On the Calibration of the SABR–Libor Market Model Correlations
Master’s Thesis
Dr. Elidon Dhamo Christ Church College
University of Oxford
Submitted in Partial Fulfillment for the MSc in
Mathematical Finance
September 2011
To Migena ...
Abstract This work is concerned with the SABR-LMM model. This is a term structure model of interest forward rates with stochastic volatility that is a natural extension of both, the LIBOR market model (Brace-Gatarek-Musiela [1997]) and the SABR stochastic volatility model of Hagan et al. [2002]. While the seminal approximation formula (developed by Hagan et al. [2002]) to implied Black volatility using the SABR model parameters allows for a successful calibration of each forward rate dynamics to the volatility smile of the respective caplets/floorlets, an adequate calibration of the rich correlation structure of SABR-LMM (correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great interest in practice. Although widely used for calibration, it is well known that swaptions’ volatilities carry only little information about correlations among the forward rates. As practically successful for the classical LMM, desirable would be to take the market swap rate correlations into account for the model calibration. In this study we develop a new approach of calibrating the model correlations, aiming at incorporating the market information about the forward rate correlations implied from more correlation-sensitive products such as CMS spread derivatives, in which also swap rate correlations are involved. To this end we derive a displaced-diffusion model for the swap rate spreads with a SABR stochastic volatility. This we achieve by applying the Markovian projection technique which approximates the dynamics of the basket of forward rates, in terms of the terminal distribution, by a univariate displaced-diffusion. The CMS spread derivatives can then be priced using the SABR formulas for the implied volatility, taking the whole market smile of CMS spread options into consideration. For the ATM values in the payoff measure of the projected SDE we use a standard smile-consistent replication of the necessary convexity adjustment with swaptions. Numerical simulations conclude the work, giving a comparison between this method and the classical one of calibrating the model correlations to swaption volatilities. Furthermore, we study the performance of different parameterizations of the correlation (sub-)matrices.
v
Acknowledgements First of all, I would like to thank Dr. Christoph Reisinger for agreeing to supervise this thesis, his support and his encouragements. I would like to express my gratitude towards my former employer, d-fine GmbH, for offering me the opportunity to take part in the Mathematical Finance course at the University of Oxford, and for providing financial support. Last but not least I am particularly indebted to my family for their understanding, their great moral support and their patience over the numerous weekends I did not spend with them.
vii
Contents Introduction
1
Chapter 1. Forward Libor and Swap Market Models
4
1.1
A Review of the Classical Libor and Swap Market Models 1.1.1 Libor Dynamics Under the Forward Measure . . . . 1.1.2 Valuation in LMM . . . . . . . . . . . . . . . . . . 1.1.3 Covariance and Correlations in LMM . . . . . . . . 1.1.4 Swap Rate Models and Measures . . . . . . . . . . 1.1.5 Incompatibility Between the LMM and the SMM .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
4 7 8 9 10 11
1.2
The Convexity Adjustment and CMS Derivatives . . . . . . . . . . . 1.2.1 Constant Maturity Swaps and Related Derivatives . . . . . . . 1.2.2 Valuation of CMS Derivatives . . . . . . . . . . . . . . . . . .
13 14 16
1.3
Parameterization and Calibration . . . . . . . . . . . . . . . . . . . . 1.3.1 Parametric Forms of the Instantaneous Volatilities . . . . . . . 1.3.2 Calibration to the Cap/Floor Market . . . . . . . . . . . . . . 1.3.3 The Structure of Instantaneous Correlations . . . . . . . . . . 1.3.4 Calibration of LMM Correlations to Swaptions Volatilities . . 1.3.5 Calibration to Correlations Implied From CMS Spread Options
22 23 24 25 30 30
Chapter 2. The SABR Model of Forward Rates
32
2.1
General Model Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.1 The Time-Homogeneous Model . . . . . . . . . . . . . . . . . 34 2.1.2 Joint Dynamics of the SABR Forward Rates and Their Volatilities 35
2.2
Valuation in the SABR Model . . . . . . . . . . . . . . . . . . . . . .
Chapter 3. Pricing CMS Derivatives in SABR
36 37
3.1
The Markovian Projection Method . . . . . . . . . . . . . . . . . . .
37
3.2
A Displaced SABR Diffusion Model for CMS Derivatives . . . . . . . 3.2.1 Projection of CMS-Spreads to Displaced SABR Diffusion . . .
38 38
ix
x
Contents 3.2.2
Pricing of CMS-Spread Options in a SABR Displaced Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 4. The SABR-LMM Model and Its Calibration
45 48
4.1
SABR–Consistent Extension of the LMM and Its Calibration . . . . .
48
4.2
Calibrating the Volatility Process . . . . . . . . . . . . . . . . . . . .
50
4.3
The SABR Correlation Structure . . . . . . . . . . . . . . . . . . . .
51
4.4
Calibration of the SABR–LMM Correlations to Swaption Implied Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5
Calibrating to Correlations Implied From CMS Spread Options
. . .
55
4.6
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Chapter 5. Conclusion and Outlook
59
Appendix A. Classical Models and SABR-LMM
60
A.1 Valuation in LMM . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
A.2 Swap Rate Dynamics and the Choice of Numeraire . . . . . . . . . .
63
A.3 Valuation in the Log-Normal Swap Market Model . . . . . . . . . . .
64
A.4 Drift Approximation in LMM and Simulations . . . . . . . . . . . . .
65
A.5 SABR Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . .
66
Appendix B. Calibration Details
69
B.1 Bootstrapping the Market Data . . . . . . . . . . . . . . . . . . . . .
69
B.2 Parameterization of SABR–LMM and Its Calibration . . . . . . . . . B.2.1 Parameterization of the SABR–LMM Model . . . . . . . . . . B.2.2 Calibration Procedure . . . . . . . . . . . . . . . . . . . . . .
71 71 72
Bibliography
77
Introduction While the Brace-Gatarek-Musiela (BGM) or Libor1 Market Model (LMM), based on the assumption that forward term rates follow lognormal processes under their corresponding forward measure, has established itself as a benchmark model for pricing interest rate derivatives, it is less successful in recovering other essential characteristics of interest rate markets, particularly volatility skews and smiles. The presence of these volatility skew and smiles in the market, however, indicates that a pure lognormal forward rate dynamics is not appropriate. In the last decade several extensions of the BGM model have been proposed, in which various versions of the volatility structures of forward Libor rates have been designed to match the observed volatility smile effects in the market. The set of extensions covers local volatility, jump-diffusion and stochastic volatility models with and without time-dependent parameters. The calibration procedures in most of these models are complicated and computationally expensive, and are performed on a bestfit basis. One of the most successful and popular extensions of the LMM, the SABR2 model, models the forward rate process under its forward measure using a correlated lognormal stochastic volatility process. Its success is indebted to two main properties of the model: the crucial property of taking into account the quality of prediction of the future dynamics of the volatility smile, meeting the observations from the market reality, and the seminal asymptotic expansion formula, developed by Hagan et al. [2002], to approximate the implied Black volatility using the SABR model parameters. Hence, prices of options, such as caps and floors, can be calculated using the well known Black pricing framework but taking the volatility smile surface via the SABR parameters into account. The SABR model and the LMM, although modeling the same assets, ”do not directly talk to each other”3 . The SABR does not link the snapshots of the caplet smiles into well-defined joint dynamics. To overcome this Rebonato [2007] introduced a natural extension of the LMM, the SABR-LMM, that recovers the SABR caplet prices almost exactly for all strikes and maturities. The dynamics of the volatility in this model is chosen so as to be consistent across expiries and to make the evolution of the implied volatilities as time-homogeneous as possible. While the approximation to implied Black volatility using the SABR model parameters allows for a successful calibration of each forward rate dynamics to the volatility 1
Libor = London Inter-Bank Offered Rates Launched by Hagan et al. [2002] 3 Rebonato [2007] 2
1
Introduction
2
smile of the respective caplets and floorlets, an adequate calibration of the rich correlation structure of SABR-LMM (comprising correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great interest in practice. Although widely used for calibration, it is well known that swaption volatilities carry only little information about correlations among the forward rates4 . Facing the richness of the correlation structure of SABR-LMM, the need for a calibration approach to more correlation-sensitive products is obvious and of wide practical interest. As already successfully applied for the classical LMM5 , it is desirable to additionally take the swap rate correlations into account for the model calibration, which consistently are to be implied from the market prices of appropriate products. A broadly known and traded class of interest rate derivatives meeting these requirements, particularly incorporating information about the swap rate correlations, is the one of Constant Maturity Swap (CMS) spread derivatives. While valuation (in all the mentioned forward rate models) of these products is typically done straight-forwardly by Monte-Carlo simulation, the calibration of the models to these products, by finding accurate and fast analytical approximations to reproduce the prices of these instruments, has been always subject to research6 , even within the simpler LMM framework7 .
Scope of the Present Work and Contribution In this work we develop a novel approach for the calibration of the rich correlation structure in the SABR-LMM model of forward rates. Given the reasons above, our scope is to extract the information about the forward rate correlations from the market prices of correlation-sensitive derivatives, such as CMS spread options, and fit the model correlation parameters to those. To this end the derivation of analytical pricing formulas for these products in the SABR framework is necessary. The Markovian projection (MP) technique8 is an effective technique to volatility calibration that seeks to optimally approximate a complex underlying process with a simpler one, keeping essential properties of the initial process, and is, in principle, applicable to any diffusion model. Starting with the SABR-LMM we apply the MP technique to the CMS spreads and derive a displaced–diffusion SABR model for the spread between the swap rates with different maturities. To achieve this we adapt the recent work of Kienitz-Wittkey [2010], carried out in a SABR swap rate framework, to our SABR-LMM model. Consequently, we can price the CMS spread options in the resulting SABR swap spread model by making use of the seminal SABR formula of Hagan et al. [2002]. In this way we can calibrate the SABR swap spread model parameters to the market implied (normal) volatilities of the corresponding CMS spread options. For the ATM 4
We refer at this point to the works of Alexander [2003], Brigo-Mercurio [2007], Rebonato [2002], Schoenmakers [2002, 2005], Schoenmakers-Coffey [2003]. 5 B¨ orger-van Heys [2010]. 6 Antonov-Arneguy [2009], Castagna-Mercurio-Tarenghi [2007], Lutz [2010], Kienitz-Wittkey [2010], etc. 7 Belomestny-Kolodko-Schoenmakers [2010], B¨ orger-van Heys [2010], etc. 8 MP has been introduced in this context by Piterbarg [2003, 2005a,b] and formalized in Piterbarg [2007].
Introduction
3
values in the expiry forward measure of the projected SDE (i.e. expiry time of the corresponding CMS spread option) we use a standard smile-consistent replication9 of the necessary convexity adjustment with swaptions. By this means we shall be able to implicitly retrieve the important information about swap rate correlations contained in the market smile of CMS spread options and embed it into the SABR-LMM model correlations. This work is concluded with the numerical implementation of this new calibration procedure and some numerical simulations. We shall discuss different parameterizations of the sub-matrices of the model correlation matrix, in particular the Doust parameterizations, and study the performance of this calibration approach, in terms of pricing errors for swaptions, in comparison to the approach of calibrating the model correlations to swaptions’ implied volatilities, given in Rebonato [2007]. The performed simulations shall also illustrate the effectiveness and robustness of this approach, and provide information about expected and possible drawbacks.
Outline The thesis is organized as follows. In the first chapter we review the classical forward LIBOR and swap rate market models. Particular focus is set on the introduction of convexity and the different approaches to carry out the convexity correction. We incorporate these methods to the pricing of CMS derivatives, in particular, the CMS caps and spread options. We also describe the calibration of LMM and introduce the different parameterizations of the correlation matrix. At the end of the chapter a recent approach to imply correlations from CMS spread options is presented. The second chapter is devoted to introduction and general properties of the SABR model. The application of the Markovian Projection (MP) method to the basket of forward rates is carried out in the third chapter. Here we derive a displaced-diffusion model for the swap rate spread with a SABR stochastic volatility. For the pricing of CMS spread caplets in the payoff forward measure we use a smile-consistent replication of the convexity adjustment for the ATM spreads via swaptions, and apply the Hagan et al. [2002] formulae. The SABR-LMM model is introduced in the fourth chapter which is mainly concerned with the parameterization and the calibration of the model. Two different approaches to correlation calibration are presented. Numerical simulations give a comparison between the these approaches, taking different parameterizations of the sub-matrices of the correlation matrix into account. We conclude the work by presenting a short summary of our analysis, and give an outlook of possible future directions of the discussed topics.
9
According to Hagan [2003].
Chapter 1
Forward Libor and Swap Market Models
1.1
A Review of the Classical Libor and Swap Market Models
Over the past two decades the Brace-Gatarek-Musiela [1997] model (BGM) has established itself as a benchmark model for pricing and risk managing interest rate derivatives. It is based on the assumption that the forward rates follow lognormal processes with deterministic (time-dependent) volatilities under their corresponding measures, and it is widely known as the lognormal forward Libor Market Model (LMM). The popularity of this model is indebted to its compatibility with the seminal Black model which establishes a direct relationship between caplets’ prices and local (implied) volatilities of forward rates and constitutes the standard market convention for quoting benchmark instruments. While the LMM model has established a standard for incorporating all available atthe-money information, it is less successful in recovering other essential characteristics of interest rate markets, particularly volatility skews and smiles1 . Various extensions of the LMM model, designed to incorporate skew and smile effects, have been proposed. The set of extensions covers local volatility, jump-diffusion and stochastic volatility models with and without time-dependent parameters. In the next chapters we shall discuss one of the most successful extensions of the LMM, the SABR model, which enjoys increasing popularity among the practitioners and academics alike, again due to its compatibility with the Black ’s framework. Nevertheless, one of the biggest challenges in using these models is the calibration of the forward rate correlations which are not covered by the caplets markets, but rather incorporated in other benchmark instruments, such as European swaptions or CMS spread derivatives. While the valuation in all these forward rate models is 1 The volatility tends to rise if the option is out of the money. This results in the so called volatility smile describing the fact that implied Black volatility is strike-dependent.
4
5
Chapter 1. Forward Libor and Swap Market Models
typically done straight-forwardly by Monte-Carlo simulation, calibration by finding accurate and fast analytical approximations to prices of these benchmark instruments has always been subject to research. In this chapter we briefly describe the construction of forward Libor models as given in Brace-Gatarek-Musiela [1997] and Jamshidian [1997], whereby we follow the approach proposed by Musiela-Rutkowski [2005]. Assuming, for the time being, that there are no smile effects present in the interest rate markets, the formal model setup we present here is based on assumptions made in Musiela-Rutkowski [2005]. In the following sections we shall present some of the mostly used parameterizations and calibration methods for the forward rate volatilities and their correlations, which will prepare the ground for introducing the SABR-LMM model in the next chapters and its calibration approaches. We shall also briefly mention the lognormal swap rate model (SMM), putting emphasis on the incompatibility between the two models. Furthermore, a separate section is dedicated to the approximation and pricing of CMS derivatives via the convexity correction technique which paves the way to calibrating the forward rate correlations to the prices of CMS spread derivatives. Let T ∗ > 0 represent a fixed time horizon. Given a filtered probability space ∗ Ω, {Ft }t∈[0,T ∗ ] , PT which satisfies the basic assumptions made in Musiela-Rutkowski ∗ [2005], let {WTt }t∈[0,T ∗ ] denote a d-dimensional standard Brownian motion (Wiener ∗ process) and assume that the filtration {Ft }t∈[0,T ∗ ] is the usual PT −augmentation of ∗ the filtration generated by {WTt } (cf. Hunt-Kennedy [2004]). In the given probability space an interest rate system formally consists of a system of zero-coupon bonds B = {B(t, T ) | 0 < t < T < T ∗ } satisfying a set of stochastic differential equations (SDEs) and defined by the following assumptions: • The system of zero-coupon bond prices B = {B(t, T ) | 0 < t < T < T ∗ } is ∗ modeled as a strictly positive continuous semi-martingale under PT . A deterministic initial set of bond prices B(0, T ), T ∈ [0, T ∗ ], is exogenously given and the bond price process satisfies the relationship B(t, T ) > B(t, S) for any t ≤ T < S with B(t, T ) ≡ 1 for any t ≥ T. • For any fixed T ∈ [0, T ∗ ) the forward rate process2 (1.1.1) F (t, T, T ∗ ) =
B(t, T ) − B(t, T ∗ ) , τ B(t, T ∗ )
0 < t < T,
τ (T, T ∗ ) = T ∗ −T,
is a strictly positive, continuous martingale under PT . ∗
The equivalent martingale measure PT can be interpreted as the time T ∗ -forward measure and implies that the bond price dynamics is arbitrage-free. It follows from the Martingale Representation Theorem (cf. Karatzas-Shreve [1991]) that for every T ∈ [0, T ∗ ) the forward interest rate process F (t, T, T ∗ ) has the representation ∗
(1.1.2)
dF (t, T, T ∗ ) = F (t, T, T ∗ )γ(t, T, T ∗ ) · dWTt , ∗
0 ≤ t ≤ T,
2
This definition is derived from the self-financing portfolio of zero bonds: at time t we sell B(t, T ) B(t,T ) B(t,T ) ∗ ∗ and buy B(t,T ∗ ) · B(t, T ) at total of zero, B(t, T ) − B(t,T ∗ ) · B(t, T ) = 0. This leads to the definition
of F (t, T, T ∗ ), satisfying 1+(T ∗ −T )F (t, T, T ∗ ) = at time T ∗ .
B(t,T ) B(t,T ∗ ) .
The latter is the interest amount received
Chapter 1. Forward Libor and Swap Market Models 6 ∗ ∗ T∗ T∗ under PT , where WTt = Wt,1 , . . . , Wt,d is a Rd -valued (element-wise independent) ∗ ∗ d 3 PT -Brownian motion and γ(t, volatility process i hR T, T ) is a R -valued, {Ft }-adapted ∗ T kγ(u, T, T ∗ )k2d du < ∞ = 1. satisfying the condition PT 0
Given these assumptions and the representation of the forward rate process (1.1.2), we can in principle construct an interest rate model with an exogenously specified volatility structure process γ(t, T, T ∗ ). The volatility structure γ(t, T, T ∗ ) might be, in general, a stochastic process. In the special case where γ(t, T, T ∗ ) is a deterministic, bounded, piecewise continuous function, the forward Libor rate F (t, T, T ∗ ) is a lognormal martingale under its equivalent martingale measure. The construction of a model of forward rates as presented by Brace-Gatarek-Musiela [1997] starts by postulating that the dynamics of the forward ∗ rates F (t, T, T ∗ ) under the equivalent martingale measure P T are governed by the stochastic differential equation (1.1.2), where the deterministic volatility function is exogenously given. The model for the forward rates (1.1.2) is referred to in the literature as BGM (Brace-Gatarek-Musiela) Model or the lognormal Libor Market Model (LMM). In practice, however, we do not model a continuum of forward rates with a fixed compounding period τ but only a finite number of simple forward rates, which in the following will be termed forward Libor rates.
Definition 1.1 (d-factor Libor Market Model). Let {T0 , . . . , TN } be the set of expiries and {B(t, T0 ), . . . , B(t, TN )} the corresponding set of zero coupon bond prices. Let d be the fixed number of independent driving Brownian motions in the model. For each i ∈ {0, . . . , N − 1} the d-factor Libor Market Model (LMM) assumes the following GBM dynamics for forward rate Fi (t) := Fi (t, Ti , Ti+1 ), under its payoff martingale measure Pi+1 := PTi+1 : (1.1.3)
dFi (t) = Fi (t)γ i (t) · dWi+1 t ,
0 ≤ t ≤ Ti ,
where Wi+1 is a standard d-dimensional Brownian motion under the forward meat i+1 i+1 i+1 sure P , with d(Wt,k , Wt,l ) = δk,l dt, k, l ∈ {1, . . . , d} (δk,l is the usual Kronecker Delta). γ i (t) is a deterministic vector process4 given by γ i (t) = (σi,1 (t), . . . , σi,d (t))T , with σi (t) := kγ i (t)kd . Using the Itˆo’s lemma, the GBM equation (1.1.3) can be solved by Z t Z 1 t i+1 2 (1.1.4) Fi (t) = Fi (0) exp γ i (s) · dWs ds − kγ i (s)kd ds , 2 0 0
0 ≤ t ≤ Ti .
The dynamics in (1.1.3) does not yet distinguish between the correlations and the volatility of the forward rates. To make this clearer we re-formulate the equation (1.1.3) in the form (cf. Rebonato [1999a] for more details) (1.1.5)
dFi (t) = Fi (t)σi (t) bi (t) · dWi+1 t , ∗
The filtration {Ft } is consistently generated by {WTt }. 4 The interpretation for γ i (t) is that it contains the responsiveness of the i’th forward rate for d different independent random shocks. 3
7
Chapter 1. Forward Libor and Swap Market Models where bi (t) ∈ S d ⊂ Rd (S d the unit hypersphere in Rd ) is given by (1.1.6)
σi,k (t) bi,k (t) = , kγ i (t)kd
d X
b2i,k = 1.
k=1
In this way (1.1.5) formally separates the volatility σi of the forward rate Fi from the correlation structure ρ between the forward rates, which can be equivalently defined via its pseudo square root b = {b0 , . . . , bN −1 }, containing the vectors bi as columns: γ i (t) · γ j (t) ⊥ , 0 ≤ t ≤ min{Ti , Tj }. (1.1.7) ρ(t) = b(t) b(t) = kγ i (t)kd kγ j (t)kd i,j Rebonato [1999a] presented a significant and efficient way to reduce to a very large extent the difficulties in the simultaneous calibration of the volatilities and the correlation matrix thanks to straightforward geometrical relationships and matrix theory. The notation so far with the introduction of loading vectors in (1.1.6) shall simplify the understanding and the usage of these results. Remark 1.2 Once the fixing time Ti is reached the forward rate becomes constant, which means that Fi (t) remains constant for all t ≥ Ti . Although obvious, let it be mentioned that the instantaneous volatility function then satisfies σi (t) = kγ i (t)kd ≡ 0,
1.1.1
∀ t ≥ Ti ,
i ∈ 1, . . . , N .
Libor Dynamics Under the Forward Measure
Of course, in order to use the LMM in practice the dynamics of all forward Libor rates have to be formulated in a single measure. In this respect convenient choices are either the terminal measure PN which is induced by taking the terminal discount bond B(t, TN ) as numeraire, or the spot measure which is defined by the numeraire given in (1.1.10). As a consequence only one of the forward Libor rates is a martingale and, according to Girsanov’s theorem, all other forward rate processes will have be modified by additional drift terms (cf. Hunt-Kennedy [2004]). In practice, the standard approaches to construct the system of forward Libor rates rely either on the forward induction as in Brace-Gatarek-Musiela [1997] or on the so-called backward induction as in Musiela-Rutkowski [2005]. Concretely, under the measure Pk+1 the forward rate process Fj reads (1.1.8)
dFj (t) = µ(t, Tj , Tk ) dt + Fj (t)γ j (t) · dWk+1 , t
where the drift term µ(t, Tj , Tk ) is determined by requiring lack of arbitrage. For 0 ≤ t ≤ Tj the drifts are given by (cf. Brace-Gatarek-Musiela [1997]) (τi = Ti+1 − Ti ): k P τi Fi (t) σi (t)σj (t)ρi,j (t) − for j < k 1+τi Fi (t) i=j+1 (1.1.9) µ(t, Tj , Tk ) = Fj (t) · 0 for j = k j P τi Fi (t) σi (t)σj (t)ρi,j (t) for j > k. 1+τi Fi (t) i=k+1
Chapter 1. Forward Libor and Swap Market Models
8
The spot measure (cf. Jamshidian [1997]) is induced by the rolling bond numeraire5 ξ(t)−1
(1.1.10)
Gt = B(t, Tξ(t) )
Y
(1 + τj Fj (t)),
j=0
where the left-continuous function ξ : [0, TN ] → {1, . . . , N } gives the next reset date at time t: ( ) k−1 n o X (1.1.11) ξ(t) = inf k ∈ N | T0 + τi ≥ t = inf k ∈ N | Tk ≥ t . i=0
The forward Libor process for Fj , j = 0, . . . , N − 1, is then given by (1.1.12)
dFj (t) = Fj (t) γ j (t) ·
dW∗t
j X τi Fi (t) σi (t)σj (t)ρi,j (t) + Fj (t) dt, 1 + τi Fi (t) i=ξ(t)
with W∗t denoting a standard Brownian motion under the spot measure P∗ . We see that in the spot Libor measure Fj in (1.1.12) contains j − ξ(t) + 1 drift terms, whereas in the terminal measure it contains N − j − 1 drift terms, cf. (1.1.9). For numerical reasons it is important to keep the calculation costs of the Libor drifts as small as possible. Therefore, for products involving only short maturity Libors the dynamics in the spot Libor measure (1.1.12), involving repeatedly the rolling of the bond with the shortest time to maturity available, is preferable, whereas for longer dated products the representation in the terminal measure may be recommended. In both cases the numeraire process remains alive throughout the time span of the tenor structure {Tn }N n=0 . This is particularly necessary for the evaluation of derivative securities that involve random payoffs at any date in the tenor structure. Our simulations in Sec. 4.6 are carried out by calculating the drifts with respect to the spot measure.
1.1.2
Valuation in LMM
The key advantage of LMM in regard to model calibration as well as to pricing of the financial interest rate products is its compatibility of the forward rates’ modeling to the Black framework, due to the assumed lognormality of the forward rate dynamics and, of course, to the assumed deterministic (time-dependent) volatility. With regard to the scope of this work, we present in Appendix A.1 the valuation formulae of the basic benchmark instruments which will be used to calibrate the LMM and other models we will consider later in the next chapters. The spectrum of interest rate products which can be priced with LMM is huge. In practice, a lot of products are being priced with LMM by taking into consideration 5
Gt represents the wealth at the time t of a portfolio that starts at time 0 with one unit of cash invested in a zero-coupon bond of maturity T0 , and whose wealth is then reinvested at each time Tj in zero-coupon bonds maturing at the next date Tj+1 , cf. Schoenmakers [2005]. The process Gt is a continuous and completely determined by the Libors at the tenor dates, such that the spot measure P∗ is then defined such that the relative bond prices B(t, Tj )/Gt , j = 1, . . . , N are local martingales.
Chapter 1. Forward Libor and Swap Market Models
9
approximations. One family of popular approximations we shall discuss in detail in Chap. 1.2. Nevertheless, the main reason to develop a market model of forward rates is however to price exotic interest rate options, whose complex payoff can be expressed in terms of market observable Libor rates. In the most cases this is done by performing joint Monte-Carlo simulations of the forward rates in a calibrated LMM. The joint distributional evolution of the forward rates with respect to e.g. a payoff measure, though, results in solving a system of stochastic differential equations, as in (1.1.8)(1.1.9), which involves state dependent drift terms. In Appendix A.4 we briefly present a standard method how to approximate the corresponding drifts in the joint evolution of forward rates.
1.1.3
Covariance and Correlations in LMM
In general, if one is interested in terminal correlations of forward rates at a future time instant (when pricing financial instruments with payoffs at future times), as implied by the LMM model, then the computation has to be based on a Monte Carlo simulation technique. Following Brigo-Mercurio [2007], let us assume we are interested in computing the terminal correlation between forward rates Fi and Fj at time Tk , k < i < j, say under the measure Py , y > k. Then we need to compute the terminal covariance h i Eyt (Fi (Tk ) − Eyt [Fi (Tk )]) (Fj (Tk ) − Eyt [Fj (Tk )]) Corry (Fi (Tk ); Fj (Tk ))(t) = r h . 2 i y h 2 i y y y Et Fi (Tk ) − Et [Fi (Tk )] Et Fj (Tk ) − Et [Fj (Tk )]
We notice that, while the instantaneous correlations do not depend on the particular probability measure or numeraire under which we are working, the terminal correlations do. Recalling the dynamics of Fi and Fj under Py , the expected values appearing in the above expression can be obtained by simulating the above dynamics of Fi and Fj up to time Tk . Fortunately, there exist approximated formulas that allow us to derive terminal correlations algebraically from the LMM parameters ρi,j (.) and σi (.). By partial freezing of the drift components in the log-normal dynamics of the forward rates with respect to Py , we can easily obtain (cf. Brigo-Mercurio [2007]): nR o Tk exp t σi (s)σj (s)ρi,j (s) ds − 1 r Corry (Fi (Tk ); Fj (Tk ))(t) ≈ r . nR o nR o Tk Tk 2 2 exp t σi (s) ds − 1 exp t σj (s) ds − 1 This approach makes the terminal correlations independent of the chosen probability measure. Notice that a first order expansion of the exponentials appearing in the above formula yields a second formula for the terminal correlations (Rebonato [2004]): R Tk σi (s)σj (s)ρi,j (s) ds y qR . (1.1.13) CorrRR (Fi (Tk ); Fj (Tk ))(t) = qR t Tk Tk 2 2 σi (s) ds t σj (s) ds t
10
Chapter 1. Forward Libor and Swap Market Models
An immediate application of Schwartz’s inequality shows that terminal correlations, when computed via Rebonato’s formula, are always smaller, in absolute value, than instantaneous correlations. In agreement with this general observation, recall that through a careful repartition of integrated volatilities (caplets) in instantaneous volatilities σi (t) and σj (t) we can make the terminal correlation CorryRR arbitrarily close to zero, even when the instantaneous correlation ρi,j is one.
1.1.4
Swap Rate Models and Measures
P A probability measure Pm,n , induced by the annuity Bm,n (t) = ni=m+1 τi−1 B(t, Ti ) ∗ and equivalent to the measure PT , is said to be the forward swap probability measure associated with the dates Tm and Tn , or simply the forward swap measure, if i) for every i = 0, . . . , N the relative bond price BB(t,T , for all t ∈ [0, Ti ∧ Tm+1 ], folm,n (t) m,n lows a local martingale process under P . Thus, the forward swap rate Sm,n (t) = B(t,Tm )−B(t,Tn ) , t ∈ [0, Tm ], is a Pm,n -martingale (cf. Appendix A.2). Bm,n (t) Definition 1.3 If the (vector-valued) volatility process t → γ m,n (t) is a deterministic function we speak of a (lognormal) Swap Market Model (SMM) for Sm,n , assuming that forward swap rates follow a lognormal diffusion process of type (1.1.14)
dSm,n (t) = Sm,n (t)γ m,n (t) · Wm,n , t
0 ≤ t ≤ Tm ,
˜ where Wm,n denotes the corresponding d-dimensional Brownian motion under Pm,n . As the correlations between the forward swap rates will not be focused on in this section, (1.1.14) can be alternatively expressed in an one-dimensional form6 as (1.1.15)
dSm,n (t) = Sm,n (t)σm,n (t)dWtm,n ,
0 ≤ t ≤ Tm ,
(t)
being an one-dimensional where σm,n (t) = kγ m,n (t)kd˜, and Wtm,n = kγ m,n(t)k ˜ · Wm,n t m,n d Brownian motion under Pm,n . As an important consequence, European options on swap contracts over [Tm , Tn ], called swaptions, can be priced exactly with the Black-Scholes formula ( see Appendix A.3). Moreover, as we will show, there exist very accurate swaption approximation formulas for swaptions in the LMM. While in the Libor model of forward rates there is only one degree of freedom for choosing the numeraire, see (1.1.9)–(1.1.12), for swap market models in general there are N degrees of freedom for a N +1 time grid. For instance, for a complete system of standard swaps it is possible to choose σ0,N , . . . , σN −1,N simultaneously deterministic (cf. discussions in Schoenmakers [2005]). In Appendix A.2 we shall briefly present some of swap rate models mostly used in practice: in particular, the co-terminal and the co-initial swap rate models. We refer to Galluccio et al. [2006] for their extensive studies on these and further swap rate models and their adequateness in practice. γ
6
We will consider the swap rate process Sm,n (t), if not otherwise explicitly specified, always in its ”natural” measure Pm,n .
11
Chapter 1. Forward Libor and Swap Market Models
1.1.5
Incompatibility Between the LMM and the SMM
The cap and swaption markets are underpinned by the same state variables, either forward rates or, equivalently, swap rates which can be transformed to each other by simple bootstrapping methods. As a corollary, the instantaneous volatilities of forward rates and swaptions cannot be assigned independently. Once the instantaneous volatilities of, and correlations among, forward rates are given, then the correlations among and volatilities of swap rates are completely specified (cf. Rebonato [1999b]). It is market practice to price both sets of instruments (caps and swaptions) using the Black [1976] formula which is inconsistent as lognormal forward and lognormal swap rate models are incompatible; if simple forward rates are lognormal, swap rates can only be approximately so, and vice versa (cf. Brace [1997]). The Black model ceases to be arbitrage–free when it is assumed that, at the same time, forward rates and swap rates are all lognormal. Further discussions about the effects of this incompatibility in practice can be found in Rebonato [1999b], Brigo-Liinev [2005], Brigo-Mercurio [2007]. The next two paragraphs show how the two models interact with each other, and how far they are compatible. Swap rate dynamics under the forward measure. Following Brigo-Mercurio [2007], the dynamics of the forward swap rate Sm,n in the SMM model (cf. (1.1.14)), under the Libor forward measure numeraire B(t, Tm ) is given (after lengthy calculations) by m (1.1.16) dSm,n (t) = µm m,n (t)Sm,n (t)dt + Sm,n (t)γ m,n (t) · Wt ,
0 ≤ t ≤ Tm .
The drift is defined by (1.1.17) n−1 P m,n νi,j (t)τi τj B(t, Tm , Ti+1 )B(t, Tm , Tj+1 )ρi,j (t)σi (t)σj (t)Fi (t)Fj (t) i,j=m m µm,n (t) = , 1 − B(t, Tm , Tn ) B(t,Tk ) where B(t, Tm , Tk ) = B(t,T denotes the price of the zero-coupon bond at time t for m) maturity Tk , as seen from expiry Tm , hence the forward price of the zero bond from Tm to Tk as seen at time t. m,n The weights νi,j are defined as
m,n νi,j (t)
=
B(t, Tm , Tn )
·
n X
Pi
Pn k=m+1 τk−1 B(t, Tm , Tk ) + k=i+1 τk−1 B(t, Tm , Tk ) 2 Pn τ B(t, T , T ) k−1 m k k=m+1
τk−1 B(t, Tm , Tk ).
k=j+1
Forward rate dynamics under the annuity measure. Symmetrically, it is possible to work out the dynamics of the forward Libor rates under the SMM numeraire Bm,n . Applying the change-of-numeraire technique we have the following dynamics
12
Chapter 1. Forward Libor and Swap Market Models for the forward rate Fi under Pm,n (cf. (1.1.15)): (1.1.18)
dFi (t) = σi (t)Fi (t) (µm,n (t)dt + Fi (t)σi (t)Wtm,n ) , i
0 ≤ t ≤ Ti .
The drift is given by (1.1.19) µm,n (t) = i
n X
(2χ{k≤i} − 1 τk−1 B(t, Tk ) Bm,n (t) k=m+1
max{i,k}−1 X τj Fj (t)σj (t)ρi,j (t) j=min{i,k}
1 + τj Fj (t)
.
The details of these derivations can be found in Brigo-Mercurio [2007]. Approximating the Swap Rate Volatility in LMM The dynamics of swap rates in a Libor market model, as seen in (1.1.16), is rather complicated due to the stochastic factors involved in the drifts. Therefore, closed form pricing of swaptions in the LMM is in general not possible, nonetheless it is possible to give surprisingly accurate swaption approximation formulas in LMM. To this end we write the swap rate again as a combination of forward rates and discount zero bonds (cf. (A.1.2)) (1.1.20) Sm,n (t) =
Pn
i=m+1 τi−1 B(t, Ti )Fi−1 (t)
Bm,n (t)
=
n−1 X
wim,n (t)Fi (t),
i=m
t ∈ [0, Tm ],
B(t,Ti+1 ) with the stochastic weights wim,n (t) = τiB . The popular freezing of these weights, m,n (t) which certainly simplifies the swap drifts in LMM, will also help us in approximating the swap rate variance in LMM. 2 Following Schoenmakers [2005], the swap rate variance σm,n (t) = kγ m,n (t)k2d˜ may be expressed in terms of the forward Libor volatilities by ! n−1 X n−1 X 1 vim,n (t)vjm,n (t)Fi (t)Fj (t)γ i (t) · γ j (t) , (1.1.21) σm,n (t)2 = 2 (t) Sm,n i=m j=m
with some weights vim,n (t) whose distance to the swap weights wim,n (t) is given via (1.1.22)
vim,n (t) − wim,n (t) = τi
0 0 (t) − Si,n (t) Bi,n (t) Sm,n =: yˆim,n (t), Bm,n (t) 1 + τi Fi (t)
0 where m ≤ i < n and Si,n denotes a system of virtual swap rates over a period [Ti , Tn ] defined as (see Schoenmakers [2005] for the details of derivation) 0 Si,n (t) =
B(t, Ti ) − B(t, Tn ) . Pn−1 Bm,n (t) k=min{l|m+l≥i} wkm,n (t)
The terms yˆim,n (t) have magnitudes comparable with differences of swap rates, hence, 0 0 they are usually rather small. They are zero when Si,n (t) = Sm,n (t) for m < i < n.
Chapter 1. Forward Libor and Swap Market Models
13
For example, this is the case for standard swaptions when the yield curve is flat (see also Rebonato-Jaeckel [2003]). Integrating (1.1.21) over time to expiry we obtain (1.1.23) Z Tm n−1 Z Tm m,n X vi (s)vjm,n (s)Fi (s)Fj (s) 1 2 γ i (s) · γ j (s) ds. σm,n (s) ds = 2 (s) Tm − t t Sm,n i,j=m t We now note that the (stochastic) fractions in the r.h.s. of (1.1.23) add up to approximately one and thus may be regarded as weights, tend to vary relatively slow in practice and therefore may be approximated by their values at t. Under this additional assumption instantaneous swap volatilities may be considered as deterministic (though model inconsistent). This technique of ”freezing” all weights and forward rates in (1.1.23) to their initial value leads to Rebonato’s formula (Rebonato [2002]): (1.1.24) Z Tm Z n−1 X vim,n (t)vjm,n (t)Fi (t)Fj (t) Tm 1 2 σm,n (s) ds = γ i (s) · γ j (s) ds, 2 (t) Tm − t t S t m,n i,j=m B(t,Ti+1 ) + yˆim,n (t). This formula can be used to calibrate of the model with vim,n (t) = τiB m,n (t) parameters to implied swaption volatilities according to (A.3.3).
1.2
The Convexity Adjustment and CMS Derivatives
In finance convexity is a broadly understood and non-specific term for nonlinear behavior of the price of an instrument as a function of evolving markets. Such convex behaviors manifest themselves as convexity corrections/adjustments to various popular interest rate derivatives. From the perspective of financial modeling they arise as the results of valuation done under the wrong martingale measure. Practitioners use various ad hoc rules to calculate convexity corrections for different products, often based on Taylor approximations (cf. Hunt-Pelsser [1998], Benhamou [2000], etc). However, Pelsser [2003] is the first to put convexity correction on a firm mathematical basis by showing that it can be interpreted as the side-effect of a change of numeraire. It can be understood as the expected value of an interest rate under a different probability measure than its own martingale measure. The well known Change of Numeraire Theorem, due to Geman et al. [1995], shows how in an arbitrage-free economy an expectation under a probability measure PN , generated by the numeraire N, can be represented as an expectation under a probability measure PM , generated by the numeraire M, times the Radon-Nikodym derivative dPN /dPM . For an expectation at time 0 of a random variable H at time T we have " # N (T )M (0) (1.2.1) EN H(T ) = EM H(T ) . N (0)M (T ) Following Pelsser [2003], suppose we are given a forward interest rate F (t, T, T ∗ ) with maturity T < T ∗ and a numeraire B(t, T ∗ ) such that the forward rate is a martingale
14
Chapter 1. Forward Libor and Swap Market Models T∗
under the associated probability measure P . Now assume we have a contract where the interest rate F (T, T, T ∗ ) is observed at T but paid at a later date S ≥ T. At time T the discounted interest payment is given by V (T ) = B(T, S)F (T, T, T ∗ ) and in PS (1.2.2) V (0) = B(0, S)ES0 F (T, T, T ∗ )
follows. However, under the measure PS the process F (t, T, T ∗ ) is in general not a martingale such that the expectation (1.2.2) can be expressed as F (0, T, T ∗ ) times a correction term. This correction term is known in the market as the convexity correction or convexity adjustment. Applying the change of numeraire technique ∗ (1.2.1) we can express (1.2.2) in terms of ET as follows " # " # S ∗ dP B(T, S)B(0, T ) ∗ ∗ ES0 F (T, T, T ∗ ) = ET0 F (T, T, T ∗ ) T ∗ = ET0 F (T, T, T ∗ ) dP B(0, S)B(T, T ∗ ) ∗ = ET0 F (T, T, T ∗ )R(T ) ,
where R denotes the Radon-Nikodym derivative which is also a martingale under ∗ the measure PT . If we know the joint probability distribution of F (T, T, T ∗ ) and R(T ) the expectation can be calculated explicitly and we obtain an expression for the convexity correction. Only for very special cases exact expressions for the convexity correction can be obtained. In these special cases the Radon-Nikodym derivative of the change of measure is equal to (a simple function of) the interest rate that determines the payoff. A prominent example where an exact expression for the convexity correction is possible is a Libor in Arrears contract, in which the payment is in arrears, i.e. at fixing time. ∗ Thus, we have for the Radon-Nikodym derivative dPT /dPT we have dPT B(T, T )B(0, T ∗ ) 1 + τ F (T, T, T ∗ ) = = , τ = T ∗ − T, dPT ∗ B(0, T )B(T, T ∗ ) 1 + τ F (0, T, T ∗ )
(1.2.3) and hence, (1.2.4)
1.2.1
ET0
∗
F (T, T, T )
RT
1 + τ F (0, T, T ∗ )e 0 σT (s) = F (0, T, T ) 1 + τ F (0, T, T ∗ ) ∗
2 ds
.
Constant Maturity Swaps and Related Derivatives
The acronym CMS stands for constant maturity swap, and it refers to a swap rate with a pre-defined length which fixes in the future. CMS rates provide a convenient alternative to Libor as a floating index, as they allow market participants to express their views on the future levels of long term rates (for example, the 10 year swap rate). There are a variety of CMS based instruments, the simplest of them being CMS swaps and CMS caps / floors. A particularly known type of exotic European interest rate contract is a (fixed for floating) CMS swap. This is a swap where at every payment date a payment calculated from a swap rate is exchanged for a fixed rate. The floating leg pays periodically a swap rate of fixed length (say, the 10 year swap rate) which fixes at the beginning of the accrual period.
Chapter 1. Forward Libor and Swap Market Models
15
A CMS cap or floor is a basket of calls or puts on a swap rate of fixed tenor (say, 10 years) structured in analogy to a Libor cap or floor, cf. Sec. A.1. For example, a 5 year cap on 10 year CMS struck at K is a basket of CMS caplets over 5 years, each of which pays max(10 year CMS rate − K; 0), where the CMS rate fixes at the start of each accrual period. Needless to say, a plethora of more sophisticated contracts are traded in the markets, which may differ from the standard ones by differences in fixing and payment frequencies, whether the floating leg fixes in arrears or in advance, whether the term and payment frequency of the swap rate may be different from the specifications of the CMS swap itself and further market particularities. Moreover, the contracts can consist of even more complicated formulas involving algebraic expressions of CMS rates of different lengths. CMS Swaps As mentioned above the floating payments of a CMS swap are not based on the Libor forward rates but on some swap rate. Formally, at the settlement dates Ti+1 , i ∈ {0, . . . , N − m − 1}7 , the fixed payment K is exchanged for the variable payment Si,i+m (Ti ) for a preassigned length m ≥ 1. Let us consider one CMS swaplet only, paying at Ti+1 and based on a notional of 1. The discounted payment on the fixed leg as of t is obviously given by B(t, Ti+1 )τi K, while for the floating leg (1.2.5) CMS(t, Ti , m) = B(t, Ti+1 )τi Ei+1 Si,i+m (Ti ) t holds. Ei+1 denotes the expectation at time t with respect to the forward measure t Pi+1 . For a fixed natural number n ≤ N − m and k ∈ {0, . . . , N − m − 1}, we denote with CMS(t, Tk , m, n), a n–period (forward starting) CMS swap rate which is defined by Pn−1 i=k CMS(t, Ti , m) K = CMS(t, Tk , m, n) = (1.2.6) . Bk,n (t) CMS Caps/Floors The CMS caplets and CMS floorlets are built up analogously to their classical pendants, i.e. the interest rate caplets and floorlets. We can write h + i k+1 (1.2.7) CMSCPL(t, Tk , m, κ) = B(t, Tk+1 )τk Et , Sk,k+m (Tk ) − κ i h + (1.2.8) , κ − Sk,k+m (Tk ) CMSFLL(t, Tk , m, κ) = B(t, Tk+1 )τk Ek+1 t
where κ is obviously the optionlet strike. Not surprisingly, this implies a put-call parity relation for the CMS rate: (1.2.9) 7
CMSCPL(t, Tk , m, κ) − CMSFLL(t, Tk , m, κ) = CMS(t, Tk , m) − κ.
We are assuming that the total time horizon in our economy is up to TN .
16
Chapter 1. Forward Libor and Swap Market Models
Analogously to the classical ones (Sec. A.1), for CMS caps (for CMS floors analogously) we have (1.2.10)
CMSCAP(t, Tk , m, n, κ) =
n−1 X
CMSCPL(t, Ti , m, κ).
i=k
CMS Spread Options A holder of a CMS spread option(let) has the right to exchange for one period of time the difference between two CMS rates minus a spread κ. Hence, the payoff at expiry time Tk equals (1.2.11) + CMSSPO(Tk , Tk , n1 , n2 , κ) := τk a1 ωSk,k+n1 (Tk ) + a2 ωSk,k+n2 (Tk ) − ωκ ,
n1 6= n2 , k + ni < N, a1 , a2 ∈ R, ω ∈ {−1, 1}. A CMS spread option(let) can be seen as a special case of a CMS basket option(let). A generic CMS basket option(let), written on M CMS rates that reset at the option’s expiry date Tk , has the payoff !+ M X ωai Sk,k+mi (Tk ) − ωκ , (1.2.12) CMSSPOB(Tk , Tk , {mi }M i=1 , κ) := τk i=1
where ai ∈ R denote weights and mi , i = 1, . . . , M, (k + mi < N ) are preassigned lengths of reference swaps. A natural step further, far beyond the scope of this work, though, is to consider caps/floors of CMS spreads or even CMS basket options with periodic expiries/fixings whose payoff at every fixing/expiry time Tk reads as in (1.2.12).
1.2.2
Valuation of CMS Derivatives
We now come to the point where we can examine the pricing of CMS products we introduced previously. The most common characteristic of these products with respect to pricing is that their payoffs are functions of one or more CMS rates f (Si,i+m1 , Si,i+m2 , . . .), which are usually fixed at Ti and paid at Ti+1 . As we know the swap rate Si,i+m (t) is a martingale with respect to the measure Pi,i+m , induced by the annuity Bi,i+m (t). The forward measure Pi+1 , associated with the payment date Ti+1 , is not its natural measure, i.e. Si,i+m (t) is not a martingale w.r.t. Pi+1 . Turning back to the CMS swaplet (1.2.5), by using the change of numeraire technique (1.2.1), we can write for i = 0, . . . , N − m − 1: Bi,i+m (t) i,i+m B(Ti , Ti+1 ) i+1 Et Si,i+m (Ti ) = (1.2.13) . E Si,i+m (Ti ) B(t, Ti+1 ) t Bi,i+m (Ti ) They are basically two ways how to deal with the expectation: either find a lognormal approximation for the CMS rate in the forward measure, by approximating the Libor drifts, or a convexity correction approach shall be applied by expressing the RadonNikodym derivative as a (simple) function of the interest rate that determines the payoff, as, for instance, in the case of the Libor in Arrears in the LMM (cf. (1.2.3)).
17
Chapter 1. Forward Libor and Swap Market Models
In this section we want to discuss some of the methods used in the practice to approximate (1.2.13). The first method exploits the idea of making the Radon-Nikodym derivative a function of the payout rate. Approximating the Radon-Nikodym Derivative and the Convexity Correction For evaluating (1.2.13) we here recall the convexity approach in Pelsser [2003], based on the assumption of a lognormal SMM, cf. Hunt-Kennedy [2004]: B(Ti , Ti+1 ) ≈ a + bi+1 Si,i+m (Ti ), Bi,i+m (Ti )
(1.2.14)
where a and bi+1 are constants which are determined as follows. As the RadonNikodym derivative is a martingale w.r.t. to the annuity measure, by taking the expectation we obtain B(t, Ti+1 ) i,i+m B(Ti , Ti+1 ) = a + bi+1 Si,i+m (t). = Et Bi,i+m (t) Bi,i+m (Ti ) Hence, (1.2.15)
1
bi+1 =
Si,i+m (t)
B(t, Ti+1 ) −a . Bi,i+m (t)
On the other hand we have by summing up 1 =
i+m−1 X k=i
τk B(t, Tk+1 ) = Bi,i+m (t)
i+m−1 X
τk (a + bk+1 Si,i+m (t)) .
k=i
Replacing bi+1 by (1.2.15), (1.2.16)
1
a = Pi+m−1 k=i
τk
,
bi+1 =
Bi+1 (t) −
Bi,i+m (t) Pi+m−1 τk k=i
Bi (t) − Bi+m (t)
hold. Finally, we can rewrite (1.2.13) as (1.2.17)
Ei+1 t
Si,i+m (Ti )
bi+1 Vari,i+m [Si,i+m (Ti )] = Si,i+m (t) 1 + Si,i+m (t)(a + bi+1 Si,i+m (t))
.
The linear approximation in (1.2.14) does seem very crude at first, but can be justified by the following argument (cf. Pelsser [2003]). Convexity corrections only become sizable for large maturities. However, for large maturities the term structure almost moves in parallel. Hence, a change in the level of the long end of the curve is well described by the swap rate. Furthermore, for parallel moves in the curve, the ratio B(Ti , Ti+1 )/Bi,i+m (Ti ) is closely approximated by a linear function of the swap rate, which is exactly what the approach does. This leads to a good approximation of the convexity correction for long maturities. With these formulas we can easily price linear CMS products like in (1.2.5) – (1.2.6).
18
Chapter 1. Forward Libor and Swap Market Models
In his seminal work Hagan [2003] discusses general approximations of functional form to the Radon-Nikodym derivative given through (1.2.13). He writes for the CMS caplets: + B(Ti , Ti+1 )/Bi,i+m (Ti ) i,i+m CMSCPL(t, Ti , m, κ) = B(t, Ti+1 )τi Et Si,i+m (Ti ) − κ B(t, Ti+1 )/Bi,i+m (t) + = B(t, Ti+1 )τi Eti,i+m Si,i+m (Ti ) − κ + B(Ti , Ti+1 )/Bi,i+m (Ti ) i,i+m + B(t, Ti+1 )τi Et Si,i+m (Ti ) − κ −1 . B(t, Ti+1 )/Bi,i+m (t)
The first term is exactly the price of a European swaption (cf. (A.3.2)) with notional B(t, Ti+1 )/Bi,i+m (t), regardless of how the swap rate is modeled. The last term is the convexity correction. Following the argumentation in Hagan [2003], since Si,i+m is a B(Ti ,Ti+1 )/Bi,i+m (Ti ) martingale in the annuity measure and B(t,T − 1 is zero on average, this i+1 )/Bi,i+m (t) term goes to zero linearly with the variance of the swap rate, and is much smaller than the first term. Giving the ratio a general form (1.2.18)
B(Ti , Ti+1 )/Bi,i+m (Ti ) = G(Si,i+m (Ti )),
for some function G, we then have B(t, Ti+1 )τi Eti,i+m
+
Si,i+m (Ti ) − κ + G(Si,i+m (Ti )) i,i+m + B(t, Ti+1 )τi Et Si,i+m (Ti ) − κ −1 . G(Si,i+m (t))
CMSCPL(t, Ti , m, κ) =
Using the general property for smooth functions f with f (κ) = 0 (integration by parts): Z ∞ + ′′ + f (S) for S > κ ′ S − x f (x) dx = (1.2.19) f (κ) S − κ + 0 for S < κ, κ
and choosing (1.2.20)
f (x) =
x−κ
we obtain by simple transformations
G(x) −1 , G(Si,i+m (t))
B(t, Ti+1 ) CMSCPL(t, Ti , m, κ) = τi 1 + f ′ (κ) PSWO(t, Ti , Ti+m , κ) Bi,i+m (t) Z ∞ ′′ (1.2.21) PSWO(t, Ti , Ti+m , x)f (x) dx . + κ
This formula replicates the value of the CMS caplet in terms of European swaptions at different strikes It takes into account the presence of a market smile, incorporating consistently the information coming from the quoted swaption Black -volatilities. We refer to Mercurio-Pallavicini [2006] for discussions about the approximation of the
19
Chapter 1. Forward Libor and Swap Market Models
integral above on a practically plausible, in general not negligible, strike interval [0, K], with K ”large enough”. We will come back to this approximation of the convexity adjustment when considering the SABR model. The formula for CMS floorlets (1.2.8) is a slight adaption of (1.2.21), replacing PSWO with RSWO (cf. (A.3.4)): B(t, Ti+1 ) CMSFLL(t, Ti , m, κ) = τi 1 + f ′ (κ) RSWO(t, Ti , Ti+m , κ) Bi,i+m (t) Z κ ′′ (1.2.22) RSWO(t, Ti , Ti+m , x)f (x) dx . − −∞
The value of the CMS swaplet is easily derived from the CMS put-call parity (1.2.9). The method of replicating the CMS caplets/floorlets by means of swaptions is opaque and computationally intensive. Hagan [2003] gives simpler approximate formulas for the convexity correction, as an alternative to the replication method. Expanding at first order the function G around Si,i+m (t) makes f quadratic (1.2.23)
f (x) ≈
G′ (Si,i+m (t)) (x − Si,i+m (t))(x − κ), G(Si,i+m (t))
and f ′′ (x) constant. Together with the equality Z
∞
PSWO(t, Ti , Ti+m , x) dx = Bi,i+m (t)Eti,i+m
κ
"Z
1 Bi,i+m (t)Eti,i+m = 2
∞ κ
Si,i+m (Ti ) − x
Si,i+m (Ti ) − κ)
+
+ 2
dx
#
,
2 we have, by considering (Si,i+m (Ti ) − κ)(Si,i+m (Ti ) − κ)+ = Si,i+m (Ti ) − κ)+ ,
B(t, Ti+1 ) PSWO(t, Ti , Ti+m , κ) Bi,i+m (t) i h i,i+m ′ + +τi G (Si,i+m (t))Bi,i+m (t)Et (Si,i+m (Ti ) − Si,i+m (t))(Si,i+m (Ti ) − κ) ,
CMSCPL(t, Ti , m, κ) = τi
B(t, Ti+1 ) RSWO(t, Ti , Ti+m , κ) Bi,i+m (t) i h −τi G′ (Si,i+m (t))Bi,i+m (t)Eti,i+m (Si,i+m (t) − Si,i+m (Ti ))(κ − Si,i+m (Ti ))+ .
CMSFLL(t, Ti , m, κ) = τi
For a CMS swaplet, (1.2.24)
CMS(t, Ti , m) = τi B(t, Ti+1 )Si,i+m (t) h i +τi G′ (Si,i+m (t))Bi,i+m (t)Eti,i+m (Si,i+m (Ti ) − Si,i+m (t))2
holds. The SMM (cf. Hunt-Kennedy [2004]) gives h i h R Ti 2 i i,i+m σi,i+m (s)ds 2 2 t Et (Si,i+m (Ti ) − Si,i+m (t)) = Si,i+m (t) e (1.2.25) −1 .
20
Chapter 1. Forward Libor and Swap Market Models
Given that G(Si,i+m (t)) approximates the ratio B(t, Ti+1 )/Bi,i+m (t) (cf. (1.2.18)) linear as in (1.2.14)), the equation (1.2.24) perfectly matches with (1.2.17). Hagan [2003] suggests to use for CMS swaps the volatility of at-the-money swaptions, since the expected value includes high and low strike swaptions equally. For out-ofthe-money CMS caplets and floorlets, the strike-specific volatility should be used, while for in-the-money options, the largest contributions come from swap rates near the mean value. Accordingly, call-put-parity should be used to evaluate in-the-money caplets and floorlets as a CMS swap payment plus an out-of-the-money CMS floorlet or caplet. The function G has been considered a general smooth and slowly varying function, regardless of the model used to obtain it. Hagan [2003] develops simpler approximate formulas for the convexity correction, by specifying G. We shall present here the market standard method for computing convexity corrections which uses bond math approximations and goes as follows. Let the yield curve be flat, fixed at a level y. Then, given an equidistant time grid with step size ∆T and discrete discounting, we can write Bi,i+m (t) =
i+m X
τk−1 B(t, Tk ) =
k=i+1
i+m X
k=i+1
∆T
B(t, Ti+1 ) . (1 + ∆T y)k−i−1
The standard formula for the geometric sum gives then " # B(t, Ti+1 ) 1 Bi,i+m (t) = (1 + ∆T Si,i+m (t)) − , Si,i+m (t) (1 + ∆T Si,i+m (t))m−1 where the par swap rate y = Si,i+m (t) was taken as discount rate, since it represents the average rate over the life of the reference swap. Thus, (1.2.26)
Gstd (Si,i+m (t)) =
Si,i+m (t) . 1 (1 + ∆T Si,i+m (t)) − (1+∆T Si,i+m m−1 (t))
A more accurate lognormal approximation of the swap rates and their correlations in the forward measure was introduced by Belomestny-Kolodko-Schoenmakers [2010], based on the method of freezing the weights (as in Section 1.1.5) but assuming a more sophisticated approximation. The approximation (1.2.17) is model independent and quite accurate especially for flat yield curves and highly correlated rates. Such constraints are not necessarily unrealistic, since adjustments are mostly relevant for long maturities (and tenors), where (forward) rates tend to be constant and to move in parallel fashion. Assuming lognormal-type dynamics for the swap rates as in SMM we obtain the classical Black-like adjustment with at-the-money implied volatilities given through (1.2.24)–(1.2.26). Pricing CMS Spread Options Once more than one CMS rate is part of a payoff, as in case of CMS spread options, the correlation between the CMS rates in the forward measure starts playing an important role in pricing these products.
21
Chapter 1. Forward Libor and Swap Market Models
Let us first focus on CMS spread option(let)s with zero strike whose payoff at expiry time Ti equals (cf. (1.2.11)) + CMSSPO(Ti , Ti , n1 , n2 , 0) := τi Si,i+n1 (Ti ) − Si,i+n2 (Ti ) .
(1.2.27)
The arbitrage-free value of the payoff (1.2.27) at time t is given by (1.2.28) CMSSPO(t, Ti , n1 , n2 , 0) := τi B(t, Ti )Eit
h
Si,i+n1 (Ti ) − Si,i+n2 (Ti )
+ i
,
which can be calculated as soon as we know the joint distribution of the pair of swap rates Si,i+n1 and Si,i+n2 under the forward measure Pi . Apart from the fact that the expectation is taken in the non-natural forward measure, this payoff is the one of an exchange option. Therefore, the simplest valuation procedure is based on assuming that the logarithms of the swap rates are jointly normally distributed as in the Black-Scholes model of two underlying assets. A formal justification of this approach is given by resorting to the SMM (cf. Def. 1.3) and suitable approximations. Thus, let assume that both swap rates evolve according to dSi,i+n1 = µi,i+n1 (t)Si,i+n1 dt + σi,i+n1 Si,i+n1 dWti ˜ ti , dSi,i+n2 = µi,i+n2 (t)Si,i+n2 dt + σi,i+n2 Si,i+n2 dW
(1.2.29) (1.2.30)
˜ ti are Brownian motions under Pi , correlated via d(Wti , W ˜ ti ) = ρˆ(t)dt, where Wti and W with ρˆ(t) assumed to be given (estimated historically or approximated, for instance, as in Belomestny-Kolodko-Schoenmakers [2010]. The drifts µi,i+n1 and µi,i+n2 of the corresponding swap rates with respect to Pi are motivated by (1.1.17) and assumed to be frozen or deterministic. The formula for pricing exchange options, developed by Margrabe [1978] using the change of numeraire technique, can now be applied to obtain: (1.2.31) h + i p i i i = BS Et [Si,i+n1 (Ti )], Et [Si,i+n2 (Ti )], σ ¯ Ti − t, 1 . Et Si,i+n1 (Ti ) − Si,i+n2 (Ti ) With the swap rate dynamics given in (1.2.29)–(1.2.30) we obtain explicitly: Eit [Si,i+nk (Ti )] = Si,i+nk (t)e
R Ti t
µi,i+nk (s)ds
,
k = 1, 2,
and (1.2.32)
σ ¯
2
1 = Ti − t
Z
Ti t
2 2 σi,i+n (s) + σi,i+n (s) − 2ˆ ρ(s) σi,i+n1 (s)σi,i+n2 (s) ds. 1 2
There are several ways how to approximate drifts deterministically under Pi , for instance: • they can be inferred from the convexity adjustments, i.e. from the approximations to Eit Si,i+nk (Ti ) , k = 1, 2 (discussed in the previous section). We note that the convexity adjustment technique does not give the correlation between the swap rates w.r.t. Pi ;
22
Chapter 1. Forward Libor and Swap Market Models
• the classical method of ”freezing the coefficients” can be applied to (1.1.17), i i to make the P -dynamics of the swap rates lognormal, i.e. Et Si,i+nk (Ti ) = i Si,i+nk (t)eµi,i+nk (t)(Ti −t) , with µii,i+nk (t) given in (1.1.17); • complex lognormal approximations, as in Belomestny-Kolodko-Schoenmakers [2010] for instance, can be applied to the swap rates under Pi . Assuming the dynamics (1.2.29)–(1.2.30) with drifts µii,i+nk (t), k = 1, 2, and the correlation ρˆ(t) between the two swap rates frozen at evaluation time t, Brigo-Mercurio [2007] gives a formula for the more general case of a time Ti payoff (1.2.11) with a strike κ 6= 0: + i Z +∞ 1 h 1 2 i √ e− 2 v f (v) dv, = (1.2.33) Et aωSi,i+n1 (Ti ) + bωSi,i+n2 (Ti ) − ωκ 2π −∞ where
f (v) = aωSi,i+n1 (t) exp
µii,i+n1 (t)
√ 1 2 2 − ρˆ(t) σi,i+n1 (t) τ + ρˆ(t)σi,i+n1 (t) τ v 2
√ 2 + µi,i+n1 (t) + ( 21 − ρˆ(t)2 )σi,i+n (t) τ + ρ ˆ (t)σ (t) τv i,i+n1 1 × Φ ω √ p 2 σi,i+n1 (t) τ 1 − ρˆ(t) √ aS (t) 1 2 1 + µ (t) − σ (t) τ + ρ ˆ (t)σ (t) ln i,i+n τv i,i+n1 i,i+n1 h(v) 2 i,i+n1 , − ωh(v)Φ ω √ p σi,i+n1 (t) τ 1 − ρˆ(t)2
ln
aSi,i+n1 (t) h(v)
1
2
h(v) = κ − bSi,i+n2 (t)e(µi,i+n2 (t)− 2 σi,i+n2 (t))τ +σi,i+n2 (t)
√ τv
,
τ = Ti − t.
A straight-forward calculation shows that for κ = 0 the equation (1.2.31) is recovered. By no log-normality of the swap rates, an analytical solution for the case κ 6= 0 is only feasible if the spread is modeled as a normal distributed random variable: (1.2.34)
¯ Si,i+n1 (t) − Si,i+n2 (t) = S(t)
with
¯ ¯ (t). dS(t) = σ ¯ dW
This framework is too simple to consistently price CMS spread options since implicitly a perfect correlation is assumed. And it is also not taking into account the smile and the skew effects. The market quotes spread options by their implied normal volatilities, similar to swaptions which are quoted by their implied Black volatility.
1.3
Parameterization and Calibration
The general form of the forward Libor model (cf. Def. 1.1) is merely a framework which becomes a model once the forward volatility structure γ i (t), i ∈ {0, . . . , N −1}, is specified, which determines both the level of the forward rates and the correlation between the forward rates via ρi,j (t) =
γ i (t) · γ j (t) , kγ i (t)kkγ j (t)k
0 ≤ t ≤ min{Ti , Tj }.
23
Chapter 1. Forward Libor and Swap Market Models
The selected covariance structure should match the observable dynamics of the Libor rates, such as the number and the shape of the underlying principal components, cf. Rebonato [2002]. Once the forward volatility structure is specified, the chosen model is calibrated to the current forward rate curve and to liquid market instruments. Since the current (at t0 ) forward rates Fi (t0 ) are initial conditions, and hence inputs for the forward LMM, the calibration to the current forward rate curve is automatic. Calibration to cap and European swaption prices is achieved by choosing the forward volatility structure such that the model prices of these derivatives match their market prices as closely as possible. As shown in (A.1.3)–(A.1.7), the lognormal assumption in the forward Libor model allows for the pricing of caplets by the ”market convention” Black-Scholes formula, and, as we will see, it enables the derivation of good closed-form approximations of European swaption prices, which then leads to efficient calibration of the model correlations to swaption market prices. Nevertheless, with regard to the valuation of correlation-sensitive products such as CMS spread options, the calibration of instantaneous Libor correlation has always been a challenging point of the LMM which has not been satisfactorily fulfilled by the classical way of approximating the swap rate implied volatilities. At the end of this section we shall present two approaches to calibrate the LMM correlations: • by approximating the swap rate volatilities implied from the swaption quotes, Sec. 1.3.4; • by approximating the swap rate correlations implied from the prices of CMS spread options, Sec. 1.3.5.
1.3.1
Parametric Forms of the Instantaneous Volatilities
Driven by empirical observations many authors and practitioners put special emphasis to the desideratum that the term structure of instantaneous volatilities should evolve in a time–homogeneous manner, assuming ”by default” that it is desirable for a instantaneous volatility function to be able to reproduce (at least approximately) the current term structure of volatilities in the future (cf. Rebonato [2002]). As a result the instantaneous volatility function should be modeled not as a function of calendar time, but rather as a function of left time to maturity σi (t) = g(Ti −t). It is important to point out that the result does not depend on the details of the functional form of the instantaneous volatility function; the future smile surface will exactly ”look like” today’s smile surface. Apart from the time-homogeneity, Rebonato [1998, 1999a] states that the volatility function should have a flexible functional form to be able to reproduce either a humped or a monotonically decreasing instantaneous volatility, and allow for an easy analytical integration of its square (facilitating the evaluation of the necessary variance and covariance elements). Rebonato suggests in his works the following parametric form: (1.3.1)
g(Ti − t) = [a + b(Ti − t)]exp{−c(Ti − t)} + d,
c, d > 0, a + d > 0,
which fulfills these criteria to an acceptable degree (see Rebonato [2002] for examples and further explanations).
Chapter 1. Forward Libor and Swap Market Models
24
(1.3.1) can be extended to a richer parametric form; the extended linear-exponential volatility model (cf. Rebonato [2002], Brigo-Capitani-Mercurio [2003]): (1.3.2)
gext (Ti − t) = ki g(Ti − t),
k(Ti ) > 0.
Rebonato [2002] models the vector k ∈ RN as ki = 1 + ǫ(Ti ), being ideally close to one and flexible enough to allow for a better fit of volatility function to the market implied volatilities of different maturities. Assuming no smile and skews in the caplet markets, any choice of the parameters {a, b, c, d} will only approximately satisfy the ATM caplet condition, (1.3.3)
Black
(σi
2
) (Ti − t) =
Z
Ti
g(u)2 du,
t
across all forward rates8 . The parameters ki then allow for the Libor rate specific adjustment to exactly fit the market implied volatility: ki2 =
(1.3.4)
(σiBlack )2 (Ti − t) . R Ti 2 du g(u) t
The caplet condition (1.3.3) is then fulfilled by construction everywhere along the curve. A good and extensive overview of the volatility parameterizations used in practice can be found in Brigo-Mercurio [2007].
1.3.2
Calibration to the Cap/Floor Market
The market convention to quote caps and floors is to use the (implied) Black- volatility which plugged into the Black formula gives the market price of the cap/floor. In a smile-less world, we know from the Black-Scholes theory that for the instantaneous volatility function σi (t) of a forward rate Fi (t) in the lognormal LMM the implied Black volatility σiBlack is given by (cf. (A.1.4)), (1.3.5)
Black
(σi
2
) (Ti − t) =
Z
Ti
σi (u)2 du.
t
Ideally, in case of given market prices of ATM caplets, their implied Black volatility constitutes the right value to fit with the model volatility parameters. The market prices are unfortunately a bit more involved. The market quotes flat volatilities for caps of different maturities, T and strikes, K. Thus, an implied volatilBlack ity surface σcap (T, K) is quoted at any point in time. So what are the implied volatilities of the caplets that make up the caps with different strikes and maturities that are consistent with the quoted cap volatility surface? Alexander [2003] gives a brief and good overview about the particularities in stripping the information out of cap market prices. For instance, each fixed strike caplet in a cap with the 8
Depending on the calibration target one can choose the model volatility parameters to meet condition (1.3.3) even not (only) for ATM Black volatilities.
Chapter 1. Forward Libor and Swap Market Models
25
ATM strike K has a different moneyness. Each Ti maturing caplet is assumed to be ATM if Fi (Ti ) = K, but since each caplet has a different underlying forward rate, it will have a different ATM strike. So the different caplets in an ATM cap are only approximately ATM. One of the popular iterative methods, used to back out these caplet volatilities from the cap market implied volatility surface, is the vega-weighted interpolation technique, for which we refer to Alexander [2003]. Several stripping algorithms9 to extracting caplet volatilities out of quoted cap volatilities are presented in detail in the technical work by Hagan-Konikov [2004]. Finally, in a smile-less world, the calibration to caplets’ and floorlets’ implied volatilities (once extracted from the market quotes of caps and floors) for the LMM model is straight-forward via (1.3.5), and the correlations between forward Libor rates have no impact on the cap/floor prices. The application of the calibration requirement (1.3.5) to the volatility parameterizations given above is straight-forward as well. Therefore, in the sequel we will focus on the parameterization given in (1.3.2)–(1.3.4). The first step in the calibration procedure is to find the solution parameters {a, b, c, d} for the minimizing problem (1.3.6) s " #2 Z Ti h N −1 i2 X p min σiBlack (Ti − t) − [a + b(Ti − t)]exp{−c(Ti − t)} + d du . a,b,c,d
t
i=0
Additionally, the free Libor rate specific parameters ki in (1.3.4) can be used to exactly fit the respective (ATM) Black caplet volatilities.
1.3.3
The Structure of Instantaneous Correlations
As discussed in Sec. 1.1.3, both, the instantaneous volatility formulation as well as the chosen instantaneous correlation, can contribute to terminal correlations. The qualities and properties an instantaneous correlation matrix ρ associated with a LMM should have are (cf. Brigo-Mercurio [2007]): • Symmetry and ones on the diagonal: ρi,j = ρj,i ,
ρi,i = 1,
for all i, j ∈ {0, . . . , N − 1}10 ;
• ρi,j ≥ 0 for all i, j, and the map i 7→ ρi,j has to be decreasing for i ≥ j, thus, moving away from the diagonal along a column or row the entries become monotonically decreasing as joint movements of far away rates are less correlated than movements of the rates with close maturity; • When moving along the yield curve, the larger the tenor, the more correlated the adjacent forward rates are. Hence, the sub-diagonals, i 7→ ρi+p,i , will be increasing for a fixed p. 9
The idea behind the boot-stripping algorithms is that if we know, for instance, the 1 and 2 year flat volatilities we know the 1 year and 2 year cap prices. Their price difference is by no arbitrage arguments the second caplet in 2 year cap contract. It is thus required to solve a volatility that implies this caplet price. The same procedure is continued iteratively further. 10 For ease of notation we will be numbering the elements of the correlation matrix by beginning N −1 , coinciding with the numbering of the forward rates and their expiries. with zero, ρ = {ρi,j }i,j=0
Chapter 1. Forward Libor and Swap Market Models
26
A variety of parameterization functions have been introduced over the past years that allow for expressing a given correlation matrix of forward rates in a functional form. There are several advantages to this: of course, it is computationally convenient to work with an analytical formula. But also noise, such as bid-ask spreads, and illiquidity are removed by focusing on general properties of correlation. Furthermore, the rank and the positive semi-definiteness of the correlation matrix can be controlled through the functional form. The parameterizations we shall present here are full-rank parameterizations. We will also discuss how to reduce their rank depending on the number of underlying Brownian motions of the model. One property that is implicitly present in all parameterizations is the desirable time-homogeneity of the correlations. Full-rank correlation parameterization In general, the full instantaneous correlation matrix is characterized by N (N − 1)/2 entries, given the symmetry and the ones on the diagonal. This number of entries may be too high for practical purposes, thus, a parsimonious parametric form with reduced number of parameters has to be found. In the literature a vast number of correlation parameterizations is presented; to be mentioned here are the works of Schoenmakers-Coffey [2003], Wu-Zhang [2003], Morini-Webber [2006], and lately the papers of B¨orger-van Heys [2010] and Lutz [2010]. We will focus in the sequel on some of the parameterizations which will be used in our model calibration later on. Three-parameters full-rank exponential parameterization. For 0 ≤ t ≤ min{Ti , Tj } Rebonato [2004] proposed a parameterization of the form h i (1.3.7) ρi,j (ρ∞ , α, β; t) = ρ∞ + (1 − ρ∞ ) exp − |Ti − Tj |(β − α max{i, j}) ,
which fulfils the desirable properties given above. Thus, it may produce for a given tenor structure realistic market correlations for properly chosen ρ∞ ∈ (−1, 1), β > 0 and (small) 0 ≤ α ≤ β/(N − 1) (cf. Rebonato [1999a]). A slight modification of (1.3.7), also given in Rebonato [2004], reads: h i (1.3.8) ρ˜i,j (ρ∞ , α, β; t) = ρ∞ + (1 − ρ∞ ) exp − β|Ti − Tj | exp −α max{i, j} , with ρ∞ ∈ (−1, 1), β > 0 and α ∈ R. A special case of (1.3.7) is the Rebonato’s two-parameters full-rank exponential parameterization: h i (1.3.9) ρi,j (ρ∞ , β; t) = ρ∞ + (1 − ρ∞ ) exp − β|Ti − Tj | , β > 0, ρ∞ ∈ (−1, 1). However, it should be noted that for a particular choice of parameters it is not directly guaranteed that (1.3.7) or (1.3.8) defines valid correlation structure indeed (it might violate the positive semi-definiteness of the correlation matrix). The special case with ρ∞ = 1 (cf. Rebonato [2002]), h i (1.3.10) ρi,j (β; t) = exp − β|Ti − Tj | , t ∈ [0, Ti ∧ Tj ],
27
Chapter 1. Forward Libor and Swap Market Models
assures a symmetric correlation matrix with positive eigenvalues. This parameterization is analytically very attractive and fulfills the basic modeling requirements. Apart from their parameter poorness which might turn out to be a handicap when fitting to market quotes, these parameterizations do not distinguish on the distance between two different forward rates such that different pairs of forward rates with the same distance to each other are correlated to the same degree. Since an unconstrained optimization is preferable to a constrained one, Schoenmakers-Coffey [2003] parameterizations might be preferred from this point of view. Nonetheless, the Rebonato’s parameterizations are widely used in practice because of their analytical tractability and the easy calibration. Doust’s multiplicative correlations (cf. Doust [2007], Rebonato-McKay-White [2009]). To overcome the above-mentioned problem that the decorrelation (brought into the model by the constant exponential decay factor β) only depends on the distance between two rates, the challenge will be to introduce a dependence of the decorrelation factor on the expiries of the forward rates, β = βi,j , in such a way that the resulting correlation matrix does not loose any of the desired properties of being a valid correlation matrix. Similar in spirit to the construction in Schoenmakers-Coffey [2003], Doust proposes the following parametric structure (cf. Doust [2007]): For ai ∈ [−1, 1],11 i = 1, . . . , N −112 the elements of the correlation matrix are defined recursively: - First define the trivial diagonal elements, ρi,i = 1, i = 0, . . . , N − 1; - Then define the elements of the first row by respecting the symmetry as ρ0,j =
j Y
j = 1, . . . , N − 1;
ak = ρj,0 ,
k=1
- By inspection, assuming that i > j, fill the lower triangle part by ρi,j =
ρ0,i = ρj,0
The upper triangle part is then defined by 1 a1 a1 1 a1 a2 a2 ρ(a1 , ..., aN −1 ; t) = .. . a1 · · · aN −2 a1 · · · aN −1
i Y
ak .
k=j+1
the symmetry relationship: a1 a2 a1 a2 a3 . . . a1 · · · aN −1 a2 a2 a3 . . . a2 · · · aN −1 1 a3 . . . a3 · · · aN −1 .. ... ... ... . . . . aN −2 1 aN −1 ... aN −1 1
.
Given the N − 1 quantities ai , Doust [2007] proves that the resulting matrix is always a real symmetric positive definite matrix which admits a simple Cholesky 11
In the most cases ai will be positive, 0 < ai ≤ 1, as, empirically, the forward rate – forward rate correlations are proven to be positive. 12 Recall that a correlation matrix has got N (N −1)/2 elements to be specified; here we are dealing with N − 1 unknown parameters.
28
Chapter 1. Forward Libor and Swap Market Models
decomposition. Given an equidistant time grid ∆T = Ti+1 − Ti , i = 0, . . . , N − 1, (i.e. constant spacing between the forward rates in the considered model) RebonatoMcKay-White [2009] suggests the following choice of the parameters ai : (1.3.11)
ak = exp{−βk ∆T },
k = 1, . . . , N − 1.
Then we have for i > j (1.3.12)
n
ρi,j (β1 , ..., βN −1 ; t) = exp −
i X
k=j+1
o βk ∆T ,
0 ≤ t ≤ min{Ti , Tj }.
The dependence of βk on k allows us to specify the degree of decorrelation between rates with same distance but different expiries. A decreasing property of βk : βk > βk+1 , is empirically evident. Obviously, for a constant βk ≡ β, the simple parameterization in (1.3.10) will be recovered. The flexibility can be even increased by introducing functional forms to describe the dependence of βk > 0 on k. Polynomial forms of degree M ≤ N − 1, PM , like (1.3.13)
βk = PM (k) =
M X
gl /k l ,
l=0
with positive parameters gl , easily guarantee the desired properties of the matrix. The correlation parameterization then takes for i > j the shape (1.3.14)
M i o n X X gl /k l , ρi,j (g0 , ..., gM ; t) = exp −∆T k=j+1 l=0
t ∈ [0, Ti ∧ Tj ].
Finally, Rebonato-McKay-White [2009] go one step further imposing an additional long-term decorrelation among the forward rates, preventing that the decorrelation goes asymptotically to zero with increasing distance between the rates, but rather to some finite economically plausible level ρ∞ > 0: (1.3.15) ρi,j (ρ∞ , g0 , ..., gM ; t) = ρ∞ + (1 − ρ∞ )ρi,j (g0 , ..., gM ; t),
t ∈ [0, Ti ∧ Tj ].
Most of the introduced parameterizations are discussed with great detail in Rebonato [2004], while Schoenmakers [2002] is a good reference for the particularities of their numerical implementation and performance. Rank-Reduced Correlations From the standard matrix calculus it is well known that any positive semi-definite symmetric matrix ρ ∈ RN ×N can be diagonalized by means of a real and orthogonal matrix P ∈ RN ×N : (1.3.16)
ρ = PDP⊥ ,
with PP⊥ = P⊥ P = I,
where D ∈ RN ×N is the diagonal matrix containing the positive eigenvalues of the original matrix √ ρ, whereas the columns of P are the eigenvectors of ρ. Setting B := P D we obtain (1.3.17)
ρ = BB⊥
and
B⊥ B = D.
29
Chapter 1. Forward Libor and Swap Market Models
Rebonato [1999a] mimics the decomposition (1.3.17) by means of a suitable matrix B ∈ RN ×d of rank d < N such that BB⊥ is a d-rank correlation matrix. In Rebonato [1999a] it is stated and proved that for any real matrix B ∈ RN ×d with rank d ≤ N, the matrix product BB⊥ is real and symmetric, and it can be diagonalized into BB⊥ = PDP⊥ , where P ∈ RN ×d is an orthogonal matrix containing the orthogonal eigenvectors of BB⊥ and D ∈ Rd×d a diagonal matrix storing the squares of the eigenvalues of BB⊥ . This result is obviously very useful in the parametrization of a d-factor Libor model. As we will see it shall allow us to reduce to a very large extent the difficulties in the simultaneous calibration to volatilities and to the correlation matrix. Moreover, it is this result which completes the model picture started in (1.1.5)– (1.1.7), where already a low-factor Brownian shock was anticipated. It can be easily seen that the introduced matrix B coincides with the matrix containing the loading vectors bi as row vectors: ⊥ B = b⊥ (t) = b0 (t), . . . bN −1 (t) at any time t.
Here we shall present the two mostly used rank reduction techniques.
The Hypersphere Decomposition. Rebonato-Jaeckel [1999] suggests the following form of the i-th column vector of the matrix b in (1.1.7): ( Qk−1 sin (θi,j (t)) if k = 1, . . . , d − 1 cos (θi,k (t)) j=1 Qk−1 (1.3.18) bi,k (t) = if k = d, j=1 sin (θi,j (t))
where the angles θi,j (t) constitute the parameters to be altered within the fitting optimization algorithm. We denote the resulting low-rank correlation matrix by ρθ .
The Spectral Decomposition. Following Rebonato-Jaeckel [1999] an alternative and effective way of rank-reduction is the so-called spectral decomposition. Given a target number of driving factors d and assuming (1.3.16) we can arrange the eigenvalues in D in descending order and rearrange in P the corresponding eigenvectors such that their numbering in columns corresponds to the order of eigenvalues in the ˜ new D: ˜ D(t) ˜ P ˜ ⊥ (t). ˜ (t) = P(t) ρ Then the smallest N − d eigenvalues will be set to zero and the corresponding eigen˜ vectors will be taken off the matrix P(t), resulting in the approximative correlation matrix ˜ (d) (t)D ˜ (d) (t)(P ˜ (d) )⊥ (t). ˆ (d) (t) = P ρ ˆ (d) (t), although positive semidefinite, does not necIn general the resulting matrix ρ ˆ (d) (t) as a covariance essary feature ones on its diagonal. The solution is to interpret ρ matrix and to derive the correlation matrix associated with it by normalizing it: (d)
(d) ρi,j (t)
= q
ρˆi,j (t) (d) (d) ρˆi,i (t)ˆ ρj,j (t)
.
Chapter 1. Forward Libor and Swap Market Models
30
By following this procedure we obtain an acceptable correlation matrix ρ(d) (t) which is a d-rank approximation and intuitively similar to the target one. This methodology can be found in the literature as the principal component analysis (PCA). Approach to Optimizing on a Low Rank Parametric Form Once the target full-rank correlation matrix ρmod is given as input, we can minimize over the angle parameters θi,j (t) (cf. (1.3.18)) the norm of the difference between the target matrix ρmod and the low-rank matrix ρθ : min
θi,j (t)
N X
i,j=1
θ 2 . |ρmod i,j (t) − ρi,j (t)|
Rebonato-Jaeckel [1999] proved empirically that the differences between the reduction over the angles and the spectral decomposition is typically very small. In general, when we calibrate the LMM to swaptions using the instantaneous correlations ρ as fitting parameters, as we will see below, we are free to select a-priori a parametric form for the correlation matrix. Once the model matrix of instantaneous correlations ρmod is defined we can use one of the introduced rank reduction algorithms to reduce the degrees of freedom for the random shocks we will use to simulate the forward term structure for future times and price interest rate derivatives.
1.3.4
Calibration of LMM Correlations to Swaptions Volatilities
The market is quoting the swaptions in terms of their Black implied volatilities. For instance, a Tm × (Tn − Tm ) ATM payer swaption with expiry at Tm on the swap rate Black Sm,n (t) is conventionally quoted as σm,n . Calibrating the LMM consists of finding the instantaneous volatility {σi (t)}i and correlation parameters ρ(t) in the LMM dynamics that reflect the swaptions prices observed in the market. Combining the equations (A.3.3) and (1.1.23) we obtain n−1 Z Tm m,n X vi (s)vjm,n (s)Fi (s)Fj (s) Black 2 (1.3.19) σm,n (Tm − t) = γ i (s) · γ j (s) ds. 2 (s) Sm,n i,j=m t A sufficiently good and market proven approximation of (1.3.19) is the following: Z n−1 X vim,n (t)vjm,n (t)Fi (t)Fj (t) Tm Black 2 (1.3.20) σm,n (Tm − t) = γ i (s) · γ j (s) ds, 2 (t) S t m,n i,j=m
which we shall consider as the standard approach.
1.3.5
Calibration to Correlations Implied From CMS Spread Options
Among practitioners and academics alike, there is consensus on the fact that even the low-parametric LMM correlation parameterizations can hardly be calibrated reliably
Chapter 1. Forward Libor and Swap Market Models
31
to market data due to the fact that swaptions carry only little information about correlations (Alexander [2003], Brigo-Mercurio [2007], Rebonato [2002], Schoenmakers [2002, 2005], Schoenmakers-Coffey [2003]). The market of structured interest rate products, in particular pay-off structures including derivatives of constant maturity swaps, has undergone an enormous growth during the past few years. As already discussed in Sec. 1.2.2, such structures equipped with call rights such as CMS spread options, require a realistic modeling of not only the development of swap rates, but also their correlation. Such correlation information is meanwhile available, because a separate market has developed for spread options used for hedging purposes almost like plain vanilla instruments. Assuming jointly lognormal swap rates we can work out an implied correlation of the involved swap rates in a CMS spread option. This correlation can then be taken as target to be reproduced by a pricing model and, therefore, has to be included in a model calibration procedure of LMM. Motivated by the general formula for the valuation of the CMS spread optionlet given in (1.2.33), assuming log-normality of the swap rates under Pi , with Ti the expiry time of the optionlet, B¨orger-van Heys [2010] propose a calibration of the parameterized instantaneous forward rate correlations to prices of CMS spread options. Regarding (1.2.33), the value of a CMS spread optionlet can be written as (1.3.21) + i h = F Si,i+nk (t), µii,i+nk (t), σi,i+nk (t), ρ(t), Ti , κ , Eit ωSi,i+n1 (Ti ) − ωSi,i+n2 (Ti ) − κ
where k = 1, 2. Applying the typical freezing for the drifts µii,i+nk (t) of the swap rates under Pi , and deriving the swap rate volatilities σi,i+nk (t) from the given market quotes of the swaption ATM-implied volatilities (as in (A.3.3)), (1.3.21) can now be considered as a target function for the swap rate correlation ρ(t), and also for the forward rate correlations through ρ(t) = (1.3.22)
1
i+n 2 −1 1 −1 i+n X X
(Ti − t)σi,i+n1 (t)σi,i+n2 (t) Z Ti × σl (s)σk (s)ρl,k (s) ds.
k=i
l=i
vki,i+n1 (t)vli,i+n2 (t)Fk (t)Fl (t) Si,i+n1 (t)Si,i+n2 (t)
t
Here we used a standard approximation, cf. Belomestny-Kolodko-Schoenmakers [2010], for the swap rate correlation. This approach to the calibration of the Libor rate correlations was treated as a improvement of the typical market practice of calibration to swaption volatilities, showing satisfactory results for certain parameterization of the instantaneous correlations, such as in Schoenmakers-Coffey [2003]. We refer for the details to B¨orger-van Heys [2010]. The latter underpins our motivation for calibrating the forward rate correlation of the more involved SABR–LMM model we shall introduce in Chap. 3.2.
Chapter 2
The SABR Model of Forward Rates
2.1
General Model Dynamics
As already mentioned in Chap. 1.1, one problem encountered when modeling derivatives like caplets in the LMM and therefore using the Black formula is, that the market prices for caplets over different strikes cannot be obtained with a constant volatility parameter as the model demands1 . The presence of these volatility skews and smiles in the market is however evidence that the underlying is driven by some process other than a lognormal one. With this in mind Dupire [1994] proposed the local volatility model, which has the advantage that the model perfectly replicates the current market situation. But the approach behaves poorly in forecasting future dynamics and option pricing is not possible in closed form. Thus, it became of practical interest to develop stochastic versions of the volatility structures of forward Libor models capable of matching the observed volatility in the markets of caps and swaptions, by considering a more general volatility process of the form γ(t, T, S; F (t, T, T ∗ ))2 . It was the seminal work of Hagan et al. [2002] who launched the so called SABR model, where the forward rate process is modeled under its forward measure using a correlated lognormal stochastic volatility process. Hagan et al. [2002] explain clearly why ”just fitting the today’s market prices” is not good enough. Taking into account the quality of prediction of the future dynamics of the volatility smile, meeting the observations from the market reality, is as crucial as the best achievable fitting to the today’s market prices. Hagan and his colleagues were not the first to equip BGM-type models with stochastic volatility, see for instance the Cox-Ingersoll-Ross (CIR)-type models of AndersenAndreasen [2002], Andersen-Brotherton-Ratcliffe [2005], Piterbarg [2003, 2005a,b] or models of jump-diffusion. All these models lack the ability to fit accurately the 1
It should be mentioned that the generic BGM framework does not necessarily require the forward volatility functions of forward rates to be deterministic functions; they may be adapted processes or some deterministic or random functions of the underlying forward Libor rates. 2 Nevertheless, the class of models of practical interest are mainly characterized by a separable volatility structure, cf. Andersen-Andreasen [2002], Andersen-Brotherton-Ratcliffe [2005], Andersen-Piterbarg [2007], Piterbarg [2003, 2005a,b], Wu-Zhang [2006], etc.
32
33
Chapter 2. The SABR Model of Forward Rates
appropriate market smile surface in a simple fast and robust manner. It is the ability to do so which constitutes one major advantage of the SABR model, as there exists an approximation formula to implied Black volatility using the SABR parameters, introduced by Hagan et al. [2002]. Hence, option prices, such as of caps and floors, can be calculated using the well known Black pricing framework but taking into account the volatility surface using a strike dependent volatility function. Nowadays the SABR model has become a reference stochastic volatility framework for modeling smiles in the financial industry, because of the described properties and its easy application. The SABR model attempts to capture the dynamics of a single forward rate. Depending on the context, this forward rate could be a Libor forward rate, a forward swap rate, the forward yield on a bond, etc. However, we shall focus in the following on the SABR model of forward Libor rates. Definition 2.1 (multifactor SABR Model). Assume that the number of model factors, that is the number of independent driving Brownian motions, is d + d˜ under the forward measure Pi+1 , for the index i ∈ {0, . . . , N − 1}. Building on the preliminary framework presented in Chap. 1 for the classical LMM, ˜ the (d × d)–factor SABR model (SABR) assumes the following dynamics for the forward rate Fi under its payoff measure Pi+1 : (2.1.1)
β (t)
dFi (t) = Fi i (t)σi (t) bi (t)·dWi+1 t ,
0 ≤ t ≤ Ti ,
where 0 ≤ βi (t) ≤ 1 and σi (t) is a stochastic variable following a diffusion process of type3 : (2.1.2)
˜i (t)·dW ˜ i+1 , dσi (t) = σi (t)νi (t) b t
0 ≤ t ≤ Ti ,
˜ ˜ i+1 a d-dimensional independent standard where Wi+1 is a d-dimensional and W t t i+1 Brownian motion under P , and νi (t) the exogenously given deterministic volatility of volatility function4 . ˜i (t) ∈ Rd˜ satisfy kbi (t)kd = kb ˜i (t)k ˜ = 1 and The loading vectors5 bi (t) ∈ Rd and b d for 0 ≤ t ≤ min{Ti , Tj }: bi (t)T bj (t) = ρi,j (t),
(2.1.3) ˜ min{d,d}
(2.1.4)
X
bik (t)˜bjk (t) = φi,j (t),
φi,i (t) = ξi (t),
˜j (t) = θi,j (t), ˜i (t)T b b
j = 0, . . . , N − 1,
k=1
(2.1.5) 3
Note that there is no mean reversion for the volatility process. Since we are looking at one forward rate at a time, this is not necessarily a problem as long as the correct terminal distribution of the forward rate is obtained. 4 While P i+1 , induced by B(t, Ti+1 ), is a natural martingale measure for the forward rate Fi (t), due to its definition in (1.1.1), the fact that the volatility process in (2.1.2) is defined to be driftless under P i+1 is rather ”artificial” and can be seen as a model assumption. 5 For ease of notation we will be numbering the elements of all correlation matrices by beginning with zero such that they coincide with the numbering of the forward rates.
Chapter 2. The SABR Model of Forward Rates
34
where the matrices ρ(t), φ(t), θ(t) ∈ RN × RN are exogenously defined, with the particularity that φ(t) has got ξi (t) = (ξ(t))i , the correlation between the forward rate and its own volatility process, on the diagonals (instead of ones)6 . Imposing the initial conditions for the forward rate process Fi (0) and its initial volatility σi (0) = σiSABR , the model becomes fully specified. Investigating the qualitative behavior of the (Black) volatility implied from the SABR model, Rebonato-McKay-White [2009] spotted the following properties for the SABR parameters: • a change (upwards) of the SABR initial volatility σiSABR causes an almost shift (upwards) of the implied volatility smile across strikes, and a modest steepening of the smile (low strikes increase more than high strikes); • when the exponent βi goes from 1 to 0, it causes a progressive steepening of the smile and introduces modest curvature to it, while, on the other hand, an increasing β lowers the level of the smile; • a similar effect as for decreasing βi can be spotted when the correlation parameter ξi moves from 0 to −0.5; the smile becomes negatively slopped, accompanied with small decrease in curvature. There seems to be a pronounced redundancy in the resulting effects in choosing the parameters βi and ξi ; the prevalent market practice is to fix the exponent βi (usually at 0.5) and optimize the fitting over the other parameters; • finally, νi caters for the curvature of the smile (increasing νi increases the curvature), with certain secondary effect on the steepness of the smile. However, only the interaction of all these parameters together makes the model successfully capable to capture the different market volatility smile surfaces. For deeper discussions about the solvability of the SABR system for different CEV exponents β and the property of the dynamics whether zero or negative rates are attainable we refer to the extensive analysis done in Rebonato-McKay-White [2009]. Rebonato-McKay-White [2009] provides a good empirical overview of the strengths of the SABR model with respect to recovering the dynamics of the smile evolution when the underlying changes, emphasizing, in particular, the aspects of hedging of the interest rate risk.
2.1.1
The Time-Homogeneous Model
Definition 2.2 The multifactor time–homogeneous SABR model is a special case of the model described in Def. 2.1. The alteration consists of the constant parameters over time: βi (t) ≡ βi ∈ [0, 1], νi (t) ≡ νi ∈ R, 6
Here we assume that the matrices ρ(t) and θ(t) are valid correlation matrices (cf. Sec. 1.3.3). ˜ ˜ The slightly modified matrix φ(t) = (φi,j (t))i6=j and (φ(t)) i,i = 1 is assumed to be a valid correlation matrix as well.
35
Chapter 2. The SABR Model of Forward Rates and time–homogeneous correlation matrices for 0 ≤ t ≤ min{Ti , Tj } (2.1.6) ˜ min{d,d}
(2.1.7)
X k=1
(2.1.8)
bi (t)T bj (t) = ρi,j (Tj − Ti ), bik (t)˜bjk (t) = φi,j (Tj − Ti ),
φi,i (0) = ξ i ,
˜j (t) = θi,j (Tj − Ti ). ˜i (t)T b b
Imposing the initial conditions for the forward rate process Fi (0) and its volatility process σi (0) = σiSABR , the model becomes fully specified.
2.1.2
Joint Dynamics of the SABR Forward Rates and Their Volatilities
Applying the same change of measure technique as in Sec. 1.1.1, in order to derive the arbitrage–free dynamics of the system of the SABR forward rates and their volatilities in a single measure, say Pk+1 , able to be implemented in practice, we obtain for j < k (j > k analogously): k B(t, Tj+1 ) B(t, Tj+1 ) X τi d = dFi (t) + O(dt2 ), B(t, Tk+1 ) B(t, Tk+1 ) i=j+1 1 + τi Fi (t)
Consequently the following general drift formula for the forward rates model at time 0 ≤ t ≤ Tj can be derived (cf. Sec. 1.1.1): (2.1.9) k β P τi Fi i (t) σi (t)ρi,j − i=j+1 1+τi Fi (t) β βj j + σj (t)Fj dt dFj (t) = Fj (t)γ j (t)·dWk+1 0 t j βi P τi Fi (t) σi (t)ρi,j 1+τi Fi (t)
in the SABR
,
j k,
i=k+1
where γ j = σj (t)bj (t). Analogously, the arbitrage free dynamics with respect to Pk+1 ˜ j (t)): of the volatility parameters in the SABR model is given by (ν j (t) = νj b (2.1.10) k β P τi Fi i (t) σi (t)φi,j , j k. 1+τi Fi (t) i=k+1
Similarly, under the spot measure P∗ the dynamics is given by the stochastic system: β
(2.1.11) dFj (t) = Fj j (t)γ j (t)·dWst + Fjβ (t)
j X τi Fiβi (t) σi (t)σj (t)ρi,j dt, 1 + τi Fi (t)
i=m(t)
(2.1.12)
s
˜ + σj (t) dσj (t) = σj (t)ν j (t)·dW t
j X τi Fiβi (t) σi (t)νj (t)φi,j dt. 1 + τi Fi (t)
i=m(t)
Chapter 2. The SABR Model of Forward Rates
2.2
36
Valuation in the SABR Model
The SABR model has meanwhile established itself as one of the most popular models for pricing and risk managing interest rate derivatives. One of the main virtues of this model is its ability to describe the smile effects in the volatility market quotations of the benchmark instruments which is the major limitation of the classical LMM. Although there are many models which try to catch the volatility smiles, the SABR’s popularity is indebted to its approximative compatibility with the Black formula (cf. Hagan et al. [2002]), the standard market practice of pricing benchmark instruments. It allows us to easily and accurately price benchmark instruments by making use of the SABR implied volatility in the Black –formula. In Appendix A.5 we give the formulas for the SABR implied Black and normal volatilities. It should be mentioned that the Hagan et al. [2002] asymptotic expansion to the SABR model to approximate the Black volatility, is an expansion in small volatility and small time, and was originally tested against short-dated Eurodollar options in a low-volatility high-rate environment. In the industry, the model is known to break down for high volatilities, high volatility-of-volatility, low rates, and long times to expiry. As mentioned above, the benchmark instruments can be accurately priced by making use of the SABR implied volatility in the Black formula. For instance, assume the SABR model of forward rates given in Def. 2.1, 2.2. The pricing of the caplets/floorlets is then approximated by7 √ (2.2.1) CPL(0, T, S, κ) = B(0, S)τ (T, S)BS F (0, T, S), κ, σ ˜TBlack (κ) T , 1 , with
σ ˜TBlack (κ) = σ ˜ Black (T, K, F (0, T, S), σTSABR , νT , βT , ξT ) given in Appendix A.5. While in LMM for a certain forward rate we were able to directly recover the price of a caplet only for certain strike, mostly ATM, to which the model volatility was calibrated, in SABR we are able to recover the prices of the entire caplet smile using the SABR implied volatility. An other example of straight-forward pricing in SABR, by means of the SABR implied volatility, are the swaptions. Assume that the evolution of the swap rate is governed by the SABR dynamics given in (4.4.1). The payoff (A.3.1) can now be priced in SABR with respect to the measure Pm,n by p Black PSWO(0, Tm , Tn , κ) = Bm,n (0)BS Sm,n (0), κ, σ ˜m,n (κ) Tm , 1 ,
with
Black SABR σ ˜m,n (κ) = σ ˜ Black (Tm , κ, Sm,n (0), σm,n , νm,n , βm,n , ξm,n )
given in Appendix A.5. Here again, the whole swaption smile can be accurately recovered by the model parameters. We will see in the next chapters that the fitting of the SABR model parameters to the respective smile constitutes the basis of any calibration of the SABR models to be considered. 7
The notation corresponds to the notation in Sec. A.1.
Chapter 3
Pricing CMS Derivatives in SABR
3.1
The Markovian Projection Method
The Markovian projection method, first introduced by Piterbarg [2003, 2005a,b] and formalized in Piterbarg [2007], is an approach to volatility calibration and represents a way of deriving efficient, analytical approximations to European-style option prices on various underlyings. This generic framework is applicable to a wide range of diffusion models and its power has been demonstrated on a number of examples, including spread and basket options, relevant to practical applications (cf. AntonovArneguy [2009], Antonov-Misirpashaev [2006], etc.). As we shall see, this method is also capable to incorporate stochastic volatility models with a correlation structure between all stochastic variables/processes. We will apply this technique to approximate a basket of SABR variables by an univariate model, aiming at pricing CMS derivatives analytically, in particular the CMS spread contracts.
The Mimicking Theorem The term Markovian projection (MP) refers to a technique that is based on a theorem by Gyoengy [1986] which explains how a complicated, usually non-Markovian process can be replaced by a Markovian process, the mimicking process, with the same onedimensional marginal distributions as the original process. Theorem 3.1 (Gyoengy [1986]) Given a filtered probability space Ω, {Ft }, P , let X(t) be an It˜ o process governed by
(3.1.1)
dX(t) = α(t)dt + β(t) · dW(t),
where W(t) is a d-dimensional Ft -Brownian motion and α(t) ∈ Rn , β(t) ∈ Rn,d are bounded measurable Ft -adapted processes. Let (3.1.1) admit a unique solution. Then there exist bounded measurable functions a : R+ × Rn → Rn and b : R+ × Rn → Rd,d , defined for every (t, y) ∈ R+ × Rn through 1/2 (3.1.2) a(t, y) = E[α(t) | X(t) = y], b(t, y) = E β(t)T β(t) | X(t) = y , 37
38
Chapter 3. Pricing CMS Derivatives in SABR such that the following SDE: (3.1.3)
dY(t) = a(t, Y(t))dt + b(t, Y(t)) · dW(t),
Y(0) = X(0),
admits a weak solution Y(t) that has the same one-dimensional marginals as X(t). Since X(.) and Y(.) have the same one-dimensional marginal distributions, the prices of European-style options on X(.) and Y(.) for all strikes K and expiries T are the same. Thus, for the purpose of European option valuation or the purpose of calibration to European options, one can replace a potentially complicated process X(.) with a much simpler Markov process Y(.). The correspondence between the processes is called Markovian projection of X(.) onto Y(.), where Y(.) follows a local volatility process. The function b(t, x) is often called Dupire’s local volatility. It should be noted that the Markovian projection is exact for European options but, of course, does not preserve the dependence structure of the underlying at different times. Thus, the prices of securities dependent on sampling at multiple times, such as barriers, American options and so on, are different between the original model and the projected model (cf. Piterbarg [2007]). A direct application of this result is however not possible, since the ”simpler” equivalent Markovian process is usually due to the calculation of the conditional expected values in (3.1.2) still too complicated to enable analytical tractability. The Markovian process then needs to be approximated by e.g. a displaced-diffusion which is a linear function of state, possibly with time-dependent parameters.
3.2
A Displaced SABR Diffusion Model for CMS Derivatives
3.2.1
Projection of CMS-Spreads to Displaced SABR Diffusion
A detailed discussion about the CMS spread options and their pricing in LMM is given in Sec. 1.2.2. The scope of this chapter is to develop a approximation formula for pricing of these derivatives in the SABR model of forward rates. In what follows we shall treat these derivatives as basket options, thus, options where the underlying is a basket of SABR forward rates. In the setting of a basket of forward price processes, an option on the basket can only be valued analytically by the formula of Margrabe [1978] and its derivations in the case of two assets. For higher dimensions the arbitrage-free price needs to be computed numerically, usually by Monte Carlo simulations, which can become in the case of stochastic volatility very time consuming and impracticable for calibration purposes. Therefore, the necessity for approximation formulas for calibration purposes can not be circumvented. Following Kienitz-Wittkey [2010], our aim is to make use of the Markovian Projection (MP) to approximate, in terms of the terminal distribution, a basket of diffusions by a univariate diffusion. In the case of multivariate SABR diffusions for a basket of
39
Chapter 3. Pricing CMS Derivatives in SABR
forward rates, as the CMS spreads can be formulated as, we show how these CMS spreads can be approximated by a displaced diffusion model of Rubinstein [1983] with a SABR stochastic volatility, aiming at valuing the spread option in closed form by taking into account the volatility cube and a full correlation structure of the SABR model of forward rates. To this end we shall apply the techniques developed by Piterbarg [2007] with the applications in Antonov-Arneguy [2009], Antonov-Misirpashaev [2006], and later adopted by Kienitz-Wittkey [2010] in the case of a more generic basket of SABR diffusions. We start with the time–homogeneous SABR model introduced in Def. 2.1 – 2.2 and make the simplification ˜ i (t) · dW ˜ i+1 , i = 0, . . . , N − 1, ˜ i+1 = b dWti+1 = bi (t) · dWi+1 dW t t , t i+1 i+1 ˜ t rate-specific one-dimensional Brownian motions under Pi+1 . The with Wt , W ˜ i will come into play when a given correlation model matrix loading factors bi and b needs to be factorized in the spirit of the Sec. 1.3.3 by a rank reduction algorithm, in ˜ to be used order to match its rank (2N ) to the number of Brownian drivers (d + d) in simulations. Summarized the model dynamics reads for i = 0, . . . , N − 1:
(3.2.1)
(3.2.2) (3.2.3) (3.2.4) (3.2.5) (3.2.6)
dFi (t) dσi (t)
i dWt , dWtj D E j i ˜ dWt , dWt D E ˜ ti , dW ˜ tj dW
= Fiβi (t)σi (t) dWti+1 , ˜ ti+1 , 0 ≤ t ≤ Ti , = νi σi (t) dW = ρi,j (Tj − Ti )dt = ρi,j dt, j = 0, . . . , N − 1
= φi,j (Tj − Ti )dt = φi,j dt, with φi,i (0) = ξi , = θi,j (Tj − Ti )dt = θi,j dt,
0 ≤ t ≤ min{Ti , Tj },
imposing the initial conditions for the forward rate process Fi (0) and its volatility process σi (0) = σiSABR . In the following we will consider the spread between two swap rates with different lengths: Sk,n1 ,n2 (t) := Sk,k+n1 (t) − Sk,k+n2 (t). As in (1.1.20) the difference between the two swap rates can be written as (3.2.7) Sk,n1 ,n2 (t) =
k+n 1 −1 X
wik,k+n1 (t)Fi (t)
i=k
with the stochastic weights wim,n (t) =
−
k+n 2 −1 X
wik,k+n2 (t)Fi (t),
i=k
t ∈ [0, Tk ],
τi B(t,Ti+1 ) . Bm,n (t)
Let us assume, without loss of generality, that n1 = max{n1 , n2 }, and the weights belonging to the shorter swap with length n2 are set to zero beyond k + n2 , i.e. wik,k+n2 = 0, for i ≥ k + n2 , such that the two weight vectors are of the same length n1 . This eases the notation for the difference in (3.2.7) to Sk,n1 ,n2 (t) = (3.2.8)
=
k+n 1 −1 X
i=k k+n 1 −1 X i=k
wik,k+n1 (t) − wik,k+n2 (t) Fi (t),
υik,n1 ,n2 (t)Fi (t),
t ∈ [0, Tk ],
40
Chapter 3. Pricing CMS Derivatives in SABR
where υik,n1 ,n2 is defined by υik,n1 ,n2 (t) := wik,k+n1 (t) − wik,k+n2 (t). The more general swap spread, given in (1.2.11), can be written as (3.2.9) k+n 1 −1 X υik,n1 ,n2 (t; a, b, ω)Fi (t), Sk,n1 ,n2 (t; a, b, ω) = aωSk,k+n1 (Tk ) + bωSk,k+n2 (Tk ) = i=k
with a, b ∈ R, ω ∈ {−1, 1} and (3.2.10)
υik,n1 ,n2 (t; a, b, ω) := aωwik,k+n1 (t) + bωwik,k+n2 (t).
For the sake of readability of what follows we will be abbreviating the notation for the swap spread in (3.2.8) or (3.2.9) to (3.2.11)
S(t) =
M −1 X
υi (t)Fi (t),
i=0
0 ≤ t ≤ T0 ,
assuming that the swap rates are fixed at time T0 and M = max{n1 , n2 }. For the more general case the formulas shall be given subsequently in Sec. 3.2.2. Using the methodology of Markovian Projection (MP) we shall project the multidimensional diffusion process of the spread basket of the SABR forward rates onto an one-dimensional displaced SABR-type diffusion model. The stochastic weights υi (t) of this basket will be frozen at the initial time, thus, for the following approximations they are constants. Formally, we approximate the diffusion of the basket (3.2.11) of forward rates, which evolve according to (3.2.2)–(3.2.6), with a displaced diffusion model with stochastic volatility under an abstract spread measure. Lemma 3.2 Given the multivariate SABR model of forward rates (3.2.2)–(3.2.6), applying the Markovian Projection technique leads to an approximation of the dynamics of the swap spread S(t) by the system (3.2.12) (3.2.13) (3.2.14) (3.2.15)
dS(t) = U (t)f (S(t))dWt , t ∈ [0, T0 ], dU (t) = U (t)ΥdZt , t ∈ [0, T0 ], hdWt , dZt i = Γdt, f (S0 ) = p, f ′ (S0 ) = q,
with the start values S(0) = S0 ,
U (0) = 1.
f denotes a deterministic function, while p and q are given in (3.2.22) and (3.2.32), respectively. The parameters Υ and Γ are defined in (3.2.36)–(3.2.37). The function f (S(t)) may be for instance a linear function f (S(t)) = p + q(S(t) − S0 ), cf. Sec. 3.2.2. Proof. Let the stochastic weights υi (t) of this basket be frozen at the initial time, υi (t) ≡ υi (0) =: υi , thus, for the following approximations they are constants. Define ui (t) :=
σi (t) , s.t. ui (0) = 1, σi (0)
41
Chapter 3. Pricing CMS Derivatives in SABR and the function φ(.) by
φ(Fi (t)) := σi (0)Fiβi (t).
Furthermore, define the parameters pi and qi by pi := φ(Fi (0)) = σi (0)Fiβi (0), qi := φ′ (Fi (0)) = σi (0)βi Fiβi −1 (0).
(3.2.16) (3.2.17)
Let us rewrite equation (3.2.2) as dFi (t) = ui (t)φ(Fi (t)) dWti+1 .
(3.2.18)
By applying the freezing the weights υi (t) ≡ υi to (3.2.11) we obtain (3.2.19)
dS(t) =
M −1 X
υi dFi (t) =
i=0
M −1 X
υi ui (t)φ(Fi (t)) dWti+1 .
i=0
We now write the dynamics of S in (3.2.19) as a single diffusion with stochastic variance. The dynamics can be seen as being given with respect to an abstract spread measure, where it becomes driftless. Define the process Wt as dWt
M −1 1 X := υi ui (t)φ(Fi (t)) dWti+1 , σ(t) i=0
with σ(t) given through 2
σ (t) =
M −1 X
υi2 u2i (t)φ(Fi (t))2
i=0
+2
M −1 X
υi υj ui (t)uj (t)φ(Fi (t))φ(Fj (t))ρi,j .
i,j=0, i
