stochastic volatility for interest rate derivatives

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Stochastic volatility for interest rate derivativesLinus Kaisajuntti a & Joanne Kennedy ba Danske Markets , Stockholm , Swedenb Department of Statistics , University of Warwick , Coventry , UKPublished online: 18 Mar 2013.

To cite this article: Linus Kaisajuntti & Joanne Kennedy , Quantitative Finance (2013): Stochastic volatility for interestrate derivatives, Quantitative Finance, DOI: 10.1080/14697688.2012.757848

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Stochastic volatility for interest rate derivatives

LINUS KAISAJUNTTI*† and JOANNE KENNEDY‡

†Danske Markets, Stockholm, Sweden‡Department of Statistics, University of Warwick, Coventry, UK

(Received 25 July 2011; in final form 29 November 2012)

This paper uses an extensive set of market data of forward swap rates and swaptions covering3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for thelevel of rates. The process is identified step by step by increasing the requirement of the modeland introducing appropriate adjustments. The first part of the paper investigates the smiledynamics of forward swap rates at their setting dates. Comparing the SABR (with different b)and Heston stochastic volatility models informs us what different specifications of the drivingSDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcomeof the analysis is that a normal SABR model (b ¼ 0) satisfactorily passes all tests and seems toprovide a good match to the market. In contrast, we find that the Heston model does not. Thenext step is to seek a model of the forward swap rates (in their own swaption measure) basedon only two Brownian motions that enables a specification with common parameters. It turnsout that this can be done by extending the SABR model with a time-dependent volatility func-tion and a mean-reverting volatility process. The performance of the extended (SABR withmean-reversion) model is analysed over several historical dates and is shown to be a stable andflexible choice that allows for good calibration across expiries and strikes. Finally, a time-homo-geneous candidate stochastic volatility process that can be used as a driver for all swap rates isidentified. This candidate process may in future work be used as a building block for a separa-ble stochastic volatility LIBOR market model or a stochastic volatility Markov-functionalmodel.

Keywords: Interest rate derivatives; Stochastic volatility; Interest rate modelling; LIBOR marketmodels; Market dynamics

JEL Classification: C1, C6, C15, C16, C63, G1, G13

1. Introduction

The objective of this paper is to perform a data-driveninvestigation in order to find a good model for pricing andhedging interest rate derivatives. This is, of course, a hugelycomplex problem in general, requiring an understanding ofthe number of factors needed in the model and whichcopula is most appropriate, amongst other things.

To keep the problem manageable, we restrict our analy-sis to the development of a model suitable for pricing andhedging products whose values are primarily determined bythe level of rates. In principle, such products are typicallycharacterized by looking at their values on their key ‘eventdates’ and noting that their ‘payoff’ on each of those datesis determined by a single swap rate. The canonical exampleis the standard Bermudan swaption. For a Bermudan at

each exercise time if we choose to exercise the payout is aswap (whose value is primarily determined by the singlecorresponding swap rate). On the other hand, note that thisdefinition excludes products such as CMS-spread options.In practice, one might decide whether any given product isof this type by valuing on both a model where swap ratesare functions of a single process (appropriately chosen toreflect the market) and a more general multi-factor modeland comparing.

Because of their ease of implementation it is commonpractice to price level-dependent derivatives using a modelwhere the driver is based on a single Brownian motion.However, it is clear that such a model does not adequatelycapture the joint distribution of the level of rates at differenttimes. Indeed, Hagan et al. (2002) showed that in order toadequately capture the distribution of a single swap rate atits setting date an extra factor is required. Our objective inthis paper is to formulate a model for the level of rates (foruse as a driver in a full term-structure model) that willimprove on this situation, without creating a model withunnecessary complexity (too many factors). It turns out that*Corresponding author. Email: [email protected]

� 2013 Taylor & Francis

Quantitative Finance, 2013

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we can do this with just one more Brownian motion whichallows for stochastic volatility.

At the outset of our investigation it was not clear that adata-driven study would result in a model for the level ofrates that needed just one more Brownian motion. We pro-ceeded by first choosing a rate to represent the overall levelof rates and investigated how to model it. We later changedthis choice to ensure that the results of our analysis werenot dependent on this somewhat arbitrary initial selection.In formulating our model we initially take as proxy for thelevel of rates process the 10-year spot rate. An importantproperty that we assume for the level of rates process is thatit is time-homogeneous. This reflects expectations we haveof the market.

Working on the assumption that a time-homogeneouslevel of rates process exists we need to consider how tobuild the model so that it is consistent with the whole arrayof data we have available. Let Z ¼ ðX ;UÞ, where X is ourcandidate level of rates process and U is the associated sto-chastic volatility. We need the model to represent correctlythe conditional distribution

ZTþS j ZT ; for all T and S: ð1Þ

However, for any time-homogeneous Markov process it isenough to capture

ZS j Z0; for any S and all values of Z0: ð2ÞNote that, for a time-homogeneous model, the second equa-tion (2) implies the first (1).

By looking on any given date at European swaptionprices of expiry S, in principle we can find the marketimplied distribution for

ZS j Z0 ¼ z0: ð3Þ

Looking at the equivalent data on different days will informus about the market implied distribution of (3) for differentvalues of z0 (i.e. (2)).

In principle, as we are assuming time homogeneity, weonly need to fit to one expiry S; but in practice, as we areunable to observe the volatility process directly, we need towork with several expiries to pick up the right signal fromthe data. In order to arrive at a model for the spot rate wefirst investigated a model for swap rates having a 10-yeartenor and start dates varying from 2 to 30 years from today.Start dates shorter than this may require a different class ofmodel to capture the behavior appropriately. By looking at10-year tenors we were purposely choosing rates that had alarge overlap so the models for different rates could beviewed as a block. From this we would expect the parame-ters driving the model to be comparable between rates andone would be hopeful of finding an appropriate commondriver for all rates. Having fitted a model with commonparameters that captured the (macro distributions of the)market well we were in a position to read off a suitablemodel for our proxy the 10-year spot rate. Later checksusing 20- and 30-year tenors show the results to bequalitatively similar.

At the end of our investigation we had identified atime-homogeneous model for the level of rates that fitted allmarginals (i.e. all expiries) and that worked from any start-ing point (so the same parameters are used for any futuretime). It is the property of time-homogeneity that ensureswe have a model whose dynamics reflect those of themarket and not just one that focuses on matching terminaldistributions as seen today.

As noted above our search for a model for the level ofrates began with first modeling swap rates of various expi-ries. Let yi denote the swap rate corresponding to start dateTi in the future and having a 10-year tenor. In seeking asuitable model for the 10-year swap rates themselves weworked within the class of stochastic volatility models spec-ified by a system of SDEs of the form

dyit ¼ f iðTi � t; yit; aitÞdWi

t ; ð4Þ

dait ¼ hiðaitÞdt þ gðaitÞdV it ; ð5Þ

under the martingale measure Si corresponding to taking Pi

(the PVBP corresponding to yi) as numeraire, and whereWi and V i are Brownian motions with

dWit dW

jt ¼ dV i

t dVjt ¼ dt; dWi

t dVit ¼ qdt:

Theoretically, it is not necessary to restrict to this class inour search for a model, but we found this class to be ade-quate for our purpose at each stage of the modeling pro-cess. In our investigations we put aside considerations oftractability and let the data dictate the precise final formof the model as we increased the modeling demands madeof it. The judgement of whether the model was successfulat any stage was based on how well it could representinformation on the macro distributions contained in thedata. Given the quality of the data available from the mar-ket we believe this was the most sensible yardstick to judgea model by.

We now review in more detail the steps taken in ouranalysis. The first part in our search for an appropriatebuilding block is to find a good model for the Si marginaldistribution

ðyiTi ; aiTi j yi0; a0Þ; ð6Þ

for each i ¼ 1; . . . ;N . Note that fitting the implied volatilitysmile for swaptions with time to expiry Ti implies fittingthe Si marginal distribution

ðyiTi ; aiTi j yi0 ¼ y; a0 ¼ aÞ: ð7Þ

This may be done with good accuracy for a large variety ofmodels in the class of stochastic volatility models. How-ever, models agreeing on (7) do not necessarily agree on(6) (where we are assuming the parameters of the modelremain fixed). To be able to analyse this we assume thatmarket data are generated by a model such that the condi-tional distribution is roughly stable over time. This enables

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us to use observations of swaption prices (implied volatilitysmiles) with time to expiry Ti at several trading dates toinform about (6).

A model assigning an appropriate distribution (6)provides a smile dynamic that is similar to that of themarket and hence the calibrated parameters would be nearlystationary across dates. Section 4 provides an investigationof the SABR model with different b as well as the Hestonmodel in order to inform about what the different specifica-tions of the SDE have to offer in terms of reflecting theavailable information on the market one step conditionaldistributions. We found that the SABR model with b ¼ 0gave very good results so we did not need to extend oursearch beyond these two popular models. We note inpassing that the results of this section are of independentinterest in terms of offering a data-driven analysis of SABRversus Heston.

What we have identified so far is a model that gives agood fit to the distribution of the swap rate at its settingdate with each start date treated separately. Having a modelwhich gives a good representation of the conditionaldistribution at Ti does not tell us anything about whetherthe model gives a good representation of the (conditional)‘forward smiles’ i.e. the distribution of ðyiTi ; aiTi j yit; aitÞ,t 2 ð0; TiÞ. Note that a stochastic volatility model for yi

does specify all the conditional distributions and hence doesimply both a certain smile dynamic and forward smile. Inparticular, the stochastic volatility models studied in thisfirst stage are time-homogeneous, whereas we expect thedistribution of yit to depend on time to maturity Ti � t. Nev-ertheless, we use the SDE identified at this stage, the SABRmodel, as our starting point for the next step and addressthe issue of the appropriate time dependence of the modelwhen fitting to data at several expiries requires it.

Our next objective in our search for a stochastic volatil-ity model for the level of rates is to seek a model of theswap rates (in their own swaption measure) based on onlytwo factors and that enables a specification of parametersthat is common to all forward swap rates having 10-yeartenor and start dates from 2 to 30 years. This simultaneousfit to all expiries is done in section 5 (for a single date) andcan be achieved by replacing geometric Brownian motionfor the volatility process in the SABR model by theexponential of an Ornstein–Uhlenbeck process and a termexponential in Ti � t in the equation for the forward rate.Section 6 provides further study of this model by analysingits performance over several historical dates and the modelis shown to be a good representation of the market smiledynamics. In particular, it is investigated if using a separa-ble volatility structure is a major limitation and to whatextent the model could be simplified in order to allowefficient implementation.

Our end goal is to identify a two-dimensionaltime-homogeneous process of stochastic volatility type forthe level of rates. Up to this point the focus has been onfinding a suitable volatility structure linking the rates yi

under their respective measures. Given that the swap rateswe are looking at have a large overlap and that we havebeen able to represent the data well with common

parameters, recalling our earlier discussion the expectationis that this procedure should give us a good gauge on aprocess which captures the level of rates.

In section 7 by considering the corresponding spot rateprocess and ignoring the effect of measure changes weidentify a candidate model for the level of rates. Denotingthis candidate process by X we have that

dXt ¼ �cXt dt þ expðUtÞ dWt; X0 ¼ 0;

dUt ¼ jUt dt þ m dVt;

dW dV ¼ q dt;

where c, m and j are positive constants, q 2 ð�1; 1Þ and Wand V are correlated Brownian motions under SN say, withTN being the final expiry.

Finally, note that having identified a process that cancapture the level of rates in the sense described above thisprocess can be incorporated in a full term-structure modelin various ways. It could be used as the motivation forspecifying a process for a short rate model. It could formthe driver for a Markov-functional model or it could beused as the driving process in a market model. For anygiven product the pricing and hedging of the product usinga low-factor model needs adequate evaluation, but ourexpectation is that a term-structure model with a drivingprocess which better reflects the dynamics of the marketwill turn out to have benefits over low-factor models whichare currently in use. This is left as a subject of futureresearch.

2. Preliminaries and market data

Consider the discrete date structure 0 ¼ T0 \ T1 \ T2 � � �TN \ TNþ1 with di ¼ Ti � Ti�1 and let DTiTj denote thezero-coupon bond price at time Ti for a unit payoff at thematurity time Tj. The time t equilibrium forward swap rate fora swap that start at time Ti and ends at time Tj is given by

ytðTi; TjÞ :¼DtTi � DtTjþ1

PtðTi; TjÞ ;

where P is the present value of a basis point (PVBP) of theswap

PtðTi; TjÞ :¼Xj

k¼i

diþ1DtTkþ1 :

For ease of notation, when the tenor of the swap is eitherobvious or irrelevant, the forward swap rate setting at dateTi will be referred to as yit and the PVBP as Pi

t.Let the uncertainty in the economy be resolved over a

complete filtered probability space ðX;F ;P; fF tgt�0Þ,t 2 ½0; TN �, where the filtration fF tgt�0 is the augmentednatural filtration generated by some Brownian motion W .Under the ‘swaption measure’ Si with numeraire Pi

t, thetime-t price of a (payers) swaption with strike K and expiryTi is given by

Stochastic volatility for interest rate derivatives 3

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CtiðKÞ ¼ Pi

tESt ½ðyiTi � KÞþ j F t�: ð8Þ

In the market, prices of swaptions are tracked using theimplied Black volatility as this allows for easier comparisonof swaptions at different strikes, tenors and expiries as wellas across dates. To refer to the implied volatility we willuse R equipped with specific arguments when needed (suchas RðK; yiÞ). Moreover, the term implied volatility smilerefers to the implied volatilities across strikes and likewisethe implied volatility surface/cube for implied volatilitiesacross strikes, tenors and/or expiries.

Finally, note that the distribution of ðyiTi j yi0Þ under Si

completely specifies the time-0 implied volatility smileacross strikes for swaptions with expiry Ti. Vice versa,market implied volatilities with expiry Ti at some strikesprovide information about the Si distribution of ðyiTi j yi0Þ.

2.1. Market data

The empirical investigations in this paper are based on anextensive data set of Euro swap rates and swaption prices.Each snapshot of data contains forward swap rates withexpiries 2Y, 3Y,… , 10Y, 12Y, 15Y, 20Y, 25Y, and 30Y andtenors 1Y, 2Y,… , 10Y, 12Y, 15Y, 20Y, 25Y, and 30Y. Allrates are based on annual payment frequencies. For eachrate the corresponding swaption implied Black volatilitiesare given for strikes –2, –1, –0.5, –0.25, 0, 0.25, 0.5, 1 and2 percent away from the ATM strike.

The data consists of two series, the first 66 roughlymonthly spaced snapshots of the market spanning the per-iod 3 July 2002 to 21 May 2007 and the second 556 dailysamples of data covering 10 March 2007 to 21 May 2009.To our knowledge this is one of the very first papers thatuses such an extensive data set. From almost daily samplesof USD market data covering June 2002 to June 2005 theperformance of a variety of models for dynamic swaptionhedging was performed by Henrard (2005). While the meanis different the goal and the conclusions of the article ispartly similar to the analysis performed insection 4. Using data of Euro swaptions covering 15December 2004 to 5 October 2007, Rebonato et al. (2010)make similar observations as we do and propose a SABR/LIBOR market model to account for the observed effects.While their aim and to some extent approach is similar toours their end goal is different and they limit the study tothe LIBOR market model framework. In a recent paper,Trolle and Schwartz (2010) use a very extensive data set ofboth Euro and USD swaptions across strikes, tenors andexpiries. Their end goal, however, is quite different fromours as they focus on displaying stylized facts of the distri-butions and its macroeconomic drivers.

Figure 1. The implied volatility smiles for 10-year tenor swaptionswith 2, 5, 10, 20, and 30 years to expiry on October 27, 2007.

Figure 2. The forward swap rates and the implied volatility smile levels, slopes and curvatures at 103 evenly sampled dates in the dataset. Swaptions with 10-year tenors. Dates are in chronological order, i.e. the left-most data points refer to 3 July 2003.


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Figure 1 displays a typical set of 10-year tenor swaptionimplied volatility smiles from 27 October 2007. Note thatthe smiles look quite similar in shape, although the curva-ture of the implied volatility smile seems to be decreasingin expiry. Figure 2 displays the forward swap rates as wellas the ATM level, slope and curvature of the impliedvolatility smiles at 101 dates spanning the available periodof data (66 dates up to 9 March 2007 and then another 35evenly spaced samples from the daily data period). As aproxy for the ATM slope and curvature, finite differenceapproximations are computed using the ATM and theplus/minus 1% strikes. These quotes are used as they are,according to traders, the most liquid strikes in the market.

First note that, beginning September 2008, the marketturmoil is clearly spotted with large movements in rates andvolatility measures and in particular the 30-year ratebecomes very low compared with previous dates. Also notethat, except for the turmoil period, the different expiriesmoves largely in parallel and that there seems to be a posi-tive correlation between the forward spot rate and the ATMlevel and curvature of the implied volatility smile, implyingthat, as rates go down, implied volatilities and curvaturebecome larger. Moreover, while in the calm period theATM level and curvature of the implied volatility smiles aredeclining in expiry, this is not the case in the turmoil periodwhere things seems much more disconnected.

For the slope of the implied volatility smiles there is noclear relation with time to expiry. There does, however,seem to be a negative correlation between the rates andslope, implying that, as rates go down, the slope of theimplied volatility smile become steeper.

Although the above analysis only provides a rough ideaconcerning the macro (large time steps) behavior of ratesand implied volatilities, it provides an idea about thechallenge and requirements on a sound model for the levelof rates. In particular, one would expect that, during theturmoil period, any model would struggle to fit the marketdata and market moves.

Section 4 provides a further and cleaner look at slopesand curvature dynamics of the smiles at one expiry at atime, and sections 5 and 6 develop and test a model for allexpiries.

3. Objective

As outlined in the Introduction, our approach to finding asuitable model for the level of rates is to build the modelone step at a time, at any stage trying to understand whereour model fails to reflect the data and adding complexity asnecessary to capture the right qualitative behavior.

As discussed in the Introduction, our first task is tospecify a model that gives a good representation of (6), theone-step marginal distributions of the swap rates at theirsetting dates. Hagan et al. (2002) make clear that, even forthe first step, we need to introduce at least a second factorto adequately reflect the behavior of the market.Technically, a second factor can only be introduced as amultiplicative factor if we are to specify our model as adiffusion. Given this, we choose to start our analysis with

an assessment of how well a stochastic volatility model cancapture the one-step macro distributions of the data wherethe volatility is specified by an autonomous equation.

The two most well-known stochastic volatility modelsare of course the SABR model, introduced by Hagan et al.(2002), and the Heston model (Heston 1993). In section 4we carefully examine the ability of each of these models toreflect the features of our data at a single expiry. This wasnot a case of deciding which model is best, but whethereither does the job. We found the SABR model to pass allthe tests we put it through and so took this as our buildingblock for the next stage. One concern regarding the SABRmodel is that its widespread use in the market in generaland its influence on practitioners in filling in the impliedcurve from liquid points in particular could make thischoice of model a self-fulfilling one. We address this pointin our study and argue why we believe it to be a soundchoice for our purposes, although there is room forimprovement in capturing behavior away from at themoney.

We should mention here that there are several stochasticvolatility models for interest rate derivatives introduced inthe literature (see, for example, Andersen and Brotherton-Ratcliffe (2005), Piterbarg (2005) and Rebonato et al.(2010) in a LIBOR/swap market model setting, or Andersenand Andreasen (2002) and Albanese and Trovato (2005) forHJM-type models). Our work differ from the above in thatwe initially put considerations of tractability aside andinvestigate historical data to dictate the final form of themodel and we only concern ourselves with capturing thelevel of rates process, a (potentially) low-dimensional prob-lem. Our approach may in future studies give some insightinto what the most appropriate building block would be fora high-dimensional full term-structure model such as aseparable LIBOR market model.

4. Smile dynamics: SABR vs. Heston

Recall our objective of formulating a model that is data dri-ven. However, we have to work within the constraints ofthe quality of the available data as we do not have availablereliable and liquid option prices for each swaption at everystrike. On any given day, apart from the ATM strike, only afew of the implied volatilities at various strikes will corre-spond to liquidly traded swaptions, with the other pricescoming from some form of interpolation carried out by thebank. Given this, we need to impose some structure tobegin the modeling process and we have chosen to workwithin the class of stochastic volatility models specified by(4) and (5).

The first test any proposed model within this class mustpass is that it should be able to fit each of the one-step mar-ginal distributions of

ðyiTi ; riTij yi0; ri

0Þ; i ¼ 1; . . . ;N ; ð9Þ

under the measures Si, respectively. In this section we willmake the assumption that, for each i, this conditionaldistribution is stable over time.


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Through this assumption we can link our model to theobserved changes in the market smile as yi0 and ri0 change,referred to as the smile dynamics. In this section we investi-gate whether the one-step marginals distributions given bythe most popular models, the SABR model (for different b)and the Heston model are consistent with the one-step mar-ginal distributions observed using historical data of forwardswap rates and implied volatility smiles. For this section wewill use the daily data series.

4.1. The SABR model

The SABR model dynamics for some underlying forwardprocess ft under some forward martingale measure is givenby

dft ¼ f bt rt dWt; f0 ¼ f ; ð10Þ

drt ¼ mrt dVt; r0 ¼ r; ð11Þ

dWt dVt ¼ q dt; ð12Þ

for b 2 ½0; 1�, q 2 ½�1; 1� and m � 0. For this model, Haganet al. (2002) derive a reasonably accurate approximation ofthe implied volatility as a function of the initial valuesof the variables f and r, the parameters b, q and m and timeto expiry T. This approximation is in widespread use tocalibrate the SABR model to option prices in an efficientmanner.

In addition to the higher-order approximation, a simplerapproximation (equation (3.1) of Hagan et al. (2002)) isalso derived from which it is easier to understand thebehavior of the model. For this ‘crude’ approximationthe implied volatility for an option with strike K when theunderlying forward is at f is given by

RðK; f Þ � rf 1�b

1� 1

21� b� qmf 1�b

r

� �log

K

f

�

þ 1

12ð1� bÞ2 þ ð2� 3q2Þm2 f

2�2b

r

� �log2

K

f

�:

ð13Þ

Note that using this approximation the ATM impliedvolatility is given by

Rð f ; f Þ ¼ rf 1�b

: ð14Þ

Inserting this in (13) gives a formula for the implied volatil-ity across strikes in terms of the ATM implied volatility

RðK; f Þ � �ð f ; f Þ 1� 1

21� b� qm

rð f ; f Þ� �

logK

f

�

þ 1

12ð1� bÞ2 þ ð2� 3q2Þ m2

r2ð f ; f Þ� �

log2K

f

�:

ð15Þ

For a pure fitting of the implied volatility smile the mostinteresting effects of the parameters are that b and q bothhave a direct affect on the slope of the smile, whereas m hasa direct affect on the curvature. The overlap of theparameters b and q implies that fitting the market smile forsome expiry across a reasonably wide range of strikes maybe done with similar precision for any value of b and hencethe exact value of b would seem redundant. This is not thecase.

Even for pure inter/extrapolating of implied volatilities bhas an effect since while SABR models with different bmay be fitted to more or less exactly agree on some strikesaround the ATM point, they do not, in general, agreeexactly on all other strikes. In particular, for large strikesthis is an issue and market prices of CMS products candepend quite strongly on the choice of b.

Another aspect on the choice of b is that, for a change inf or r, the change in the implied volatility smile dependson b. This difference in ‘smile dynamics’ will have aneffect on the hedging or pricing of anything more exoticthan vanilla swaptions. As our data set does not containmarket prices for high strikes or prices of CMS productsthe focus of our investigations is on this latter aspect of thechoice of b.

In terms of distributions, recall that fitting the impliedvolatility smile for some particular values of f and r fixesthe distribution of

ð fT ; rT j f0 ¼ f ; r0 ¼ rÞ: ð16Þ

While this can be done with similar precision for anyb 2 ½0; 1� it does not imply that the distributions of

ð fT ; rT j f0; r0Þ ð17Þ

are the same (for a fixed set of parameters). This is readilyseen by matching up the implied volatility smiles fordifferent b and then varying the variables f and r. Anatural question to ask is hence: Is the SABR model consis-tent with observed market ‘martingale measure’ marginaldistributions and, if so, which b is appropriate?

Hagan et al. (2002) suggest choosing b by fitting the‘backbone’ to historical data, where the backbone is definedas the curve traversed by the ATM volatility, Rð f ; f Þ, as theunderlying f varies. Indeed, as shown in figure 3, there is

Figure 3. Daily quotes of ATM implied volatility plotted versusthe forward swap rate for the period 8 March 2007 to 21 May2009. Swaptions on the 10Y� 10Y forward swap rate.


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typically an inverse relation between the ATM volatilityand the underlying. In the SABR model, Hagan identifiesthe backbone as approximately r=f 1�b and proposes toestimate b by a linear regression of the relation

logRð f ; f Þ ¼ log r� ð1� bÞ log f ð18Þ

using historical observations of ðf ;Rð f ; f ÞÞ pairs. This esti-mation procedure is unfortunately flawed, as within a sto-chastic volatility model, as f changes, so does r. If themarket data was really from a SABR model we cannot treatr as a fixed parameter, since it is highly linked to f throughthe correlation q. That is, observations of ðf ;Rð f ; f ÞÞ donot provide sufficient information to estimate b via themacro approach (i.e. via considering the distribution of theforward at time T ).

Let us now consider how observations of ðf ;Rð f ; f ÞÞcould be used to estimate b by taking an infinitesimalapproach. First note that one may rewrite the SDE for thevolatility as

drt ¼ qmdft

f btþ mr

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

pd Zt; ð19Þ

where Zt is a Brownian motion uncorrelated to W and V .Letting Rt :¼ Rðft; ftÞ and using Ito’s formula on theapproximation (14) gives (after a few manipulations)

dRt ¼ ½qmþ ðb� 1ÞRt� dftft þ mffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

pRt dZt

þ ðb� 1ÞRtb� 2

2R2

t þ qm

� �dt: ð20Þ

Dividing through by Rt gives that qm and ðb� 1Þ may beestimated by regressing on the factors dft=Rt ft and dft=ftusing a set of historical observations. Using a large amountof simulated data from a SABR model we have found thisprocedure to work decently. However, for estimation usingmarket data, one would need a lot more data than we haveavailable and, moreover, as we cannot expect market datato be generated exactly by a SABR model the estimationwould probably be a rather noisy exercise.

Returning to the macro approach, note that from (13) theATM slope of the implied volatility smile is approximately

@RðK; f Þ@K

��K¼f

� 1

2fðRð f ; f Þðb� 1Þ þ qmÞ: ð21Þ

Estimating b from a regression of

2f@RðK; f Þ

@K

��K¼f

¼ Rð f ; f Þðb� 1Þ þ qm ð22Þ

requires estimation of the ATM slope of the implied volatil-ity smile and thus information about the implied volatilityat more than the ATM strike is needed. While ATM swap-tions are liquidly traded this is not the case for all strikesand in particular not necessarily the strikes that are veryclose to ATM and needed for an accurate computation of

the ATM slope. The next subsection deals with informingabout b (also, if needed, q and m) from the slope and curva-ture of the implied volatility smile-like relations.

4.2. Estimating β for the SABR model

As argued above we would not expect to be able to esti-mate b from historical observations of ATM implied volatil-ities and forward swap rates with good precision. To extractinformation about b we instead propose to investigate theslope of the implied volatility smile. Note that from (15)the first-order difference between the implied volatility atstrikes f þ h and f � h is approximately

Rð f þ h; f Þ � Rð f � h; f Þ � 1

2½ðb� 1ÞRð f ; f Þ þ qm�

� logf þ h

f� log

f � h

f

� :

ð23Þ

After some manipulations, define

Sðh; f Þ :¼ 2½Rð f þ h; f Þ � Rð f � h; f Þ�

log½ð f þ hÞ=ð f � hÞ�� ðb� 1ÞRð f ; f Þ þ qm: ð24Þ

Since, for small h, log½ð f þ hÞ=ð f � hÞ� � 2ðh=f Þ the l.h.s. of the above relation is tending to 2f times the ATMimplied volatility slope as h tends to 0. This means thatthe above relation is relating a (suitably scaled) finite dif-ference approximation of the ATM implied volatility slopeto the parameters of the SABR model. Hence, if theSABR model is a good representation of the market, thenb (and the product qm) could be estimated by a linearregression of the l.h.s. on Rð f ; f Þ from a set of marketsnapshots.

To further investigate (or indeed estimate the parametersq and m) whether the SABR model fits market data, onemay study a relation linked to the curvature of theimplied volatility smile. Using the implied volatility atthe strikes f þ 2h; f and f � 2h and (15) up to secondorder gives

Cðh; f Þ :¼ 2Rð f ; f ÞF2

½Rð f þ 2h; f Þ þ Rðf � 2h; f Þ� 2Rð f ; f Þ�

� ð1� bÞ26

� ð1� bÞF1

F2

" #R2ð f ; f Þ

þ qmF1

F2Rð f ; f Þ þ 2� 3q2

6m2; ð25Þ

where F1 ¼ log½1þ ð2h=f Þ� þ log½1� ð2h=f Þ� and F2 ¼log2½1þ ð2h=f Þ� þ log2½1� ð2h=f Þ�.

Note that, for small h, F2 � 2ð2h=f Þ2 and hence the l.h.s.above is closely related to the finite difference approxima-tion of the ATM curvature of the implied volatility smilescaled with the factor f 2Rð f ; f Þ. Moreover, for small


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h, F1=F2 � �1=2 and hence as h tends to 0 the above rela-tion tends to the (suitably scaled) ATM curvature of theimplied volatility smile calculated from the approximation(15).

To sum up, if the SABR model is a good representationof the market, then the parameters of the quadratic polyno-mial in Rð f ; f Þ on the r.h.s. of (25) may be estimated usinga set of market snapshots. Alternatively, b may be fixedfrom (24) and a linear regression may be performed towork out q and m.

4.3. Investigating the SABR model using market data

The previous subsection outlines a method to estimate theparameters of a SABR model using a set of implied volatil-ities for the ATM plus two other strikes. However, typically,for a fixed b, q and m are calibrated using current marketdata and there is hence no need to estimate them using his-torical data. What we are mainly interested in is investigat-ing whether there is a b such that the marginal distributionsof (9) from the SABR model is consistent with the market.As we cannot extract much information from using only theATM implied volatilities we will use the dynamics of there-scaled slope and curvature relations to inform aboutappropriate marginal distributions.

This subsection tests the SABR model using the set ofdaily market data of swaptions covering the period 9 March2007 to 21 May 2009. To evaluate the SABR model wehave chosen to use the 10-year tenors and the strikes ATM,ATM� 1% and ATMþ 1%. According to traders these arethe most liquid strikes and tenors and hence the most reli-able quotes. The main focus is on the 10 years expiry, butwe also provides results for the 2, 5 and 30 years expiries.

4.3.1. Slope. Recall that, if the market is well representedby a SABR model, then daily values of the l.h.s. of (24)plotted versus the ATM implied volatility would give,approximately, an affine function. Figure 4 plotsSð0:01; y100 Þ vs. rðy100 ; y100 Þ for the 10Y � 10Y swaptions atall available dates. Note that the points seems to form‘lines’ between which there is a jump at some dates. Interms of equation (24), ðb� 1Þ would then be the slope ofthe ‘lines’ and the jumps would correspond to a change inthe product qm. By inspection, the slope of the lines seemsto correspond to a b of around 0.

To be able to make a more precise investigation, onemay look at the difference between the approximate sloperelations between dates. Define the change in ‘slope’between date ti and tj as

DSðh; i; jÞ :¼ Sðh; ftjÞ � Sðh; ftiÞ; ð26Þ

and the change in ATM implied volatility as

DRði; jÞ :¼ Rðftj ; ftjÞ � Rðfti ; ftiÞ: ð27Þ

For a fixed b, but allowing a change in qm, the differencein ‘slope’ between dates i and j is then (using (24))approximately

DSðh; i; jÞ � ðb� 1ÞDRði; jÞ þ qð jÞmð jÞ � qðiÞmðiÞ: ð28Þ

Hence, if the SABR model is a good representation of themarket marginal distributions, by using market data to plotthe above l.h.s. versus the change in ATM implied volatilitywe would expect to see points lining up along a line withslope b� 1 as well as potentially some scattered points cor-responding to changes in qm. Figure 5 confirms that this isthe case for 10Y � 10Y swaptions. In this case a linearregression using (28) would estimate b to be �0.04 with anR2 of 0.84 and a 95% confidence interval of ½�0:08; 0�.

The accuracy of informing about an appropriate b fromthe above procedure depends mainly on two things. First,one needs a reasonably large sample of reliable market data.The historical data set of swaption data used in this paperconsists of 556 dates. Hence, using daily steps to obtaindata for (28) gives 555 points. By stepping more than onedate forward, one would obtain more points (as well aslarger differences) but, in our experiments, one date forwardgives enough points to provide estimates with decentaccuracy.

Second, the approximation leading to (28) needs to bereasonably accurate. As this is based on a rather crudeapproximation this might of course be far from the case but

Figure 4. Market values of the l.h.s. of the ‘slope’ relation (24)versus the ATM implied volatility, h ¼ 0:01. Daily values fromthe period 2007-03-08 to 2009-05-21 of 10Y� 10Y swaptions.

Figure 5. Market values of the ‘slope’ difference for 10Y� 10Yswaptions plotted versus the ATM implied volatility differenceestimated using daily steps from 2007-03-09 to 2009-05-21.The linear regression lines of the market values are also dis-played, implying a b estimate of about �0.04.


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there might also be hope for the approximation errors tocancel out as we focus on differences. To address this wefirst fixed a set of parameters ðb; m; qÞ as well as a set offorward swap rates and implied volatilities ð f ð jÞ;Rð f ð jÞ;f ð jÞÞÞ. Then, using both the full approximation of Haganet al. (2002) as well as accurate Monte Carlo values the‘slope’ changes going between the dates may be computedand compared with the relation (28). Our finding points tothe fact that while expression (28) is quite accurate and thehigher-order terms have rather small effects, it slightlyunderestimates the value of b by about 0–0.1, depending onthe level of b. It should at this point be noted that eventhough we do produce point estimates for b we are mainlyinterested in informing about a reasonably good choice of band for our purposes we are hence satisfied with this levelof accuracy.

Table 1 displays the estimates of b and its standard error,qm and the R2 obtained by regressing on (28) using data of2Y � 10Y, 5Y � 10Y, 10Y � 10Y, 20Y � 10Y and 30Y� 10Y swaptions. The regression is performed for both allhistorical observations as well as the first one and a halfyears of the data that covers a period of quite calm markets.The table also displays bc which is just the estimate of bwith a correction term added to allow for the underestima-tion mentioned above.

Note that the 2-, 5- and 10-year expiries suggests a b ofabout zero irrespective of estimation period. For the 20-and 30-year expiries b zero seems appropriate during thecalm period, whereas, during the turmoil, b seems to haveincreased to 0.2 and 0.5, respectively. Actually, as theturmoil period provides most of the variation in the data,estimation during the turmoil period gives an estimate veryclose to estimation over the complete set of data. Othertenors were checked as well and give similar estimates andaccuracy.

The change in b for the longer expiries during the periodof turmoil does not appear to have its basis in a short-termchange in perception of the market (where we might expectb to alter for the shorter expiries). Rather it is indicative ofa period of market instability and we would need a richermodel than SABR to capture anything useful. Even with aricher model it would be difficult to extract a coherentsignal. Indeed, this period corresponds to the subprimecredit crunch. From the plots in section 2.1 it can be seen

that the period of turmoil covers the last six months of theseven-year period of the monthly series. Our focus in thispaper will be on building a model for the period when themarkets are stable.

Remark 1: The plot of the slope relation in figure 5 looksremarkably like what one would expect if the data camefrom a SABR model where the parameters q and m werealtered from time to time. In searching for an appropriatemodel it is important to be careful not to carry out an inves-tigation whose conclusions are a consequence of a SABR-like implied volatility curve being used by traders to inter-polate between liquid strikes. That is why we worked withthe most liquid strikes in studying the data.

Remark 2: It is interesting to note, however, that from talk-ing to traders they seem to conclude that while b ¼ 0 isappropriate they tend to prefer using a larger b for day-to-day use. The reason for this is that although any b providesa good fit near at the money, the prices of CMS productssuggest that away from the money b closer to one tends tobe more appropriate. The chosen value of b is hence a com-promise between these cases. This would suggest that if itwere possible to identify the liquid points away from ATM,one might choose a more general functional form than thatof a CEV model in the equation for the forward or choos-ing a different scaling of volatility of volatility.

4.3.2. Curvature. As reported in the previous subsection aSABR model with an appropriately chosen b provides agood reflection of the market dynamics of the slope of theimplied volatility smile. As a further check of whetherthe SABR model with a suitably chosen b provides appro-priate marginal distributions, one may study the dynamicsof the curvature. As shown above by looking at scaled‘curvature’-type relations, one could also estimate q and musing historical data, provided sufficient accuracy of theapproximation leading to (25). In general, however, this isnot needed since q and m will be decided by fitting the modelto option values across strikes and this relation is only used toprovide further intuition for the behavior of the model.

As a first check of whether the SABR model provides areasonable reflection of the curvature it is instructive torelate the market changes in curvature to the model

Table 1. Regressing (28) on Rðf ; f Þ to estimate b for different expiries and estimation periods. C.I. refers to the limits of the 95%confidence interval.

Expiry Period b qm R2 C.I.

2 Full �0.15 2E–4 0.63 [–0.23, –0.08]2 Calm 0.01 8E–05 0.58 [–0.07, 0.09]5 Full �0.09 1E–4 0.87 [–0.13, �0.05]5 Calm �0.07 3E�05 0.81 [� 0.12, � 0.02]10 Full �0.04 2E�4 0.84 [� 0.08,0]10 Calm �0.07 5E�05 0.82 [� 0.12, � 0.02]20 Full 0.2 2E�4 0.88 [0.17,0.23]20 Calm �0.07 4E�05 0.82 [� 0.12, � 0.02]30 Full 0.46 9E�05 0.78 [0.44,0.48]30 Calm �0.05 5E�05 0.8 [� 0.1,0]


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changes. Note from (25) that, for a SABR model, thechange in the re-scaled finite difference approximation ofthe curvature will depend on b. Hence, plotting the marketchanges on the l.h.s. of (25) versus the model changes fordifferent b will show how well the model reflects the curva-ture dynamics.

To obtain the SABR model values the following proce-dure is used. First, the higher-order approximation of theSABR model is calibrated at date i to the strikes K ¼ f ,f þ 0:01 and f � 0:01. Note that this can be done perfectlyfor all b and will hence exactly agree with the market val-ues. Then, at date j the model is recalibrated to the ATMimplied volatility by updating (the variable) r. All parame-ters (b, m and q) are kept fixed at the date i values. Themodel values of the change on the l.h.s. of (25) may thenbe computed. Figure 6 displays this for the 10Y � 10Yswaptions. Changes are computed during the calm periodand by stepping one date forward.

Note that, for b ¼ 0, most points line up along the45 degree line, implying a close relation between themarket and the model. For b ¼ 1 the points line up along aline with a significantly less steep slope and b ¼ 0:5 is inbetween. The reason for choosing the calm period is thatthe scatter is quite small and hence good for displaying theeffect without too much noise. However, other periods andexpiries display the same effect, albeit in some cases with abit more scatter. In general, the b estimated from the ‘slope’relations above seems to work best also for the curvature.

As a further test it is instructive to plot the curvaturechanges versus the ATM implied volatility changes as madein the ‘slope’ case above. Recall from (25) that a finite dif-ference approximation of the re-scaled curvature is approxi-mately given by a quadratic polynomial in the ATMimplied volatility. Hence, one cannot expect a linear relationbetween curvature and ATM implied volatility changes.However, as for d small the difference ðxþ dÞ2 � x2 isapproximately linear it would still be instructive to plot thechange in curvature versus the change in the ATM impliedvolatility for a period of small changes. Figure 7 plots themarket change and the model changes with b ¼ 0, 0.5 and1 during the calm period for the 10Y � 10Y swaption.

Note that b ¼ 0 gives a pattern similar to the market,whereas for b ¼ 0:5 and 1 the points line up with a slopethat is too fat.

4.4. The Heston model

This section investigates whether the Heston model (Heston1993) is consistent with the market marginal distributions.The Heston model is defined by the system of SDEs

d ft ¼ ftrt dWt; f0 ¼ f ; ð29Þ

dr2t ¼ jð�r2 � r2

t Þ dt þ mrt dVt; r20 ¼ r2; ð30Þ

dWt dVt ¼ q dt: ð31Þ

For the above system option prices may be found bynumerical inversion of an associated Fourier transformimplying reasonably efficient pricing and calibration. Interms of fitting option prices across strikes the effects of qand m are similar as in the SABR model. In addition to theSABR model there is also a mean-reversion term where themean-reversion level �r determines the long-run ATMimplied volatility. The mean-reversion speed j also has aneffect on the ATM implied volatility but the more interest-ing effect is a decline in curvature of the smile which isroughly inversely proportional to the time to expiry. Hence,m and j have similar, but opposite, effects. Through thecombination of m and j, one may hence control the curva-ture of the smile across several expiries. However, in termsof fitting one expiry only this may be done at virtuallysimilar accuracy for any �r and j.

In the standard Heston model there is no b term. Extend-ing the model with a CEV-type local volatility function isof course in theory an easy task, however this implies thatthe Fourier transform technique can no longer be used andhence efficient pricing of European-type options is lost. Toget around this issue, one may use a local volatilityfunction of displaced diffusion type for which the Fouriertransform technique is still applicable (see Andersen andPiterbarg (2010) for details). To limit the extent of thisstudy we have however chosen to only investigate the stan-dard Heston model and the fairest comparison with theSABR model is hence the b ¼ 1 case.

For the Heston model there does not, to our knowledge,exist a simple and reasonably accurate approximation ofimplied volatilities as is the case for the SABR model. Forshort expiries and strikes close to ATM, Durrleman (2004)

Figure 6. Scatter plots of changes in ‘curvature’ as defined in (25) for the market (x-axis) and the SABR model (y-axis) with differentb. Changes are for the 10Y� 10Y swaption and taken over one date during the calm period, implying a total of 388 points. Seethe text for a more thorough explanation.


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provides an expression that is of similar complexity as (13).Expanding this expression in logð f =KÞ and a few manipu-lations give

RðK; f Þ � Rð f ; f Þ þ qm4Rð f ; f Þ log

K

f

þ m2

24Rð f ; f Þr21� 7

4q2

� � m2q2

32�3ð f ; f Þ�

log2K

f:

ð32ÞNote that this expression does not contain �r or j and ishence only expected to work for very short expiries, if atall. While this approximation is a lot worse than the SABRapproximation in terms of pricing it does provide a reason-ably accurate intuition for the behavior of the Hestonmodel. For the ‘slope’ expression Sðh; f Þ defined in (24),one obtains that the ‘slope’ is inversely proportional toRð f ; f Þ. Note that since interest rate implied volatilitysmiles are typically (always) downward-sloping, q is typi-cally negative. Hence, for an increase in implied volatility,the implied volatility slope will also increase (less nega-tive). Note that this is the opposite behavior to what isobserved in the market.

For the curvature expression (25) the story is similar tothat for the slope. To analyse this case it is instructive tomake a further approximation. Using that, for short expiries,r2 � R2ð f ; f Þ (Durrleman 2004), the ‘curvature’ expressionCðh; f Þ defined in (25) is for the Heston model

Cðh; f Þ � mqF1

F2þ m2

96R2ð f ; f Þð4� 10q2Þ: ð33Þ

Hence, the effect of an increase in Rð f ; f Þ depends on thesign of ð4� 10q2Þ. It turns out that, across all dates andexpiries in our sample, the calibrated ð4� 10q2Þ is negativeand hence an increase in Rð f ; f Þ leads to an increase in the‘curvature’ Cðh; f Þ of the implied volatility smile. Also, thisis opposite to what is observed in the market.

Since the above approximation is quite crude it is notcertain that this effect is valid for the true Heston model.As a check, consider figure 8 in which 10Y � 10Y swap-tions are investigated during the calm period of the dailydata set. The upper plots in the figure display the Hestonmodel ‘slope’ and ‘curvature’ changes plotted versus themarket ‘slope’ and ‘curvature’ changes. As for the SABRmodel, the Heston changes are produced by fitting theparameters at one date and then at the next date, updatingr2 such that the ATM implied volatility is matched. Thelower plots display the changes in ‘slope’ and ‘curvature’versus the changes in the ATM implied volatility. Note thatthe ‘curvature’ changes are almost exactly opposite to themarket changes and that the ‘slope’ changes have the wrongsign.

To isolate the effect of �r and j these parameters are fixedat the initial value (r) as well as 0.01, respectively, for alldates. A j of 0.01 is a typical value resulting from a cali-bration of the Heston model to expiries ranging from 2 to

Figure 7. Changes in ‘curvature’ as defined in (25) for the market and the SABR model with different b plotted against the change inthe ATM implied volatility. Changes are for the 10Y� 10Y swaption and taken over one date during the calm period, implying atotal of 388 points. See the text for a more thorough explanation.


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30 years simultaneously. Different values of �r and j weretested and provided very similar results.

In general, for shorter expiries, the results are a bit lessnoisy and more pronounced than those displayed in the fig-ure, whereas for longer expiries investigated during the fullperiod of available historical observation there is morenoise. However, all performed tests reveal the same issue:the Heston model provides wrong changes in slope and cur-vature. Compared with SABR using b ¼ 1, the curvaturecase is the most different.

Remark 3: Following Durrleman (2004) it may be seenthat extending the Heston model with a local volatilityfunction of CEV type (or a matched up displaced diffusiontransformation) adds the same b terms that are present inthe ‘crude’ approximation of implied volatilities in theSABR model. Hence, approximately, one may expect tomodify the base case (b ¼ 1) behavior in a similar magni-tude as in the SABR model. However, as the base case(b ¼ 1) is a lot closer to the market behavior in the SABRmodel compared with the Heston model we conjecture thatmatching the market smile dynamics would also be chal-lenging when using an extended Heston model. In particu-lar, as the b terms have a smaller effect on the curvature,the market curvature changes would seem hard to match.

4.5. Summing up

The tests performed in this section provide evidence thatthe SABR model with an appropriately chosen b is a good

reflection of the marginal distributions of swaptionsobserved in the market. The Heston model, on the otherhand, seems to provide a worse representation of themarket.

Note that these two models are of course not the onlyones in the class (4)–(5) and there could be other stochasticvolatility models that perform as well as or better thanSABR. In particular, one may consider using different localvolatility functions for the forward or, for example, a powerfunction in r in the SDE for the stochastic volatilityprocess. For the latter choice note that, in the Hestonmodel, the power constant would be 0.5, whereas in theSABR model it would be 1. Again, Durrleman (2004) canbe used to obtain intuition about how the dynamics of the‘slope’ and ‘curvature’ would look in this more generalcase and it might be possible to find a structure thatmatches up both the dynamics of the smile and matchesmarket prices of CMS products.

However, given the constraints of our data and how wellthe SABR model with an appropriate b performs in the out-lined tests, we chose the SABR model as a building blockfor (9).

5. One date, many expiries: The SABR-MR model

In the previous section we found that the SABR model withappropriate b provides a good fit to the distribution of aforward swap rate at its start date. For our objective offinding a stochastic volatility model for the level of ratesthe contribution of this section is to engineer a model

Figure 8. Changes in ‘slope’ and ‘curvature’ for the Heston model for 10Y� 10Y swaptions investigated during the calm period.The tests were performed in the same manner as for the SABR model.


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linking the forward swap rates (although still under theirown swaption measure) based on only two factors thatenables a specification of parameters that is common to allforward swap rates having 10-year tenor and start datesfrom 2 to 30 years.

Note that a model with a good representation of the con-ditional distribution at Ti does not necessarily provide agood representation of the (conditional) forward smiles, i.e.the distribution of ðyiTi ; aiTi j yit; aitÞ, t 2 ð0; TiÞ. Using theSABR model as our starting point we address in this sec-tion the issue of appropriate time dependence as fitting todata at several expiries requires it.

For our general model setup,

dyit ¼ f iðTi � t; yit; ritÞ dWi

t ;

drit ¼ hiðri

tÞ dt þ giðritÞ dV i

t ;

we start by assuming the function f i is of the form

f iðTi � t; yit; ritÞ ¼ liðTi � tÞ/iðyitÞri

t: ð34Þ

Note that this setup is not particularly restrictive and stillcovers most of the previously introduced stochastic volatil-ity models in the literature. The following sections discussthe implications of various choices of the functions f i, gi

and hi and introduce the SABR with mean-reversionmodel.

5.1. Many expiries with the SABR model

Table 2 displays the parameters of disconnected SABRmodels calibrated to expiries ranging from 2 up to 30years. Note that, for both b ¼ 0 and b ¼ 0:5, the cali-brated parameter values of r0 and m are steadily decreas-ing in expiry, whereas the q values, although scattered,seem less dependent on expiry. The patterns in r0 and mare a general feature of our data sample (albeit thedeclines are at some dates less smooth), whereas for qthere are also dates with more monotone patterns. In gen-eral, though, the dependence between expiry and q is, ifat all there, less clear and less pronounced than that in

r0 and m. As we are after a model for the level of ratesand hence need common parameters for all swap rateswe need to modify our model to avoid such systematicchanges in parameters.

5.2. Extending the SABR model

This section puts the issues of calibrating the SABR modelto several expiries simultaneously into context and discussessome potential extensions. To obtain insight into how theSABR model can be suitably modified, the displaced diffu-sion SABR model (DD-SABR) will be helpful. In thismodel the dynamics of yit under S

i are given as

dyit ¼ ðyit þ diÞrit dW

it ; ð35Þ

drit ¼ miri

t dVit ; ð36Þ

dWit dV

it ¼ qi dt; ð37Þ

for some displacement constant di 2 <. For models withdeterministic volatility, Marris (1999) showed that, bychoosing (for b 2 ð0; 1�)

di ¼ yi01� bb

; ð38Þ

rit ¼ rcev;i

t bðyi0Þb�1; ð39Þ

the CEV and displaced diffusions models produce very sim-ilar prices of European calls across strikes. In a stochasticvolatility setting, Kennedy et al. (2012) find that the stan-dard SABR (CEV-SABR) and DD-SABR models continueto be very close qualitatively (matched up using (38) and(39) and taking the same mi and qi parameters). To keep thelink between the CEV-SABR and DD-SABR models wewill continue to refer to the parameter b also for the DD-SABR model and will hence implicitly assume that di fromequation (38) is used.

Table 2. Calibration of the SABR model on 27 October 2007. Expiry in years. Options to enter 10�year swaps.

Expiry b r0 m q b r0 m q

2Y 0.5 0.026 0.326 �0.160 0 0.0057 0.309 0.0603Y 0.5 0.025 0.309 �0.183 0 0.0056 0.290 0.0424Y 0.5 0.025 0.299 �0.204 0 0.0056 0.278 0.0235Y 0.5 0.025 0.297 �0.220 0 0.0055 0.275 0.0036Y 0.5 0.024 0.289 �0.215 0 0.0054 0.268 0.0077Y 0.5 0.024 0.283 �0.211 0 0.0054 0.263 0.0128Y 0.5 0.024 0.276 �0.209 0 0.0053 0.255 0.0179Y 0.5 0.023 0.269 �0.207 0 0.0053 0.249 0.02210Y 0.5 0.023 0.263 �0.205 0 0.0052 0.243 0.02712Y 0.5 0.022 0.253 �0.207 0 0.0050 0.233 0.02815Y 0.5 0.022 0.241 �0.211 0 0.0048 0.221 0.03120Y 0.5 0.020 0.223 �0.219 0 0.0044 0.203 0.03525Y 0.5 0.019 0.212 �0.221 0 0.0041 0.192 0.03830Y 0.5 0.018 0.203 �0.223 0 0.0038 0.183 0.041


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For the DD-SABR model with b 2 ð0; 1� the distributionof yiTi can be written (Kennedy et al. 2012) (see ourappendix A) as an exponential of a function of

V iTi¼ R Ti

0 ðrisÞ2 ds and an independent standard normal ran-dom variable G

yiTi ¼d ðyi0 þ diÞ exp qi

miðri

Ti� ri

0Þ �1

2V iTiþ

ffiffiffiffiffiffiV iTi

qG

� � di:

ð40Þ

For b ¼ 0 the equivalent relation is

yiTi ¼d

yi0 þqmðri

Ti� ri

0Þ þffiffiffiffiffiffiV iTi

qG: ð41Þ

To analyse the (DD-) SABR model we will consider thecase qi ¼ 0 for simplicity. The case qi – 0 is similar inessence, but with greasier expressions. Note that, for b ¼ 0,taking qi ¼ 0 is not a too bad choice compared with theoutcome of a market calibration (see table 2).

Note that, for b 2 ð0; 1�, conditional on V iTi, the distribu-

tion of yiTi is displaced log-normal and the hence the priceof a swaption with strike K can be written as an expectationof the standard Black’s pricing formula where the impliedvolatility is a function of V i

Ti. For b ¼ 0 the conditional

distribution of yiTi is instead Gaussian and so the Bachelierformula applies in this case. Without stochastic volatilitythe implied volatility smiles are monotone, downward-sloping (steepness depending on b) and with no or verylittle curvature. Adding stochastic volatility provides anincrease in the overall level of the smile as well as anincrease in curvature (and if q is not zero a tilt in theslope). Hence the particular distribution of V i

Tiis linked to

these features.Although we do not know the distribution of V i

Tiit is

informative to study its first two moments:

ESi ½V iTi� ¼

Z Ti

0

ðri0Þ2 expððmiÞ2sÞ ds

¼ ðri0Þ2

ðmiÞ2 ðexpððmiÞ2TiÞ � 1Þ; ð42Þ

ESi ½ðV iTiÞ2� ¼ 2

Z Ti

0

Z t

0

ESi½ðrisÞ2ðri

tÞ2� ds dt

¼ ðri0Þ4

15ðmiÞ4½expððmiÞ2TiÞðexpð5ðmiÞ2TiÞ � 6Þ þ 5�:

ð43ÞNote that, in the limit m ! 0, the first moment collapses to

ðri0Þ2Ti and the second moment is zero and we are back atthe Black/Bachelier models. In terms of the implied volatil-ity smile a rough rule of thumb is that the first momentcontrols the overall level of the smile and the secondmoment is linked to the curvature of the smile.

Now recall that the calibration of the CEV-SABR modelto market data returned decreasing ri0 and mi with time toexpiry. If we calibrate a DD-SABR model instead we obtain

similar parameter values and, in particular, qualitativelythe results will be exactly the same. For the purpose of thisdiscussion we can hence assume at this point thatthe calibrated parameters in table 2 are from a calibratedDD-SABR model.

For our objective of finding a model with commonparameters for all forward swap rates the first and secondmoments are helpful in understanding how the SABRmodel should be extended. Both moments in (42) and (43)are growing at exponential pace in expiry (or really in

ðmiÞ2Ti). Compared with market data, exponential growthseems to be too fast (as calibration returns decreasing mi)and in particular one would not expect this to be reasonablefor long expiries.

To match the (market-induced) moments of V iTi

we startwith some intuition from the deterministic case. Put mi ¼ 0and let r̂i0 denote the outcome of a calibration to the marketprice of an expiry-Ti ATM swaption. If we takerit ¼ lðTi � tÞ, then equations (40) and (41) still hold andwe need to choose the parameters of lðTi � tÞ such that

V iTi¼ R Ti

0 l2ðTi � tÞ dt ¼ ðr̂i0Þ2Ti for all i to calibrate themodel to the market prices of ATM swaptions acrossexpiries.

If we extend the model in the stochastic volatility settingnow with

dyit ¼ ðyit þ diÞlðTi � tÞrit dW

it ; ð44Þ

with rit as in the usual SABR model but with ri0 ¼ 1 and

V iTi:¼ R Ti

0 l2ðTi � tÞrit dt, the distribution of yiTi can still bewritten as in equations (40) and (41) with the first andsecond moments of V i

Tigiven by

ESi ½V iTi� ¼

Z Ti

0

l2ðTi � tÞ expððmiÞ2sÞ ds; ð45Þ

ESi ½ðV iTiÞ2� ¼ 2

Z Ti

0

l2ðTi � tÞ expððmiÞ2tÞZ t

0

l2ðTi � sÞ

� expð5ðmiÞ2sÞ ds dt: ð46Þ

Note that the function lðTi � tÞ affects both moments in asimilar fashion. However, as mi does not have the sameeffect on each of the moments the extended model is stillnot powerful enough to match up both moments and, inparticular, it would struggle with matching up the secondmoment. At an intuitive level the function lðTi � tÞ can bethought of as targeted to fitting the first moment (or theATM level of the smiles) under the respective measures Si.

To match both moments we hence need to extend themodel further. To account for the time-to-expiry effect of m,Rebonato et al. (2010) use a similar approach as for ri0 alsofor the volatility of volatility and extend the SABR modelconstant m with a function nðTi � tÞ. By using rather generaland flexible functions lðTi � tÞ and nðTi � tÞ they manageto fit their model quite well across strikes and expiries andthen construct a LIBOR/SABR market model. Althoughthis approach is adequate for the (high-dimensional) LIBORmarket model it does not enable the use of a common


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stochastic volatility driver for the level of rates and is hencenot suitable for our purposes. Instead, we choose to focuson an alternative route that alters the stochastic volatilityprocess and introduces mean-reversion.

While mean-reversion may be added to the SABR modelin several ways we aim for a specification that is similar tothe standard SABR model but at the same time flexibleenough to be able to fully appreciate the extra degree offreedom. Our approach builds on a model by Fouque et al.(2000) and is based on taking

rit ¼ ri

0 expðUit Þ; ð47Þ

dUit ¼ �jUi

t dt þ m dV it ; U0 ¼ 0: ð48Þ

This modeling choice was also made by Jackel and Kahl(2007), although they chose to focus on a minor modifica-tion of the exponential function of hyperbolic type. Withthis choice of equation for the volatility process replacingthe usual log-normal equation in the SABR model, againwe have the distribution of yiTi as in (40) and (41) (seeappendix A). If we allow for i-dependent ri0 but choosecommon m and j parameters we obtain

ESi ½V iTi� ¼ ðri

0Þ2Z Ti

0

expm2

jð1� e�2jtÞ

� dt; ð49Þ

ESi ½ðV iTiÞ2� ¼ 2ðri

0Þ4Z Ti

0

Z t

0

� expm2

jð5� 4 e�2js � e�2jðt�sÞÞ

� ds dt:

ð50Þ

As both moments are controlled by the combination of mand j there is an extra degree of freedom that will aid infitting both moments across expiries with a single set ofparameters. Note that m directly affects the level of themoments, whereas j controls the dampening over time.Finally, note that the growth of the first moment in Ti isnow linear and the second quadratic compared to the expo-nential growth for these moments in the SABR model.

5.3. The DD-SABR-MR model

The previous two subsections argued that the standardSABR model is not adequate as a choice of model for thelevel of rates and reported two types of extensions of theSABR model; deterministic time-to-expiry-dependentinstantaneous volatility and mean-reversion. Combiningboth approaches gives the displaced diffusion SABR withmean-reversion (DD-SABR-MR) model where each forwardswap rate is modeled as

dyit ¼ liðTi � tÞ expðUit Þðyit þ diÞ dWi

t ; yi0 ¼ yi0; ð51Þ

dUit ¼ �jUi

t dt þ m dV it ; Ui

0 ¼ 0; ð52Þ

dWit dW

jt ¼ dV i

t dVjt ¼ dt; dWi

t dVit ¼ q dt;

under Si corresponding to Pi as numeraire and with d 2 <.Recall at this point the link (38) between the CEV and dis-placed diffusion formulations. For the remainder of thispaper we will continue to refer to the parameter b, whichmeans that the corresponding di from equation (38) is beingused. Moreover, note from (38) that, for large d, b is closeto zero (or indeed zero as d ! 1) and hence in this casewe use the stochastic volatility Gaussian model (the b ¼ 0case in the CEV formulation) with yi dynamics

dyit ¼ liðTi � tÞ expðUit Þ dWi

t ; yi0 ¼ yi0: ð53Þ

To obtain an intuitive feeling for the model note that Umean-reverts around zero, implying that the volatility of yitmean-reverts around the function liðTi � tÞ. Indeed, Ito’sformula gives

drt ¼ d expðUtÞ

¼ �jrt lnrt

r0

� dt þ mrt dZt þ rt

m2

2dt; ð54Þ

which is why we have chosen to denote the model as theSABR with mean-reversion model.

Note that we have chosen to use the displaced diffusionformulation instead of the CEV formulation in the model.The main reason for this is that the CEV-type local volatil-ity function /ðxÞ ¼ xb is quite cumbersome from an imple-mentation point of view. In particular, for b 2 ð0; 1Þ, thebehavior around x ¼ 0 is tricky to deal with due to theproperty of absorption at 0. For example, for the MonteCarlo method the behavior close to zero needs special(ad-hoc) treatment and, in particular for longer expiries,different (but all reasonable) choices can imply rather differ-ent values of certain moments and products.

The displaced diffusion setting leads to a much more sta-ble and effective implementation as, in this case, one maywrite out the distribution of each of the yit in explicit form(see appendix A for details). The direct disadvantage withthis choice is that the domain of the underlying is ð�d;1Þ,implying the possibility of negative rates. Note, however,that while we will focus on the DD case, most results aregeneric in the sense that they would look very similar if theCEV-type formulation was used instead.

5.3.1. Modeling liðTi � tÞ. In models with deterministicvolatility, for example the LIBOR market model, a popularchoice of liðTi � tÞ is to model this by an instantaneous vol-atility function of the type (see, for example, Brigo andMercurio (2006))

l̂iðTi � tÞ :¼ ðaþ bðTi � tÞÞ expð�cðTi � tÞÞ þ d: ð55Þ

The function l̂i links the different rates and induces appro-priate time-dependence by calibrating the constants a, b, cand d and is a rather powerful and flexible choice. It is also


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the choice of Rebonato et al. (2010) in their LMM/SABRmodel.

However, for our end goal of finding a commonstochastic volatility driver for all rates we need the volatilitystructure to be separable. Separability appears in the litera-ture when requiring some high-dimensional system to berepresented by a low-dimensional Markov process (see, forexample, Pietersz et al. (2004) for the use of separability torepresent an approximate LIBOR market model by a one-or two-dimensional driving Markov process). Separabilityis, in this context, achieved if we represent each of thetime-dependent functions liðTi � tÞ as

liðTi � tÞ ¼ KðTiÞkt: ð56Þ

Putting b ¼ d ¼ 0 in (55) gives

liðTi � tÞ ¼ KðTiÞkt :¼ a expð�cTiÞ expðctÞ; ð57Þ

which is of separable form. Note that if we are willing toadd another factor we could achieve the flexibility of (55)while retaining separability. However, by using market datait is shown in sections 5.4 and 6 that this is not necessary as(57) does a good job provided the the ‘right’ choice of b.

To be able to perfectly calibrate to ATM implied volatili-ties we have furthermore chosen to equip the model witheither i-dependent constants ki or use a piecewise constantfunction

at ¼ ai; Ti�1 \ t � Ti: ð58Þ

Ideally, for good model performance, all ki should be closeto 1 or all ai of a similar level. While we do believe thatthe piecewise constant formulation is a better choice inpractice, the constant ki setting is good for showing theintuition and behavior of the model.

5.3.2. Comment on smile dynamics. Recall that in section4 we deduced that the smile dynamics of the SABR modelwith an appropriate b matched the market dynamics quitewell. Even though the DD-SABR-MR model is supposedlyclose to the SABR model in terms of dynamics it is notclear that it would inherit these properties. To address thisfirst, note that applying the results of Durrleman (2004)shows that the first-order ‘small time’ approximation (15) isin fact the same in this and the standard SABR model. Asargued in section 4 in connection with the Heston model,note that while this means the mean-reversion term is noteven present in this expression, and hence the expression isnot appropriate for pricing, it is a reasonably good approxi-mation for the dynamical relations studied in section 4.Numerical investigations of the same type as performedwhen testing the appropriateness of using the approximate‘slope’ expression (28) reveal that the ‘slope’ and ‘curva-ture’ dynamics are almost exactly the same in the SABRand DD-SABR-MR models. Hence we believe that theresults and intuition from section 4 may be transferred tothe DD-SABR-MR model.

5.4. Calibration results

This section reports the results from calibrating theDD-SABR-MR model to market data of options to enter10-year swaps on 27 October 2007 and 9 March 2003across the same expiries as in section 5.1.

The calibrated parameters on 27 October 2007 usingb ¼ 0, 0.5 and 1 are given in tables 3 and 4 and thecalibration errors (market minus model implied volatility)are given in figure 9. First note that b ¼ 0 seems toproduce a very good fit to the market implied volatilities.For b ¼ 0:5 the calibration is also very satisfactory, whereasfor b ¼ 1 the errors are (although still quite acceptable)about three times larger than for b ¼ 0. In fact, on this date,for b ¼ 0 the calibration errors are about the same size asthe outcome from standard SABR models calibrated at eachexpiry individually.

Note that, with a single set of parameters, one wouldexpect the slopes of the implied volatility smiles to be toosteep/flat and have too much/little curvature at someexpiries. This effect is shown in figure 9 by studying theplus/minus 2% offset strikes. Moreover, note that, on thisdate, the constants ki are all close to 1 and the piecewiseconstant function, at, is close to constant, implying a verygood fit to the full implied volatility surface.

As the results look remarkably good it should be notedthat this particular date is one of the better dates in oursample. As a comparison, figure 10 displays the calibrationresults on 9 March 2003. Also on this date b ¼ 0 performsbest in terms of fitting errors with two to three timessmaller errors than the other two cases. Moreover, note that,for b ¼ 0, it seems like a single q works very well, whereasthe shorter (longer) expiries would need slightly smaller(larger) m with the medium-term expiries displaying close tozero fitting errors. For b ¼ 1, one would, on the other hand,need to change both q and m individually at almost allexpiries to effectively reduce the errors.

For this date, studying the constants ki provides moreinformation about the appropriate choice for b and whychoosing to use (57) instead of (55) as the parametrizationof lt is not necessarily particularly restrictive. Figure 11displays the forward swap rates, the ATM implied volatili-ties and the calibrated ki for the different b. First note that,on this date, the ATM implied volatilities are steadilydeclining until the 15-year expiry, where it starts increasing.As, approximately, for b ¼ 1 the function lt needs to fit thisdependence, the restricted function in (57) is not flexibleenough, something that is clearly displayed by the widelyvarying constants ki.

However, as seen in figure 11 for b ¼ 0 or 0.5 theconstants ki are all close to 1. To understand why this is the

Table 3. Calibrated parameters of the SABR-MR model on 27October 2007; 2- to 30-year options to enter 10-year swaps.

b a c j m q

1 0.122 0.078 0.015 0.283 �0.4760.5 0.027 0.068 0.050 0.278 �0.2330 0.006 0.065 0.098 0.317 0.033


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case recall that, in the SABR model (as well as the deter-ministic CEV model or a matched up displaced diffusiontransformation), the ATM volatility is approximately givenby R ¼ r=yb. Hence, there is a direct relation between theparameters ri, the forward swap rates and the marketimplied ATM volatilities and it turns out that using anappropriate b stabilizes ai and makes the function (57) flex-ible enough to fit the model.

Remark 4: Note that the size of the calibrated parametersin table 3 depends on the choice of b. As we know fromsection 4 that the slope and curvature of the smiles in theSABR model depend on all parameters b, q and m, this isnot surprising. Naturally, the different parameter specifica-tions yield different model dynamics (and hence hedges andprices of exotic derivatives in a full term-structure modelbased on this specification). Recall that, under the

Table 4. Calibrated ki and levels of the piecewise constant function at for the SABR-MR model on 27 October 2007. Expiry inyears. Options to enter 10-year swaps.

Expiryki ai

b ¼ 0 b ¼ 0:5 b ¼ 1 b ¼ 0 b ¼ 0:5 b ¼ 1

2Y 0.999 1.008 1.016 0.00579 0.02684 0.12413Y 0.996 1.004 1.011 0.00574 0.02655 0.12244Y 0.996 1.003 1.007 0.00578 0.02659 0.12215Y 0.998 1.002 1.004 0.00582 0.02661 0.12166Y 0.995 0.995 0.995 0.00571 0.02589 0.11777Y 0.997 0.994 0.991 0.00583 0.02636 0.11958Y 1.003 0.997 0.992 0.00595 0.02685 0.12169Y 1.005 0.997 0.991 0.00588 0.02653 0.120510Y 1.005 0.996 0.989 0.00583 0.02636 0.120012Y 1.004 0.994 0.988 0.00580 0.02635 0.120515Y 1.004 0.997 0.996 0.00582 0.02665 0.122920Y 1.000 1.000 1.005 0.00578 0.02668 0.123525Y 0.994 0.999 1.003 0.00573 0.02654 0.122430Y 0.996 1.008 1.011 0.00579 0.02706 0.1243

Figure 9. Calibration errors (market minus model implied volatilities) for the SABR-MR model on 27 October 2007; 2- to 30-yearoptions to enter 10-year swaps.


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assumption of time homogeneity (across historical dates inthe sample), the analysis in section 4 indicates that b ¼ 0provides superior smile dynamics compared with b ¼ 0:5and 1. Based on the results for the two dates in this subsec-tion it seems that using b ¼ 0 enables accurate calibrationusing common parameters. This is a requirement for amodel that reflects the right dynamics in the sense of

forward smiles and a key feature in order to be able toidentify a time-homogeneous process for the level of rates.The next subsection provides further analysis by calibratingto all expiries at several dates throughout the sample. Asb ¼ 0:5 is a popular choice by traders this is also included,but b ¼ 1 is dropped as it is clearly inferior in both smiledynamics and forward smiles.

Figure 11. The forward swap rates, the ATM implied volatilities and the resulting constants ki on 9 March 2003; 2- to 30-year optionsto enter 10-year swaps.

Figure 10. Calibration errors (market minus model implied volatilities) for the SABR-MR model on 9 March 2003; 2- to 30-yearoptions to enter 10-year swaps.


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6. Testing the model on all dates

Recall that a well-calibrated model has small calibrationerrors across expiries and strikes as well as close to con-stant function at or all constants ki close to 1. As reportedin the previous section, a SABR-MR model with appropri-ate b seems to be able to fulfill both of of these propertieswith good accuracy. To test if this is a general feature andwhether b ¼ 0 performs well in general, this section teststhe calibration performance of the SABR-MR model for the101 dates plotted in figure 2 covering the period 3 July2002 to 21 May 2009. As the calibration errors during theturmoil period will inevitably be larger and provide a toughchallenge for any model, the data set is divided into twoparts with 89 dates up to the turmoil and 12 dates duringthe turmoil.

6.1. Calibration errors, ‘normal’ markets

Table 5 provides summary statistics of the calibrated ki val-ues for the first 89 dates up to the start of the turmoil. Notethat the means of all ki (in total, 89 dates with 14 expiries,implying 1246 values) are close to 1 (as they should begiven appropriate calibration). Moreover, for both modelsthe standard deviation and the maximum and minimum val-ues are quite close to zero and one, respectively, implying asound model from this perspective across all dates. This isan appealing feature as is tells us that the sacrifice of usingthe simple form for liðTi � tÞ given in (57) is minor andhence separability may be imposed without sacrificing thecalibration performance of the model.

Summary statistics for the calibration errors in impliedvolatility are given in table 6. As expected, the mean isquite close to zero, however the standard deviation, meanof the absolute errors and the maximum and minimumerrors are a bit smaller for b ¼ 0 compared with b ¼ 0:5.Even though plus/minus a couple of percent in implied vol-atility is not too bad, note that if only studying the plus/minus 1% strike offset or expiries ranging from 5 to 20years, then all fitting errors are within 1% for b ¼ 0.

At first sight the maximum and minimum fitting errorsmight seem a bit prohibitive. However, as the overall levelof the implied volatility surface changes over time it is inrelative terms not too bad. The date with the maximumerror has ATM implied volatilities ranging from 15 to 21%and while 2.76% is a bit off it is not much worse in relativeterms compared with other dates (for example, comparedwith the 9 March 2003 date studied in the previous sec-tion). Moreover, the particular point with an error of 2.76%is for the 2-year expiry with �2% strike and a marketimplied volatility of 26%.

Recall that, for each date, the model is trying to fit14� 9 ¼ 126 data points with rather few parameters. Asthe parameters a and c and the constants ki or the functionat mainly focus on the ATM part, there is in effect onlythree parameters to play with for the away-from-the-moneystrikes. In principle, the differences between the market andmodel implied volatilities are due to either slope or curva-ture mismatch of the smiles.

Recall that, for a fixed b, q fits the slope of the smile. Ifthe slopes are ‘different’ for different expiries, one parame-ter is not enough to fit all expiries perfectly and the calibra-tion will return an ‘average’ q. For some dates this is notmuch of an issue, whereas for others it is worse. Note that,for the second calibration date displayed in section 5, this isan issue for b ¼ 1, whereas b ¼ 0 has curvature errors.

Also recall that the combination of m and j controls thecurvature of the smiles. The parametrization of the SABR-MR model gives a specific type of control of the curvaturethat seems to be a decent one for most dates. The b ¼ 0case in the calibration on 9 March 2003 displayed in

Table 5. Summary statistics of calibrated constants ki on 89 datesfrom 2002 up to the turmoil period; 2- to 30-year options to

enter 10-year swaps.

b Mean Stdev Max Min

0 1.0002 0.0108 1.0455 0.95930.5 1.0002 0.0113 1.0389 0.9567

Table 6. Summary statistics of the errors (market minus model) in implied volatility for calibrated SABR-MR models on 89 datescovering 3 July 2002 up to the start of the turmoil in September 2008; 2- to 30-year options to enter 10-year swaps. Strike offset awayfrom ATM and the implied volatilities are reported in percent. The row Mean(abs) refers to the mean of the absolute values of the fitting

errors.

Strike offset

�2 �1 �0.5 �0.25 0 0.25 0.5 1 2

b =0Mean �0.05 0.16 0.10 0.05 0.00 0.01 0.03 0.04 0.07Stdev 0.47 0.19 0.09 0.05 0.00 0.04 0.07 0.16 0.34Mean(abs) 0.37 0.16 0.08 0.04 0.00 0.03 0.05 0.12 0.26Max 2.76 1.08 0.43 0.21 0.00 0.19 0.39 0.74 1.59Min �1.85 �0.51 �0.24 �0.12 0.00 �0.12 �0.15 �0.36 �0.82

b =0.5Mean 0.15 0.10 0.05 0.01 0.00 0.04 0.08 0.10 0.04Stdev 0.62 0.24 0.10 0.05 0.00 0.05 0.10 0.25 0.55Mean(abs) 0.48 0.18 0.07 0.03 0.00 0.04 0.08 0.18 0.38Max 3.48 1.24 0.47 0.19 0.00 0.23 0.49 0.91 1.69Min �1.90 �0.56 �0.28 �0.14 0.00 �0.11 �0.27 �0.78 �1.72


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figure 9 is a good example of what is going on with a cali-bration returning too little curvature for the 2-year smileand too much curvature for the 30-year smile. For somedates this effect is larger, for some dates smaller.

With only three parameters to fit all away-from-the-money strikes we conclude that the SABR-MR does a goodjob in terms of fitting and seems to be well specified andparametrized. In particular, we find no major reason to gointo complications concerning extending the model withtime-dependent parameters q, j and m, a route that has beensuggested by some authors.

6.2. Calibration errors: Turmoil period

As discussed above the period of market turmoil provides atough challenge for any interest rate model. While of coursemodels with more parameters will perform well in terms offitting, there is a risk of over-fitting when market prices arenot necessarily internally sound and consistent. On the otherhand, while a model with few parameters may struggle interms of fitting, it could be a more solid and stable choicewhen times are tough.

As can be seen from figure 2 the yield curve as well asthe ATM level, slope and curvature of the implied volatilitysmiles are almost inverted and some values are very differ-ent compared with ‘normal’ markets. As rates are very lowwe will disregard the �2% strike offset since this strike isclose to zero, implying very high implied volatilities thatwill affect the calibration and choice of b too much andhence blur the results.

Summary statistics for the calibrated constants ki aregiven in table 7. Note that, in this case, the errors are sig-nificantly larger (about twice as large) than before and thatb ¼ 0 performs far better than b ¼ 0:5.

Summary statistics for the calibration errors in impliedvolatility are given in table 8. Note that, in this period, theerrors are quite a lot larger than before and that bothchoices of b have similar performance. Part of the increasein the errors is, of course, due to the fact that the level ofimplied volatility is larger (note from figure 2 that the 2-and 30-year ATM implied volatilities are about twice aslarge as on previous dates and, moreover, for some strikesthe implied volatility is above 40%) and actually the rela-tive errors are not that much larger compared with previousresults. Given the large magnitude and the vastly differentslopes and curvatures of the smiles across expiries com-pared with previous dates, the results are perhaps betterthan one could expect. However, although we can fit thedata using the model, the results of section 4 suggest that itis not consistent with how the market is behaving.

6.3. Calibrated parameters

To further investigate the performance of the SABR-MRmodel, figure 12 plots the time series of calibratedparameters for the 101 reference dates. Note thatalthough parameter stability is one measure of a soundmodel it is the combination of the parameters that pro-vides the distribution. This means that rather differentcombinations of some parameters (for example, m and jcombinations) may produce similar distributions andhence one should not pay too much attention to simplystudying parameter stability. Nevertheless, it does providesome further intuition and evidence concerning modelperformance.

First note that the parameter variations are very similarfor both choices of b. Although the parameters do varyover time the variation is not particularly large. For exam-ple, for b ¼ 0, minor tilts of m of about 0.33, j of 0.08 andq of 0 seem to do the job. Moreover, note that, during the

Table 7. Summary statistics of calibrated constants ki on 12evenly spaced dates during the turmoil period; 2- to 30-year

options to enter 10-year swaps.

b Mean Stdev Max Min

0 1.004 0.026 1.085 0.9620.5 1.008 0.063 1.211 0.906

Table 8. Summary statistics of the errors (market minus model) in implied volatility for calibrated SABR-MR models on 12 evenlyspaced dates during the turmoil period; 2- to 30-year options to enter 10-year swaps. Strike offset away from ATM and the implied

volatilities are reported in percent. The row Mean(abs) refers to the mean of the absolute values of the fitting errors.

Strike offset

�1 �0.5 �0.25 0 0.25 0.5 1 2

b =0Mean �0.08 0.10 0.06 0.00 0.06 0.09 0.05 �0.13Stdev 1.24 0.34 0.13 0.00 0.15 0.25 0.42 0.66Mean(abs) 0.98 0.22 0.08 0.00 0.13 0.22 0.32 0.44Max 1.60 0.67 0.31 0.00 0.68 1.21 1.98 2.87Min �5.21 �1.33 �0.48 0.00 �0.18 �0.33 �0.62 �1.21

b =0.5Mean 0.04 0.07 0.04 0.00 0.09 0.09 �0.04 �0.50Stdev 0.97 0.26 0.09 0.00 0.07 0.12 0.36 0.75Mean(abs) 0.74 0.19 0.07 0.00 0.06 0.07 0.24 0.65Max 1.93 0.76 0.33 0.00 0.35 0.29 0.63 1.41Min �4.66 �0.98 �0.30 0.00 �0.16 �0.42 �1.19 �2.45


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turmoil period, the parameter values are not much differentfrom earlier periods, which is a bit surprising bearing inmind the vastly different ATM levels, slopes and curvaturesof the smiles observed in this period. This suggests that themodel does a good job in explaining the implied volatilitysmiles conditional on the values of the forward swap ratesand the instantaneous stochastic volatility.

Finally, note that, for about a third of the dates, q is alsonegative for b ¼ 0 and recall that the combination of b andq determines the slope of the implied volatility smile. Thismeans that, in general, one cannot put q ¼ 0 (which maybe desired from an implementation point of view) andchange b to fit the smile, as then the slope of the smile willbe too flat.

7. A stochastic volatility model for the level of rates andpricing under a single measure

The output of our investigations so far is a model for each10-year swap rate in its own swaption measure that is con-sistent with market implied distributions for the correspond-ing swaptions and which is based on two factors and a setof parameters that are common for all swap start dates. Inthis section our task is to identify a candidate for modelingthe level of rates process which is of low dimension. Wewill specify our candidate process in the measure Sn corre-sponding to the final expiry Tn. The intention is to concludethe paper by identifying a time-homogeneous stochasticvolatility process from the volatility structures of the previ-ous section, reporting the issues arising when putting every-thing under a single measure and provide inspiration forfuture work.

In order to understand how to link the swap rates under asingle measure we begin by studying the case correspond-ing to b ¼ 0 where, for each i under Si, we have

yit ¼ yi0 þ a

Z t

0

expð�cðTi � uÞÞ expðUiuÞ dWi

u;

¼ yi0 þ expð�cðTi � tÞÞX it ;

where Wi and V i are correlated Brownian motions and Ui

is an Ornstein–Uhlenbeck process as specified in equation(52), and we have defined

X it :¼ a

Z t

0

expð�cðt � uÞÞ expðUiuÞ dWi

u:

Note that the process X i in the above equation provides arepresentation for the spot process at time Ti under S

i, sinceyiTi ¼ yi0 þ X i

Ti.

Now further assume that q ¼ 0. If we were to specify anexpression for the process X i in some equivalent martingalemeasure (here the drift of V i and hence of Ui is left unal-tered as we are working with bond-based numeraires) wewould have

X it ¼ a

Z t

0

expð�cðt � uÞÞ expðUuÞ dWu þ Fit ;

Ut ¼ m expð�jtÞZ t

0

expðjsÞ dVt;

where ðW ;V Þ is a standard Brownian motion under ourchosen EMM and Fi is a finite variation process dependent

Figure 12. Calibrated parameters for all 103 dates. Date 1 is the most recent (21 May 2009).


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on the particular measure we have chosen. Assuming thecommon diffusion part is what is significant for modelingsuggests a candidate for our level of rates process. Note thatif q is not equal to zero, moving to the one measure wouldintroduce an i-dependent drift change into the equation forU . Ignoring this complication we take our candidateprocess for representing the level of rates when b ¼ 0 as

dXt ¼ �cXt dt þ expðUtÞ dWt; X0 ¼ 0;

dUt ¼ jUt dt þ m dVt;

dW dV ¼ q dt;

where W and V are correlated Brownian motions under Sn

say, with Tn being the final expiry.

Remark 5: Observe that our candidate model ismean-reverting. Recall that the short rate in a one-factorHull–White model is mean-reverting and if the instanta-neous volatility of the forward rate ftT is of the formexpð�cðT � tÞÞrt, then the instantaneous volatility of theshort rate r is rt. It is interesting to note that if we view yitas a discrete analogue of ftTi , and Xt as playing the role ofr, then we have the same relationship between the instanta-neous volatility of the two processes as in the Hull–Whitemodel despite the presence of stochastic volatility.

For the case when b – 0, observe that taking a mono-tonic increasing function of the swap rates will not alter theinformation about the level of rates. For the CEV form ofthe model, define

FðxÞ ¼ ð1� bÞx1�b; 0 � b\ 1;

¼ log x; b ¼ 1;

and for the displaced diffusion version take

FðxÞ ¼ logðxþ dÞ:

In the CEV version for each i under Si, by Ito’s formulawe have

FðyitÞ ¼ Fðyi0Þ þ expð�cðTi � tÞÞX it �

ab2

Z t

0

ðyiuÞb�1

� expð�2ðcðTi � uÞ expð2UiuÞ du:

The displaced diffusion version for a given d gives the sameformula for FðyitÞ as above but with b ¼ 1. Hence, for thetransformed rate we are back at the b ¼ 0 case, albeit withthe addition of a finite variation term. Again assuming thatthe common diffusion part is what is significant for modelingsuggests the same candidate for our level of rates process.

Building a full term-structure model based on the candi-date process is beyond the scope of this paper. At this point

we just want to note that, in order to test that the simplify-ing assumption made above concerning the role of the driftis reasonable, we set up a terminal swap rate Markov-func-tional-type model and checked that we may recover theprices of vanilla swaptions under a single joint measure(see Kaisajuntti (2011) for more details).

8. Conclusion

This paper has used an extensive set of historical marketdata of swap rates and swaptions to identify a two-dimen-sional stochastic volatility process for the level of rates. Theprocedure of the paper has been to identify this processfrom the bottom up, step by step by increasing the require-ments of the model, and discusses how to adjust the processto take this into account.

We started off in section 4 by identifying a suitabletime-homogeneous model for swap rates at their settingdates and it turned out that the SABR model with b ¼ 0satisfactorily passed all our tests (assigning appropriatesmile dynamics) and that also b ¼ 0:5 seemed to performreasonably well. This section is of independent interest interms of offering a data-driven analysis for the appropri-ate choice of b, a comparison of SABR versus Hestonand intuition for model behavior for different specifica-tions of the driving processes in a stochastic volatilitymodel.

Section 5 identified a model of 10-year swap rates undertheir own measures based on common parameters for allswap start dates, and section 6 tested it on market data cov-ering 3 July 2002 to 21 May 2009. The tests demonstratedthat the model is a stable and flexible choice that allowsfor good calibration across expiries and strikes as well asassigning appropriate forward smiles. It also turns out thatusing a separable volatility structure poses no major limita-tion in terms of fitting to several expiries and acrossstrikes.

At this point the developed DD-SABR-MR model couldbe used directly as a building block for a stochastic volatil-ity LIBOR market model, and the only complication inwriting down the model would be to find the induced driftterms under a single joint measure. However, as we haveput some effort into keeping the dimension low and allow-ing for separability we believe it could also be a suitablebuilding block for a stochastic volatility Markov-functionalmodel. With this in mind, section 7 identified and put intocontext a time-homogeneous candidate stochastic volatilityprocess that can be used as a driver for all swap rates undera joint measure. This process can then be used in aMarkov-functional-type model or as motivation for specify-ing a short-rate-type model.

For any given product, pricing and hedging using a low-factor model needs adequate evaluation. It is our expecta-tion that a term-structure model based on a driving processthat better reflects the dynamics of the market will turn outto have benefits over the low-factor models that arecurrently in use. Developing and testing full term-structuremodels based on the outcome of this paper is left as asubject of future research.


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Finally, recall that, at the outset of our investigations, the10-year swap rates were chosen in order to have someoverlap between the rates and hence provide hope that find-ing a model of the above type would be possible. We alsoperformed the above tests on the 5-, 20- and 30-year swaprates and observed similar results as for the 10-year case.Moreover, we calibrated the model to co-terminalswaptions, as is required when pricing some Bermudanswaption-type products, and our findings suggest that thisalso may be done with good accuracy.

Acknowledgements

We are most grateful to an anonymous referee for helpfulcomments. Kaisajuntti gratefully acknowledges financialsupport from the Jan Wallander and Tom HedeliusFoundation.

References

Albanese, C. and Trovato, M., A stochastic volatility model forswaptions and callable swaps. Imperial College, 2005.

Andersen, L. and Andreasen, J., Volatile volatilities. Risk, 2002,15(12), 163–168.

Andersen, L. and Brotherton-Ratcliffe, R., Extended LIBORmarket models with stochastic volatility. J. Comput. Finance,2005, 9(1), 1.

Andersen, L. and Piterbarg, V., Interest Rate Modeling, 1st ed.,2010 (Atlantic Financial Press).

Brigo, D. and Mercurio, F., Interest Rate Models – Theory andPractice, 2nd ed., 2006 (Springer: Berlin).

Durrleman, V., From implied to spot volatilities. PhD thesis,Princeton University, 2004.

Fouque, J.P., Papanicolaou, G. and Sircar, K.R., Derivatives inFinancial Markets with Stochastic Volatility, 2000 (CambridgeUniversity Press: Cambridge).

Henrard, M., Swaptions: 1 price, 10 Deltas, and...6 1/2 Gammas.Wilmott Mag., 2005, November, 48–57.

Heston, S.L., A closed-form solution of options with stochasticvolatility with applications to bond and currency options. Rev.Financ. Stud., 1993, 6(2), 327–343.

Jackel, P. and Kahl, C., Hyp hyp hooray. Working paper, 2007.Available online at: http://www.jaeckel.org.

Kaisajuntti, L., Multidimensional Markov-functional and stochasticvolatility interest rate modelling. PhD thesis, Stockholm Schoolof Economics, 2011.

Kennedy, J.E., Mitra, S. and Pham, D., On the approximation ofthe SABR model: a probabilistic approach. Appl. Math.Finance, 2012, 19(6), 553–586.

Marris, D., Financial option pricing and skewed volatility. M. Phil.thesis, University of Oxford, 1999.

Pietersz, R., Pelsser, A. and Van Regenmortel, M., Fast drift-approximated pricing in the BGM model. J. Comput. Finance,2004, 8, 93–124.

Piterbarg, V., Stochastic volatility model with time-dependent skew. Appl. Math. Finance, 2005, 12(2), 147–185.

Rebonato, R., McKay, K. and White, R., The SABR/LIBOR MarketModel: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives, 2010 (Wiley: New York).

Trolle, A.B. and Schwartz, E.S., An empirical analysisof the swaption cube. National Bureau of Economic Research,2010.

Appendix A: Implementation and calibration

A.1. Monte Carlo implementation

This section outlines a comparably fast and efficient way ofimplementing the DD-SABR-MR model. Recall the dynamics(without the constants ki or piecewise constant function at forclarity) under each of the measures Si

d logðyit þ dÞ ¼ Fðlit;UtÞðyit þ dÞ dWit �

F2ðlit;Uit Þ

2dt; ðA1Þ

dUit ¼ �jUi

t dt þ m dZit ; ðA2Þ

dWit dZ

it ¼ q dt; ðA3Þ

with

lit ¼ a expð�cðTi � tÞÞ; ðA4Þ

Fðl; uÞ ¼ l expðuÞ: ðA5Þ

First note that Wit may be rewritten as Wi

t ¼ qZitþ


pBit,

where Bi is a Brownian motion independent of Wi and Zi. Thisgives

d logðyit þ dÞ ¼ Fðlit;Uit Þðq dZt þ


pdBi

tÞ

� F2ðlit;Uit Þ

2dt

¼ qmFðlit ;Ui

t Þð dUit þ jUi

t dtÞ

þ Fðlit;Uit Þ


pdBi

t

� F2ðlit;Uit Þ

2dt

¼ qmFðlit ;Ui

t Þ dFðlit;Uit Þ

� qmcFðlit;Ui

t Þ dt �qm2Fðlit;Ui

t Þ dt

þ qjmFðlit ;Ui

t ÞUit dt �

F2ðlit;Uit Þ

2dt

þ Fðlit;Uit Þ


pdBi

t: ðA6Þ

Now note that, conditional on the r-algebra generated by Zi

over ½0; T �, the Si conditional moment-generating function of

Z T

0

Fðlit;Uit Þ dBi

t

is equivalent to that of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV ið0; TÞ

pG;

with V iðt; TÞ: ¼ R Tt F2ðlis;Ui

sÞ ds and G a standard normalrandom variable that is independent of Bi. By the towerproperty this holds also for the unconditional Si moment-generating functions and hence

Z T

t

Fðlis;UisÞ


pdBi

s ¼dffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

p ffiffiffiffiffiV i

pG: ðA7Þ


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http://www.jaeckel.org

Finally, integrating (A6) from t � 0 to T � Ti and again com-paring the moment-generating functions under Si and using thetower law it is possible to show that

yiT ¼d ðyit þ dÞ exp qmðFðlT ;Ui

T Þ � Fðlt;Uit ÞÞ þ qI iðt;TÞ

� 1

2V iðt; TÞ þ


p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV iðt; TÞ

pG

� d; ðA8Þ

with

I iðt; TÞ ¼Z T

t

jmFðlis;Ui

sÞUis �

c

mFðlis;Ui

sÞ �m2Fðlis;Ui

sÞ �

ds:

ðA9Þ

Hence, to simulate a realization of yiT , one only needs onestandard normal random variable plus the integrals I i and V i

along a particular path of Ui. For a simulated path of Ui theintegrals I iðt; TÞ and V iðt; TÞ may be computed by numericalquadrature using, for example, the Simpson rule.

Moreover, note that, conditional on the full path of U from0 to Ti, yiTi is a displaced log-normal random variable andhence prices of swaptions are given in closed form. Hence, ina Monte Carlo implementation to price swaptions it is onlyrequired to simulate realizations of Ui, compute the values ofFi, I i and V i using numerical quadrature and finally insert inthe displaced diffusion Black formula. Finally, note that, forb ¼ 0, the above calculations proceed with just a minor modifi-cation.

To obtain an idea of the accuracy of using this method com-pared with the (log) Euler method, figure A1 shows the pricesof a 10-year ATM swaption computed using a different numberof steps per year. Note that while the Euler method requiresmany steps per year to eliminate the discretization error, thenew method allows for very long steps.

A.2. Calibration

To calibrate the SABR-MR model we minimize the squaredrelative error between the market and model implied volatilitiesat all strikes and expiries. Note that while in principle there areonly five parameters to calibrate (a; c; j; m; q) we also need tofind the parameters of the function at or the constants ki. As afull global optimization of all parameters is too time-consum-ing to perform in practice, we perform the calibration in twosteps

(1) First calibrate (a; c; j; m; q). Since eventually we willmake sure that the ATM levels are perfectlymatched, the goal function is structured such that,first, the ATM errors are found, then the model isadjusted to fit the ATM perfectly by using the con-stants ki and finally the errors at the other strikes arefound.

(2) In the second step we determine the piecewise con-stants needed for at and then the constants ki startingfrom time T1.

As there is no sufficiently good approximation of the SABR-MR model available and deriving one is out of the scope of thispaper we use the Monte Carlo method in the calibration routine.To simulate yiTi we use the method outlined in the previous sub-section with 10,000 Sobol paths and 30 steps out to eachexpiry. Note that, in principle, it is the number of pointsbetween time 0 and expiry that is important, not the number ofsteps per year, and we have found that 30 steps gives goodaccuracy for all expiries. Compared with more accurate choicesthat reduce the numerical errors we found that the abovechoices are correct within a few basis points. On a standardcomputer this means that a full implied volatility surface of 14expiries and nine strikes per expiry is priced in about 1 secondand a calibration is made in at most a few minutes.

While this is, of course, not fast enough to be used in a mar-ket environment, the above structure gives the hope of findinga good approximation for calibration (see Kennedy et al.(2012) for a similar approximation for the standard SABRmodel).

Figure A1. A comparison of the new implementation method of yi with the standard Euler method for pricing a 10-year ATM swap-tion. For the one-step method the prices plus/minus two standard deviations are also reported. Four million pseudo-random pathsare used, implying a very small statistical error. b ¼ 0, a ¼ 0:009, c ¼ 0:1, j ¼ 0:05, m ¼ 0:36 and q ¼ 0:2.


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stochastic volatility for interest rate derivatives

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