud be near to one if the fit is good. Notice, in (6.5) the caibration of the vo-vo functions h i depends on the g i as we. So the vo-vo impied by the h i is higher weighted if the genera voatiity eve, given by the g i, is higher and vice versa The Caibration of the Correation to Swaps Simiar to the case of the caibration of the voatiity structure in chapter 6.2, we use for the caibration some coefficient from the simpe SABR mode as target vaues. The correation is caibrated on a set of cotermina swaps {S i,n } i. Those swaps have different expiry dates T i and tenors T N T i, but a mature in T N. So a set of cotermina swaps can be seen as a set of swaps in which a swaps end at the same date. We choose cotermina swaps, because, depending on the picked expiry dates, we can take amost every entry of correation matrices ρ, r and R into account during the caibration. To reaize this we remember the sum structure of swaps from (2.5) S i,n N 1 t = ω mi,n F i. i=1 62

71 6.3. The Caibration of the Correation to Swaps Hence, in a swap rate the forward rates with index in {i,..., N 1} and, especiay, a the correations d W, W j t = ρ j, d Z, Z j t = r j and d W, Z j t = R j. for, j {i,..., N 1} are invoved. If we consider a possibe expiries for cotermina swap rates we impy for a entries of the super correation matrix P a condition. However in practice it is enough to choose ony the quoted expiry dates Expiry 2 Expiry Expiry Expiry Figure 6.1.: This graphic shows the entries of ρ which are invoved in cotermina swap dynamics. Back means invoved and white means not invoved. On the right we see the case if we use every possibe cotermina swap and on the eft we see the case in which we ony used the cotermina swaps whose expires are quoted in the market. Here the tenor structure is (T,..., T 4 ) with δ i.5. The pot was done in Matab. The first step in the caibration is to caibrate the SABR mode for swaps from definition (9) in chapter 4.1 to swaption prices of cotermina swaps. Those prices quoted in impied voatiity. Here we use an extended version of the minimization probem in (3.5) (σ S i, ρ S i, ν S i ) = arg min α,ρ,ν + ϕ(σ, ρ, ν) ( ) j [ 1 (ɛ3, ) σm (S i,n, Kj i) σ I(S i,n, Kj i, βs i, α, ν, ρ, T i) ] 2, (6.6) where ɛ 3 > is the required accuracy of the genera optimization probem and ϕ a penaty function for the possibe soutions. The σ M (S i,n, Kj i ) are the observed swaption voatiities in the market, the strikes (Kj i) j are the avaiabe strikes for the i-th cotermina swap and the swap betas β S i are cacuated as in (4.2). The penaty 63

72 6. Caibration of the SABR-LMM to Swaptions function contros the freedom given through ɛ 3 >. In practice ɛ 3 is in dimension of 1 6 to 1 5, so we disturb the optimization probem ony sighty. Basing on the estimated parameters in (6.6) we wi setup an optimization probems for each submatrix of P. In chapter 4.2 we approximated the dynamics of swaps in the SABR-LMM assuming that the swaps foow a SABR mode. Now, we wi use the resuts therein. Approximation (4.17) eads to ρ = arg min ρ as a way to estimate ρ. ( i ( [ 1 σi S T mi N 1 ρ k W mi,n k k,=m i k k k Tmi ()W m i,n () g k t g tdt] 1/2 ) 2 ) 1/2 (6.7) Next we use (4.18) to set up the foowing minimization probem, where we assume that ρ was aready estimated: r = arg min r ( i [ ν S i 1 σ S i T m i [ 2 N 1 ρ k r k W mi,n k k,=m i Tmi gt k gt (ĥk t ()W m i,n ()k k k ) 2tdt ] 1/2 ] 2 ) 1/2, (6.8) where ĥ k t := 1 t t h k sh sds. Our ast optimization probem to estimate the cross skew matrix R is obtained by using (4.19) and the idea of we wanting to regain the SABR dynamics as cose as possibe, is the foowing where R = arg min R Ω i k := 2ρ kr k W m i,n j i N 1 (ρ S i ()W m i,n ()k j k k,=m i Ω i k R k Tmi g j ) 2tdt t g t(ĥj ( ν S i σi ST ) 2. m i ) 2, (6.9) 64

73 6.3. The Caibration of the Correation to Swaps and we force R ii = ρ i, (6.1) where ρ i is the skew from (6.2) of the forward rate F i. The step (6.1) forces the correation of each forward rate and its voatiity exacty to the same eve as in the simpe SABR mode. By doing this, we once more try to keep the dynamics of the F i in the SABR-LMM as cosey as possibe to the simpe SABR mode. 65

74

75 7. CMS Spread Options and Swaps A constant maturity swap (CMS) is a swap consisting of foating egs given by swap rates of constant ength and some fixed eg. In those contracts the ong party is obigated to pay a fixed amount F at each settement date T i and to receive a swap rate S fixed at time over some predefined period. From the ong party s point of view, and thus simiar to a simpe swap, the CMS over the period [T i, ] has a vaue of j =i τ B(, T )E [ ] S,+c T F, where c is the constant ength of a swaps. A CMS can be seen as a norma swap where swap rates with constant ength are exchanged instead of forwards. Constant maturity swaps, and options on those, enabe market participants to hedge for certain periods against risks due to changes in the genera shape of the underying forward-curve for the swaps. For exampe, CMS can be used to hedge against changes of the 1 year swap rates by buying a ca, put or stradde on a CMS. An extension of CMS is a CMS spread, where one swap rate is exchanged for another one. The exchanged swap rates have the same expiry and underying forward curve but different engths. If the CMS spread consists of two swap rates of ength c 1 and c 2, both expiring in T i, the contract has the foowing payoff profie for the ong-party S i,c 1,c 2 T i := S i,i+c1 T i S i,i+c2 T i F, (7.1) where F is some fixed rate and has to be paid by the ong-party. Therefore, today s abstract, undiscounted vaue under the CMS measure P CMSs, under which the spread S i,c 1,c 2 is a martingae, is E CMSs[ ] S i,c 1,c 2 T i F = E CMSs[ S i,i+c1 T i ] S i,i+c2 T i F. 67

76 7. CMS Spread Options and Swaps CMS spread prices depend more strongy on the correation between the underying forward rates than ordinary swaps [29]. If we are abe to evauate spreads in the SABR-LMM context, we are capabe of caibrating the correation structure which is induced by spread market prices. We hope that this caibration is quaitativey better than the caibration on swaps due to the higher sensitivity of CMS spread prices to correation forwards. First we observe: The CMS spread dynamics of S i,c 1,c 2 can be written in direct dependency on forward rates. By assuming without imitation c 1 > c 2 we write S i,c 1,c 2 t = S i,i+c1 t = i+c 1 1 =i where ω i,i+c 1 (t) := τ kp (t,t +1 ) A i,i+c1 t we define and S i,i+c2 t F i+c ω i,i+c (t)ft =i ω i,i+c 2 (t)f t F, are the we-known stochastic weights from (2.6). Now ω i,i+c 2 := for a > i + c 2 1 v i,c1,c2 (t) := ω i,i+c 1 (t) ω i,i+c 2 (t) for a {i,..., i + c1 1}. Hence, for the CMS spread we can write i+c S i,c 1 1 1,c 2 t = v i,c 1,c 2 (t)ft F. (7.2) =i Therefore, a CMS spread is a portfoio of severa forward rates and each rate is weighted with v i,c1,c2. This notation further reveas the possibiity of gaining insights into the correation of forwards through CMS. In this section we want to show how the correation structure can be caibrated on prices of European options on CMS spreads. First, we anayze the dynamics of CMS and give approximations for those in a SABR-ike environment under the abstract spread measure. Second, we show how to transform CMS prices evauated under the spread measure to prices under a forward measure. Finay, we expain the caibration in detai. 68

77 7.1. Markovian Projections of CMS spreads 7.1. Markovian Projections of CMS spreads If we want to examine the CMS dynamics (7.2), we first have to simpify it. The reason for that is that the process in (7.2) is the sum of correated SABR processes, which are weighted stochasticay. Therefore, the resuting process may again be a SABR process, as foreshadowed in [4], but it is impossibe to cacuate the exact distribution or dynamics. In the foowing we wi project a portfoio dynamics driven by a number of random processes (7.2) on a dispaced SABR-proces. Here an important too is the technique of Markovian Projection, which was deveoped by Gyöngy in [19]. This technique was extended by Brunick and Shreve in [15]. We use the atter to give the foowing proposition Proposition 5. (Markovian Projection) Let (Ω, F, (F t ) t, P) be a fitrated probabiity space fufiing the usua conditions and on which (W t ) t is a d-dimensiona Wiener-process. Further, et b and (σ t ) t adapted processes with vaues in R d and R d d, respectivey. Let X be a R d -vaued process given through the SDE dx t = b t dt + σ t dw t, t, X = x. (7.3) Then there exists a R d -vaued measurabe function b, a R d d -vaued measurabe function σ and a Lebesgue-nu-set N, so that ] b(t, Xt ) = E [b t Xt [ σ(t, X t ) σ tr (t, X t ) = E σ t σt tr (7.4) ] X t P-a.s. and a t N (7.5) hods. Here tr is the trace operator. Furthermore, there exits a fitrated probabiity space ( Ω, F, ( F t ) t, P) that supports a R d -vaued adapted process X and a d- dimensiona Wiener-process Ŵ, satisfying d X t = b(t, X t )d t + σ(t, X t )dŵt, t, X = x (7.6) and such that for a t the distribution of X t under P agrees with the distribution of X t under P. Proof. A proof can be found in [15]. 69

78 7. CMS Spread Options and Swaps The proposition enabes us to project a process of the form (7.3) on a process of the form (7.6) with coefficients given through (7.4) and (7.5). This new process is then in every case weak Markovian. Further, the distribution of X t and X t is the same for each t. However, it is not the case that the distribution of individua paths coincide. So the distribution of (X t ) t I and ( X t ) t I for some index set I may not agree. For this reason, the Markovian Projection is not suited to simpify a process first and then evauating a path dependent options afterwards. In the case of European options we don t face those probems since they ony depend on the distribution of the underying process at some fixed point T. So, we don t run into any probems if we simpify a CMS spread to evauate European options. The coefficients in (7.4) and (7.5) obtained by using the proposition on CMS spreads can be easier understood and approximated then the origina ones. Our goa is not to cacuate the resuting process in (7.6) accuratey. For that the process under which we have to condition wi be too compicated. We rather want to approximate condition expectations in the frame of another mode, the dispaced SABR mode. We wi introduce this mode in the foowing, but first et s take a ook at a simpe exampe for a Markovian Projection to get a better feeing for it. In addition, the exampe shows that not every projected process is strong Markovian. Exampe (From Brunick and Shreve [15]). Let in proposition 5 the parameters be d = 1, b = and the initia process dx t = σ t dw t with σ t := 1 (1, ) 1 {W1 >}, and X =. Ceary it hods X t = 1 (1, ) (t)1 {W1 >}[ Wt W 1 ]. Now we cacuate the Markovian Projection as [ σ t 2 (t, X t ) = E σt 2 X t ] = 1, fas X t =, fas X t =. 7

79 7.1. Markovian Projections of CMS spreads It foows X (1) t and X (2) t = 1 (1, ) (t) [ W t W 1 ] agree with X on {X 1 < } and {X 1 > }, respectivey. However, X1 t and X 2 t don t have the right distribution to be a projection of X on both subsets. To obtain a proper projection consider Y Ber(, 1 2 ) independent of (W t) t and F measurabe. So F is non-trivia. Define X t := Y (1) (2) X t + (1 Y ) X t. Taking into account W 1 N (, 1) and therefore P(W 1 < ) = 2 1 foows X d t = Xt for a t. However, X is not strong Markovian. This can be seen by defining the stopping time τ 2 := inf{t 2 X t > } This stopping time is not independent from F because of Y and therefore ( X τ2 +s) s conditioned on τ 2 < is not independent from F. Now that we we have a better understanding for the Markovian Projection we use the technique on CMS spreads. The foowing theorem gives a approximation of the spread dynamics (7.2) in a context of a dispaced SABR mode in dependence of forward rates F i which foow a SABR-LMM mode. The dispaced SABR mode coefficients depend on the correation matrix P and therefore the theorem gives the ink between forward correations and spread dynamics. 71

80 7. CMS Spread Options and Swaps Theorem 4 (The Dispaced SABR Mode for CMS spreads). For a strike price F = the dynamics of the CMS spread i+c S i,c 1 1 1,c 2 t = v i,c 1,c 2 (t)ft. (7.7) can be approximated under the spread measure P CMSs via where G is defined via =i i+c ds i,c 1,c 2 t = u t G(S i,c 1,c 2 t )dŵt, S i,c 1 1 1,c 2 = v i,c 1,c 2 F (7.8) du t = A p,v (ρ, r)d Z t, u = 1 (7.9) d S i,c 1,c 2, u t = X (ρ, r, R)dt, (7.1) G(x) = (x S i,c 1,c 2 + pi,c 1,c 2 q i,c 1,c 2 )q i,c 1,c 2. (7.11) The coefficient a given in the proof at (7.4), (7.41), (7.36), (7.37), (7.42), (7.43), (7.33) and (7.34). The proof is spit into severa parts. The first step is to project the dynamics (7.7) with the hep of the Markovian Projection on a simper process. The coefficients of the projection are then approximated such that we are abe to express the spread dynamics in a dispaced SABR mode. This mode is defined as Definition 1. (Dispaced SABR mode) The dispaced SABR mode is given through =i ds t = u t G(S t )dw t, für ae t, S = x, du t = ηu t dz t, für ae t, u = 1, (7.12) d W, Z t = γdt, where η > and γ [ 1, 1]. The processes W and Z are one-dimensiona Wiener processes. Further, G : R R is a Bore measurabe function. In the proof the form of G is given through the approximation of the projections parameters. Then the dynamics of u and therefore the parameter η and γ, respectivey, are cacuated. We wi see that a parameters wi depend on the correation structure 72

81 7.1. Markovian Projections of CMS spreads of the SABR-LMM mode. The theorem 4 was originay proven by Kienitz and Wittke in [22] for a simper SABR-LMM mode with constant parameters with methods from [41]. The version stated here is a rea extension, since we approximate the spread dynamics in a SABR- LMM environment with time-dependent parameters gt i and hi t. Further we dea with the more compicated voatiity processes σt i = gi t ki t. Remark. In the proof we wi assume that the g i and h i are aready caibrated to capets as in chapter 6. So the parametrization of the g i fufi (6.3) and the one of the h i (6.5). This doesn t imitate our approach, since we ony want to caibrate the suppercorreation matrix P based on CMS spreads. As in the case for the caibration to swaptions, the voatiity structure is caibrated to capets. Proof of Theorem 4. First, we define and u t := σ t σ for a {1,..., N 1} ϕ(f t ) := σ (F t ) β for a {1,..., N 1}. Therefore u = 1 and df t = µ t dt + u t ϕ(f t )dw t under the measure P CMSs - the measure under which the CMS spread is a oca martingae - for a {1,..., N 1}, where µ t are drifts coming from the change of measures. Further we set p k := ϕ(f) = σ k (F) β q k := ϕ(f) = σβ(f k ) β 1. Now, we use the dynamics in (7.7) and freeze the weights v i,c 1,c 2 at their initia vaues 73

82 7. CMS Spread Options and Swaps in t =. Under P CMSs we obtain ds i,c1,c2 t i+c1 1 = d v i,c 1,c 2 (t)ft =1 i+c1 1 =1 i+c1 1 =1 v i,c 1,c 2 ()dft v i,c 1,c 2 ()u tϕ(f t )dw t, where in the ast approximation the drifts µ t are negected. Now define and σ 2 t := + 2 i+c1 1 dŵt := σt 1 v i,c 1,c 2 ()u tϕ(ft )dwt (7.13) i+c1 1 =i i+c1 1 k<=i+1 =i ( v i,c 1,c 2 () ) 2( u t ) 2ϕ(F t ) 2 v i,c 1,c 2 k ()v i,c 1,c 2 ()u k t u tϕ(f k t )ϕ(f t ). Hence, the Levy characterisation [2] gives us: Ŵ is a Wiener-Process, since Ŵ =, Ŵ t = t and Ŵ is continuous. The spread dynamics can now be written as ds i,c 1,c 2 t = σ t dŵt. (7.14) Our goa is to describe the function G in (7.12) using (7.2). In order to achieve that we define u 2 t := 1 [ i+c 1 1 p k=i i+c 1 1 k<=i+1 ( v k () ) 2 p 2 k ( u k t ) 2 (7.15) p k p u k t u tv i,c 1,c 2 k ()v i,c 1,c 2 ()ρ k ], 74

83 7.1. Markovian Projections of CMS spreads with p := [ i+c 1 1 k=i ( v k () ) 2 p 2 k + 2 i+c 1 1 k<=i+1 p k p v i,c 1,c 2 k ()v i,c 1,c 2 ()ρ k ] 1 2. In particuar, this impies u = 1 as needed in the dispaced SABR mode. If we appy proposition 5 to (7.14), we get σ(t, x) = E CMSs[ σ 2 t ] S i,c 1,c 2 t = x (7.16) and on the other hand, if we assume that G(S i,c 1,c 2 t ) and u 2 t are under σ(s i,c 1,c 2 t ) approximatey stochastic independent, we get for the projection of our dispaced SABR dynamics σ(t, x) = E CMSs[ u 2 t For (7.16) and (7.17) being equivaent it has to hod E CMSs[ σ G 2 t 2 (x) = E [u CMSs 2 t which impies the exact structure of G. ] S i,c 1,c 2 t = x G 2 (x). (7.17) S i,c 1,c 2 t ] = x S i,c 1,c 2 t = x ], (7.18) Our next goa wi be to approximate the conditiona expectations in (7.18) as we as possibe. For that we wi rewrite σ 2 t and u 2 t. We define f k, := ϕ(f k t )ϕ(f t )u k t u t and Hence σ 2 t = i+c 1 1 =i f, (t) ( v i,c 1,c 2 g k, (t) := p kp u k t u t p 2. () ) i+c k<=i+1 f k, v i,c 1,c 2 k ()v i,c 1,c 2 ()ρ k (7.19) 75

84 7. CMS Spread Options and Swaps and u 2 t = i+c 1 1 =i g k,k (t) ( v i,c 1,c 2 () ) i+c k<=i+1 g k, v i,c 1,c 2 () k v i,c 1,c 2 ()ρ k. (7.2) If we interpret f k, as a function depending on Ft k, F t, uk t and u t we get f k, (t) = f(f k t, F t, u k t, u t), with f(x, y, z, w) = ϕ(x)ϕ(y)zw. Therefore a Tayor series extension around the point (F, F k, 1, 1) in direction of (F t, F k t, u t, uk t ) gives f k, (t) p k p ( 1 + q k p (F k t F k ) + q p (F t F ) + (u k t 1) + (u t 1) ) (7.21) and it foows in the same manners that g k, p kp p 2 ( 1 + (u k t 1) + (u t 1) ). (7.22) To cacuate the conditiona expectations in (7.18) we just need to hande expressions of the form E CMSs[ F t F ] S i,c 1,c 2 t = x (7.23) and E CMSs[ u t 1 ] S i,c 1,c 2 t = x. (7.24) Our pan is to cacuate (7.23) via the definition of the conditiona expectation as a orthogona projection. For that we approximate our SABR-LMM ike in [22] as foows dft i = p i d W t i, F i = F i () du i t = ν i d Z t, i u i = 1 d W, i W j t = ρ ij dt (7.25) d Z, i Z j t = r ij dt d W, i Z j t = R ij dt, where the (ν i ) i {1,...,N 1} are the vo-vo parameters for the forwards obtained from 76

85 7.1. Markovian Projections of CMS spreads ordinary SABR fitting and the ( W i ) i {1,...,N 1}, ( Z i ) i {1,...,N 1} Wiener processes. In addition, we approximate the spread dynamic as ds i,c 1,c 2 t = pd W t. So we just freeze the coefficients in (7.14) at t =. That impies Now we get and i+c1 1 d W t = σ 1 v i,c 1,c 2 ()u ϕ(f)d W t =i i+c1 1 = p 1 =i i+c d W, W 1 1 i t = p 1 v i,c 1,c 2 ()p d W t. (7.26) =i ν i,c 1,c 2 p ρ i dt =: ρ W i (ρ)dt (7.27) i+c d W, Z 1 1 i t = p 1 =i ν i,c 1,c 2 p R i dt =: ρ Z i (R)dt. (7.28) We estimate (7.23) with the hep of (7.27) and whie keeping (7.25) in mind as E CMSs[ F t F and with the hep of (7.28) we get E CMSs[ u t 1 ] S i,c 1,c 2 t = x = F, i S i,c 1,c 2 t S i,c 1,c 2, S i,c (x S i,c 1,c 2 1,c 2 t t ) = p iρ W i (ρ) (x S i,c 1,c 2 t ) (7.29) p ] S i,c 1,c 2 t = x = ui, S i,c 1,c 2 t S i,c 1,c 2, S i,c (x S i,c 1,c 2 1,c 2 t t ) = ν iρ Z i (R) (x S i,c 1,c 2 t ). (7.3) p 77

86 7. CMS Spread Options and Swaps With (7.19) and (7.21) in addition with the hep of (7.29) and (7.3) it foows E CMSs[ σ 2 t ] i+c S i,c 1 1 [ 1,c 2 t = x v i,c 1,c 2 () 2 p q p ρ W p p =i + 2 ν ρ Z (R) ] (x S i,c 1,c 2 t ) p + 2 i+c 1 1 j<=i+1 [ 1 + q j p j ρ W j p j p (ρ) ρ j v i,c 1,c 2 ()v i,c 1,c 2 j ()p j p (ρ) (x S i,c 1,c 2 ) (x S i,c 1,c 2 ) + q p ρ W (ρ) (x S i,c 1,c 2 ) + ν jρ Z j (R) (x S i,c 1,c 2 t ) p p p + ν ρ Z (R) ] (x S i,c 1,c 2 t ) p = i+c 1 1 =i i+c 1 1 j<=i+1 [ i+c1 1 =i i+c 1 1 j<=i+1 v i,c 1,c 2 () 2 p 2 ρ j v i,c 1,c 2 ()v i,c 1,c 2 j ()p j p 2p 2 vi,c 1,c 2 () 2 (A (ρ) + B (ρ)) ρ j v i,c 1,c 2 j ()v i,c 1,c 2 ()p j p (A j (ρ) + A (ρ) + B j (ρ) + B (ρ)) ] (x S i,c 1,c 2 ), (7.31) where we define A (ρ) := q ρ W (ρ) p (7.32) and B (ρ) := ν ρ Z (R) (x S i,c 1,c 2 t ). p 78

87 7.1. Markovian Projections of CMS spreads Further we set A (ρ) := + 2 i+c 1 1 =i i+c 1 1 j<=i+1 2p 2 vi,c 1,c 2 () 2 (A (ρ) + B (ρ)) v i,c 1,c 2 j ()v i,c 1,c 2 ()p j p ( A j (ρ) + A (ρ) + B j (ρ) + B (ρ) ). (7.33) Overa, we get E CMSs[ σ 2 t ] S i,c 1,c 2 t = x p 2 + A o (ρ)(x S i,c 1,c 2 ). Now we have evauated the enumerator of equation (7.18). To approximate the denominator we use (7.2) and (7.22) together with (7.3) and obtain E CMSs[ u 2 t ] i+c S i,c 1 1 1,c 2 t = x v i,c 1,c 2 () 2 p2 ( p = [ =i i+c 1 1 j<=i+1 ρ j v i,c 1,c 2 j ()v i,c 1,c ν iρ Z i (R) (x S i,c 1,c 2 p i+c 1 1 =i i+c 1 1 j<=i+1 [ i+c1 1 =i i+c 1 1 j<=i+1 (x S i,c 1,c 2 ) v i,c 1,c 2 () 2 p2 k p 2 ρ j v i,c 1,c 2 j ()v i,c 1,c 2 v i,c 1,c 2 () 2 p2 k p 2 2B ρ j v i,c 1,c 2 j ()v i,c 1,c 2 = 1 + A u (ρ)(x S i,c 1,c 2 ), ν iρ Z i (R) p () p jp p 2 ) + ν iρ Z i (R) p () p jp p 2 ) (x S i,c 1,c 2 ) (x S i,c 1,c 2 ) ] () p [ ] ] jp p 2 B j (ρ) + B (ρ) 79

88 7. CMS Spread Options and Swaps where A u (ρ) := + 2 [ i+c1 1 =i i+c 1 1 j<=i+1 v i,c 1,c 2 () 2 p2 k p 2 2B (ρ) ρ j v i,c 1,c 2 j ()v i,c 1,c 2 Now we can use (7.18) to get the form of G. We get Hence () p jp p 2 [ B j (ρ) + B (ρ)] ]. (7.34) G 2 (x) p2 + A o (ρ, R)(x S i,c 1,c 2 ) 1 + A u (ρ, R)(x S i,c 1,c 2 ). (7.35) and for the derivative G 2 (S i,c 1,c 2 ) p 2 (7.36) d dx G(Si,c 1,c 2 ) = A o(ρ, R) p 2 A u (ρ) 2p = p 1( i+c i+c 1 1 j<=i+1 =i v i,c 1,c 2 p 2 vi,c 1,c 2 () 2 A (ρ) j ()v i,c 1,c 2 ()p j p ( A j (ρ) + A (ρ) )). (7.37) Now we wi cacuate the dynamic of u t, whie approximating uj t u t u t and (u t )2 u t via their expectation vaues as in [22]. The expectation vaues are E CMSs[ u j ] t u t u t 1 E CMSs[ (u t )2 u t ]. We obtain using du t = 1 σ dσ g t σ dkt = g t σ ktζ h tdzt u tν dzt 8

89 7.1. Markovian Projections of CMS spreads the foowing du 2 t = 1 p 2 (2 + 2 i+c 1 1 =i 1 p 2 (2 + 2 i+c 1 1 =i i+c 1 1 j<=i+1 p j p v i,c 1,c 2 j ()v i,c 1,c 2 ( v i,c 1,c 2 () ) 2 p 2 u tdu t i+c 1 1 j<=i+1 [ = 2 i+c1 1 p 2 =i ) ρ j p j p v i,c 1,c 2 j ()v i,c 1,c 2 ( v i,c 1,c 2 ()) 2p 2 ( u t ) 2ν dz t [ ()ρ j u j t du t + u tdu k ] t ) [( 2p i+c v i,c 1 1 1,c 2 2 ()) + =j=i ()u j t u t Since u > and u never changes its sign we obtain Now (7.38) eads to du t u t = d u 2 t 1 2 u t 1 u 2 du 2 t t [ 1 i+c1 1 p 2 i+c =j=i =i [ ] ν dzt + ν j dz j t ] ] p j p v i,c 1,c 2 j ()v i,c 1,c 2 ()ρ j u j t u t ν dzt. [( v i,c 1,c 2 ()) 2p 2 p j p v i,c 1,c 2 j ()v i,c 1,c 2 ()ρ j ]ν dz t du t u t A p,v (ρ, r)d Z t, ]. (7.38) 81

90 7. CMS Spread Options and Swaps where with λ (ρ) := Z t = 1 A p,v (ρ, r)p 2 i+c = =j=i [ i+c1 1 =i [( v i,c 1,c 2 ()) 2p 2 p j p v i,c 1,c 2 j ()v i,c 1,c 2 ()ρ j ]ν dz t i+c A p,v (ρ, r)p 2 λ (ρ)ν dzt, (7.39) =i ( 2p i+c v i,c 1 1 1,c 2 2 ()) + p j p v i,c 1,c 2 j ()v i,c 1,c 2 ()ρ j. =j=i The variabe A p,v (ρ, r) is cacuated such that Z is a Wiener process ( ) 2 A p,v 1 (ρ, r) := t 1 i+c 1 1 p 2 ε (ρ)ν dz t = 1 i+c 1 1 p 4 j,=i =i ] λ j (ρ)λ (ρ)r j ν j ν. (7.4) Our ast step is to cacuate the correation between S i,c 1,c 2 t and u t. We get using (7.13), (7.39) and σ = p d Ŵ, Z t = 1 A p,v (ρ, r) 1 1 σ t i+c1 1 p 2 =i =i i+c v i,c 1 1 1,c 2 ()u ϕ(f )dw, i+c1 1 1 A p,v (ρ, r)p 3 i+c v i,c 1 1 1,c 2 ()σ FdW, λ (ρ)ν dz t = =i i+c A p,v (ρ, r)p 3 v i,c 1,c 2 j ()σ j F j λ (ρ)ν R j j,=i =i λ (ρ)ν dz t =: X (ρ, r, R). (7.41) Therefore we have approximative for the CMS spread dynamics with the resuts in 82

91 7.1. Markovian Projections of CMS spreads (7.4), (7.41), (7.36), (7.37), (7.33) and (7.34) i+c ds i,c 1,c 2 t = u t G(S i,c 1,c 2 t )dŵt, S i,c 1 1 1,c 2 = v i,c 1,c 2 F du t = A p,v (ρ, r)d Z t, u = 1 d S i,c 1,c 2, u t = X (ρ, r, R)dt, =i where and d dx G(Si,c 1,c 2 So G can be approximated via G(S i,c 1,c 2 ) = p =: p j,c 1,c 2 (7.42) ) = p 1( i+c i+c 1 1 j<=i+1 =i v i,c 1,c 2 p 2 vi,c 1,c 2 () 2 A (ρ) j ()v i,c 1,c 2 ()p j p ( A j (ρ) + A (ρ) )) =: q j,c 1,c 2. (7.43) G(x) = (x S i,c 1,c 2 + pj,c 1,c 2 q j,c 1,c 2 )q j,c 1,c 2. Remark. In Theorem 4 the coefficients p j,c 1,c 2 and q j,c 1,c 2 are obtained by using a freezing procedure to approximate the expected vaues as in Theorem 5. For short terms the approximation is vaid, but the onger the expiry of the spread the more inaccurate the approximation gets. Further, the parameters A p,v t and X (ρ, r, R) are taiored to our SABR-LMM with time-dependent coefficients and a voatiity structure caibrated to capets. Simiar to the case of the approximation of the swap dynamics in chapter 4.2 we froze in the beginning of the proof the CMS spread weights v i,c 1,c 2 j (). The freezing is needed 83

92 7. CMS Spread Options and Swaps since the voatiity of CMS spreads in the SABR-LMM is stochastic again, because the weights are ceary random, and we want to have deterministic voatiity for the dispaced SABR mode in the end Convexity Correction for CMS spreads In Theorem 4 we approximate the dynamics of CMS spreads under the spread measure P CMSs under which we assume a spread as drift-free. However our goa is to estimate the correation of forward rates under the forward measure P i to be consistent with our estimations. Since we don t assume deterministic voatiities for the processes of interest, the measure under which we cacuate the correations indeed matters. In addition, a spread option fixed in T i pays in T i+1. Therefore we want to discount this payment by the numéraire B(t, T i+1 ) which beongs to P i. The market quotes prices of CMS spread options and we mode the underying processes as drift-ess under P CMSs. Hence, the quoted prices are not cacuated under P i and we have to do a so-caed convexity correction, which heps us to transfer prices gained under one measure to another measure. Historicay a convexity correction is understood as an adjustment to the risk exposure of an asset to changes of the forward or yied curve. The idea was: If the forward curve changes over time, the vaue of the asset somehow comoves and therefore the expected future price of an asset depends on the yied curve of interest. We wi use this idea to cacuate the convexity correction for CMS spreads. But first we introduce the mathematica way of thinking about those corrections. A convexity correction enabes us to transform prices in one measure into prices in an other measure by adding a correction term. The correction term is then caed convexity correction. The added term can be interpreted as some drift correction due to a measure change, just as in Girsanovs Theorem [4]. To describe the convexity correction in formuas we remember that if we have to random variabes X and Y then and a integras are we defined, it hods that Cov(X, Y ) = E[XY ] E[X]E[Y ] E[XY ] = E[X]E[Y ] + Cov(X, Y ). Now, et s consider two different measures P and Q with numéraire B and A, respectivey. Then it hods due to the change of numéraire technique, ike in the proof of the forwardrate dynamics under different measures in Theorem 1, for a caim S T which 84

93 7.2. Convexity Correction for CMS spreads pays in T that E P[ S T ] = A B E Q[ S T B T A T ] = A B E Q[ B T A T ] E Q [ S T ] + A B Cov Q[ S T, B T A T ] = E Q[ S T ] + A B Cov Q[ S T, B T A T ]. (7.44) Here the term is the convexity correction. A B Cov Q[ S T, B T A T ] To cacuate the convexity correction for an CMS spread S j,c 1,c 2 up a genera framework. For a spread S j,c 1,c 2 with zero strike we can write S j,c 1,c 2 = S (1) S (2), we first want to set where the S (i) are swap rates over the time interva [, +ci ], where j N is some index and c i N the tenor of the j-th swap rate. Let P CMSs be the spread measure with numéraire SN, et P j,j+c i the swap measures for the i -th swap rate with numéraire A j,j+c i and define P j as the forward measure beonging to the rate F j. We want to evauate the spread under P j at time, since we have to price European 85

94 7. CMS Spread Options and Swaps options expiring at ater on. By foowing [22] we write E j[ S j,c 1,c 2 ] [ = E CMSs S j,c 1,c 2 B(, +1 ) SN ] SN Tj B(, +1 ) = S (1) S (2) ( [ + E j,j+c 1 S (1) ( j,j+c B(Tj, +1 ) A 1 )] A j,j+c 1 B(, T +1 ) 1 j ) [ E j,j+c 2 S (2) ( j,j+c B(Tj, +1 ) A 2 )] A j,j+c 2 B(, T +1 ) 1 j ( [ E j,j+c 1 S (1) ( SNTj j,j+c A 1 )] A j,j+c 1 1 SN [ E j,j+c 2 S (2) ( SNTj [ E j,j+c 1 ( + E j,j+c 1 A j,j+c 2 S (1) ] E j,j+c 2 )] ) 1 SN [ S (2) ] j,j+c A 2 [ S (1) ( B(Tj, +1 ) A j,j+c 1 [ E j,j+c 2 S (2) ( B(Tj, +1 ) A j,j+c 2 j,j+c A 1 )] B(, +1 ) 1 ) j,j+c A 2 )] B(, +1 ) 1, (7.45) where we used the martingae property of S (i) under P j,j+c i and negected in the ast approximation the difference [ E j,j+c 1 S (1) ( SNTj A j,j+c 1 j,j+c A 1 )] [ 1 E j,j+c 2 S (2) SN ( SNTj A j,j+c 2 j,j+c A 2 )] 1. (7.46) SN According to [22] the term (7.46) is amost zero, since the swap measure and the spread measure evauate the spreads neary identica. In order to cacuate E j[ S Tj ] we just have to approximate the convexity corrections [ E j,j+c i S (i) ( B(Tj, +1 ) A j,j+c i j,j+c A i )] B(, +1 ) 1 (7.47) 86

95 7.2. Convexity Correction for CMS spreads as we as possibe. Using the definition of ( ) + we obtain a aternative form for (7.47). We get [ E j,j+c i S (i) ( B(Tj, +1 ) A j,j+c i j,j+c A i )] B(, +1 ) 1 [( + E j,j+c i S (i) = S (i) ( B(, Tj+1 ) A j,j+c i S (i) ) + ( B(Tj, +1 ) A j,j+c i [( E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i ) + ( B(Tj, +1 ) [( = E j,j+c i S (i) S (i) A j,j+c i [( E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i A j,j+c i ) B(, +1 ) 1 j,j+c A i )] B(, +1 ) 1 j,j+c A i )] B(, +1 ) 1 j,j+c A i )] B(, +1 ) 1 j,j+c A i )] B(, +1 ) 1. (7.48) So we find there a ca part and a put part. Further, if we appy the ca-put parity with strike S (i) on the asset S (i), we get [ E j,j+c i S (i) ] = S (i) [ + E j,j+c i (S (i) S (i) [ )+] E j,j+c i (S (i) S (i) ) +]. (7.49) Together with (7.45) and (7.48) we can concude the foowing: To evauate the convexity correction we just have to cacuate the expected vaues of ca/put ike options. The ca ike options are [( E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i [ = E j,j+c i (S (i) S (i) )+] [( + E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i j,j+c A i )] B(, +1 ) j,j+c A i )] B(, +1 ) 1 (7.5) 87

96 7. CMS Spread Options and Swaps and the put ike options are where [( E j,j+c i S (i) S (i) ) + ( j,j+c B(Tj, +1 ) A i )] A j,j+c i B(, T +1 ) j [ = E j,j+c i (S (i) S (i) ) +] [( + E j,j+c i S (i) S (i) ) + ( j,j+c B(Tj, +1 ) A i )] A j,j+c i B(, T +1 ) 1, (7.51) j B(, +1 ) A j,j+c i j,j+c A i B(, +1 ) 1 vanishes on average and goes, according to Hagan [32], to zero ineary with the variance of the swap rate S (i). Therefore the second expected vaue in (7.5) and (7.51), respectivey, shoud be much smaer than the first one. To evauate the expressions in (7.5) and (7.51) we have to do two things. First, we have to express the moving of the underying forward curve in terms of the swap rate S (i). Then we have to use this dependency to evauate the expected vaues. Here, the SABR formua for impied voatiity wi pay a major roe. To tacke the first probem we want to write B(t, +1 ) A j,j+c i t = H(S (i) t ) (7.52) for some function H. To obtain a proper form for H we assume that there are ony parae shifts in the yied curve and, by foowing [32], we obtain A j,j+c i t = j+c i k=j+1 = B(t, ) δ k B(t, T k ) j+c i k=j+1 δ k B(t, T k ) B(t, ). 88

97 7.2. Convexity Correction for CMS spreads If we assume δ k = δ for a k and interpret S (i) t period [, +c1 ], we get as the average discount rate over the A j,j+c i t B(t, ) j+c i k=j+1 c i = B(t, ) k=1 δ (1 + δs (i) t ) k j δ (1 + δs (i) t ). k Hence with the formua for geometric sum and some cacuations it foows that A j,j+c i In a simiar way we estimate t B(t, ) S (i) t ( 1 1 (1 + δs (i) t ) c i B(t, +1 ) B(t, ) 1 + δs (i). t To achieve (7.52) it makes now sense to define Hence H(S (i) t ) := S (i) t (1 + δs (i) t )(1 1 (1+δS (i) t H (S (i) t ) = (δs(i) t + 1) ci 2 ((δs (i) t + 1) c i c i δs (i) t 1) ((δs (i). t + 1) c i 1) 2 ). ). (7.53) ) c i Using (7.53) we are abe to evauate the expected vaues in (7.5). We write [( E j,j+c i S (i) [( = E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i S (i) ) + (H(S (i) ) )] H(S (i) ) 1 j,j+c A i )] B(, +1 ) 1 (7.54) 89

98 7. CMS Spread Options and Swaps To evauate (7.54) we define the smooth function f(x) := (x S (i) (H(S (i) ) ) ) H(S (i) ) 1 (7.55) and cacuate by using partia integration f (S (i) )(S(i) S (i) )+ + S (i) If we use this on (7.54), we obtain [( E j,j+c i S (i) S (i) ) + (H(S (i) ) (S (i) x) + f (x)dx = H(S (i) f (S (i) [ )Ej,j+c i (S (i) S (i) )+] + )] ) 1 = S (i) f(s (i) ), if S (i), if S (i) S (i) < S (i) [ E j,j+c i (S (i) x) +] f (x)dx. (7.56). Now we define [ C(K) := E j,j+c i (S (i) t K) +] (7.57) and get for (7.5) by using (7.56) [( E j,j+c i S (i) = S (i) ) + ( B(Tj, +1 ) A j,j+c i ( 1 + f (S (i) ) ) C(S (i) ) + S (i) j,j+c A i )] B(, +1 ) C(x)f (x)dx. (7.58) Therefore (7.58) gives us the expected vaue of a ca under P j with strike S (i) on the underying S (i) t by integration of a possibe ca prices greater then the strike. To see that, just remember [( E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i j,j+c A i )] = E j[( S (i) B(, +1 ) S (i) ) + ]. 9

99 7.2. Convexity Correction for CMS spreads In the same fashion foows for (7.51) [( E j,j+c i S (i) S (i) ) + ( B(Tj, +1 ) A j,j+c i = E j[( S (i) S (i) ) + ] = ( 1 + f (S (i) ) (i) S ) P (S (i) ) j,j+c A i )] B(, +1 ) P (x)f (x)dx, (7.59) where [ P (K) := E j,j+c i (K S (i) t ) +]. (7.6) Up to now the function f depends through H on the swap rate S (i) at time T i. This is highy probematic when it comes to impementation since S (i) T i is random. To circumvent this issue we use a Taior-expansion on H at S (i) in direction of x. Hence and therefore Further we obtain and H(x) H(S (i) ) + H (S (i) )(x S(i) ) +... (7.61) f(x) H (S (i) ) H(S (i) (x S(i) ) )2. (7.62) f (x) 2 H (S (i) ) H(S (i) (x S(i) ) ) (7.63) f (x) 2 H (S (i) ) H(S (i) ). (7.64) According to Hagan [32] is the approximation in (7.61) fairy good since H is a smooth function which varies very sowy. The foowing Theorem summarizes the resuts in this chapter. 91

100 7. CMS Spread Options and Swaps Theorem 5 (Convexity Correction for CMS spreads). Let S j,c 1,c 2 be a CMS spread with zero strike defined via Then the expected vaue E j[ S j,c 1,c 2 under the forward measure P j can be approximated as S j,c 1,c 2 t = S j,j+c 1 t S j,j+c 2 t. ] E j[ ] S j,c 1,c 2 = S j,c 1,c 2 + E j[( ) + ] S j,j+c 1 S j,j+c 1 E j[( ) + ] S j,j+c 1 S j,j+c 1 E j[( ) + ] S j,j+c 2 S j,j+c 2 + E j[( ) + ] S j,j+c 2 S j,j+c 2 ( ) = S j,c 1,c f (S j,j+c 1 ) C(S j,j+c 1 ) + C(x)f (x)dx where f is defined in (7.55). ( ) 1 + f (S j,j+c 1 ) P (S j,j+c 1 ) + ( ) 1 + f (S j,j+c 2 ) C(S j,j+c 2 ) ( ) f (S j,j+c 2 ) P (S j,j+c 2 ) S j,j+c 1 S j,j+c 1 S j,j+c 2 S j,j+c 2 P (x)f (x)dx C(x)f (x)dx P (x)f (x)dx, Proof. The genera structure of the approximation is given at (7.45) and the ca/put spits are shown in (7.48) and (7.49). The approximation through the integration over expected ca and put vaues for different strikes is given in (7.58) and (7.59), where we approximate f through (7.62), (7.63) and (7.64). The convexity correction can be seen as a measure for the skew difference of P j and P CMSs. The skew of a probabiity measure describes where it has most of its mass. Under martingae measure P CMSs beonging to S j,c 1,c 2 it hods E CMSs[ (S j,j+c 1 S j,j+c 1 ) +] E CMSs[ (S j,j+c 1 S j,j+c 1 ) ] =E CMSs[ ] S j,j+c 1 S j,j+c 1 =. (7.65) If P j and P CMSs have the same skew and shape we woud obtain E j[ ] S j,c 1,c 2 S j,c 1,c 2, since the integras in theorem 5 woud cance each other out as we. However in most cases the size of the expected vaues under the two measures is nonzero. This = 92

101 7.3. Caibration to CMS Spread Options means that P j vaues undiscounted cas or put different from P CMSs. So under P j high payoffs are more ikey or ess ikey then under P CMSs. Therefore the measures differ in their skew, which is expressed by the vaue of the correction. For competeness we mention that is possibe to cacuate the drift correction the intuitive way given at (7.44), but it is rather difficut and unknown if the accuracy can be improved significanty. However the interested reader can find more about this approach in [5] Caibration to CMS Spread Options With Theorem 4 from chapter 7.1 and Theorem 5 from chapter 7.2 we have everything together to caibrate our supercorreation matrix P which contains a the correations of the processes in the SABR-LMM mode on CMS spread options. Theorem 4 gives us a proper dynamic for the spreads in a SABR-ike environment, whereas Theorem 5 enabes us to evauate under the forward measures P j. We wi use the formua for impied voatiity (3.4) to cacuate the integras in Theorem 5. To use the formua or some other workaround is necessary for two reasons. First, ony ca and put prices for strikes K in a certain range [K min, K max ] are quoted in the market. Second, there is ony a finite amount of strikes, whereas we need a continuum. In the mode caibration based on CMS spreads the very first step is the same as in the mode caibration based on Swaps. The voatiity functions g j and h j are caibrated to caps, just as the coefficients k j and ζ j. We refer to chapter 6.2. However, to caibrate the correation we first focus on how to sove the integras for the convexity correction in Theorem 5 and how to mode the dynamics of the invoved swaps. We assume that under the swap measure P j,j+c i the swap rate S j,j+c i evoves ike a SABR process. Therefore, the dynamics are ds j,j+c i t dσ j,j+c i t ( = σ j,j+c i t = σ j,j+c i t d W j,j+c i, Z j,j+c i t = ρ j,j+c i dt, S j,j+c i) βj,j+ci dw j,j+c i t, S j,j+c i = S j,j+c i () ν j,j+c i dz j,j+c i t, σ j,j+c i = σ j,j+c i () (7.66) 93

102 7. CMS Spread Options and Swaps where as usuay β j,j+ci [, 1], ν j,j+c i, σ j,j+c i() > and ρ j,j+c i [ 1, 1]. From this we get the impied Back voatiity (3.4) basing on the mode parameters σ j,j+c i B (K) := σ I (S j,j+c i (), K, β j,j+ci, σ j,j+c i (), ν j,j+c i, ρ j,j+c i, ), (7.67) which gives us the right voatiity to price cas and puts with strike K on the underying S j,j+c i. With that we can write for the expected vaues in (7.57) and (7.6) C(K) = S j,j+c i ()N (d 1 ) KN (d 2 ) (7.68) and P (K) = KN ( d 2 ) S j,j+c i ()N ( d 1 ), (7.69) where and d 1 = n ( ) S j,j+c i () K (σj,j+c i B (K)) 2 σ j,j+c i B (K) d 2 = d 1 σ j,j+c i B (K). As suggested and tested in [13] we can use (7.68) and (7.69) together with (7.67) to evauate the integras in Theorem 5 from S j,j+c i() to and to S j,j+c i(), respectivey. To incorporate the market data and to obtain the mode parameters of the swap dynamics in (7.66) we sove an ordinary, unweighted east-square probem (3.5) min σ j,j+c i (),ν j,j+c i,ρ j,j+c i K ( σ j,j+c i B (K) σ M (K)) 2, (7.7) where σ M (K) is the impied voatiity quoted in the market for the strike K of a ca or put. As in chapter 6, which treated the caibration of the SABR dynamics to swaps, we set β j,j+ci = j+c i 1 k=j ω j,j+c i k β k, (7.71) 94

103 7.3. Caibration to CMS Spread Options where β k are the betas in the SABR dynamics of the forward rates F k. The techniques above enabe us to cacuate the convexity corrections propery. The CMS spread dynamic for the spread S j,c 1,c 2 is modeed equivaent to Theorem 4 as where ( ) d S j,c 1,c 2 t + B i,c 1,c 2 = u t (S j,c 1,c 2 t + B i,c 1,c 2 )dw i,c 1,c 2 t du t = u t A p,ν j,c 1,c 2 (ρ, r)dz i,c 1,c 2 t, u = q j,c 1,c 2 (7.72) d W i,c 1,c 2, Z i,c 1,c 2 t = X j,c 1,c 2 (ρ, r, R)dt, B j,c 1,c 2 := pj,c 1,c 2 q j,c 1,c 2 S j,c 1,c 2 () and p j,c 1,c 2, q j,c 1,c 2 are given in the proof of the Theorem at (7.42) and (7.43). The parameterization of A p,ν j,c 1,c 2 (ρ, r) is given in (7.4) whereas the one of X j,c 1,c 2 (ρ, r, R) is given in (7.41). To see the equivaence of both formuation for the spread dynamics in definition (4) and this one here just notice ( ) d S j,c 1,c 2 t + B i,c 1,c 2 = ds j,c 1,c 2, since B i,c 1,c 2 is constant and that we just mutipied u t by q i,c 1,c 2. Before we can start with the caibration of the correation to CMS spreads we have to consider the way how the market quotes options on spreads. Different to swaptions, which are quoted in impied Back voatiities, options on CMS spreads are quoted in impied norma voatiities. This is because CMS spreads can get negative as differences of two swap rates. If the price is given in norma voatiity σ n, it is assumed that the underying S foows a Bacheier mode. That is ds = σ n dw t, S = S() 95

104 7. CMS Spread Options and Swaps for σ n > and a Wiener process (W t ) t. Therefore a ca on a underying S with strike K, expiry T exp can be evauated as [42] E [ (S K) +] ( S K ) = (S K)N σ n + σ n ( S K ) T exp ϕ T exp σ n, T exp where ϕ is the density function of the norma distribution. In the same fashion as for impied Back voatiity there exists a anaytic formua for the impied norma voatiity σ n of the SABR mode (3.1). Again, the formua depends ony on the current underying price, strike of the ca or put and the mode parameters. In [17] Hagan approximates the impied norma voatiity as foows where σ n I (F, K, β, α, ν, ρ, T exp) := α ( F K ) β/ ( ) 24 og2 F K + 1 ( ) 192 og4 F K 1 + (1 β)2 24 og 2 ( ) F K + (1 β) 2 ( ) 192 og4 F K [ 1 + ( β(2 β)α 2 24(F K) 1 β + ρανβ 4(F K) 1 β /2 ( ζ ) X(ζ) + 2 3ρ2 ν 2) T exp ], (7.73) 24 ζ := ν α (F K)(1 β)/2 og(f /K) and ( 1 2ρζ + ζ X(ζ) := og 2 ρ + ζ ). 1 ρ As in the case of the formua for the impied Back voatiity the formua in (7.73) is purey anaytic and highy tractabe in regards of impementation. Now, in order to fit the correations under the forward measure P j basing on European options on CMS we define σ j n(k) := σ n I ( B(, +1 )E j [S j,c 1,c 2 ] + B i,c 1,c 2, K + B i,c 1,c 2, 1, q j,c 1,c 2, A p,ν j,c 1,c 2 (ρ, r), X j,c 1,c 2 (ρ, r, R), ). (7.74) 96

105 7.3. Caibration to CMS Spread Options This makes sense since the price of a ca option with expiry T i and settement date on a CMS S i,c 1,c 2 is B(, )E i[ (S i,c 1,c 2 T i K) +] = B(, )E i[ ((S i,c 1,c 2 T i + B i,c 1,c 2 ) (K + B i,c 1,c 2 )) +]. In (7.74) we cacuate E j [S j,c 1,c 2 ] with the convexity correction as shown above. Now, we take cas and puts on CMS spreads for different expiry dates and strikes into account to caibrate to market data. Our goa is to caibrate the CMS spread dynamics (7.72) as cose as possibe to a market prices. We do this by an ordinary east-square probem and using (7.74) min q j,c 1,c 2,A p,ν j,c 1,c (ρ,r),x j,c 1,c 2 (ρ,r,r) 2 2, (σn(k j j ) σ j M j)) (K (7.75) K j j where we sum over a expiry dates and avaiabe strikes K j for this date. Here σ j M (K j) is the impied norma voatiity observed in the market for a ca or put on S j,c 1,c 2 with underying price S j,c 1,c 2 and strike K j. Since a the variabes in the minimization probem depend on P through ρ, r or R we achieve our goa to caibrate the supercorreation matrix. Remark. In the same fashion as above it is possibe to caibrate the CMS spread dynamics on straddes on CMS spreads. A stradde is the sum of a ca and put option with the same strike. A stradde with strike K on S i,c 1,c 2 has in T i+1 the payoff S i,c 1,c 2 T i K. (7.76) Now et s denote by C(S i,c 1,c 2, K) the expected vaue of a ca and P (S i,c 1,c 2, K) the expected vaue of a put with underying price S i,c 1,c 2 and strike K. From the definition we know that the expected vaue of a stradde can be cacuated as E [ Stradde ] := C(S i,c 1,c 2, K) + P (S i,c 1,c 2, K) = 2C(S i,c 1,c 2, K) S i,c 1,c 2 + K (7.77) where we used the ca put parity. By assuming that S i,c 1,c 2 foows a Bachier mode with voatiity σ n >, the representation in (7.77) eads to the foowing expected 97

106 7. CMS Spread Options and Swaps vaue E [ Stradde ] ( = 2(S i,c 1,c 2 S i,c 1,c 2 K ) K)N σ n + 2σ n ( S i,c 1,c 2 K ) T i ϕ T i σ n T i S i,c 1,c 2 + K. (7.78) Now, CMS spread straddes are quoted in impied norma voatiity consistent with formua (7.78) and the impied norma voatiity beongs to a ca/put on S i,c 1,c 2 with strike K and exipiry T i. This enabes us indeed to caibrate on CMS stradde prices in the same way as we caibrate to cas and puts to CMS spreads. 98

107 8. Impementation and Empirica Study 8.1. Impementation The impementation of the caibration on swaps and CMS spreads was carried out by the author himsef, for which the programming anguage F# was used. F# is a free, functiona.net anguage, which is supported by a great number of routines provided by packages ike Math.Net and Microsoft s Sover Foundation. A minimization probems from chapter 6 and chapter 7.3 were soved by the Neder-Mead agorithm from the Sover Foundation package and for numerica integrations as in the case of generating random numbers the Numerics package from Math.Net was used Empirica Study The caibration was tested for the Euribor as underying yied curve. We used data from two different dates and carried out the caibration for a SABR-LMM which covers 2 years. We set δ i.5 and used the tenor structure (T i ) i {,...,4}. Hence, we caibrated the dynamic for haf year forward rates whose first expiring date is in.5 years and the ast in 19.5 years. Afterwards we tested the quaity of the caibration by doing Monte Caro simuations. The simuation was impemented by Cresnik [28], where discretization of the mode SDE s was done by using the Mistein scheme The Data The caibration of the SABR-LMM was tested for the caibration on swaptions and CMS spread options. In both cases we compared the (2SC) and (5L) parametrization for ρ. Further, for the swaptions we compared two dates. The first data set is from the and represents quiet market situation in an environment of ow voatiity and high interest rates. The second data set is from the and represents a 99

108 8. Impementation and Empirica Study rough market with high uncertainty and ow interest rates. The caibration on CMS spreads was carried out on data as of For the caibration we foowed the methods described in chapter 6 and chapter 7.3. The swaption prices are from Boomberg. We thank Peter Krierer for providing the CMS spread option prices from BGC Market Data Caibration to Swaption Prices as of For this date we used a cap cube with 18 strikes ranging from ATM over 1% to up to 14% and maturities given in fu years starting from 1 year and ranging up to 2 years. We interpoated the missing haf-year maturities by using spine interpoation for the voatiities and strikes. The given swaption cube for cotermina swaps consists of 22 strikes, ranging from ATM-2% to ATM+3%, and expiry dates in 1 2, 1, 2,..., 1, 12 and 15 years. However not a tenors for cotermina swaps were avaiabe. For exampe, there are no options on 9x11 swaptions. We got the missing tenors by spine interpoation for strikes and voatiities. After setting up the data we foowed chapter 6 to caibrate first the voatiity structure to capet voatiities and second the correations, based on the swaption prices. The parameters of the caibration and pots of the supercorreation matrices can be found in the appendix. To describe the distinction between the supercorreation matrices P by using the two different parameterizations for ρ, ets focus first on the difference in the ρ s itsef. The genera shape of ρ using the different parameterizations amost the same. Nevertheess, there are differences, which are visuaized in figure 8.1. First, the wings of the matrices differ. The (5L) matrix goes in areas for the correation of rates more then 2 years apart down to 5% and therefore around 1% ower then the (2SC) parametrized matrix. Second, the correation for rates which are ess than 2 years apart is a itte bit higher. However, the other submatrices of P under the different approaches for ρ cannot be distinguished, since the parameters are amost the same. That is a bit surprising since the differences in the outer areas of ρ are pronounced. To anayze the quaity of our caibration we repriced the target products, which were used in the caibration process, and cacuated the reative pricing errors. First, we 1

109 8.2. Empirica Study used the guing agorithm from chapter 5.3 to estimate the nearest correation matrix P of rang k = 1. Then, in each simuation we did 3. runs with 1 Steps for the whoe time interva [T, T 39 ]. Pots of the reative errors can be found in the appendix. We repriced a capets on a strikes. It did not matter, if we used the supercorreation matrix basing on (5L) or (2SC). In both cases, the picture was the same. Overa the resuts are satisfying and getting more accurate with higher expiry, as can be seen on the mean errors in tabe 8.1. For capets with any expiry we observed, that the reative error for strikes greater then 6% expodes. In this area the errors begin to sky rock starting at 2% and went up to 5%. Therefore, we excuded strikes greater then 6% from tabe 8.1. One way to expain the errors is as foows: We tried to preserve the simpe SABR mode for forward rates in the SABR-LMM as we as possibe. Further, capets are amost not effected by the correations of forward rates given through the submatrices ρ and r of P, but they rey on the skews given through R. The diagona of R pays the roe of the ρ s from the simpe SABR mode, since it hods R ii = ρ i. If ρ i in the simpe SABR is sighty disturbed the impied mode voatiity for this forward rate is disturbed. As figure 3.1 shows the reative error gets bigger with greater strikes and the induced reative pricing error through Back s formua is even higher in this area. Now, to be abe to do the Monte Caro simuation we need to approximation mode supercorreation matrix P by a rea correation matrix P. In genera P and P wi deviate by a sma amount and therefore the diagonas of R wi sighty differ as we. This eads to differences of the individua skews for each forward rate in the simpe SABR compared with the SABR-LMM. The approximation of P was introduced in chapter 5.3. A simiar expanation hods for capets with sma expiry and strikes away from the money. The market prices of those options are in the area from.1bp to 1bp and therefore reay ow. If the impied mode voatiity is disturbed sighty, the effect on the prices is tremendous and expains the reative errors in a area of 1%. To verify the correation structure we repriced a used cotermina swaptions. We compared the resuts we get by using the two different parameterizations (5L) and (2SC) for ρ. As in the case for capets the resuts are overa satisfying. The absoute reative mean-pricing errors in tabe 8.2 show, that the (5L) parametrization works sighty better then the (2SC) parametrization. However, in both cases we face huge 11

110 8. Impementation and Empirica Study Tabe 8.1.: Absoute Reative Pricing Errors for Capets with Strikes in {1%,..., 6%} as of i max min mean i max min mean pricing errors for swaptions with sma expiry and strikes far out-the-money this visibe in the pots in the appendix. In the end of chapter 4.2 we mentioned that Rebonato found that the used approximation for the swap dynamics breaks down in exacty those cases. The SABR-LMM does not reproduce the impied smie for the swap dynamics correcty which eads to tremendous pricing errors for high strikes, if the rea prices and expiries of the options is ow. 12

111 8.2. Empirica Study Tabe 8.2.: Absoute Reative Pricing Errors for Cotermina Swaptions with Strikes in {ATM 2%,..., ATM + 3%} as of P basing on (5L) P basing on (2SC) i max min mean i max min mean Caibration to Market Prices as of Caibration of the Voatiity Structure To caibrate the voatiity as expained in chapter 6.2 we again used a cap cube with 18 strikes ranging from ATM over 1% to up to 14% and maturities given in fu years starting from 1 year and ranging up to 2 years. We interpoated the missing haf year maturities by using spine interpoation for the voatiities and strikes, as we. The parameters for the voatiity functions can be found in the appendix. Caibration of the Correation Structure to Swaption Prices For the caibration of the correations we used a swaption cube consisting of 13 strikes, ranging from ATM-.3% to ATM+2.5% and swaptions with expiries in 1 2, 1, 2,..., 1, 12 and 15 years. Again, not a tenors for cotermina swaps were avaiabe and cacuated the missing tenors by spine interpoation for strikes and voatiities. The parameters for the correation matrices and the pots can be found in the appendix. 13

112 8. Impementation and Empirica Study Figure 8.1.: Here we see the difference of Lutz (5L) parameterization and Schoenmakers & Coffey s (2SC) parametrization for The (5L) parametrized matrix is ower at the wings and higher near the main diagona. Notice, the curved difference surface shows the higher fexibiity of the (5L) parametrization as we. The pot was done in Matab. In contrary to the caibration on the data from year 26 the shapes of the correation matrix ρ by a great amount depending on the parametrization. The (5L) matrix goes down to zero in the front and yieds a tiny correation ony for forward rates with sma expiries to any other forward rate. For these correation the situation is differenty under the (2SC) parametrization. Here the correation of the short-term rates is bounded downwards by roughy 3%. This phenomena can be expained by the higher infexibiity of the parametrization. For the (2SC) parametrization hods: If the correation in the front goes down the correation in the back is drawn down as we. Therefore, in the caibration process some trade off between high correations in the back and ow in the front has to be found and the boundary is the resut. In this case the impact is even visibe in the submatrix r. In the case of the (5L) parametrization the minimum of r is 9%, whereas in case of the (2SC) parametrization r is amost constant one. But, R is in both cases the same. 14

113 8.2. Empirica Study Numerica Resuts for Capets To verify the caibrations we repriced again the target options via Monte Caro Simuations. Here we doubed the number of runs to 6., because the simuations converged sower. We sticked to 1 steps for the whoe time interva [T, T 39 ]. Here we choose the parameterization for ρ and caibration approach whose resuting mode supercorreation matrix P with the owest distance to the approximated supercorreation matrix P from chapter 5.3. Thus, we used the correation matrix based on (5L) and caibrated to swaption prices. We simuated capet prices with strikes in a range of {1%,..., 14%}. Differenty from 26, the reative error for strikes over 6% went up to ony 2%. But again, a gradua increase of the error for those strikes was observabe and the expanation is the same as for the previous data. For this reason, we ony consider strikes of maxima 6% in tabe 8.3. The pricing errors for short expiries are more pronounced than in 26. We beieve this is a numerica issue resuting from the higher overa voatiity and much ower initia forward rates F, which are beow one percent. The probematic is further ampified by the missing drift or refection barrier in the SABR-LMM. In 26 the initia rates were above two percent whereas in 214 there were by around.3%. A pot of the error surface can be found in the appendix. Numerica Resuts for Swaptions when Caibrated to Swaps The simuation of swaption prices shows that the parametrization of ρ reay matters. In the case of a (5L) parametrization of ρ the reative errors are ony a itte bit higher than 26 as can be seen in tabe 8.4 and is on a satisfying eve if we consider the increase of pricing errors of the capets. However, in the case of a (2SC) parametrization for ρ the reative error sky rocks. One expanation for this phenomena coud be the higher correations between forward rates with onger expiry, when we use the (5L) parametrization. Further, in the case of a (2SC) parametrization for ρ the distance of the approximating matrix P for P and P was amost twice as high as for the (5L) parametrization for ρ. The approximation of P was expained in chapter

114 8. Impementation and Empirica Study Tabe 8.3.: Absoute reative Pricing Errors for Capets with Strikes in {1%,..., 6%} as of i max min mean i max min mean Caibration of the Correation Structure to CMS Spread Option Prices The caibration to CMS spread option was done by using European ca and put prices in norma voatiity on 1 year vs. 2 year (1y/2y) CMS spreads with strikes ranging from ATM-.25% to ATM+1.5% and expiries of 1, 2, 3, 4, 5, 7 and 1 years. We interpoated the data inear in voatiities and strikes to get haf year expiries starting at 1/2 years. To caibrate the two SABR modes for swaps to cacuate the convexity correction derived in chapter 7.2, we used quoted market data for swaptions. Those swaptions had the standard expiries 1 2, 1, 2,..., 1, 12 and 15 years and were avaiabe for 13 strikes, ranging from ATM-.3% to ATM+2.5% and tenors of 2 years and 1 years, respectivey. We interpoated the market data by spine interpoation to get the voatiities and strikes for options with haf yeary-expiry dates. For the caibration procedure we then foowed chapter

115 8.2. Empirica Study Tabe 8.4.: Absoute Reative Pricing Errors for Cotermina Swaptions with Strikes in {ATM.3%,..., ATM + 2.5%} as of , if the correation structure is caibrated to swaps. P basing on (5L) P basing on (2SC) i max min mean i max min mean Numerica Resuts for Swaptions when Caibrated to CMS spreads As in the case for the caibration to swaption prices we cacuated swaption prices via Monte Caro simuations using 6 runs and 1 steps. For both possibe parameterizations for ρ we got errors simiar to the case of the caibration to swaptions, if we choose the (2SC) parametrization for ρ. This is a bit surprising since in the case of a (5L) parametrization for ρ the caibrated supercorreation matrix has the same characteristics ike the supercorreation matrix obtained by caibrating to swaptions and using the same parametrization. However, as in the case of the caibration to swaptions the (5L) parametrization works much better then the (2SC) parametrization. A pot of the reative pricing errors can be found in the appendix. One reason for the high errors may be the unusua high distance of the estimated correation matrix P, which is used for the simuations, and the mode correation matrix P. Another reason coud be that, different to cotermina swaps, CMS spread options aone do not induce a condition on each entry of P. In chapter we discuss 17

116 8. Impementation and Empirica Study this issue thoroughy Figure 8.2.: This pot visuaizes the difference between Lutz and Schoenmakers & Coffey s (2SC) parametrization, if we caibrate on swaption prices. The difference shows: The (5L) matrix dictates a higher correation of ong-term forward rates and a much stronger decorreation of short-term forwards. The pot was done in Matab. 18

117 8.2. Empirica Study Tabe 8.5.: Absoute Reative Pricing Errors for Cotermina Swaptions with Strikes in {ATM.3%,..., ATM + 2.5%} as of , if P is caibrated to CMS spread options. P basing on (5L) P basing on (2SC) i max min mean i max min mean

118 8. Impementation and Empirica Study Critique on Caibration Soey to CMS Spread Options CMS spreads depend stronger on correation between the underying forward rates than swaps [29], [22] but nevertheess we do not recommend to caibrate to spreads aone. If we consider the genera structure of a 1y/2y spread and δ i.5, which was given in (7.2) as S i,2,4 i+19 t = v i,2,4 (t)ft, =i it is obvious that in each spread ony 2 forward rates are invoved. Now we assume that, we have for each tenor point T i the prices for a range of European options on CMS spreads. Then, for each i it is ony possibe to estimate 2 of the 39 i + 1 possibe correations of the forward rates expiring in i or ater. Therefore, ony the first 2 eft and right main-diagonas of the correation matrices ρ and r are directy affected by the spread dynamics. This means during the caibration on spreads no conditions for the wings are given, that is a quarter of both matrices. The behavior of the outer areas of the matrix have to be extrapoated from the midde part. This can ead to overa ower correations for forward rates which are further apart. This is because, the correation for forward rates with expiries cose to each other goes down rapidy in the beginning and induces, if we do not know any termina minima correation, too ow correations for the out part of the matrix. For caibration to a set of cotermina swaptions on {S i,4 } i the situation is different. According to (2.5) a cotermina swap expiring in T i can be written as S i,4 t = 39 k=i ω i,4 k (t)f k t. This shows, that the swap depends on a forward rates that expire in T i or ater and not, as in the case of a CMS expiring in T i, on ony a sma part. Hence, using swaption prices on a cotermina swap S i,4 we can induces a condition on each entry of both the i-th row and coumn of the correation matrix ρ and r, respectivey. If the correation of the forward rates fa rapidy around the main diagona, a termina minima correation is sti given through the condition on the outer entries. To incorporate CMS spreads in a genera caibration to dynamics impied by the mar- 11

119 8.2. Empirica Study Expiry 2 Expiry Expiry Expiry Figure 8.3.: Here we see which entries of ρ and r, respectivey, are governed by options on cotermina swaps or CMS spreads. On the right, we see the entries which depend on a set of cotermina swaptions with the expiries which are quoted in the market. On the right, we see the entries which depend on a set of CMS spreads options with expires every haf year. Notice the wings the entries without any condition. ket, we suggest a joint caibration to swaptions and CMS spread options, to grantee a reiabe caibration. In the caibration process the options shoud be weighted, say 8% swaptions and 2% CMS spread options, since derivatives on swaps are more iquid then on CMS spreads. 111

120

121 9. Concusion We introduced the SABR-LMM to give a market mode that can describe the dynamics of a number of forward rates in an arbitrage free framework. The Libor-Market mode has its origin in the simpe SABR mode introduced by Hagan [17]. The goa of this work was to caibrate the SABR-LMM, effectivey that means we had to estimate the voatiity structures as we as the correations between the voatiity processes and the forward rates. For the voatiity caibration we kept the dynamics of the singe forward rates as cose to the SABR-modes as possibe. We achieved this with specia caibration techniques to caibrate the time-dependent voatiity functions to capet prices. For the caibration of the correation structure we foowed two approaches. In one approach we used swaps and the other one we used CMS spreads as target products. In both cases we estimated the assumed SABR dynamics depending on the correation matrix P. To estimate the swap dynamics we foowed the cassica approach of freezing swap weights [36], whereas in the case of CMS spreads we used Markovian Projection to simpify the target processes. Here we extended the work of Kienitz & Wittke [22] to our SABR-LMM with time-dependent coefficients. In the empirica part we got supporting evidence for that the caibration to swap dynamics works quite we. This is because we amost obtained the rea prices of swaptions by Monte Caro Simuations. Our target was to capture the assumed simpe SABR swap dynamics in the SABR-LMM as accuratey as possibe. The ow deviation of the swaption prices show that we hit the smie for swaps very we. Now, since the smie is directy inked to SABR dynamics via formua for impied voatiity, this shows we regain the swap dynamics in the SABR-LMM with the desired accuracy. In contrast, the caibration to CMS spread option prices aone did not provide the desired accuracy, because the deviations of simuated swaption prices were too big. We expained possibe reasons for this. Further, we saw that, at east in an environment of high voatiity, the parametriza- 113

122 9. Concusion tion of ρ can have a big impact on the quaity of the caibration. This is what we can concude from that the parametrization of Lutz worked much better than the one of Schoenmakers & Coeffey. In this work we did not take a ook at the hedging performance of the SABR-LMM or, in genera, how hedging works. The SABR-LMM is strongy inked to simpe SABR modes due to its caibration. It woud be interesting to compare the performance of SABR-LMM hedges with hedges done in the simper SABR environment. For exampe hedges of derivatives on swaps, CMS or forwards coud be researched. Note that for those underyings we aready gave an approximation for the SABR dynamics in a SABR-LMM word. So everything is provided. 114

123 A. Appendix A.1. Parameters Obtained from the Caibration on Data as of Tabe A.1.: The Parameters for g and h Function a b c d g h Tabe A.2.: The Parameters k i i k i i k i i k i i k i

124 A. Appendix Tabe A.3.: The Parameters ζ i i ζ i i ζ i i ζ i i ζ i Tabe A.4.: The Parameters of the Submatrices of P, if the (5L) Parametrization is used for ρ Submatrix α β γ η ρ ρ β ρ r λ 1 λ 2 R Tabe A.5.: The Parameters of the Submatrices of P, if the (2SC) Parametrization is used for ρ Submatrix γ η ρ.466. β ρ r λ 1 λ 2 R

125 A.1. Parameters Obtained from the Caibration on Data as of Supercorreation Matrix based on (5L) ρ (5L) r R Figure A.1.: The correation matrices from the caibration for , if we use Lutz (5L) parametrization for ρ. The pot was done in Matab. Supercorreation Matrix based on (2SC) ρ (2SC) r R Figure A.2.: The correation matrices from the caibration for , if we use Schoenmakers & Coffey s (2SC) parametrization for ρ. The pot was done in Matab. 117

126 A. Appendix.5 Reative Pricing Error % 4% 3% Strike 2% 1% 1 2 Expiry 3 4 Figure A.3.: The reative pricing errors for capets. The pricing error for high strikes and short expiries is ceary visibe. The pot was done in Matab. 1 Reative Pricing Error % +1% Strike ATM 1% 2% Expiry Figure A.4.: The reative pricing errors for swaptions of if for ρ the (5L) parametrization is used. The swaption prices are reproduced we. Ony swpations with short expiry and high strikes are probematic. The pot was done in Matab. 118

127 A.1. Parameters Obtained from the Caibration on Data as of Reative Pricing Error % +1% Strike ATM 1% 2% Expiry Figure A.5.: The reative pricing errors for swaptions of if for ρ the (2SC) parametrization is used. The errors ook identica to the one obtained by using the (5L) parametrization. The pot was done in Matab. 119

128 A. Appendix A.2. Parameters Obtained from the Caibration on Data as of Tabe A.6.: The Parameters for g and h Function a b c d g h Tabe A.7.: The Parameters k i i k i i k i i k i i k i

129 A.2. Parameters Obtained from the Caibration on Data as of Tabe A.8.: The Parameters ζ i i ζ i i ζ i i ζ i i ζ i Tabe A.9.: The Parameters of the Submatrices of P for the Caibration to Swaptions, if the (5L) Parametrization is used for ρ Submatrix α β γ η ρ ρ β ρ r λ 1 λ 2 R Tabe A.1.: The Parameters of the Submatrices of P for the Caibration to Swaptions, if the (2SC) Parametrization is used for ρ Submatrix γ η ρ β ρ r λ 1 λ 2 R

130 A. Appendix Tabe A.11.: The Parameters of the Submatrices of P for the Caibration to CMS Spread Options, if the (5L) Parametrization is used for ρ Submatrix α β γ η ρ ρ β ρ r λ 1 λ 2 R Tabe A.12.: The Parameters of the Submatrices of P for the Caibration to CMS Spread Options, if the (2SC) Parametrization is used for ρ Submatrix γ η ρ β ρ r.3.75 λ 1 λ 2 R

131 A.2. Parameters Obtained from the Caibration on Data as of Supercorreation Matrix based on (5L) ρ (5L) r R Figure A.6.: The caibrated correation matrices, if we use Lutz (5L) parametrization for ρ and caibrate to swaptions. Observe how the genera shape reative to 26 has changed. The short-term forward rates are now way ess correated then before. The pot was done in Matab. Supercorreation Matrix based on (2SC) ρ (2SC) r R Figure A.7.: The caibrated correation matrices, if we use the simper Schoenmakers & Coffey s (2SC) parametrization for ρ and caibrate to swaptions. As for the (5L) parametrization the short term forward rates are ess correated then in 26, but the eve of correation is around 3%. This is a resut of the infexibiity of the (2SC) parametrization. The pot was done in Matab. 123

132 A. Appendix Supercorreation Matrix based on (5L) ρ (5L) r R Figure A.8.: The caibrated correation matrices, if we use Lutz (5L) parametrization for ρ and caibrate to CMS spread options. The shape of ρ is amost identica to the shape from the caibration to swaptions. How ever the voatiity/voatiity correation is in genera ower and the surface of R seems a itte bit smoother. The pot was done in Matab. Supercorreation Matrix based on (2SC) ρ (2SC) r R Figure A.9.: The caibrated correation matrices, if we use the simper Schoenmakers & Coffey s (2SC) parametrization for ρ and caibrate to CMS spread options. The forward rates in generay are ess correated compared with the (5L) parametrization. The pot was done in Matab. 124

133 A.2. Parameters Obtained from the Caibration on Data as of Reative Pricing Error.5.5 6% 4% 3% Strike 2% 1% 1 Expiry Figure A.1.: The reative pricing errors for capets. Simiar to the error becomes ower with growing expiry. The pot was done in Matab. 1 Reative Pricing Error % +1.5% +.75% Strike ATM.3% Expiry Figure A.11.: The reative pricing errors for swaptions, if the mode is caibrated to swaptions and for ρ the (5L) parametrization is used. Besides the error peak for cotermina swaptions with expiries of 2 years and 3 years the surfaces ooks simiar to the one from 26. The pot was done in Matab. 125

134 A. Appendix 1 Reative Pricing Error % +1.5% +.75% Strikes ATM.3% Expiry Figure A.12.: The reative pricing errors for swaptions, if the mode is caibrated to swaptions and for ρ the (2SC) parametrization is used. The eve of errors is higher than for the (5L) parametrization. The pot was done in Matab. Reative Pricing Error % +1.5% +.75% Strike ATM.3% Expiry Figure A.13.: The reative pricing errors for swaptions, if the mode is caibrated to CMS spread options and the (5L) parametrization is used for ρ. The surface ooks amost ike the one obtained by caibration to swaps, but the genera eve is a bit higher. The pot was done in Matab. 126

135 A.2. Parameters Obtained from the Caibration on Data as of Reative Pricing Error % +1.5% +.75% Strike ATM.3% Expiry Figure A.14.: The reative pricing errors for swaptions, if the mode is caibrated to CMS spread options and for ρ the (2SC) parametrization is used. The pot was done in Matab. 127

136

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