calibrating cap_floors volatilities

8/9/2019 Calibrating Cap_Floors Volatilities

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THE BUSINESS SCHOOLFOR FINANCIAL MARKETS

The University of Reading

Common Correlation Structuresfor Calibrating the LIBOR Model

ISMA Discussion Papers in Finance 2002-18

First Version: 1 June 2002

This version: 20 June 2002

Carol Alexander

ISMA Centre, University of Reading, UK

Copyright 2002 Carol Alexander. All rights reserved.

The University of Reading ISMA Centre Whiteknights PO Box 242 Reading RG6 6BA UKTel: +44 (0)118 931 8239 Fax: +44 (0)118 931 4741

Email: [email protected] Web: www.ismacentre.rdg.ac.uk

Director: Professor Brian Scott-Quinn, ISMA Chair in Investment BankingThe ISMA Centre is supported by the International Securities Market Association


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This discussion paper is a preliminary version designed to generate ideas and constructive comment. The

contents of the paper are presented to the reader in good faith, and neither the author, the ISMA Centre, nor

the University, will be held responsible for any losses, financial or otherwise, resulting from actions taken

on the basis of its content. Any persons reading the paper are deemed to have accepted this.

Summary

In 1997 three papers that introduced very similar lognormal diffusion processes for interest ratesappeared virtuously simultaneously. These models, now commonly called the 'LIBOR models'are based on either lognormal diffusions of forward rates as in Brace, Gatarek & Musiela (1997)and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as inJamshidian (1997). The consequent research interest in the calibration of the LIBOR models has

engendered a growing empirical literature, including many papers by Brigo and Mercurio, andRiccardo Rebonato (www.fabiomercurio.it and www.damianobrigo.it and www.rebonato.com).The art of model calibration requires a reasonable knowledge of option pricing and a thorough

background in statistics techniques that are quite different to those required to design no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and theforthcoming book by Rebonato (2002) invaluable aids to their understanding.

I aim to provide an accessible account of some interesting problems in LIBOR model calibration,but the ideas are complex, so notational complexities are reduced to a minimum. We take freshlook at three important modelling decisions when calibrating the LIBOR model: theparameterization of the correlation matrix for semi-annual forward rates; the use of principalcomponent analysis in the orthogonal transform of the log normal forward rate model; and theiterations for recovering caplet volatilities from 'flat' cap volatilities.

The first section provides the briefest of introductions to the lognormal forward rate version ofthe LIBOR model. Section two considers the calibration of the model to the cap market, where itis shown that the iteration of caplet volatilities from the 'flat' cap volatilities quoted in the market,should be performed by equating a vega weighted sum of caplet volatilities to the cap volatility.Section 3 discusses the more difficult problem of calibration to the swaption market. Two fullrank parsimonious parameterizations of the semi-annual correlation matrix are specified, wherecorrelations between annual forward rates are determined by the same parameters.

Section 4 reconsiders calibration to the swaption market where the rank of forward ratecorrelation matrices is reduced by setting all but the three largest eigenvalues to zero. Rebonato(1999a), Rebonato and Joshi (2001) Hull and White (1999, 2000) and Logstaff, Santa-Clara andSchwartz (1999) have all found that this technique is useful for performing the simulations that

are necessary for pricing path dependent options. The implication of zeroing eigenvalues is atransformation of the lognormal forward rate model where each forward rate is driven by threeorthogonal factors that are derived from a principal component analysis. We show that, in fact, itis the common principal components model of Flury (1988) that should be used. That is, the sameeigenvectors should be calibrated to all swaptions of the same tenor, and we advocate the use ofmarket data rather than historical data for the calibration of these common eigenvectors.

Contacting Author:Prof. Carol AlexanderChair of Risk Management and Director of ResearchISMA Centre, School of Business, University of Reading, Reading RG6 [email protected]


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ISMA Centre Discussion Papers in Finance 2002-18

Copyright 2002, Carol Alexander. 3

1. The Lognormal Forward Rate Model

To ease the notation we assume that day counts are constant. That is, year-fractions betweenpayment (or reset) dates are constant for all forward rates, and the basic forward rate is a semi-annual rate. Denote by fi the semi-annual forward rate that is fixed at time t i but stochastic up tothat point in time. Each forward rate has its own 'natural' measure, which is the measure withnumeraire Pi+1 where Pi is the value of a zero coupon bond maturing at date ti. Under it's naturalmeasure each forward rate is a martingale and therefore has zero drift in its dynamics. The lognormal forward rate model is therefore

dfi(t)/ fi(t) = i (t) dWi [ i = 1, .., m; 0 < t < ti ] (1)

where dW1, , dWm are Brownian motions with correlations ij(t). That is,

E[dWidWj] = ij(t) dt (2)

Calibration of the model requires using current market (and/or historical) data to estimate the

parameters i (t) and ij(t) [i, j = 1, .., m]

2. Calibration to the Cap Market

Consider a T maturity cap with strike K as a set of caplets from ti to ti+1[i = 1,.., m 1 and tm =T]. Each caplet pay-off = max (Li K, 0) where Li is the LIBOR rate revealed at time ti and at ti

we have Li = fi. Assume for the moment that i(t) = i . Then the Black-Scholes type formula foreach caplet value at time 0 is:

B-S (Caplet, ti , K, i) = Pi+1 [fi(0)(x) - K(y)] = Ci(i), say

and x = ln (fi(0)/K) /iti + iti /2; y = x - iti

The B-S cap price is the sum Ci(i) of all prices of the caplets.

If we further assume that i = for every i then we can back-out a single flat volatility from the observed market price of a cap. Note that each fixed strike caplet in the cap with strike Khas a different moneyness. For example, consider an at-the-money (ATM) cap. Each ti+1maturity caplet is ATM if fi(0) = K, but since each caplet has a different underlying forward rate,each caplet would have a different ATM strike. We normally define the ATM cap strike as thecurrent value of the swap rate for the period of the cap, so the different caplets in an ATM cap areonly approximately ATM.

f3f2f1

t4t3t2t1Today

f3f2f1

t4t3t2t1Today


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Figure 1 shows the typical 'humped shape' term structure for flat volatilities of caps.1

This is the

curve of(n) plotted against maturity, n where (1) is the volatility of a single caplet [say for 3months]; (2) is the volatility of a 6 months cap [the sum of two 3 month caplets] and so forth. Inthe simple version of the LFR model, with i(t) = i i, the implied forward rate volatilities arethe caplet volatilities that are stripped out from the flat volatility term structure for caps usingsome iterative method. These volatilities will display a more pronounced humped shape than thecap volatilities and may even be negative if the hump is very pronounced.

The market quotes flat volatilities for caps of different maturities, T andstrikes, K; thus an

implied volatility surface (K, T) is quoted at any point in time. So what are the impliedvolatilities of the caplets that make up the caps with different strikes and maturities that areconsistent with the quoted cap volatility surface?

The first proposition in the paper concerns the iterative method used to back out these capletvolatilities from the cap market implied volatility surface. In option markets where the underlyingis a single asset whose dynamics are governed by a standard lognormal Brownian motion, theaverage variance of log returns over a period T = T1 + T2 is the sum of the average variance of logreturns over period T1 and the average variance of log returns over period T2. The additivity ofvariances is a consequence of the independent increments in the stochastic process for a singleunderlying asset, and because variances are additive it is standard practice to 'back out' volatilitiesfrom a term structure using the iterative method:

Set 1= (1); then solve for2from [12 + 22]/2 = (2)2 and so forth.

However, for a cap there is no single stochastic process for the underlying. The underlyingforward rate changes for every caplet in the cap and therefore an iteration on variances is notappropriate. In fact, the iteration should be performed on the volatilities. The followingproposition shows that the cap volatility is approximately equal to the vega weighted sum of thecaplet volatilities:

1 Note that the market prices of caps may imply inconsistent caplet volatilities; that is, caps with common

caplets may imply different volatilit ies for the same caplets.

Figure 1: Typically 'humped shape' flat cap volatility term structure. The ATM flat cap

volatility is the volatility that should be substituted into the B-S Cap formula with strike

equal to the swap rate, so that the model price equals the market prices of caps.

Maturity

Volatility

Maturity

Volatility


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Proposition 1:

Denote by (K, T) the market quote of a 'flat' implied volatility for a cap of strike K and maturityT. Denote by i(K) the implied volatility and by and v i(K) the vega of the ith caplet in the cap ofstrike K and maturity T (thus v i = Ci(i)/i ). Then

(K, T) [vi(K)i(K)]/ vi(K)

That is, the flat cap volatility (K, T) is approximately equal to the vega weighted sum of thecaplet volatilities.

Proof:

For ease of notation we drop the explicit mention of the dependence of implied volatilities on

strike and maturity. For a fixed maturity T and strike K, the flat volatility is defined as the

volatility such that the B-S cap price, (denoted C() and written as a function just of volatility) isequal to the sum of the caplets priced at the caplet constant volatilities; that is C() = Ci(i).Expanding each Ci(i) using a first order Taylor approximation about gives:

C() = Ci(i) [Ci() + (i )Ci(i)/i ]

Where the derivative Ci(i)/i = vi is evaluated at i= . This implies

(i )Ci(i)/i 0

and so the flat volatility is approximately a vega weighted sum of each caplet volatility:

wii

where wi = vi / vi so ai = 1.

The cap vega is the sum of the caplet vegas, so caplet vegas can be backed out from estimates ofthe cap vegas. They will normally decrease with maturity. Note that if one assumes that all

caplets have the same vega the caplet volatilities will be found by setting 1= (1) as before andthen iterating for 2using [1 + 2]/2 = (2) and so forth. However more generally the capletvolatilities should be backed out using the following iteration:

Set1= (1); then solve for2from [v11 + v22 ]/[[v1 + v2] = (2)

, and so forth.

The intuition behind proposition 1 may be illustrated by considering the notion of an ATM flatcap volatility term structure. The ATM cap is a sum of approximatelyATM caplets, and ATMoptions are approximately linear in volatility[but not in variance!] so it is the volatilities of thecap and the caplets that have an approximate linear relation, not their variances.

A vega weighted iteration is less likely to give negative forward rate volatilities than a varianceiteration; also note that there will be no problem with imaginary volatilities, but that imaginaryvolatilities could arise under the variance iteration when iterated variances are negative. Thefollowing simple example will illustrate this point.


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Suppose that a flat cap volatility term structure is parameterised as At[exp(-Bt)] + C, which is asimple hump shaped curve, and t = 0.5, 1, 1.5, .10 years. We consider the two cases (i) A = 0.2;B = 0.6; C = 0.2 and (ii) A = 0.2; B = 0.6; C = 0.1. Figures 2(i) and 2(ii) illustrate the cap

volatility curve and the caplet volatilities that are obtained by three different methods: thevariance iteration, the volatility iteration (which is proposition 1 with the assumption that allvegas are equal) and the vega iteration of proposition 1.

The cap volatilities are uniformly greater in case (i), and all three methods lead to real, non-negative caplet volatilities; however in case (ii) - with lower cap volatilities which are perhapsmore realistic of current market conditions in the US and Europe - the variance weighted methodgives imaginary volatilities from 4 years onwards. Note that the effect of vega weighting is toreduce the longer maturity caplet volatilities in both cases.

Figure 2: Cap volatilities and iterated caplet volatilities under the three methods

Cap Vol: A = 0.2; B = 0.6; C = 0.2

20.00%

25.00%

30.00%

35.00%

0.00%

10.00%

20.00%

30.00%

40.00%

Volatility Iteration Vega Iteration Variance Iteration

Figure 2(ii)

Figure 2(i)

Cap Vol: A = 0.2; B = 0.6; C = 0.1

10.00%

15.00%

20.00%

25.00%

0.00%

10.00%

20.00%

30.00%

Volatility Iteration Vega Iteration Variance Iteration


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Moving now to the LFR model (1) with time-varying but deterministic volatilities, it is standardto assume

i(t) = ih(t) (3)

where the parametric form h(t) that is common to all volatilities is a simple form that can capturethe hump in volatility: for example, the parametric form:

h(t) = [(a + b(T t)exp(-c(T t)) + d] (4)

was introduced by Rebonato (1999a) and has since been used by many others. The parameters a,

b, c, d define a common volatility structure, and the individual parameters i are there so thatinstantaneous volatilities can be adjusted upwards or downwards to exactly match the prices inthe cap market.

The parameters a, b, c, d and

may be calibrated to implied caplet volatilities. We have theaverage caplet volatilities implied from the market iimp and to these we need to calibrate the modelinstantaneous volatilities i (t) = i h(t). Considering the calibration objective, calibration is to(approximately) ATM option prices and ATM options are (approximately) linear in volatility.Thus an objective that minimizes the (weighted) sum of the squared pricing errors translates to anobjective that minimizes the (weighted) sum of the squared volatility differences not variancedifferences. A simple calibration objective is therefore:

The weights iare there to account for the uncertainty that surrounds different option prices (andvolatilities). Normally the weights will be a decreasing function of maturity (and tenor, in thecase of a swaption). How should these weights be chosen? The bid-offer spread is a goodindication of the uncertainty in pricing; it normally increases with maturity. Uncertainty in pricingalso translates to uncertainty in volatility through the caplet vega: high vega means that largeprice errors will induce small volatility errors and low vega means that small price errors willinduce large volatility errors. Thus, to place more weight on the more certain volatilities, set thecalibration weights ito be directly proportional to vega and inversely proportional to the bid-offer spread:

With so many parameters, of course the model will fit the market prices of caps perfectly. In factif we set

then the model will price caps perfectly. This may be used when calibrating to swaptionvolatilities, as we shall see below.

2

i

t

ii

imp

ii

i

dth(t)tmin

0

2

it

i

imp

i

i

dth(t)

t

0

2

=


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3. Calibration to the Swaption Market

Unlike caps, a swaption pay-off [max (value of swap, 0)] cannot be written as a simple sum ofoptions, so forward rate correlations as well as their volatilities will affect the value of a swaption.

We shall also need to work in a single measure, but forward rates are only drift-less under theirnatural measure. Hull and White (1999, 2000) take as numeraire the discretely re-invested moneymarket account. This gives the spot LIBOR measure, where the appropriate rate for discountingan expected cash flow at time ti+1 to time ti is the forward rate fi. This is intuitive and it leads to a(relatively) tractable specification of the drift terms, so we shall adopt this here.

2

Under the spot LIBOR measure the forward rate fi has dynamics given by

dfi(t)/ fi(t) =i(t) dt +i (t) dWiwhere

(5)

Here m(t) is the number of the accrual period. The drift becomes important when using the LFRmodel to price path dependent options, where the resolution method will, most likely, be MonteCarlo simulation, and the drift will need updating every time step. We shall return to this in thenext section, but in this section we are considering the calibration of forward rate volatilities andcorrelations to swaption volatilities, where the drift is not important.

We aim to derive an expression for the (instantaneous) volatility of the swap rate in terms of the(instantaneous) volatilities and correlations of the semi-annual forward rates underlying the swap.To do this we follow Rebonato (1998) and write SR2,3as an approximate linear function of

annual forward rates Fi below. Before following Rebonato's derivation, it should be noted thatother relationships between swap and forward rates have been developed by several authors. Hulland White (1999) write the log swap rate as a difference in logs of products of forward rates toderive an exact expression for the volatility of the swap rate. Longstaff, Santa-Clara and Schwartz(1999) use a least square regression technique to write the swap rate as an approximate linearfunction of forward rates. Jckel and Rebonato (2000) express the swap rate variance as aweighted sum of forward rate covariances and thus derive an approximation for the volatility ofthe swap rate. Andersen and Andreasen (2000) refine the relationship between forward rates andswap rates in the presence of a volatility skew.

The forward swap rate SR2,3 for a 3-year swap starting at time t2 and ending at time t5 is

illustrated in figure 3. It is given by

SR2,3 = [P2 P5]/[P3 + P4 + P5]

which, after some calculations, becomes

SR2,3 = [P3F2+ P4F3+ P5F4]/[P3 + P4 + P5]

2 See Brigo and Mercurio (2001) for the specification of the drifts under other choices of numeraire.

= +

=i

m(t)j j

jjij

ii (t)f1

(t)(t)f(t)(t)(t)


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In general for an m-year swap starting at tn (that is, an "n into m year" swap) we have:

SRn,m = w1Fn + .. + wmFn+m-1

where the weights wi = Pn+i/[Pn+1 + . + Pn+m] are assumed constant[at their current value].

Consequently the variance of the swap rate may be written as a quadratic form in the 3 x3covariance matrix of annual forward rates. Thus swap rate volatilities will be linked to annualforward rate volatilities and correlations, as shown in figure 4.

For the algebra, write w = (w1, .,wn) and let Vn,m(t)be the instantaneous covariance matrix of theforward rates Fn,Fn+m-1 for 0 < t < tn . Then SRn.m has instantaneous variance given by

n,m(t)2 = wVn,m (t)w

The average volatility during the interval [0, tn ] is (6)

and this volatility may be used with the well-known B-S type swaption pricing formula.

F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3

P2 P5

F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3

P2 P5

Figure 3: Swap rates, annual forward rates and discount bond prices

2(t)

3

(t)

4(t)

2,3

(t)

2,4

(t)

3,4

(t)

SR2,3(t)

F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3

2(t)

3

(t)

4(t)

2,3

(t)

2,4

(t)

3,4

(t)

SR2,3(t)

F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3F2

t2t0 t1 t3 t4 t5

F3 F4

SR2,3

Figure 4: Swap rates volatilities and

forward rate volatilities and correlations

dt(t)t

nt

2mn,

n

modelmn,

0

1=


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If reliable (B-S) swaption prices are available in the market, from these we can immediately

imply the average swaption volatility n, mimp . Otherwise we shall need to smooth the volatilitysurface using a 2-dimensional smoothing algorithm. It is well known that calibration results willbe sensitive to the choice of smoothing algorithm (see for example Brigo and Mercurio, 2001).

We know the weights w [from current values of discount bonds] and we want to infer theparameters ofVn,m(t) by equating the model volatilities (that are defined by equations (6)) to the

(smoothed) implied swaption volatilities n,mimp. The covariance matrix of annual forward ratesVn,m(t) contains (instantaneous) volatilities and correlations of annual rates. Indeed it may be

factored as

Vn,m(t)= Dn,m(t)n,m(t) Dn,m(t) (7)

where n,m(t)is the correlation matrix and Dn,m(t) is the diagonal matrix of the (instantaneous)

volatilities of the m annualforward rates Fn , .Fn+m-1underlying the n into m year swap.

Now we shall:

Express the annual rate volatilities (swaption volatilities) in Dn, m(t)in terms of the semi-annual rate volatilities that we know how to calibrate from the cap market, and

Derive a simple parameterization of the annual rate correlation matrix n, m(t) from a

parameterization of the semi-annual rate correlation matrix .

It is simple to express an annual forward rate F in terms of the two underlying semi-annual ratesf1 and f2:

(1 + F) = (1 + f1/2)(1 + f2 /2) so F = f1 /2 + f2/2 + f1f2 /4 (8)

Differentiating:dF = df1/2+ df2/2 + f2df1/4 + f1df2/4 = x1 (df1/f1) + x2 (df2/f2)

where x1 = f1/2 + f1f2/4 and x2 = f2/2 + f1f2/4 . Now m1= x1/F and m2= x2/F are assumed constant[at a

value given by the current estimates of forward rates]. Then the volatility of the annual rate is

expressed in terms of volatilities of semi-annual rates and their correlation 12 as:

2 m1212 + m2222 + 212m1m212

More generally, if (1 + F i) = (1 + f2i 1/2)(1 + f2i/2) then the annual rate volatility is

i2 m2i 122i 12 + m2i 22i 2 + 22i 1, 2i m2i 1m2i 2i 12i (9)?

We have expressed annual rate volatilities in terms of the semi-annual volatilities that may be

calibrated to the cap market. But since the expression involves the correlation of adjacent semi-annualforward rates, these parameters will also enter Dn,m(t).

Now to our second objective, which is to derive a simple parameterization of the annual rate

correlation matrix n,m(t) from a parameterization of the semi-annual rate correlation matrix. Someresearchers [e.g. Brigo and Mercurio (2001), Rebonato and Joshi (2001), Jckel (2002)] assume

that the annual rate correlation matrix is n,m= {i,j} where


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i,j =exp( i j) (10)

Variants of this, such as i,j =exp( ij) are discussed in Brigo (2001) but these

parameterizations do not lend themselves to a straightforward relationship between the semi-

annual rate correlations and the annual correlations. The following proposition shows that if weuse the parameterization (10) for semi-annual forward rates then there is a simple relationshipbetween semi-annual correlations and annual correlations.

Write = exp( ), so that (10) may be written i,j = i j

and we observe that the correlation

matrix is a 'circulant' matrix (written out in full below). Then we have:

Proposition 2:

Suppose the 2m x2m correlation matrix of semi-annual forward rates underlying an n into m yearswaption, denotedn, m _s emi , is given by the circulant correlation matrix:

Then the mxm correlation matrix of the annual forward rates underlying the same swaption,

denotedn, m is approximated by the correlation matrix:

where = (1+)/2.Proof:

Let be the correlation between any two adjacent semi-annual forward rates, and we assume that thecorrelation between the two semi-annual rates fi and fj is

ij . With this 'circulant' structure forsemi-annual rates the correlation between two adjacent annual forward rates is approximately

(1+)/2and more generally, the correlation between two annual forward rates F i and Fj isapproximately 2|i-j |-1(1+) / 2. The approximation is based on the assumptions that (i) annual ratesare simply the average of two semi-annual rates and (ii) semi-annual volatilities are equal.

Under these assumptions

1

111

12

322

222

122

......

.

.

.

.

.

......

.....

.......

m

m

m

m

1

11

1

22

422

322

222

......

.

.

.

.

.

......

.....

.......

)m(

)m(

)m(

)m(


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Fi = (f2i 1 + f2i )/2,

V(Fi) = 2[1 + ] / 2and

Cov(Fi, Fj) = 2 2|i-j|-1[1 + ] 2 / 4

so the result follows.

An interesting parameterization of Schoenmakers and Coffey (2000) allows for the empirical

observation that the correlation of adjacent forward rates tends to increase with maturity.3

Schoenmakers and Coffey's semi-parametric form for a correlation matrix of M forward rates is based

on and increasing sequence of M real numbers. The next proposition shows that the increasing

correlation of adjacent forward rates with maturity may be captured by a two parameter matrix for

semi-annual rates, where again there is a simple extension from the semi-annual correlation matrix to

a correlation matrix for annual rates that has the same two parameters:

Proposition 3:

Suppose the 2m x2m correlation matrix of semi-annual forward rates underlying an n into m yearswaption, denotedn, m _semi , is given by the two parameter correlation matrix (ij) where:

ij = (j-1)ij

i < j, 0 < < 1 and < 2m1 < 1.

Then the mxm correlation matrix of the annual forward rates underlying the same swaption,

denotedn, m is approximated by the correlation matrix (ij) where:

ij =)(1)(12

))()((1)(

1)2(j1)2(i

)j()j(1ji21)2(j

++

++1+

21212

Note also that an alternative parameterization of semi-annual rate correlations as ij = j-1ijleads to the approximated annual correlation matrix (ij) where:

ij =)(1)(12

)/)(1(1

1)2(j1)2(i

12j1ji2

++

++

Proof:

The results follow after some algebra using similar assumptions and methods as in proposition 2. For

example

V(Fi) = 2[1 + 2(i -1)] / 2and with the first parameterization:

Cov(Fi, Fj) = 2 (2(j -1) ) 2|i-j |-1[1 + (1+ )2(j -1) + (2(j -1))2] / 4

3Note that any empirical observation on forward rates must necessarily be based on an assumed historical

data period and an assumed method of yield curve interpolation. See Alexander and Lvov (2002) for more

information about the empirical behaviour of correlations in forward rates.


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Turning now to the calibration objective corresponding to the parsimonious parameterizations ofpropositions 2 or 3, note that in the covariance matrix of annual forward rates given by (7), the

correlation matrix n,m(t)is defined in terms of a single parameter [proposition 2] or it isdefined in terms of two parameters and [proposition 3]. Also, by (9), the diagonal matrix ofthe annual forward rate volatilities Dn,m(t) is defined in terms of the semi-annual forward ratevolatilities and the same parameters of the semi-annual correlation matrix (i.e. or, underproposition 3, and ).4

ThereforeVn,m(t)is determined bythe parameters: a, b, c, d,, and .Note that we maysuppose that is itself parameterized in terms of a, b, c, and d, serving only to equate model andmarket prices for caps as described above. Note also that we could assume time-varying

parametric forms for and . Now we can state that the objective for the calibration is to chooseparameters a, b, c, d,and and (or parameters of(t) and (t))to:

where the model volatilities are given by (6) and the implied volatilities are from the market.Again, the weights should be directly proportional to the swaption vega and inverselyproportional to the bid-ask spread.

4. Calibration for Pricing Interest Rate Options

The LFR model may be used to price path dependent options such as Bermudan swaptions. No

analytic pricing formulae exist and resolution methods are used. Some preferred methods based onMonte Carlo simulation are described in Jckel (2002). Whichever method is chosen the drifts that are

induced by the change of numeraire will be important, and they will need updating with the current

value of the instantaneous volatilities and correlations at every time step.

For example, consider the drift term for f3. By (5), and dropping the time variable temporarily, toshorten notation: when t0 < t < t1:

When t1 < t < t2:

When t2 < t < t3:

4 Note that 2i 1, 2i 2 = under proposition 2 and 2i 1, 2i 2 = 2(i 1) under proposition 3.

( ) 2

mn,

modelmn,

impmn,mn,min

][ 3

33

2

2232

1

11313

3

1=

333 +1

++1+

+1=

+1

= f

f

f

f

f

f

f

f

j j

jjj

][ 3

33

2

22323

3

2=

333 +1

++1=

+1

= f

f

f

f

f

f

j j

jjj

][3

3333 +1

=f

f


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Thus the instantaneous drift term in the ith forward rate fi depends on the forward rates oflower

maturity fi-1 , fi-2, etc., but only if these are still random variables at the time that the drift is

estimated. For example, see figure 5. In the accrual period t1 to t2, f1is known. Thus only the

volatilities and the correlation off2and f3will affect the drift of f3in that period.

The correlation matrix that one needs to consider for the calibration of the drift term thusdecreases in dimension (by 1) as every payment /reset date passes. This is not the onlycomplication for the simulation. If m is large the dimension of the simulation will be large andcomputationally burdensome. However it is possible to reduce dimensions by reducing the rankof the correlation matrix. A review of rank reduction methods is given in Brigo (2001).

Following Rebonato (1999c) and Hull and White (1999, 2000), we now consider a naturalmethod for rank reduction, that is to use an orthogonal transformation of the correlated Brownianmotions in (5). The forward rate dynamics are express in terms of three uncorrelatedstochasticprocesses that are common to all forward rates:

dfi(t)/ fi(t) = i(t) dt + i,1(t) dZ1 + i,2(t) dZ2 + i,3(t) dZ3

where dZ1, dZ2, dZ3 are uncorrelated Brownian motions and

i(t) dWi = i,1(t) dZ1+,2(t) dZ2 + i,3(t) dZ3

From this it follows that:

i(t) = [i,1(t)2+ i,2(t)2 + i,3(t)2]and

ij (t) = [i,1(t) j,1(t)+i,2(t) j,2(t)+i,3(t) j,3(t)]/i(t) j(t)

Thus the forward rate volatilities and correlations are completely determined by three volatilitycomponents for each forward rate, i,1(t), i,2(t) andi,3(t).

Suppose (for the moment) that the implied forward rate volatilities i(t) have been calibratedfrom cap prices. The volatility components 1 , 2 and 3 could then be determined from thespectral decomposition of the correlation matrix of the m forward rates

5underlying the derivative,

that is

n,m(t)= W(t)W

5Or more precisely, the correlation matrix of thefirst differences in the logarithms of the forward rates fn , .fn+m-1

(as is standard for lognormal diffusions).

Figure 5: Correlations of forward rates and the drift

t0 t1 t2 t3

m(t) = 1 m(t) = 2 m(t) = 3f3

t0 t1 t2 t3t0 t1 t2 t3

m(t) = 1 m(t) = 2 m(t) = 3f3


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where(t) is the (time-varying) diagonal matrix of eigenvalues and W is the m x m matrix ofeigenvectors (assumed constant).

To see how this spectral decomposition does indeed determine the volatility components, denote

by 1(t), 2(t), 3(t), the three largest eigenvalues of n,m(t)and denote their eigenvectors by1,2, 3. Set

M(t)2 =i(t)2 /Vi (t)where

Vi (t) = i121 (t) + i222 (t) + i323 (t)V(ln fi)Then set

i,1(t) = M(t) i11 (t)i,2(t) = M(t) i22 (t)

i,3(t) = M(t) i33 (t)So thati(t)2 = i,1(t)2+i,2(t)2 +i,3(t)2

as required.

A number of researchers, notably Rebonato (1999a), Rebonato and Joshi (2001) Hull and White(1999, 2000) and Logstaff, Santa-Clara and Schwartz (1999) have advocated the use of historicaldata for the calibration of the volatility components. The calibration is then based on a principalcomponent analysis (PCA) of historical data on forward rates.

6The general finding that the first

eigenvector is relatively flat, the second eigenvector is monotonically decreasing (or increasing)and the third eigenvector is convex - all as a function of the maturity of the forward rate - comesas no surprise. Forward rates are a highly correlated term structure and the interpretation of

eigenvectors as trend, tilt and curvature components is one of the stylised facts of all termstructures, particularly when they are highly correlated (see Alexander, 2001).

Unfortunately, a LFR model calibration that is based on any historical data will depend verymuch on which forward rates are used and the historical period chosen. Should one use theforward rates obtained from and ordinary cubic spline interpolation (as in Longstaff, Santa-Claraand Schwartz, 1999), a B-spline interpolation for the yield curve, or those obtained from theNelson-Siegal or Svensson parametric forms for the yield curve? And should one use daily dataover the past year, weekly data over the past three years, or what ? It must be stressed thatwhen historical data on forward rates are used, the calibration results will be critically dependenton both the choice of yield curve model and the historical period chosen. See Alexander andLvov (2002) for further details and empirical results.

An alternative approach is to parameterize the eigenvectors and then calibrate these and theeigenvalues using current market implied swaption volatilities.

7The eigenvalues are simply the

dot product of the forward rate volatility vector with the corresponding eigenvector:

6Note that Hull and White (2000) use the first difference in forward rates (not the first difference in their

logarithms) for the PCA.7 Rebonato (2002) does not advise one to attempt simultaneous calibration to both swaption and caplet

volatilities. Forcing the model to fit market prices of caps and swaptions is not necessary, since apparent

arbitrages arising from the model mis -pricing of either instrument would be too risky to trade upon, and

the simultaneous calibration constraint is too restrictive.


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1(t) = 11 n(t)2+ . + m1 n+m-1(t)2 = 1(t)2

and similarly 2(t)= 2(t)2 ; 3(t)= 3(t)2

Thus when the (instantaneous) forward rate volatilities are parameterized by the 'hump' (4), thetime dependence in the eigenvalues is a function of this 'hump' and the parameters of theeigenvectors. We assume the standard form for the eigenvectors of a term structure correlationmatrix, that is:

1 = (1 , 1 , .. ,1)

the first eigenvector in W is constant, and

2 = (2 + 2, 2 + 22, 2 + 32 ,.. ,2 + m2)

the second eigenvector in W is linear: i2 = 2 + i2and

3 = (3 + 3 + 3, 3 + 23 + 43 ,.. ,2 + m3 + m23)

the third eigenvector in W is quadratic: i3 = 3 + i 3 + i23

For reasons mentioned above we do not employ historical values for the eigenvectors. Howeverthe historical analysis in Alexander and Lvov (2002) demonstrates that a commonparameterization of the eigenvectors for all swaptions with underlying swaps of the same tenorshould be taken. The implementation of the common principal component model of Flury (1988)yields results that are robust to the various yield curve models used to compute the forward ratesand to the choice of historic data period. They indicate that we should define afamily of m x m

forward rate correlation matrices

n,m(t) = Amn(t) Am

wheren(t)is the 3 x 3 matrix of the three largest eigenvalues ofn,m(t)and Amis the (constant)m x 3 matrix of eigenvectors with columns 1, 23 .

To summarize and formulate the calibration objective, under the orthogonal transformation of the

LFR model, the calibration for the parameters (1 , 2 , 2, 3 , 3,3) of the eigenvectors1,2and3should be performed using market prices ofall swaptions of tenor m.The commonprincipal component model of Flury (1988) should be employed to calibrate commoneigenvectors to all swaps of the same tenor. That is, Amis the same for all n. Writing

w*(t) = Dn,m(t)w

for short, the model volatility (6) hasinstantaneous variance given by

and the average volatility in the calibration objective is

(t)(t)(t)'(t) mnm2

mn,

**'wAAw =


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As before, the calibration objective will be a vega/(bid-ask spread) weighted sum of squaredvolatility differences, and the calibration problem is:

5. Conclusion

In the lognormal forward rate LIBOR model for interest rate option pricing, forward ratevolatilities and correlations may be calibrated to a (smoothed) swaption volatility surface.Following Rebonato (1998) we write swap rates as a linear sum of annual forward rates and linkannual forward rates with semi-annual forward rates. We consider two simple parametric formsfor the semi-annual forward rate correlations: a single parameter 'circulant' form, which is basedon the parametric form for annual rate correlations that has been advocated by Brigo andMercurio (2001), Rebonato and Joshi (2001), Jckel (2002) and others, and a two parameter formthat allows the adjacent semi-annual forward rate correlations to increase with maturity, as inSchoenmakers and Coffey (2000).

We have shown that both these parametric forms imply a simple form for the annual forward ratecorrelations that depend on the same parameter(s). This construction allows one to define

calibration objectives that are used with swaption volatilities, but whose parameters are those ofthe semi-annual rate volatilities and correlations. We have also proposed a vega weightediteration for computing the semi-annual rate volatilities from the cap market volatility surfacewhich mitigates the problem of negative (or even imaginary) volatilities that may be encounteredwhen calibrating to the swaption market.

We often need to use numerical resolution (e.g. simulation) for the forward rate dynamics, wherethe drift terms that are induced by the change to a single measure will become important. Sincethese change at every time step, the computations are burdensome and it may be desirable toreduce dimensions in the numerical method. For this, an orthogonal transformation is describedthat is based on a common principal component analysis (PCA) of the forward rates. Severalresearchers have advocated the use of PCA in conjunction with historic data, for example, forcalibrating the eigenvectors of the forward rate correlation matrix. However the parsimoniousparametric form for common eigenvectors that is proposed here can be calibrated to the currentmarket data without resorting to (potentially very unreliable) historical data on the unobservableforward rates. This form, which is based on the common PCA model of Flury (1988), should beused to calibrate common eigenvectors to all swaptions of the same tenor.

AcknowledgementsI would like to thank Damiano Brigo and Fabio Mercurio of Banca IMI, Milan, Riccardo Rebonato of the

Royal Bank of Scotland, London, Peter Jckel of Commerz Bank, London and my husband Jacques Pezier,

all of whom provided very helpful comments and suggestions on preliminary drafts of this paper.

( ) 2

mn,

model

mn,

imp

mn,mn,min

dt(t)t

nt

2mn,

n

modelmn,

0

1=


18/18



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calibrating cap_floors volatilities

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