Calibration of a Libor Market Model with Stochastic Volatility

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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

28

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

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