Piterbarg s FL-TSS vs. SABR/LMM: A comparative study

Transcription

1 Piterbarg s FL-TSS vs. SABR/LMM: A comparative study University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance September 29, 2014

2 A life dedicated to the pursuit of knowledge is a life worth living

3 Acknowledgements I would like to express my gratitude to Dr Jeff Dewynne for his helpful advice and comments on my dissertation.

4 Abstract The LIBOR market model (LMM) is an established method for encoding all at-the-money volatility information for swaptions across all swaption expiries and maturities, however it is not able to recover the volatility smile. As a result, many extensions to the LMM have been proposed with the aim of extending it to encode volatility smile information. Two models of interest are Vladimir Piterbarg s forward LIBOR model with time-dependent skew (FL-TSS), and the SABR/LIBOR market model (SABR/LMM) developed by Riccardo Rebonato et al. FL-TSS assumes that forward rates follow a shifted log-normal diffusion. Forward rate volatility is modelled as a mean reverting process, and no correlation is assumed between the volatility and the forward rate dynamics. SABR/LMM uses the industry standard SABR model to provide accurate analytic prices for European options. By combining SABR and LMM, it s able to bring both forward rates and forward rate stochastic volatilities under the same measure, such that for all underlyings the dynamics are simultaneously valid and complex derivatives can be priced. By constructing both models in this study, we examine and compare various features such as their ability to calibrate to different market states. We investigate how well both models predict future volatility smiles, we explore their implementation details and we make observations on their computational aspects. We find the time-homogeneity that s inherent to the SABR/LMM model allows for better fits and future predictions of swaption implied volatility smiles overall. We find the calibration of SABR/LMM to be more stable and more computationally efficient.

5 Contents 1 Introduction Markets & Pricing of Caplets & Swaptions Notations & Definitions The SABR/LIBOR market model The LIBOR market model Volatility functions Splitting Correlation and Volatility Forward-Forward Correlation The SABR Model The SABR/LMM Model Definition The SABR/LMM Model for Caplets The SABR/LMM Model for Swaptions Implementing SABR/LMM Calibrating g( ) Calibrating h( ) Selecting β and ρ Approximating Σ 0 and V Approximating B Approximating R SABR Piterbarg s term structure of skew forward LIBOR model Introduction A stochastic volatility forward LIBOR model for European Swaptions A simple model A forward LIBOR market model with stochastic volatility Piterbarg s term structure of skew LIBOR market model The FL-TSS model set-up Pricing European swaptions via parameter averaging techniques Effective Skew Effective Volatility i

6 3.5 Calibration Finding λ mkt n,m and b mkt n,m Finding {β(t; n), t 0} n and {σ k (t; n), t 0} n,k Methodology Smile Calibrations Smile Predictions The Experimental Set-up The Dataset SABR/LMM FL-TSS Theoretical Analysis Smile Calibrations Under normal market conditions Under excited market conditions Smile Predictions normal normal normal excited excited excited Results and discussion Intermediate Calibrations SABR/LMM FL-TSS Smile Calibrations Under normal market conditions Under excited market conditions Computational considerations Smile Predictions normal normal normal excited excited excited Conclusion & further research 53 Bibliography 58 Appendix A Data 59 Appendix B Matlab code listing 61 ii

7 List of Figures 2.1 A Doust correlation surface that shows convexity Caplet calibration fits for 23-Nov Caplet calibration fits for 4-Mar Filon-Levy fits for 23-Nov Fits for expression (3.27) Implied volatility smiles generated by calibrated SABR/LMM and FL-TSS models, compared to the market for 23-Nov Implied volatility smiles generated by calibrated SABR/LMM and FL-TSS models, compared to the market for 4-Mar Predicted volatility smiles vs. Actual smile for 1Y expiry and different maturities from a normal market to a normal market Predicted volatility smiles vs. Actual smile for 5Y expiry and different maturities from a normal market to a normal market Predicted volatility smiles vs. Actual smile for 10Y expiry and different maturities from a normal market to a normal market Predicted volatility smiles vs. Actual smile for 1Y expiry and different maturities from a normal market to an excited market Predicted volatility smiles vs. Actual smile for 5Y expiry and different maturities from a normal market to an excited market Predicted volatility smiles vs. Actual smile for 10Y expiry and different maturities from a normal market to an excited market Predicted volatility smiles vs. Actual smile for 1Y expiry and different maturities from a excited market to an excited market Predicted volatility smiles vs. Actual smile for 5Y expiry and different maturities from a excited market to an excited market Predicted volatility smiles vs. Actual smile for 10Y expiry and different maturities from a excited market to an excited market iii

8 List of Tables A.1 Market Caplet data for 23-Nov A.2 Market Caplet data for 04-Mar A.3 g( ) calibrated parameters for 23-Nov A.4 h( ) calibrated parameters for 23-Nov A.5 g( ) calibrated parameters for 4-Mar A.6 h( ) calibrated parameters for 4-Mar A.7 b mkt n,m for 23-Nov-2006 found using Filon-Levy approximation A.8 λ mkt n,m for 23-Nov-2006 found using Filon-Levy approximation A.9 Parameters found for (3.63), for swaption expiry 10Y on 23-Nov A.10 Parameters found for (3.64), for swaption expiry 10Y on 23-Nov A.11 Parameters found for (3.63), for swaption expiry 5Y on 4-Mar A.12 Parameters found for (3.64), for swaption expiry 5Y on 4-Mar iv

9 Chapter 1 Introduction Before volatility smiles existed in the interest rate market, the LIBOR market model [ABM97] was the de facto standard for pricing complex interest rate derivatives. Since then there has been a considerable amount of work done to create models that are able to obtain non-monotonic smiles. These models come in a variety of forms, for example Levy market models [Klu05] or two-state Markov-chain volatility models [RK03] that take into account the overall regime of a market. The largest body of work however is in the area of extending the current LIBOR market model to incorporate stochastic volatility. Stochastic volatility models also come in various flavours, some assume that the underlying forward rate volatility follows a CEV process, others assume a displaceddiffusion process. It is argued in [RRW09] that stochastic volatility LIBOR market models are fairly troublesome to implement, and there are a number of variables that allow tweaking of the model that don t have any real economic meaning. Another way to obtain non-monotonic smiles has been proposed by Riccardo Rebonato in [Reb07] and further extended by Rebonato et al in [RRW09]. The idea is to marry together two industry standard frameworks the LIBOR market model and the SABR model [PSHW02]. The SABR model provides analytical approximations to the true price of European options, it produces stable fitted parameters and it s easy to calibrate. The smile dynamics it describes are fundamentally correct as stated in [RRW09]. The drawback of just using SABR is that it doesn t take into account correlation between forward rates and hence views European options in isolation. This is where the LIBOR market model comes in. On its own it brought forward rates under a single measure so that dynamics were valid simultaneously for all underlyings. Combining it with SABR as in [RRW09] allows both forward rates and forward rate stochastic volatilities to be brought under the same measure so that complex derivatives can be priced. The SABR/LMM model now has to describe many correlation terms such as the forward rate/forward rate volatility correlation 1

10 structure or the volatility/volatility correlation structure. It can be argued that this provides too many degrees of freedom. The aim of this study is to compare and contrast this SABR/LMM model with a stochastic volatility forward LIBOR model developed by Vladimir Piterbarg in [Pit03] and [Pit05a]. The model, known in this dissertation as FL-TSS, assumes that forward rates follow a shifted log-normal diffusion. It models volatility as a mean reverting process and assumes no correlation between volatility and the rate dynamics. Using generalised time-dependant parameters, FL-TSS allows for fast and accurate European option prices. The model links these time-dependent parameters to effective (constant) parameters that describe the smile for each swaption in the swaption cube (expiry maturity strike). Using this link the model can be calibrated accurately to swaptions. We compare the models abilities to calibrate to caplets and swaptions in both normal and excited market conditions (the definition for normal and excited market conditions can be found in [RRW09]). We look at how the models reproduce future implied volatility smiles in different market conditions and we investigate computational aspects and implementation details of both models. In Chapter 2 we describe in detail the SABR/LMM model, and in Chapter 3 we do the same for the FL-TSS model. Chapter 4 then describes the tests that are run and how the tests will be set up. Chapter 5 provides a theoretical comparison of both models which we use to make predictions on the outcomes of the tests. In Chapter 6 we analyse and discuss the results, referring back to Chapter 5 to see if our predictions are correct, and we attempt to explain observations made. Finally in Chapter 7 we give conclusions based on the results of the tests, and suggest further research. 1.1 Markets & Pricing of Caplets & Swaptions A caplet is a European call option on an interest rate, and a floorlet is a European put. Market participants may pay or receive cash flows based on a floating interest rate such as LIBOR caplets or floorlets can be used to hedge the risk on these cash flows. They re not traded directly, instead strings of caplets or floorlets are chained together into products called a cap or a floor. The cap(floor)lets in a cap (floor) typically have the same strike price but different expiries. In most cases the expiries match the underlying LIBOR rate, so for example a ten year cap on three-month LIBOR will generate a portfolio of 39 caplets which will have expiries that range from six months to ten years in steps of 3 months. A derivative known as a swap allows two counterparties to exchange cash flows on instruments they own, where each cash flow is known as a leg. Swaps can be based 2

11 on different types of instruments such as bonds, interest rates or foreign exchange rates. Specifically interest rate swaps allow two counterparties to exchange the interest payments on a fixed rate and variable rate loan. A swaption is an option on a swap. The swaption expiry denotes when the option itself expires and the swap starts. The swaption maturity denotes when the underlying swap makes/receives its final payment from the time it started. They are often referred to in the format expiry x maturity. So a 5Y x 10Y swaption expires in 5 years and contains an underlying swap which has a maturity of 10 years. Swaptions can be exercised as European, Bermudan or American and can come in two basic forms the payer swaption and the receiver swaption. Options on the market are quoted by their implied volatility. Hagan et al [PSHW02] show that the implied Black volatility ˆσ(K, T ) of an option can be found using ( ) z ˆσ(K, f, T ) = A B (1.1) X(z) A = (fk) (1 β)2 2 σ T 0 [ 1 + (1 β)2 24 ln 2 f K + (1 β) ln4 f K + ] (1.2) B = [ 1 + ( (1 β) 2 (σ0 T ) 2 24 (fk) + ρβνσt 0 1 β 4(fK) 1 β 2 X(z) = ln z = ν (fk) 1 β σ0 T 2 ln f K ( ) 1 2ρz + z2 + z ρ 1 ρ ) ] + 2 3ρ2 ν 2 T + 24 (1.3) (1.4) (1.5) where K is the strike, T is the expiry, f is the initial forward rate, σ0 T is the initial volatility, ν is the vol-vol of the underlying rate, ρ is the correlation between the forward price process and the volatility process, and β is the skewness parameter. 1.2 Notations & Definitions The notations and definitions here will be used throughout this dissertation (unless otherwise stated) and describe some of the basic building blocks of both the SABR/LMM and FL-TSS models. Take the following tenor structure, where the sequence of N tenors is approximately equally spaced 0 = T 0 < T 1 <... < T N, τ i = T i+1 T i (1.6) 3

12 where τ i is the time in between each tenor. Next consider a forward rate indexed by i at an arbitrary time t denoted by f i t. f i t = f(t, T i, T i+1 ), i = 1, 2,..., N (1.7) This represents a forward rate at time t resetting at time T i and then paying at time T i + τ i where, in the case of spanning forward rates, T i + τ i = T i+1. The payment is made upon reset of the next forward rate. Reset times are denoted by (1.6). Both FL-TSS and SABR/LMM make use of a discrete set of zero coupon bonds P i t for time t indexed by i. Each bond has an associated f i t. The numéraire used to discount the forward rates can be chosen to be the zero coupon bonds upon which the forward rate is based. This is beneficial as the zero coupon bond does not pay coupons/dividends and its price is strictly positive. terms of its corresponding zero coupon bond is shown in (1.9) The forward rate written in f i t = P i t = P (t, T i ) (1.8) ( ) P i t 1 Pt i+1 1, τ i = T i+1 T i (1.9) τ i P i t can be further defined in terms of just forward rates and τ i+1 in i 1 Pt i = j= τ j+1 f j t Next, the swap rate is defined as the following: (1.10) S n,m (t) = P t n Pt m m i=n+1 τ ip i t (1.11) where n is the first fixing date of the swap (and hence the expiry of the swaption) and m is the last payment date of the swap (hence the maturity of a swaption on this swap). All other definitions and assumptions will be stated in the relevant sections. 4

13 Chapter 2 The SABR/LIBOR market model 2.1 The LIBOR market model The LIBOR Market Model (LMM) is based on a series of no arbitrage conditions between discount bonds or forward rates in a deterministic volatility setting. The model itself depends upon the chosen numéraire, that is, the basic standard by which instrument values are measured. Constructing these no arbitrage conditions means eliminating the covariance between the instrument and numéraire The payoff must be completely independent of the way the instrument has been discounted so that one can obtain a price that doesn t depend on the numéraire that was used. This follows the argument and definitions laid out in [RRW09]. Here, we use the definitions made in Section 1.2 and use further notation to describe LMM and SABR/LMM. We start by defining instantaneous volatility which can be written as σ(t, T i ) = σ i t (2.1) The forward rate i and the forward rate j have an instantaneous correlation which can be written as ρ(t, T i, T j ) = ρ t i,j i, j = 1, 2,..., N (2.2) As discussed in Section 1.2, the numéraire chosen to discount the forward rates ft i in (1.9) is the zero coupon bond in (1.8). These pieces of the framework now need to be tied together to create an expression describing the evolution of forward rates in a deterministic volatility setting. It s important to note that this framework describes a discrete collection of forward rates referenced by a continuous time index. and df i t f i t = µ i (f t, σ t, ρ, t)dt + σ i (t, T i )dz i t (2.3) 5

14 E [ ] dztdz i j t = ρ(t, Ti, T j )dt (2.4) In (2.3) a vector of spanning forward rates and their associated volatilities are denoted by f t and σ t respectively, and ρ represents the matrix of correlations between the forward rates. As [RRW09] explains, (2.3) allows the possibility of different volatility functions for σ i (t, T i ) by inclusion of the superscript i. σ i allows for the definition of a different volatility function per forward rate. If, however, the volatility functions are the same, then the model is said to be time-homogeneous. For this to be the case the volatility function must be of the form in (2.5) where the i superscript is dropped. σ(t, T i ) = σ(t i t) (2.5) This defines σ as being a function of T i t. As in [RRW09] the i superscript is now reintroduced with a different meaning where it represents the dependence on a specific reset time T i, rather than representing a specific volatility function as it denoted previously in (2.1). σ(t i t) = σt i (2.6) This notation emphasises the importance of the expiry T i in the time-homogeneous volatility function it means that for two forward rates, the volatilities are different only because of their differing times to expiry (the value of (T i t)). The drifts µ i (f t, σ t, ρ, t) that are stated in (2.3) are defined later in Section Volatility functions [Reb02] and [Reb04] both explain that (2.7) is a good functional form which satisfies (2.6), where a, b, c and d are constants. There are several justifications for this as outlined in [RRW09]. σt i = (a + bˆτ i )) exp( cˆτ i ) + d ˆτ i = T i t (2.7) (2.7) is square-integrable over the interval 0 and T i. This is important because it allows for closed-formed solutions to be found for the integral of its square, which is used to price caplets and swaptions. The parametrisation also picks out key features of the volatility curve such as lim ˆτ i 0 σi t = a + d (2.8) As ˆτ i goes to zero, a + d represents the instantaneous volatility of the forward rate. Looking at the case where ˆτ i 6

15 lim ˆτ i inf σi t = d (2.9) which allows control over the instantaneous volatility for very long expiries, and shows that the function converges at these high values of ˆτ i. By differentiating to find the maximum of the function, which is found at a value of ˆτ i equal to τ i, we get τ i = 1 c a b (2.10) which shows that the position of the hump in the instantaneous volatility curve can be controlled by a, b and c. This shows that (2.7) allows two types of volatility functions to be modelled humped shape and monotonically decreasing. [Reb02] and [Reb04] explain that a humped shape is important for normal market conditions and a monotonically decreasing volatility correctly models excited conditions. The function (2.7) is able to achieve humped shaped curves and monotonically decreasing curves by using a, b and c to adjust the location of the peak. A monotonically decreasing curve will have a peak at very short expiries. Note that d doesn t effect the position of the peak and therefore only effects the volatility close to 0 and as ˆτ i. [RRW09] and [Reb04] detail the financial justification for having a humped or monotonically decreasing instantaneous volatility curve. For very short expiries, monetary authorities tend to indicate their intentions well before any rate decisions. This effects short-term deposit rates, and as a consequence pins the prices of futures contracts. Therefore in these normal market situations, volatility is low as there aren t any surprises. Looking at longer expiries, the driving factor behind changing rates is long-term inflation. This is also controlled by monetary authorities, and therefore the expectation is that the banks and authorities will be influencing inflation to hit a specific target. Therefore the most uncertainty in rates for normal market conditions, as [Reb04] explains, is in the region of 6 to 18 months (hence for expiries in this period, instantaneous volatilities will be higher). In excited conditions, the lack of any consensus in the decisions that monetary authorities are going to make regarding rates will effect the earliest expiring forward rates, and therefore the volatility for short expiries is high. Hence at the short end the curve is steep, causing the hump to disappear and generating a monotonically decreasing curve. 7

16 2.1.2 Splitting Correlation and Volatility It is now useful to split the stochastic term in (2.3) into two parts a term relating only to the volatility which can be related to caplet prices, and a term relating only to correlation. It s possible to rewrite (2.3) as where df i t f i t m = µ i (f t, σ t, t)dt + σ ik dz k (2.11) k=1 E [dz j dz k ] = δ jk dt (2.12) This has essentially split the volatility of the Brownian increments up into a series m factors, where m N. Another way to look at this is that σ ik represents the weight of the kth factor on the ith forward rate. δ jk represents a value known as the Kronecker delta where δ jk = 0 for j k and δ jk = 1 otherwise. This relationship between σt i and loadings σ ik for time t (denoted by the addition of the t subscript in (2.13)) is given by the caplet pricing condition σt i = m (σ ik,t ) 2 (2.13) k=1 In order for the caplet prices to be correctly calculated, the implied Black volatilities ˆσ i must be related to the volatility functions σt i by Ti 0 (σ i t) 2 dt = ˆσ 2 i T i (2.14) Taking this further, (2.11) needs to be split into a volatility term and a correlation term. By multiplying and dividing the stochastic term by the volatility σ i of the ith forward rate (notice we have dropped the t subscript and will also be dropping the t subscript from σ ik,t because their dependence on t is not important for this derivation) we get df i t f i t = µ i (f t, σ t, t)dt + σ i m k=1 σ ik dz k σ i (2.15) By substituting the caplet pricing (2.13) condition into (2.15) the following expression is obtained By defining b ik to be df i t f i t m = µ i (f t, σ t, t)dt + σ i σ ik dz k (2.16) k=1 m (σ ik ) 2 k =1 8

17 we can write σ ik b ik (2.17) m σik 2 k =1 df i t f i t m = µ i (f t, σ t, t)dt + σ i b ik dz k (2.18) If we take b to be a matrix of size N x m (remembering that m is the number of volatility factors and N is the total number of forward rates), where each element of b is (2.17), [RRW09] states that it can be readily shown that k=1 bb T = ρ (2.19) where ρ is the same as in (2.3). Both (2.18) and (2.19) show that there is an expression for ρ in (2.3) that is independent of volatility, and an expression for σ i which has no references to the correlation. This is important as it allows hedging with caplets. This definition can now be used to link the implied volatility with caplet prices Forward-Forward Correlation As stated in [RRW09], in a deterministic volatility setting the LMM has never placed as much emphasis on the correlation structure as it does the volatility function. When moving to SABR/LMM, more care must be taken to correctly specify the forward-forward correlation matrix. A simple form for the correlation matrix is ρ(t, T i, T j ) = exp[ β T i T j ] t min(t i, T j ) (2.20) where β is a positive constant, and the ith and jth forward rates have reset times denoted by T i and T j. Expression (2.20) shows that the correlation is a function of the time distance between forward rate reset times. The further apart they are, the less correlated they will be. Although this is a desirable feature, the simple model assumes that 1Y and 2Y forwards will have the same correlation as 25Y and 26Y forwards. [RRW09] shows using empirical results that this is simply not the case. We must ensure that the correlation matrix is positive definite and all elements are between 1 and 1. One way to achieve this is shown in [RRW09] using L constants with values a s where s = 1, 2,..., L. Given a 5 x 5 matrix, using 4 constants a 1, a 2, a 3 and a 4 all between 1 and 1, it s possible to form the correlation matrix 9

18 1 a 1 a 1 a 2 a 1 a 2 a 3 a 1 a 2 a 3 a 4 a 1 1 a 2 a 2 a 3 a 2 a 3 a 4 a 1 a 2 a 2 1 a 3 a 3 a 4 a 1 a 2 a 3 a 2 a 3 a 3 1 a 4 a 1 a 2 a 3 a 4 a 2 a 3 a 4 a 3 a 4 a 4 1 (2.21) [RRW09] suggests selecting a s via the following expression a s = exp( β s T ) (2.22) This allows complete flexibility in selecting the right correlation shape by choosing an appropriate β s for each factor in the correlation matrix. An example of a Doust correlation function that will be used in this dissertation can be seen in Figure 2.1. It s clear that when looking down the peak of the surface, it s convex towards the front and concave towards the rear. This means that forwards with shorter reset times will be more correlated than forwards with longer reset times. [RRW09] shows that this is what empirical evidence indicates. Figure 2.1: A Doust correlation surface that shows convexity 2.2 The SABR Model The SABR model is described fully in [PSHW02] and here we will state results and cover the key aspects which are useful to this study. 10

19 Given a forward rate f i t, the SABR model defines its dynamics as df t i = (ft i ) βi σtdz i Qi t (2.23) dσt i = ν i dw Qi σt i t (2.24) E Qi [dztdw i t] i = ρ i (2.25) Here ν i,β i and ρ i are constants and are not functions of time. They are all specific to a forward rate and hence have been given the superscript i. Once f i 0 and σ i 0 are added to expressions (2.23) (2.25), the CEV model with stochastic volatility is fully specified. The forward rate is working under the terminal measure Q i. Under this measure both the volatility and the forward rate have no drift (is a martingale). This is the case when looking at forward rates in isolation, however a different forward rate and it s volatility under this same measure would not be driftless. These drifts are calculated when deriving the SABR/LMM model. We should also note that dz Qi t and dw Qi t are increments of Brownian motion in the Q i measure. Within the SABR model framework, it s impossible for forward rates to interact with each other. This means that the payoff for a path-dependent option cannot be calculated using SABR. Each forward rate works in its own measure, and therefore the dynamics of a yield curve are unable to be described by this model. The following bullets briefly describe the purpose of the 3 constants defined in this model. The i superscript is dropped as the same definition applies to the constants for all forward rates. β : Is an exponent that [RRW09] refers to as chosen by the market. It has 3 common effects on the volatility smile As β goes from 0 to 1, a steepening of the smile can be observed. As β increases, the level of the smile is lowered. The curvature is increased as β goes from 0 to 1. ρ : Is the correlation between the forward rate process and the volatility process. As ρ is reduced, the smile obtains a more negative slope. It also has the small effect of decreasing its curvature. ν : It is observed that increasing ν increases the curvature of the smile. SABR is modelled on a stochastic CEV process which has the specific advantage that negative rates are avoided. However this creates subtle issues for low rates and high volatilities. [RRW09] shows the need to create a zero rate as an absorbing 11

20 barrier and this has implications on the choice of β in the CEV process once a rate hits zero, it stays at zero. 2.3 The SABR/LMM Model Definition The complete SABR/LMM model is displayed here. Given that i = 1, 2,..., N, the joint dynamics for N forward rates and their instantaneous volatilities are defined as df i t = µ i t(f, s, ρ)dt + (f i t ) βi s i t dk i t k i t N F j=1 b ij dẑ j (2.26) ds i t = g(t, T i )dk i t (2.27) = µ k i t (f, s)dt + h i t N V j=1 c ij dŵ j (2.28) N V and N F represent the number of factors driving the volatility and the forward rates respectively where N F N and N V N. s i t is the volatility process where g(t, T i ) is the instantaneous volatility. β i is defined in the SABR model and is a constant value for a given forward rate. b ij is defined in (2.17) and is the correlation between forward rates. ρ is the same value as defined in (2.19). k i t is the stochastic process driving the volatility, and h i t represents the volatility of volatility function. c ij is the correlation between the instantaneous volatilities. µ i t(f, s) and µ k i t (f, s) both represent the fact that no-arbitrage forward rate drifts and no-arbitrage volatility drifts are both dependent on all of the forward rates and volatilities. The model specification is completed by setting N F j=1 N V j=1 b 2 ij = 1 (2.29) c 2 ij = 1 (2.30) (2.31) and defining E[dẑ j dẑ k ] = δ jk dt (2.32) E[dŵ j dŵ k ] = δ jk dt (2.33) E[dẑ j dŵ k ] = x jk dt (2.34) 12

21 where dẑ j and dŵ k are independent Brownian increments and δ jk is the Kronecker delta as defined earlier. x jk is correlation between the forward rate and volatility Brownian increments. These definitions now mean that each forward rate ft i will have an instantaneous CEV volatility s i t. It also means that each volatility factor kt i will have an instantaneous log-normal volatility h i t. 2.4 The SABR/LMM Model for Caplets Here we present a model for caplets which re-expresses the SABR model for f i t under the SABR/LMM framework. The forward rate resets at time T i and pays at T i+1 = T i + τ i as defined previously. The forward rate dynamics are specified under its terminal measure T i as df i t = (f i t ) βi s i tdz i t (2.35) ds i t = g i tdk i t (2.36) dk i t k i t = µ i kdt + h i tdw i t (2.37) E [ dz i tdz j t ] = ρij dt ρ ij 1 (2.38) E [ dw i tdw j t ] = rij dt r ij 1 (2.39) E [ ] dwtdz i j t = Rij dt R ij 1 (2.40) Here β i is the same as in the SABR framework. s i t and kt i are both defined in Section 2.3. µ i k is the volatility drift term and is non-zero for any arbitrary numéraire for all forward rate volatilities. ρ ij is the correlation between forward rates i and j. r ij is the correlation between forward rate volatilities, and R ij is the correlation between forward rates and forward rate volatilities. However, caplets only depend on a single forward rate, its volatility, and the correlation between its forward rate and its forward rate volatility. gt i = g(t i t) and h i t = h(t i t) are time-homogeneous. Their only dependence is the time left until expiry. [RRW09] explains that this is an important property for normal market conditions due to the belief that, in this situation, the market is self-similar and autonomous. 13

22 What about periods of market stress? Both SABR and SABR/LMM assume that the market is in one of two states excited or normal and the market can be described only in one of these states at a time. The model doesn t support regime switching between pricing of options with different expiries. In times of market excitement short expiry volatilities will be high and medium expiry volatility will be higher than normal. If this model state is then applied in a normal market, short expiry volatilities will be a lot higher than expected, the hump in the volatility curve will be closer to lower expiries. This is a shortcoming of time-homogeneous models. Riccardo Rebonato and Dherminder Kainth propose a solution to this problem, described further in [RK03], which suggests having a two stage Markov-chain with a transition probability that switches the model from one regime into another. The same time-homogeneous functional form described in (2.7) is a good definition for the functions g( ) and h( ) above for the reasons discussed in Subsection 2.1. and g(τ i ) = (a + bτ i ) exp( cτ i ) + d, τ i = T i t (2.41) h(τ i ) = (α + βτ i ) exp( γτ i ) + δ (2.42) Completing the definition, using the time-homogeneous forms above, the process for kt T and s T t can be written as and k T t [ t = k0 T exp ( 12 )] h(t s)2 ds + h(t s)dw s (2.43) 0 [ t s T t = gt T kt T = gt T k0 T exp ( 12 )] h(t s)2 ds + h(t s)dw s 0 (2.44) 2.5 The SABR/LMM Model for Swaptions The forward rate dynamics for the full SABR/LMM model is presented here. df i t = µ i tdt + (f i t ) βi s i tdz i t (2.45) ds i t = g i tdk i t (2.46) dk i t k i t = µ k i t dt + h i tdw i t (2.47) where the following expectation are repeated again here for ease of reference: 14

23 E [ ] dztdz i j t = ρij dt (2.48) E [ ] dwtdw i j t = rij dt (2.49) E [ ] dztdw i j t = Rij dt (2.50) As previously stated i and j act as indices to the appropriate forward rates, dz is the process driving the forward rate and dw is the process driving the forward volatility and ρ, r and R are part of the super correlation matrix P. [ ] ρ R P = R T (2.51) r The price of a caplet depends on the following 3 things A single forward rate, its volatility and the correlation between the forward rate and its volatility. Swaptions however depend on multiple forward rates and multiple volatilities, therefore the model needs to capture the correlations between these forward rates (ρ ij ) and the correlation between these volatilities (r ij ). Correlations across forward rates and volatility also need to be captured (R ij where i j). SABR market prices of caplets provide data for the diagonal of R ij. The matrices ρ, r and R can be parametrised in the following way as discussed extensively in [RRW09] ρ ij = η 1 + (1 η 1 ) exp[ λ 1 ( T i T j )] (2.52) r ij = η 2 + (1 η 2 ) exp[ λ 2 ( T i T j )] (2.53) R ij = sign(r ii ) R ii R jj exp [ λ 3 (T i T j ) + λ 4 (T j T i ) +] (2.54) where η 1, η 2, λ 1, λ 2, λ 3 and λ 4 are all constants. Next recall the definition for the swap rate S n,m (t) in (1.11) with expiry n and maturity m. [RRW09] assumes the SABR dynamics of S n,m (t) are ds n,m (t) = S n,m (t) B Σ t dz t (2.55) dσ t = V dw t Σ t (2.56) E[dZ t dw t ] = R SABR dt (2.57) If we are able to find the initial values in the dynamics above, then we can use the SABR model to give prices of swaptions. 15

24 2.6 Implementing SABR/LMM The following subsections explain how to calibrate g( ) and h( ) for caplets. Then they explain how to approximate the four SABR values for swaption based prices (shown in (2.55) and (2.57)) in terms of the forward rate parameters g i t(τ i ), β i, the super correlation matrix P, h i t(τ i ) and f i 0 for i = n,..., m 1. The four SABR values are: The exponent B Σ 0, the initial swap rate volatility R SABR the correlation between the swap rate and the volatility of the swap rate V the vol-vol of the swap rate Calibrating g( ) In order to find the parameters for the deterministic function in (2.41), we have to determine an equivalent value in the SABR model to calibrate to. To do so we write the caplet pricing equation (2.14) as T ˆσ T 2 T = (k T ) 2 g(t u) 2 du (2.58) 0 where the T superscript on k implies taking the forward rate volatility (k) for the forward rate with expiry T. By defining the root mean square of g( ) as Then (2.58) can be rewritten as 1 ĝ(t ) = T k T = T 0 g(t u) 2 du (2.59) ˆσ T ĝ(t ) (2.60) Expression (2.60) doesn t hold in a stochastic volatility situation, so in order to determine k T 0 it s important to note that E[σ(t)] = σ 0 (2.61) The reason for this is because, for SABR, the stochastic volatility is equal to [ 1 σ(t) = σ 0 exp 2 So for (2.61) to be true, we must have t 0 16 σ 2 sds + t 0 σ(s)dz s ] (2.62)

25 [ ( 1 E exp 2 t 0 σ 2 sds + t 0 σ(s)dz s )] = 1 (2.63) Hence it follows that ĝ( ) should be chosen such that it matches as closely as possible σ i 0 at time t = 0 for the SABR volatility σ i s. discrepancy X 2 should be minimised X 2 = N [ σ i 0 ĝ(t i ) ] 2 i Therefore the sum of the squared Where the minimisation is over all caplet expiries and the ĝ parametrisation is 1 ĝ(t i ) = Calibrating h( ) T i Ti 0 (2.64) [(a + bτ i ) exp( cτ i ) + d] 2 dτ i (2.65) Note here that the superscript T or T i denotes a parameter for a specific forward rate with reset time T or T i. As stated in [RRW09] it was previously suggested that the volatility of the volatility h T t could be parametrised as (2.42) and could simply be calibrated to ν T i by minimising its squared difference with the root mean squared value of h T t. Where the root mean square of h t is defined as 1 ĥ t = T t 0 (h s ) 2 ds (2.66) [RRW09] shows that it is incorrect to calibrate h t in this way because it actually matters when the vol-vol occurs. To say when the vol-vol occurs means that it makes a difference to the option price when high or low periods of vol-vol occur. The terminal distribution of the forward rates isn t uniquely determined by the terminal volatility or the average vol-vol. For instance if the volatility is low and the vol-vol is high, [RRW09] suggests that this stochasticity goes to waste. For another example, imagine a flat volatility, a high concentration of vol-vol would essentially add curvature at the point where it occurs, creating skew and having a small effect on the option price. Chapter 5 in [RRW09] shows a different way to match SABR and SABR/LMM parameters. The matching of the two models assumes that, since the same errors are made in both approximations, then the errors will cancel out to give accurate relationships. The proof detailed in [RRW09] yields the expression ν T SABR = k 0 σ 0 T ( T 1/2 2 g(t) 2 ĥ 2 t tdt) (2.67) 0 17

26 To help with the implementation of this calibration, (2.67) can be modified to νsabr T = 1 ( T 1/2 2 g(t) 2 ĥ 2 t tdt) (2.68) ĝ(t )T 0 Looking at (2.68) it s clear that in the integral, the time when the vol-vol occurs is taken into account. This allows the implementation to control when the vol-vol occurs so that an accurate representation of νsabr T can be made Selecting β and ρ Specifically for caplets it can be assumed that ρ LMM ρ SABR. The exponent β is a parameter which is chosen by the market and it is therefore reasonable to assume that β LMM = β SABR Approximating Σ 0 and V Approximations for Σ 0 and V both rest on an approximation of the instantaneous swap rate volatility S t. This can achieved by using a freezing strategy described in [Reb02] and [RRW09] where further details and derivations can be found. The resulting expression is where S t = k,m=1,n j W 0 k W 0 ms k t s m t ρ k,m (2.69) W t k = ω k (f k t ) β k (S n,m ) B (2.70) and the swap-rate between two points n and m is denoted by S nm and defined in (1.11). ω i represents a weighting for a particular forward rate i. τ i Pt i+1 ω i (t) = m 1 i=n τ (2.71) ipt i+1 where τ i is the time spacing between forward rates. In this thesis, 6M LIBOR forwards are used, therefore τ i = 0.5. Using these definitions, the initial swap rate volatility is defined by (2.72) and the volatility of volatility is defined by (2.73). Again, the proof of these can be found in [RRW09]. V = 1 Σ 0 T Σ 0 = 1 ( T ) ρ ij Wi 0 T W j 0ki 0k j 0 gtg i j t dt 0 2 i,j i,j ( T ) ρ ij r ij Wi 0W j 0ki 0k j 0 0 gtg i j ˆ t h ij (t) 2 tdt (2.72) (2.73) 18

27 2.6.5 Approximating B B represents the swap rate exponent. It is defined by B = ω k β k (2.74) k=1,n j [RRW09] suggests that reason for this is heuristic because the sum of CEV values with exponent β is generally not a CEV value with the same exponent. However [RRW09] suggests that for a log-normal case this approximation is generally good, and for the normal case it s exact. This suggests that for a value of CEV β between 0 and 1 the approximation should be as good as the log-normal case, and it will get better as β Approximating R SABR R SABR is the correlation between the swap-rate and its volatility. found in [RRW09] that Proof can be R SABR = i,j Ω ij R ij (2.75) where Ω ij = 2ρ ijwi 0 Wj 0 k0k i j T 0 0 gi g j hij ˆ (t) 2 tdt (2.76) (V Σ 0 T ) 2 19

28 Chapter 3 Piterbarg s term structure of skew forward LIBOR model 3.1 Introduction The log-normal forward LIBOR model was the first model that made it easy to calibrate ATM swaption volatilities for all expiries and maturities. It became the de facto standard, but (as mentioned in the introduction) a major issue was that it was unable to reproduce the volatility smile all swaptions of the same maturity and expiry but of different strikes had the same Black volatility which isn t what is seen in the market. There have been many attempts to encode volatility skew information, but the major building block of Piterbarg s model has been the Stochastic Volatility Forward LIBOR model (from here on known as FL-SV). [AA02] and [ABR01] both present the FL-SV model. Its main features are the application of a local stochastic volatility function, which is time-independent, that is applied to all forward LIBOR rates. Higher volatility of this stochastic variance process means greater curvature of the volatility smile. This stochastic volatility component is the same for all LIBOR forwards regardless of its maturity. This leads to the issue that FL-SV generates volatility smiles with very similar curvatures which again is not consistent with the market, therefore FL-SV is unable to match all volatility smiles for the whole swaption grid. In [Pit05a] Piterbarg proposes a new model which from here on will be called FL-TSS (term structure of skews). This model allows skews to vary across forward LIBOR rates and times, however the same stochastic volatility process is used for all forwards. Piterbarg explains that it is possible to use swaption-specific volatility drivers, however he argues in [Pit05a] that this is not necessary. In [Pit05a] Piterbarg goes on to relate time-dependent volatilities and skews to model term volatilities and skews. This relation is important as it provides a direct 20

29 relationship between the time-dependent slope of a local volatility function and the total amount of skew that the model itself produces. 3.2 A stochastic volatility forward LIBOR model for European Swaptions Here we use the definitions in Section 1.2, in particular that of the swap rate S n,m (t) in (1.11). Reminding the reader that n is the swaption expiry, m is the swaption maturity and i is an index to an arbitrary forward rate. T n is therefore the first fixing date and T m is the last payment date. Looking at the swap rate measure Q n,m, equation (3.1) is the numéraire. Under this measure the swap rate S n,m (t) is a martingale, and Brownian motions under this measure are denoted by dw n,m. N t = m i=n+1 τ i P i t τ i = T i+1 T i (3.1) where P i t is defined by function (1.8). Under this swap rate measure, Q n,m, a European swaption can be seen as a European option on the underlying rate S n,m. The price of the swaption is where K is the strike A simple model [ ] (Sn,m (T n ) K) + Swaption n,m (t) = N t E N Tn (3.2) A possible method to price Swaption n,m is to only look at the dynamics of S n,m. The model is coined simple because it only describes the term distribution of a single swap rate, and not the evolution of the rate curve as a whole. Here it s modelled as a displaced diffusion process with stochastic volatility. Piterbarg s paper [Pit05a] follows the method used in [AA02] which is to describe the distribution of the swap rate with 3 parameters: b: The swap rate skew λ: The stochastic volatility η: The volatility of variance The swap rate is expected to follow the process 21

30 ds n,m (t) = λ(bs n,m (t) + (1 b)s n,m (0)) z(t)dw n,m t dz(t) = θ(z 0 z(t))dt + η z(t)dv t z(0) = z 0 dv, dw = 0 (3.3) The dynamics of the swap rate through time are described by this model, however the only relevant aspect is its terminal distribution. dv t is a scalar Brownian motion that is also working under the swap rate measure Q n,m. The zero correlation is there to maintain analytic tractability and to ensure that the distribution of z(t) is the same under all annuity measures. To ensure z(t) > 0 we make the requirement that θ, z 0, η > 0. The model is calibrated to each swap rate S n,m which results in a grid of triplets which is generally referred to as the swaption grid. Each S n,m has associated with it a triplet {λ n,m, b n,m, η n,m } N n,m=1 where the relevant grid points are at n + m < N. Since the model is calibrated to each smile across all strikes, expiries and maturities, the swaption grid encodes all available market information in each triplet grid point. Taking a closer look at the parameters of the model, [Pit05a] states that λ is similar to, but not equal to the Black volatility. They do become equal, however, when η = 0 and b = 1; η controls the curvature of the smile and b controls the slope specifically of the at-the-money volatility smile. θ is a global parameter and is defined as the strength of mean reversion of variance. It controls how fast the smile flattens out as time increases. A good choice of θ means that η n,m can be chosen as constant η, therefore rather than having triplets for each grid point, they will be tuples {λ n,m, b n,m } with constant η. The FL-TSS model developed later forms a relation between these parameters and time dependent skew and volatility, which is why these parameters are important. They will be used for calibration of the resulting FL-TSS model A forward LIBOR market model with stochastic volatility The simple model only describes each swap rate separately, but many exotic interest rate derivatives require the full term structure to be present in order to price and adequately risk-manage. Here the LIBOR market model with stochastic volatility (FL-SV) will be formally introduced. A definition for spanning LIBOR rates is first constructed using a zero coupon bond with price P i t at time t and pays $1 at time T i. The same definitions in Section 22

31 1.2 for the forward rate f i t (indexed by i) and zero coupon bonds are used here. We model the LIBOR rates under some measure P which is further detailed below. The dynamics of the spanning LIBOR rates can be described as df n t = (βf n t + (1 β)f n 0 ) z(t) n = 1,..., N 1 K ( ) σ k (t; n) z(t)µk (t; n) + dwt k k=1 (3.4) with the stochastic variance process z(t) defined by Here: dz(t) = θ(z 0 z(t))dt + η z(t)dv t (3.5) z(0) = z 0 (µ k (t; n), t T n ) k=1:k n=1:n 1 are K-dimensional drifts that are specific to the measure P and ensures that there is no arbitrage present in the model. In this forward measure P under numéraire P n+1 t we know that µ k (t; n) = 0; dw t = (dwt 1,..., dwt K ) is a K-dimensional Brownian motion which is in the forward measure P and is completely independent of dv t ; (σ k (t; n), t T n ) n=1:n 1 k=1:k are the instantaneous volatility functions; Under the FL-SV model, S n,m approximately follows the dynamics ds n,m (t) = σ(t)(βs n,m (t) + (1 β)s n,m (0)) z(t)du t (3.6) where σ(t) is a volatility function dependent on time and U t is a Brownian motion under the swap rate measure Q n,m. Although this looks very similar to (3.3) it s crucial to remember that (3.3) is for a specific swap rate, therefore β and η in (3.6) are global parameters which means all swap rates will have the same values. It may be realistic to assume η is the same for all swap rates given a good choice of θ, but it s not realistic to assume β is the same for all swap rates. This means that FL-SV cannot reproduce volatility smiles for all maturities and expiries. The model needs to be improved to take into account the change of the swaption skews between different maturities and expiries. 23

32 3.3 Piterbarg s term structure of skew LIBOR market model To give some intuition behind this, the main goal is to make the FL-SV model more flexible so that the changeability of swaption skews are accounted for. This is done by assuming a time-dependent skew which is implemented by taking {(β(t; n), t 0)} N 1 n= The FL-TSS model set-up The dynamics of FL-TSS given this new time dependent β is df n t = (β(t; n)f n t + (1 β(t; n))f n 0 ) z(t) n = 1,..., N 1 K ( ) σ k (t; n) z(t)µk (t; n) + dwt k k=1 (3.7) The dynamics of df t n are in the forward measure for which Pt n+1 is the numéraire, therefore (3.7) can be simplified. ft n is a martingale under the forward measure hence the drift µ k (t; n) becomes zero yielding the expression df n t = (β(t; n)f n t + (1 β(t; n))f n 0 ) z(t) n = 1,..., N 1 K k=1 σ k (t; n)dw k,n+1 t (3.8) In [Pit05a] Piterbarg develops an approximation to European option values that is accurate and speedy to calculate which helps with calibration of the FL-TSS model. Calibration involves calibrating to the market-inferred parameters {λ mkt n,m, b mkt n,m}. This means the model parameters {λ mod n,m, b mod n,m } need to be estimated, therefore a relation to connect them to the time dependent β(t) and σ k (t) in (3.8) is required. The first step is to derive an approximation for the dynamics of the swap rate S n,m (t) under the swap measure Q n,m that causes S n,m (t) to be a martingale. As defined in Section 1.2 n is the expiry of the swaption and m is the maturity. The dynamics are stated as where ds n,m (t) = (β(t; n, m)s n,m (t) + (1 β(t; n, m))s n,m (0)) z(t) K σ k (t; n, m)dw k,n,m t (3.9) k=1 24