Choice of Interest Rate Term StructureModels for Pricing and

Hedging Bermudan Swaptions -AnALM Perspective

Zhenke GuanBing Gan

Aisha KhanSer-Huang Poon ∗

January 15, 2008

∗Zhenke Guan (zhenke.guan@postgrad.manchester.ac.uk) and Ser-Huang Poon (ser-huang.poon@mbs.ac.uk) are at Manchester Business School, Crawford House, University ofManchester, Oxford Road, Manchester M13 9PL, UK. Tel: +44 161 275 0431, Fax: +44 161275 4023. We would like to thank Peter van der Wal, Vincent van Bergen and Jan Remmerswaalfrom ABN AMRO for manu useful suggestions and data, Michael Croucher for NAG softwaresupport, and Dick Stapleton and Marti Subrahmanyam for many helpful advice.

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Abstract

Asset Liability Management (ALM) departments in big financial institutionshave the problem of choosing the “right ”interest rate model for managing theirbooks, which typically consists of assets and liabilities denominated in differentcurrencies. Should a bank adopt a single term structure (TS) model or is itappropriate to allow different branches use different models for ALM? Each TSmodel is based on different measure of interest rates (e.g. spot, forward or swaprates) with a different assumed dynamics for the key rate. Is it appropriate to usea single TS model for ALM when the bank is subject to interest rate exposurein very different types of economy? This paper aims to address these questionsby comparing , empirically, four one-factor TS models, viz. Hull-White, Black-Karasinski, Swap Market Model and Libor Market Model, for pricing and hedginglong term Bermudan swaptions which resemble mortgage loans in banks’ book.In contrast to pricing need in the front office, a single-factor model is preferredfor ALM purpose because of its focus in long term risk management. Managinglong term interest rate risk using a multi-factor models is cumbersome and themulti-factors could potentially introduce more input errors over long horizon.The test involved calibrating the four TS models to European swaption pricesfor EURO and USD over the period February 2005 to September 2007. Thecalibrated models are then used to price and hedge 11-year Bermudan swaptionwith a 1-year holding period. The empirical results show that, the calibratedparameters of all four models are very stable and their pricing error is small. Thepricing performance of four models is indistinguishable in both currencies. Thehedging P&L from the four models is similar for the Euro market. For the USDmarket, the short rate models preforms marginally better than SMM and LMM.The performances of HW and BK are indistinguishable.

1 Introduction

In this paper, we study the choice of interest rate term structure models for asset-liability management in a global bank. In particular, we compare the performanceof Hull-White, Black-Karasinski, Swap market model and Libor Market Modelfor hedging a 11-year Bermudan swaption on an annual basis from February2005 to September 2006. The 11-year Bermudan swaption is chosen because itresembles a loan portfolio with early redemption feature, an important productfor most banks. Unlike the short-term pricing problem, the one-factor modelis often preferred for the longer term ALM purpose because of its simplicity.For long horizon hedging, the multi-factor model could produce more noise as itrequires more parameter input. Also, for this ALM study, we do not considerthe “smile ”effect for the same reason. Pricing performance measures a model’scapability of capturing the current term structure and market prices of interestrate sensitive instruments. Pricing performance can always be improved, in analmost sure sense, by adding more explanatory variables and complexities to thedynamics. However, pricing performance alone cannot reflect the model’s abilityin capturing the true term structure dynamics. To assess the appropriateness ofmodel dynamics, one has to study model forecasting and hedging performance.

The last few decades have seen the development of a great variety of interestrate models for estimating prices and risk sensitivities of interest rate derivatives.These models can be broadly divided into short rate, forward rate and marketmodels. The class of short-rate models, among others, includes Vasicek (1977),Hull and White (1990), and Black and Karasinski (1991). A generalized frame-work for arbitrage free forward-rate modelling originates from the work of Heath,Jarrow and Morton (HJM, 1992). Market models are a class of models within theHJM framework that model the evolution of rates that are directly observable inthe market. Brace, Gatarek and Musiela (1997) firstly introduced the arbitragefree process for forward libor rates which lead to the Libor Market Model (LMM,also known as BGM model) under the HJM framework. Miltersen, Sandmannand Sondermann (1997) derive a unified interest rates term structure which givesclosed form solution for caplet under the assumption of log-normally distributedinterest rates. Different methods of parameterization and calibration of LMMare examined in Brigo, Mercurio and Mrini (2003). The term structure of swaprates is firstly developed in Jamshidian (1997) which is known as Swap Mar-ket Model (SMM). Jamshidian (1997) also proposed the concept of co-terminalmarket model at the earliest. Base on the research of the above papers, Galluc-cio, Huang, Ly and Scaillet (2007) use graph theory to classify the “admissible”market models into three subclasses named co-initial, co-sliding and co-terminal.Among other things, they show that the LMM is the only admissible model forswaps of a co-sliding type.

All these models have their own strengths and weaknesses. Short-rate modelsare tractable, easy to understand and implement but do not provide complete

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freedom in choosing the volatility structure. The HJM framework is popular dueto its flexibility in terms of the number of factors that can be used and it permitsdifferent volatility structures for forward rates with different maturities. Despitethese attractions the key problem associated with the HJM is that instantaneousforward rates are not directly observable in the market and hence models underthis framework are difficult to calibrate. The market models overcome these limi-tations but are complex and computationally expensive when compared with theshort rate models. The question is whether one should use the simple model likeGaussian HW model or the lognormal BK model, or should one use complicatedmodel like SMM and LMM. In this paper, we look at the difference between thefour interest rate models for longer term asset liability management in two inter-est rate regimes (i.e. Euro and USD). The choice of HW and BK is simple: atthe time of writing, they are the most important and popular short rate modelsused by the industry; the choice of swap market model and Libor market modelis also simple because they are new or more recent and are widely used amongpractitioners.

More recently, a number of previous studies have examined various term struc-ture models for pricing and hedging interest rate derivatives. A large part of theterm structure literature has focused on the performance of term structure mod-els for bond pricing (see, for example, Dai and Singleton 2000, Pearson and Sun1994). Andersen and Andreasen (2001) use mean-reverting Gaussian model andlognormal Libor Market Model for pricing Bermudan swaption. They find thatfor both models, Bermudan swaption prices change only moderately when thenumber of factors in the underlying interest rate model is increased from one totwo. Pietersz and Pelsser (2005) compare single factor Markov-functional andmulti-factor market models for hedging Bermudan swaptions. They find that onmost trade days the Bermudan swaption prices estimated from these two mod-els are similar and co-move together. Their results also show that delta anddelta-vega hedging performances of both models are comparable. Gupta andSubrahmanyam (2005) compare various one- and two-factor models based on theout-of-sample pricing performance, and the models’ ability to delta-hedge capsand floors. They find that one-factor short rate models with time varying pa-rameters and the two-factor model produce similar size pricing errors. But interms of hedging caps and floors, the two-factor models are more effective. Forboth pricing and hedging of caps and floors, the BK model is better than HWmodel. With regard to both pricing and hedging, their results are in line withthose obtained by Fan, Gupta and Ritchken (2006). Driessen, Klaassen and Me-lenberg (2003) use a range of term structure models to price and hedge caps and(European) swaptions. Regarding hedging, their results show that if the numberof hedge instruments is equal to the number of factors then multi-factor modelsoutperform one-factor models in hedging caps and swaptions. However, whena large set of hedge instruments is used then both one-factor and multi-factormodels perform well in terms of delta hedging of European swaptions. Fan,

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Gupta and Ritchken (2006) show that one- and two-factor models are as capa-ble of accurately price swaptions as higher order multi-factor models. However,regarding hedging , their results show that multi-factor models are significantlybetter than one-factor models. In addition, their results for swaptions show thatusing multiple instruments within a lower order model does not improve hedgingperformance. These results differ from Driessen et al (2003) and Andersen andAndreasen (2001), who lend support for the one-factor models. This paper con-tributes to the literature and the existing empirical test in: i) By comparing awide choice of models including the latest SMM, ii) by hedging the more compli-cated Bermudan swaption (instead of caplet or floorlet) and iii) by focusing onALM perspective.

Specifically, the objective is to select a interest rate term structure modelwhich best price and delta hedge a 11-year Bermudan swaption in differentcurrencies-EUR, USD. On the contract starting date, we constructed a hedgedportfolio for the 11-year Bermudan swaption; one year later, we get the new valueof the hedged portfolio. The hedging profit and loss is calculated as the differencebetween the value of the options and portfolios. The data are from Datastreamand ABN AMRO.

This remaining of the paper is organized as follows. Next section brieflydescribes the models we use and the implementation design. Data is described inSection 3. Section 4 deals with calibration procedures of all models. Calibrationinstruments and calibration algorithm are discussed. In Section 5, Bermudanswaption prices are compared within four models. Hedging performance andProfit and Loss are reported in Section 6 and finally, we conclude in Section 7.

2 Term Structure Models

We are interested in comparing the pricing and hedging performance of fourwidely used models-HW Model, BK model, Swap market model and Libor Mar-ket model. In this section, we briefly describe the four models and there charac-teristics.

There are numerous models for pricing interest rate derivatives,which, broadlyspeaking, can be divided into two categories: sport rate models and forward ratemodels. Both HW and BK model are short rate models which specify the behaviorof short-term interest rate, r. Short rate model, as it evolved in the literature,can be classified into equilibrium and no-arbitrage models. Equilibrium modelsare also referred to as “endogenous term structure models” because the termstructure of interest rates is an output of, rather than an input to, these models.If we have the initial zero-coupon bond curve from the market, the parameters ofthe equilibrium models are chosen such that the models produce a zero-couponbond curve as close as possible to the one observed in the market. Vasicek (1977)is the earliest and most famous general equilibrium short rate model. Since the

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equilibrium models cannot reproduce exactly the initial yield curve, most tradershave very little confidence in using these models to price complex interest ratederivatives. Hence, no-arbitrage models designed to exactly match the currentterm structure of interest rates are more popular. It is not possible to arbitrageusing simple interest rate instruments in this type of no-arbitrage models. Twoof the most important no-arbitrage short-rate models are the Hull-White model(1990) and the Black-Karasinski (1991) model.

2.1 The Hull-White and BK Model

Hull and White (1990) propose an extension of the Vasicek model so that it canbe consistent with both the current term structure of spot interest rates andthe current term structure of interest-rate volatilities. According to the Hull-White model, also referred to as the extended-Vasicek model, the instantaneousshort-rate process evolves under the risk-neutral measure as follows:

drt = [θt − atrt]dt + σtdz, (1)

where θ, a and σ are deterministic functions of time. The function θt is chosen sothat the model fits the initial term structure of interest rates. The other two time-varying parameters, at and σt, enable the model to be fitted to the initial volatilityof all zero coupon rates and to the volatility of short rate at all future times.1

Hull and White (1994) note, however, that while at and σt allow the model tobe fitted to the volatility structure at time zero, the resulting volatility termstructure could be non-stationary in the sense that the future volatility structureimplied by the model can be quite different from the volatility structure today.On the contrary, when these two parameters are kept constant, the volatilitystructure stays stationary but model’s consistency with market prices of e.g. capsor swaptions can suffer considerably. Thus there is a trade-off between tighter fitand model stationarity.

In HW model the distribution of short rate is Gaussian. Gaussian distributionleads to a theoretical possibility of short rate going below zero. Like the Vasicekmodel the possibility of a negative interest rate is a major drawback of this model.

A model that addresses the negative interest rate issue of the Hull-Whitemodel is the Black and Karasinski (1991) model. In this model, the risk neutralprocess for logarithm of the instantaneous spot rate, ln rt is

d ln rt = [θt − at ln rt] dt + σtdz, (2)

where r0 (at t = 0) is a positive constant, θt, at and σt are deterministic functionsof time. Equation (2) shows that the instantaneous short rate evolves as the

1The initial volatility of all rates depends on σ(0) and a(t). The volatility of short rate atfuture times is determined by σ(t) (Hull and White 1996, p.9).

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exponential of an Ornstein-Uhlenbeck process with time-dependent coefficients.The function θt is chosen so that the model fits the initial term structure ofinterest rates. Functions at and σt are chosen so that the model can be fitted tothe chosen market volatility.

Both HW and BK model are implemented by constructing a recombiningtrinomial lattice for the short-term interest rate.

2.2 Libor Market Model

In the next two subsections, we are going to describe the two most advancedmodel in interest rates-Swap Market Model and Libor Market Model. They haveattracted a lot of interests from both academic and practitioners mostly becauseof their characteristics - they are consistent with market standard - using Blackformula to price (European) swaption and caplet.

Let Li(t), i = 1, 2, ...N as the forward Libor rates. δ as the time intervalbetween two Libor rates. The dynamics of forward Libor rate under the Tn+1

terminal measure is by

dLi(t)

Li(t)= −

N∑j=i+1

δjLj(t)σij(t)

1 + δjLj(t)dt + σi(t) · dWQT

n+1(t) (3)

for 0 ≤ t ≤ min (Ti, Tn+1) , i = 1, · · · , n, · · · , N. Note that Ln(t) is driftless underthe Tn+1 terminal measure when Bn+1 (t) is used as the numeraire.

2.2.1 Model Implementation

Unlike caplet valuation, to implement LMM for pricing swaptions, it is neces-sary to simulate all forward rates under one measure. This means we need thedynamics (the drift term) for all Ln(t) under the terminal measure.

Here, we use a Euler discretization scheme for the forward rate simulation.While Glasserman and Zhao (1999) note that forward prices of bonds are notarbitrage free under the Euler discretization scheme, we adopt the Euler schemenevertheless due to its simplicity. The approximation error is not severe as longas the step size is small. Following Glasserman and Zhao (1999), the discretizedversion of the LMM dynamics under the TN+1 terminal measure in (3) is, for atime step size h,

Li((k + 1)h) = Li(kh) exp

[(µi(kh)− 1

2σi(kh)2)h + σi(kh)

√hεj+1

],

where

µi(kh) = −N∑

j=i+1

δLj(kh)σij(kh)

1 + δLj(kh).

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Here, we have chosen n = N and TN+1 as the terminal measure. Note that underthe TN+1 terminal measure, when i = N , the drift of Li = LN is zero and LN islog-normally distributed.

Predictor-corrector scheme is used here for more accuracy(See Joshi andStancy 2005 for more details).

2.3 Swap Market Model

Similar as Libor Market Model, swap market model has the same good property.However, it is not as intensively studied as LMM. Recently, people begin to realizethe usefulness of co-terminal swap market model. Given the tenor structure, aco-terminal Swap Market Model refers to a model which assigns the arbitrage freedynamic to a set of forward swap rates that have different swap starting date Tn

(n = 1, · · · ,M − 1) but conclude on the same maturity date TM . One importantfeature about the co-terminal swaption is that it is internally consistent with aBermudan swaption that gives the holder the right to enter into a swap at eachreset date during period T1 to TM−1 with TM being the terminal maturity of theunderlying swap. When considering at each tenor date whether or not to exercisethe option to enter into a swap contract, the holder need to consider the forwardswap rate dynamics from that tenor date till final maturity, which is actuallydriven by the volatility prevailing at that time. The advantage of co-terminalSMM over other market models in pricing Bermudan swaptions has already beennoted and discussed in Jamshidian (1997) and Galluccio et al (2007).

In Galluccio et al (2007) the co-terminal SMM is defined by introducing acollection of mutually equivalent probability measures and a family of Brownianmotions such that for any the forward swap rate satisfies a SDE for all . Here wefollow their approach.

2.3.1 Drift of Co-terminal SMM under the Terminal Measure

Let Cn,m (t) =∑m

i=n+1 τBi (t) represent the value of the annuity from Tn+1 toTm. The co-terminal forward swap rates satisfy the following general SDE underthe terminal measure:

dSn,m (t) = Sn,m (t) σn,m (t) dWM−1,Mt + drift for n = 1, · · · , M − 1 (4)

where WM−1,Mt is the Wiener process under the terminal measure QM−1,M .

We follow Joshi and Liesch (2006), who recommend using the cross-variationin assessing impact of changing numeraire on a drift. The drift of a forward swaprate Sn,M (t), n = 1, · · · ,M − 1 under the terminal measure QM−1,M is given by:

EM−1,M [dSn,M (t)] = µn,M = −CM−1,M (t)

Cn,M (t)

⟨Bn (t)−Bm (t)

Cn,M (t),

Cn,M (t)

CM−1,M (t)

⟩

= −CM−1,M (t)

Cn,M (t)

⟨Sn,M (t) ,

Cn,M (t)

CM−1,M (t)

⟩(5)

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Note that when n = M − 1, µM−1 = 0 because 〈Sn,M (t) , 1〉 = 0.In Joshi and Liesch (2006), the Sn,M (t) term inside the square brackets is

simplified into independent Wiener process Wk, which explains why their driftterm does not contain the correlation terms ρn,m. Although for one factor model,the correlation terms ρn,m = 1 can be ignored, it cannot be omitted when thedimension of Wk is two or more. Expanding the cross-variation term in equation(5), the drift becomes a complicated function of numeraires and cross-variationof two forward swap rates. By induction, the general form of drift under terminalmeasure can also be written as:

µn,M =M−2∑i=n

(τi+1

Ci+1,M

CM−1,M

ρn,i+1Sn,Mσn,MSi+1,Mσi+1,Mij=n+1 (1 + τjSj,M)

)(6)

Details of the derivation are given in the working paper by Gan et al. However,the above equation is not an explicit function of Sn,M (t) for n = 1, · · · , M − 1,because the Cn,M term is derived from a set of Si,M (t) for n ≤ i ≤ M − 1. Thisdrift term is complicated in appearance but not time-consuming in computation.

Similar as Libor market model, SMM is also implemented using Monte Carlosimulation, in the interest of computational efficiency. Predictor-corrector driftapproximation is employed for accuracy.

3 Research Design and Data

Let t denote a particular month in the period from February 2005 to September2007. The procedure for calibrating, hedging and unwinding a 10× 1 Bermudanswaption are as follows:

(i) At month t, the interest rate model is calibrated to 10 ATM co-terminalEuropean swaptions underlying the 10 × 1 ATM Bermudan swaption byminimising the root mean square of the pricing errors.

(ii) The calibrated model from (i) is then used to price the 10×1 ATM Bermu-dan swaption at t, and to calculate the hedge ratios using, as hedge instru-ments, 1-year, 5-year and 11-year swaps, all with zero initial swap value attime t.

(iii) A delta hedged portfolio is formed by minimising the amount of delta mis-match.

(iv) At time t + 1 (i.e. one year later), the short rate model is calibrated to9 co-terminal ATM European swaptions (9× 1, 8× 2, ...) underlying theBermudan swaption from (ii) which is now 9× 1.

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(v) The calibrated model from (iv) is used to price the 9×1 Bermudan swaption,and the time t+1 yield curve is used to price the three swaps in (ii), whichare now 0 year, 4 years and 10 years to maturity.

(vi) The profit and loss is calculated for the delta hedged portfolio formed at tand unwound at t + 1.

(vii) Steps (i) to (vi) are repeated every month for t + 1, t + 2, · · · till T − 1where T is the last month of the sample period.

To perform the valuation and hedging analyses described above, the followingdata sets are collected from Datastream:

(a) Monthly prices (quoted in Black implied volatility) of ATM European swap-tions in Euro and USD. Two sets of implied volatilities were collected: fromFebruary 2005 to September 2006, prices of co-terminal ATM Europeanswaptions underlying the 10 × 1 Bermudan swaption, and from February2006 to September 2007, prices of co-terminal ATM European swaptionsunderlying the 9× 1 Bermudan swaption. These prices are quoted in Blackimplied volatility. The implied volatility matrix downloaded has a numberof missing entries especially in the earlier part of the sample period. Themissing entries were filled in using log-linear interpolation following Brigoand Morini (2005, p 9, 24 and 25).

(b) Annual yields, R0,t, for maturities up to 11 years are downloaded fromDatastream.2 All other yields needed for producing the trinomial tree arecalculated using linear interpolation. These annual yields are converted tocontinuously compounded yields, r0,t = ln(1 + R0,t).

(c) Monthly data of the annual yield curve for the period January 1999 to July2007, i.e. total 103 observations are downloaded for Euro and USD. Thisdata was transformed into discrete forward rates (as in LMM) for use inthe principal component analysis. Also, Swap rate can be derived from theforward libor rates.

(iv) Monthly data of 1-month yield for the period January 2000 to July 2007was downloaded for estimating the “mean-reversion” parameter. In theimplementation, a time step (∆t) of 0.1 year is used for constructing thetrinomial tress, which means that the rates on nodes of the tree are con-tinuously compounded ∆t-period rates. Here we have used the one-monthyield as a proxy for the first 0.1-year short rate.

2The time step (∆t) for the trinomial tress in the C++ program is 0.1 year. The C++program linearly interpolates all the required yields.

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4 Model Calibration

Before we price Bermudan swaptions, we need to find the parameters that give thecorrect European swaptions to avoid any arbitrage opportunities. This sectiondiscusses different calibration method for each model.

4.1 HW/BK Model Calibration

In this section, we discuss HW/BK model calibration. As we know, for bothmodels, we have the ”mean reversion rate”, time-dependent volatility σt and theconstant rate of mean reversion a.

The “mean reversion rate” parameter for the two models has been extractedfrom historical interest rate data. Practitioners and econometricians often usehistorical data for inferring the “rate of mean reversion” (Bertrand Candelonand Luis A. Gil-Alana (2006)). In this study we calibrate the two models withtime-dependent short rate volatility, σ(t), and constant rate of mean reversion,a.

4.1.1 Parameterizations of σ(t)

There can be different ways to parameterise the time-dependent parameter: it canbe piecewise linear, piecewise constant or some other parametric functional formcan be chosen. In this study, the volatility parameter has been parameterisedas follows: The last payoff for the all the instruments that need to be pricedin this study would be at 11 years point of their life. We explicitly decidedthese three points, because values of σ(t), t = 0, 3, 11 on these three pointscan be interpreted as instantaneous, short term and long term volatility. Thesethree volatility parameters are estimated through the calibration process. Thevolatilities for the time periods in between these points are linearly interpolated.

4.1.2 Choosing calibration instruments

A common financial practice is to calibrate the interest rate model using theinstruments that are as similar as possible to the instrument being valued andhedged, rather than attempting to fit the models to all available market data.In this study the problem at hand is to price and hedge 10 × 1 Bermudan. Forthis 10 × 1 Bermudan swaption the most relevant calibrating instruments arethe 1 × 10, 2 × 9, 3 × 8, · · · ., 10 × 1 co-terminal European swaptions (A n ×mswaption is an n-year European option to enter into a swap lasting for m yearsafter option maturity.). The intuition behind this strategy is that the model whenused with the parameters that minimize the pricing error of these individualinstruments would price any related instrument correctly. Therefore these 10European swaptions are used for calibrating the two models for pricing 10 × 1

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Bermudan swaption.3 When the models are used for pricing 9 × 1 Bermudanswaption we use nine European swaptions; 1× 9, 2× 8, · · · ., 9× 1 for calibratingthe models. (σ0, σ3 and σ11 for the 10x1 Bermudan swaption and σ0, σ2 and σ10

for the 9× 1 Bermudan swaption)

4.1.3 Goodness-of-fit measure for the calibration

The models are calibrated by minimizing the sum of squared percentage pricingerrors between the model and the market prices of the co-terminal Europeanswaptions, i.e. the goodness-of-fit measure is

minn∑

i=1

(Pi,n,model

Pi,n,market

− 1

)2

where Pi,n,market is the market price and Pi,n,model is the model generated price ofthe i × (n− i) European swaptions, with n = 11 when models are calibrated toprice 10× 1 year Bermudan swaption and n = 10 when models are calibrated toprice 9 × 1 year Bermudan swaption. Instead of minimising the sum of squaredpercentage price errors alternatively we could have minimised the sum of squarederrors in prices. However, such a minimization strategy would place more weighton the expensive instruments. Minimization of squared percentage pricing erroris typically used as a goodness-of-fit measure for similar calibrations in literatureand by practitioners.

4.1.4 NAG routine used for calibration

We need to use some optimization technique to solve the minimization problemmentioned in the last section. Various off-the shelf implementations are availablefor the commonly uses optimization algorithms. We have used an optimisationroutine provided by the Numerical Algorithm Group (NAG) C library.

The NAG routine (e04unc) solves the non-linear least-squares problems usingthe sequential quadratic programming (SQP) method. The problem is assumedto be stated in the following form:

minx∈Rn

F (x) =1

2

n∑i=1

{yi − fi(x)}2

where F (x) (the objective function) is a nonlinear function which can be repre-sented as the sum of squares of m sub-functions (y1 − f1 (x)), (y2 − f2 (x)), · · · ,(yn − fn (x)). The ys are constant. The user supplies an initial estimate of thesolution, together with functions that define f(x) = (f1 (x) , f2 (x) , · · · , fn (x))and as many first partial derivatives as possible; unspecified derivatives are ap-proximated by finite differences.

3Pietersz and Pelsser (2005) followed the same approach.

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In order to use this routine, for our calibration purpose we did following:

(i) Set ’s to 1.

(ii) n = 10/9 , depending on the number of calibration instruments

(iii) fi (x) =Pi,n,model

Pi,n,market

(iv) Set initial estimates for all the three parameters as 0.01 i.e. 1%.

(v) Set a lower bound of 0.0001 and upper bound of 1 for the three parametersto be estimated.

Because of the complexity of our objective function partial derivatives are notspecified and therefore the routine itself approximates partial derivatives by finitedifferences.

4.1.5 Estimating mean reversion parameter (a)

As stated in the start of this section, the mean-reversion parameter has beenestimated from historical data of interest rates. To measure the presence of meanreversion in interest rates we need large data set. This study spans over a periodof one year, and within this one-year period models are calibrated every month.Hence it does not make sense to estimate this parameter for any economy on amonthly basis. Therefore, using historical data, once we estimate the value ofthis parameter and then use this value in all the tests.

For the rate of mean reversion parameter ‘a’ first order autocorrelation ofthe 1 month interest rate series has been used. Here we present the basic ideabehind the estimation procedure used. Under the HW model, the continuoustime representation of the short rate process is

drt = [θt − a rt]dt + σtdz,

The discrete-time version of this process would be

rt+1 − rt = [θt − art] + εt+1,

rt+1 = θt + (1− a)rt + εt+146 (7)

where εt+1 is a drawing from a normal distribution. Equation (46) represents anAR(1) process.

An autoregressive (AR) process is one, where the current values of a variabledepends only upon the values that variable took in previous periods plus an errorterm. A process yt is autoregressive of order p if

yt = φ0 + φ1yt−1 + φ2yt−2 + .... + εt, εt ∼ N(0, σ2).

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An ordinary least square (OLS) estimate of coefficient (1− a) in equation (46) β̂would be4

1− a = β̂ =ρσ(rt+1)σ(rt)

σ2(rt)= ρ

a = 1− ρ

where ρ is the correlation coefficient between rt+1 and rt and can be easily calcu-lated using Excel. For the BK model we perform this regression using time seriesof ln(r), where as before r, is 1M interest rate.

4.2 Libor Market Model Calibration

This section deals with the issue of Libor Market Model calibration. We observedin the previous section that, in LMM, the joint evolution of forward Libor ratesunder the pricing measure is fully determined by their instantaneous volatilitiesand correlations. Therefore, a well specified parameters is essential for obtainingcorrect prices and hedge ratios for exotic interest rate derivatives. Unfortunately,in general, they are not directly observable. Therefore, model parameters mustbe “calibrated ”, i.e., they must be inferred either from time series of forwardrates or from market prices of vanilla derivative contracts, or from a combinationof both.

Here we follow Lvov(2005) the diagonal recursive calibration procedure tothe co-terminal European swaptions. The diagonal recursive calibration need tomake assumptions of volatility of forward rates.

Assumption: The instantaneous volatility term structure has the following form

σk(t) = φkν(Tk − t, α) (8)

where Tk is the expiry time of rate k, ν is a function imposing a qualitative shapeon the volatility term structure, α is a vector of volatility parameters and φk isa rate-specific ”multiplier”. Brigo and Mercurio (2001) demonstrated that thisform has a potential to produce an economically meaningful volatility structurewhile allowing for a satisfactory calibration to market data.

4.2.1 Tested Parameterizations

We could have different parameterizations of the volatility term structure. Thisvolatility parametric form was suggested by Rebonato (1999) which is testedextensively in the literature and is proved to possess very good properties.

4For the regression yi = α + βxi + εi,, the OLS estimate of β is

β̂xy =ρxyσxσy

σ2xy

13

The parametric form as in Rebonato(1999) as

σi(t) = φ(i)[(a + b(Ti − t))exp(−c(Ti − t)) + d] (9)

In our test, we make φ(i) = 1. This parametric specification of the instantaneousvolatility, in contrast with a piecewise-const form, preserves the qualitative shapeof the implied volatility observed in the market, i.e., a hump at around two years.

4.3 Swap Market Model Calibration

Brigo, Mercurio and Morini(2003), Rebonato(2003), and other literatures at-tribute a lot to the parameterisation and calibration of LMM. Based on theirresearch, Galluccio et al (2007) test calibration of co-terminal SMM s using sim-ilar methodology as the one advocated by Rebonato(2003) in the case of LMM.By calibrating to swaption and caplet ATM volatilities, they finally get differentset of parameters {φj, aj, bj, cj, dj} where j = 1, · · · ,M − 1 for M − 1 numberof co-terminal swaptions under different forward measure. Since the model builtin this paper is under one measure and the volatility of different co-terminalswap rate will affect the drift terms in a complicated way, it is unrealistic to usedifferent parameter set for different co-terminal swaption. Therefore, a similarparameter formula is followed here, but four common parameters will be allocatedto volatilities of all co-terminal swaptions.

In order to capture the term structure of swaption volatilities which are peri-odically deterministic, the linear-exponential formulation proposed in Brigo, Mer-curio and Morini (2003) is adopted here as swaption volatility function. Furthermore the swaption volatility is formulated as

σn,m (t) = φn,Mψ (Tn − t, a, b, c, d)

ψ (Tn − t, a, b, c, d) = [a + b (Tn − t)] e−c(Tn−t) + d14 (10)

The term ψ can preserve the well-known humped shape of market quoted Blackimplied volatility.

The method adopted in this paper is to calibrate parameter a, b, c, and d tothe implied volatilities of a set of co-terminal swaptions with different maturitiesbut associated with same length of swaps. The parametric form is as below

v (t,M) = ψ (t, a, b, c, d) = [a + bt] e−ct + d for T0 < t < TM

where v (t,M) is the market implied volatility, and t denotes the maturity ofswaption. Because of its simple log-linear character, calibration algorithm usingthis parametric formula is quite fast and robust.

When performing calibration with DRC method, we use Sequential QuadraticProgramming algorithm from NAG C library to find the optimal parameters.This routine is based on the algorithm suggested in Gill et al (1986). Due to the

14

complexity of our target function, we approximate all partial derivatives usingfinite differences. This however does not appear to preclude the convergence ofthe optimization routine.

4.4 Calibration Results

Calibration results are summarized in Table(1). The calibration is performedon the last business date of each month from February 2005 to September 2006.Table 1 , Figure 1 and Figure 2 describe the errors by the calibration from differentpoint of view. Generally, the root mean square error is quite small and we saythat the model is very well calibrated.

Insert Table 1 ,Figure 1,Figure 2 Here

We also want to have the calibrated parameters to be stationary. In Table 2and Figure 3, we find that given our parametric form of volatility, the parametersare indeed stable.

Insert Table 2(a) and 2(b) and Figure 3 here

Several conclusions could be drawn at this stage. First, it has been shownthat all four models could calibrate to co-terminal European swaption impliedvolatilities matrix pretty well. The calibration methods are quite stable for allthe testing dates in our data sets. This gives us firm ground to proceed withpricing and hedging Bermudan swaptions.

5 Pricing Bermudan Swaption

We have recombining trinomial tree for pricing Bermudans swaptions with shortrate models. For Swap Market Model and Libor Market Model, we perform MonteCarlo simulation with Longstaff-Schwartz Least Square Method.As Bermudanswaption are exotic interest rate derivative product, there is no market quotedprice for it. We compare the price calculated from HW, BK, SMM and LMM.In theory, if calibrated appropriately, all models could give similar prices forthe same Bermudan swaption. The first part, we are pricing a 11-year Bermudanswaption whereas in the second part we are pricing a 10-year Bermudan swaption(with the same strike as 1-year before). We price the same Bermudan swaptionon the date of holding the option.

Insert Table 3(a) and 3(b) and Figure 4 here

The results are summarized in Table (3) and Figure 4. As we could seefrom the graph and tables, on most dates, the prices given by four models areindistinguishable.

15

6 Hedging Bermudan Swaption

Changes in the term structure can adversely affect the value of any interest ratebased asset or liability. Therefore, protecting fixed income securities from un-favorable term structure movements or hedging is one of the most demandingtasks for any financial institutions and for the ALM group in particular. Ineffi-cient hedging strategies can cost big prices to these institutes. In order to protecta liability from possible future interest rate changes first one need to generate re-alistic scenarios and then need to know how the impacts of these scenarios canbe neutralized. Thus, two important issues to be addressed by any interest raterisk management strategy are: (i) how to perturb the term structure to imitatepossible term structure movements (ii) how to immunise the portfolio againstthese movements.

Next sections explain the methodologies applied in this study for

(i) estimating the perturbations by which the input term structure had beenbumped to simulate the possible future changes in the input term structure

(ii) selecting hedge instruments

(iii) delta-hedging the underlying Bermudan swaption using the selected instru-ments.

(iv) calculating possible profits and losses (P&L).

In each section we have examples from literature have been referred to justifythe choices made.

6.1 Perturbing the term structure

Over the years researchers and practitioners have been using duration analysis forinterest rate risk management, i.e. they shift the entire yield curve upward anddownward in a parallel manner and then estimate how the value of their portfoliois affected as a result of these parallel perturbations. They then hedge themselvesagainst these risks. Parallel shifts are unambiguously the most important kindof yield curve shift but alone cannot explain completely explain the variations ofyield curve observed in market. Three most commonly observed term structureshifts are: Parallel Shift where the entire curve goes up or down by same amount;Tilt, also known as slope shift, in which short yields fall and long yields rise (orvice versa); Curvature shift in which short and long yields rise while mid-rangeyields fall (or vice versa). These three shifts together can explain almost all thevariance present in any term-structure and thus and one should not completelyrely on duration and convexity measures for estimating the risk sensitivity of afixed income security. There are numerous examples in literature to support this

16

argument. Here, we mention a few studies that lead to this conclusion. Littermanand Scheikman (1991) performed principal components analysis (PCA) and foundthat on average three factors, referred to as level (roughly parallel shift), slope,and curvature, can explain 98.4% of the variation on Treasury bond returns.They suggested that “by considering the effects of each of these three factors ona portfolio, one can achieve a better hedged position than by holding a only a zero-duration portfolio”.5 Knez, Litterman, and Scheikman (1994) investigated thecommon factors in money markets and again they found that on average three-factors can explain 86% of the total variation in most money market returnswhereas on average four factors can explain 90% of this variation. Chen andFu (2002) performed a PCA on yield curve and found that the first four factorscapture over 99.99% of the yield curve variation. They too claimed that hedgingagainst these factors would lead to a more stable portfolio and thus superiorhedging performance.

Insert Figure4, Figure 5 and Figure 6 Here

Based on the findings of these studies, in this study we performed PCA onhistorical data for estimating more realistic term structure shifts. In this studywe performed PCA on annual changes of the forward rates and used scores of thefirst three principal components for estimating the shifts by which we bumped theforward rate curves. There are not many examples of estimating price sensitivitiesw.r.t. multiple factors (like 3 principal components here) with a one-factor termstructure model. Generally risk sensitivities are calculated by perturbing only themodel intrinsic factors i.e. for the one-factor model, only one-factor is perturbedand so on.

PCA has been performed on annual changes of forward Libor rates. Annualchanges have been used because each hedge is maintained for one year. Thereason for using forward rates rather than yield curve for doing PCA is two folds:first forward Libor rates are directly observable in market. Second using forwardcurves easily we can construct zero-curve and swap curve (needed for estimatinginterest rate sensitivities of swaps that are used for hedging). Without going intothe mathematical details of this procedure, here, we briefly describe how PCAhas been done for estimating the term structure shifts/bumps in the study:6

Monthly observations of 11 forward Libor rates (f0,0,1, f0,1,2, f0,2,3, ..., f0,10,11) forthe period Jan 1999 to Dec 2006 have been used for PCA. Explicitly theserates have been used because we give annual zero curve of maturity till 11years as input to our short rat models and we want to estimate bumps forall these maturities.7

5Litterman and Scheikman (1991, 54)6For details on PCA refer http://csnet.otago.ac.nz/cosc453/student tutorials/principal components.pdf7As we know that this dissertation is a part of a big project. The models to be implemented

are LMM and SMM. These maturity forward rates are also needed by these two models.

17

Using these monthly observations of the 11 forward rates we calculate annualchanges for each of the 11 forward rates as follows. Suppose we havemonthly observations of the 11 forward rates for a period of n years i.e. in to-tal we have 12×n monthly observations of forward Libor rates f0,τj ,τj+1(ti),where i = 1, 2, · · · , 12×n and τj = 0, 1, 2, .., 10. When the observations arearranged in ascending order (by date) then annual change for the forwardLibor rates can be calculated as

cLi,j = f0,τj ,τj+1(ti + 12)− f0,τj ,τj+1(ti)

now i = 1, 2, . . . , 12 × (n − 1). These values would form a 12×(n-1) x 11matrix i.e. we have 12 less entries. To clear this, say ti= Jan 1999, andτj = 0. Then f0,0,1(ti)is value of f0,0,1 on Jan 1999; f0,0,1(ti +12)is the valueof f0,0,1 on Jan 2000 and cL is the one-year change in the market observedvalue of f0,0,1.

The matrix cLi,j is given as input to the NAG routine (g03aac) that performs“principal component analysis” on the input data matrix and returns prin-cipal component loadings and the principal component scores. The otherimportant statistics of the principal component analysis reported by theroutine are : the eigen values associated with each of the principal com-ponents included in analysis and the proportion of variation explained byeach principal component.

We used the scores of first three factors to calculate the three types of shifts forthe 11 forward Libor rates using following regression:

∆f0,τj ,τj+1 = α + βτj ,1P1 + βτj ,2P2 + βτj ,2P3 + ε

Where τj = 0, 1, 2, .., 10 and Pk is the vector of scores for the kth factork = 1, 2, 3. Nag routine (g02dac) has been employed to perform this re-gression. The routine computes parameter estimates, the standard errors ofthe parameter estimates, the variance–covariance matrix of the parameterestimates and the residual sum of squares.

After performing the regressions specified above, the 11 forward rates are bumpedby shocks corresponding to the first three factors as:

f±0,τj ,τj+1 = f0,τj ,τj+1 ± βτj,k∆Pk

where τj = 0, 1, 2, .., 10, and k = 1, 2, 3.

6.2 Choosing the hedge instruments

Selecting appropriate hedge instruments is a critical part of a successful hedg-ing strategy. In literature there are evidences of two hedging strategies, factor

18

hedging and bucket hedging. For factor hedging, in a K-factor model, K differentinstruments (together with the money market account) are used to hedge anyderivative. The choice of hedge instruments is independent of the derivative tobe hedged i.e., the same K hedging instruments can be used for hedging anyderivative in a K-factor model, and depend only on the number of factors in themodel. For bucket hedging the choice of hedge instruments depend on the instru-ment to be hedged and not on the factors in the model. In this hedging strategynumber of hedge instruments is equal to the number of total payoffs provided bythe instrument. The hedge instruments are chosen so their maturities correspondto different payment or decision dates of the underlying derivative.

On using any other criteria for selecting the hedge instruments, the num-ber of hedge instruments will lie between the numbers of hedge instruments forthese two hedging strategies. Before discussing the instruments that are used todelta-hedge the Bermudan swaption in this study, in this section first we brieflydiscuss a few examples from literature. Driessen, Klaassen and Melenberg (2002)used (delta-) hedging of caps and swaptions as criteria for comparing the hedgeperformance of HJM class models and Libor market models. They used zerocoupon bonds as hedge instruments. For each model, they considered factor andbucket hedging strategies. DKM show that when bucket strategies are used forhedging, the performance of the one-factor models improves significantly Fan,Gupta and Ritchken (2006) also used effectiveness of delta neutral hedges (forswaptions) as criteria for comparing the hedging performance of single factor andmulti-factor factor term structure models. They used discount bonds to delta-hedge swaptions. In this study they first applied factor hedging for choosing thehedge instruments and found that in context of hedging performance, multifac-tor models outperform single factor models. Next in light of the DKM results,they repeated their experiments using additional hedging instruments. Theyfound that for the one-factor and two factor models adding more instrumentsdid not result into better hedge results. Pietersz and Pelsser (2005) comparedjoint delta-vega hedging performance (for 10x1 Bermudan swaption) of singlefactor Markov-functional and multi-factor market models. They used the buckethedging strategy and set up hedge portfolios using 11 discount bonds, one dis-count bond for each tenor time associated with the deal. They found that jointdelta-vega hedging performance of both models is comparable

The general implications of these examples are:

(i) Effectiveness of delta neutral hedges is often used to evaluate the hedgingperformance of term structure models.

(ii) Using multiple instruments can improve the hedge performance of one factormodels. Practitioners also favour this practice

Therefore, in this study, we decide to use two different swaps of maturities 5and 11years as hedge instruments. Maturity of 11-year swap coincides with the

19

maturity of the co-terminal Bermudan swaption to be hedged and the length ofother swap is almost half way the life of this Bermudan swaption. We could haveused discount bonds to hedge this Bermudan swaption8 but use of swap is morein line with the general practitioners practice. Studies suggest that large bankstend to use interest rate swaps more intensively for hedging.

6.3 Constructing delta hedged portfolio

Delta hedging is the process of keeping the delta of a portfolio equal to or asclose as possible to zero. Since delta measures the exposure of a derivative tochanges in the value of the underlying, the overall value of a portfolio remainsunchanged for small changes in the price of its underlying instrument. A deltahedged portfolio is established by buying or selling an amount of the underlierthat corresponds to the delta of the portfolio.

For interest rate derivatives,if the entire initial term structure is perturbedby same amount say ε (parallel shift), then the risk sensitivity of a fixed incomesecurity w.r.t. this perturbation can be estimated as

V (ε)− V

ε

where V is the value of the derivative calculated using initial term structure andV (ε) is the value of the derivative after the initial term structure is perturbed byε. If we first increase the entire initial term structure by ε and then next decreaseit by ε, then the risk sensitivity can estimated as

V (ε+)− V (ε−)

2ε(11)

where V (ε+) is the value of the derivative calculated after initial term structurehas been shifted up by ε and V (ε−) is the value of the derivative after the initialterm structure has been shifted down by ε. In our case we have bumped theinitial forward rate curve by three factors. Also for each factor, we have bumpedthe forward rate curve both up and down. Using the idea presented in equation(11), we estimate the sensitivity (delta) of the Bermudan swaption and the twoswaps w.r.t the three factors as follows

∆Bk =

∂B

∂Pk

=B+

k −B−k

P+k − P−

k

=B+

k −B−k

2∆Pk

∆S5k =

∂S5

∂Pk

=(S5)

+k −B−

k

P+k − P−

k

=(S5)

+k −B−

k

2∆Pk

∆S11k =

∂S11

∂Pk

=(S11)

+k −B−

k

P+k − P−

k

=(S11)

+k −B−

k

2∆Pk

8Pietersz and Pelsser (2005) used discount bonds to hedge Bermudan swaption

20

for k = 1, 2, 3, and (.)+k and (.)−k respectively are the prices of derivative after the

initial forward curve has been bumped up and down by the kth factor. Now ifwe consider a portfolio, consisting of one 10 × 1 Bermudan swaption, x11 unitsof 11-year swap and x5 units of 5-year swap, then total delta mismatch of thisportfolio w.r.t. kth factor, εk is

εk = ∆Bk − x11∆

S11k − x5∆

S5k (12)

x11 and x5 need not be whole numbers. This is the usual assumption of perfectlydivisible securities. From equation (12), we can see that when we use more thanone hedging instrument in a one-factor model, the hedge ratios would not beunique, and some rule must be applied for constructing the hedge portfolio usingthe chosen hedge instruments. Here to obtain the hedge ratios, x11 and x5, weuse the basic idea behind the delta hedging. The two hedge ratios x11 and x5 areobtained by minimising the total delta-mismatch of the portfolio w.r.t. the firstthree PCA factors i.e.

minx11,x5

∑3

k=1ε2

k

where εk is given by equation (12). Some optimization technique needs to beapplied to solve this minimisation problem. We use the “C ” implementation of“Downhill Simplex Method” provided by NAG to estimate x11 and x5.

9 The nagroutine (e04ccc) minimises a general function F (x) of n independent variablesx = (x1, x2, .., xn)T .

6.4 Calculating P&L

Once hedge has been established on say day t, the hedge error can be evaluatedone year later on day t + 1 as follows

P&Lt+1 = (Bt + x11,tS11,t + x5,tS5,t)× (1 + i0,t)

−(Bt+1 + x11,tS11,t+1 + x5,tS5,t+1) (13)

where Bt is the value of a 10 × 1 Bermudan swaption on day t; xk,t are units ofk-year swap in the hedge portfolio; Sk,t is the value of k-year swap on day t; andi0,t: is the forward Libor f0,0,1 on day t.

At the point of initiation the value of any swap is zero. This means S11,t = 0and also S5,t = 0. Therefore, equation (13) can be written as

P&Lt+1 = Bt × (1 + i0,t)−Bt+1 − x11,tS11,t+1 − x5,tS5,t+1. (14)

One year later, on day t+1, when hedge portfolio is unwound the 10×1 Bermudanswaption is a 9× 1 Bermudan swaption. On day t + 1, we recalibrate the model

9The downhill simplex method (Spendley 1962, Nelder and Mead 1965, Press et al. 1992)is an efficient algorithm for solving unconstrained minimisation problems.

21

and using new parameters calculate the model price of this away-from-money9 × 1 Bermudan swaption Bt+1. Strike rate for the 9 × 1 Bermudan swaption iskept same as it was for the 10 × 1 Bermudan swaption on day t, as objective isto find the current value of that old swaption. We also calculate the values ofthe two swaps on day t + 1 Using the example of 5-year swap we show, how theswaps are re-evaluated one year later on day t + 1.

At the time of swap initialization, i0,t is known

i0,t =

{1

P (0, 1)− 1

}

and the swap rate k5,t for a 5-year swap on day t and can be calculated as

k5,t =1− P (0, 5)

P (0, 1) + P (0, 2) + ... + P (0, 5)

At time t + 1, this 5-year swap is just a 4-year swap and also floating rate i0,t+1

for the next reset is now known. The value of this swap at t + 1 would be

S5,t+1 = (i0,t − k5) + (i0,t+1 − k5)P (0, 1) + (̃ı1,t+1 − k5)P (0, 2)

+(̃ı2,t+1 − k5)P (0, 3) + (̃ı3,t+1 − k5)P (0, 4)

= (i0,t − k5) +

[{1

P (0, 1)− 1

}− k5

]P (0, 1)

+

[{P (0, 1)

P (0, 2)− 1

}− k5

]P (0, 2) +

[{P (0, 2)

P (0, 3)− 1

}− k5

]P (0, 3)

+

[{P (0, 3)

P (0, 4)− 1

}− k5

]P (0, 4)

= (i0,t − k5) + [1− P (0, 4)]− k5 [P (0, 1) + P (0, 2) + P (0, 3) + P (0, 4)]

= (i0,t − k5) +

[{1− P (0, 4)

P (0, 1) + P (0, 2) + P (0, 3) + P (0, 4)

}− k5

]

[P (0, 1) + P (0, 2) + P (0, 3) + P (0, 4)]

= (i0,t − k5) + [k4,t+1 − k5] [P (0, 1) + P (0, 2) + P (0, 3) + P (0, 4)]

where k4,t+1 would be the swap rate for any 4-year swap that would be initiatedon day t + 1. When the swap is re-evaluated on day t + 1 all the discount bonds’prices are computed using the yield curve observed on that day.

The hedging profit and loss is reported in Table 6 and Figure 7. As in Euromarket, the hedging P &L from the four models are very similar to each other.In USD market, HW and BK model is marginally better than more complicatedSMM/LMM model. This would lead us to conclude that, from ALM perspective,we actually could use simple short rate models, like HW and BK model, tomanagement the Bermudan swaption portfolios. The performance of HW andBK are indistinguishable.

Insert Table 6 and Figure 7 Here

22

7 Conclusion

The goal of this paper is to provide an empirical analysis and comparison offour one-factor interest rate term structure models from ALM perspective. Incontrast to previous research, we have compared four very different models andwe performed the pricing and hedging tests for Bermudan swaptions. We alsouse a more refined method in delta ratio calculation during the hedging process.

For model calibration, HW and BK model use time-dependent volatility forbetter calibration result and LMM and SMM model use market standard para-metric form for volatility. Though the modelling approaches are vastly different,the error from calibration is quite small for all four models and all the calibratedparameters are quite stable.

Bermudan swaptions are priced via recombining trinomial tree with HW/BKmodel while for SMM and LMM, Monte Carlo simulation with predictor-correctordrift approximation is used. Most of the time, the prices from four models arewithin the bid-ask spread. Thus we conclude that, for one factor model, we couldchoose any of the four models for pricing as long as they calibrated to the sameinstruments.

Minimizing the variations of hedging profit and loss is the objective for ALM.In Euro market, the hedging P&L from the four models are very similar to eachother. In USD market, HW and BK model is marginally better than more com-plicated SMM/LMM model. This would lead us to conclude that, from ALMperspective, we actually could use simple short rate models, like HW and BKmodel, to management the Bermudan swaption portfolios. The performance ofHW and BK are indistinguishable.

This analysis could have an immediate impact on the decision making ofthe ALM department of big financial institution on choosing the right interestrate term structure model to manage their balance sheet. Future research couldapplying the same test procedure on interest rate from a different economic regimesuch as that of emerging market or a developing country.

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[36] Ritchken, P., Chuang, I., 1999. Interest rate option pricing with volatility-humps. Review of Derivatives Research 3, 237¨C262.

[37] Ritchken, P., Sankarasubramanian, L., 1995. Volatilitystructures of forwardrates and the dynamics of the term structure. Mathematical Finance 5,55¨C72.

[38] Vasicek, O., 1977. An equilibrium characterization of the term structure.Journal of Financial Economics

26

Table 1: Root Mean Square Errors from Model Calibration to European co-terminal swaption (11Y and 10Y) in EURand USD Markets from Feb 2005 to Sep 2007

EUR 11Y Date HW BK SMM LMM USD 11Y Date HW BK SMM LMM2005-2-28 0.4% 0.4% 0.2% 1.0% 2005-2-28 0.6% 0.7% 0.5% 0.4%2005-3-31 0.5% 0.6% 0.6% 0.7% 2005-3-31 0.5% 0.7% 0.4% 0.4%2005-4-29 0.5% 0.7% 0.5% 0.6% 2005-4-29 0.5% 0.7% 0.5% 0.5%2005-5-31 0.6% 0.7% 0.5% 0.4% 2005-5-31 0.6% 0.6% 0.4% 0.4%2005-6-30 0.3% 0.3% 0.2% 0.2% 2005-6-30 0.0% 0.8% 0.3% 0.4%2005-7-29 0.3% 0.5% 0.3% 0.3% 2005-7-29 0.9% 0.8% 0.5% 0.4%2005-8-31 0.5% 0.8% 0.6% 0.5% 2005-8-31 0.6% 0.7% 0.5% 0.1%2005-9-30 0.6% 0.8% 0.5% 0.5% 2005-9-30 0.6% 0.7% 0.4% 0.8%

2005-10-31 0.9% 1.1% 0.9% 0.9% 2005-10-31 0.4% 0.5% 0.4% 0.3%2005-11-30 0.5% 0.7% 0.3% 0.2% 2005-11-30 0.6% 0.6% 0.5% 0.5%2005-12-30 0.4% 0.5% 0.3% 0.3% 2005-12-30 0.5% 0.5% 0.2% 0.2%

2006-1-31 0.6% 0.6% 0.6% 0.4% 2006-1-31 0.5% 0.5% 0.4% 0.5%2006-2-28 0.5% 0.5% 0.5% 0.5% 2006-2-28 0.8% 0.8% 0.2% 0.2%2006-3-31 0.4% 0.5% 0.3% 0.3% 2006-3-31 0.7% 0.7% 0.3% 0.2%2006-4-28 0.6% 0.7% 0.6% 0.7% 2006-4-28 0.7% 0.8% 0.3% 0.3%2006-5-31 0.4% 0.5% 0.4% 0.3% 2006-5-31 0.9% 0.9% 0.3% 0.2%2006-6-30 0.5% 0.5% 0.4% 0.3% 2006-6-30 0.6% 0.7% 0.1% 0.1%2006-7-31 0.4% 0.5% 0.3% 0.3% 2006-7-31 0.8% 0.8% 0.4% 0.3%2006-8-31 0.4% 0.6% 0.4% 0.3% 2006-8-31 0.9% 0.9% 0.3% 0.2%2006-9-29 0.3% 0.2% 0.3% 0.3% 2006-9-29 1.0% 1.0% 0.2% 0.2%

Total 9.3% 11.9% 8.6% 9.0% Total 12.8% 14.5% 7.1% 6.6%

EUR 10Y Date HW BK SMM LMM USD 10Y Date HW BK SMM LMM2006-2-28 0.3% 0.3% 0.6% 1.2% 2006-2-28 0.8% 0.8% 0.6% 0.8%2006-3-31 0.2% 0.3% 1.1% 0.5% 2006-3-31 0.7% 0.7% 1.1% 0.9%2006-4-28 0.5% 0.6% 0.9% 1.2% 2006-4-28 0.5% 0.6% 0.9% 1.0%2006-5-31 0.3% 0.3% 0.5% 0.4% 2006-5-31 0.7% 0.9% 0.5% 0.6%2006-6-30 0.4% 0.5% 0.7% 1.2% 2006-6-30 0.6% 0.6% 0.7% 0.8%2006-7-31 0.3% 0.3% 0.8% 1.0% 2006-7-31 0.8% 0.9% 0.8% 0.8%2006-8-31 0.4% 0.5% 0.6% 0.9% 2006-8-31 0.7% 0.7% 0.6% 0.6%2006-9-29 0.2% 0.2% 0.7% 0.9% 2006-9-29 0.7% 0.8% 0.7% 0.7%2006-10-31 0.7% 0.6% 0.6% 1.7% 2006-10-31 0.8% 0.8% 0.6% 0.6%2006-11-30 0.4% 0.4% 0.6% 1.5% 2006-11-30 0.4% 0.4% 0.6% 0.3%2006-12-29 0.6% 0.5% 0.7% 2.0% 2006-12-29 0.5% 0.6% 0.7% 0.5%2007-1-30 0.4% 0.6% 2.0% 0.7% 2007-1-30 0.2% 0.3% 2.0% 0.7%2007-2-28 0.3% 0.3% 0.5% 0.9% 2007-2-28 0.3% 0.3% 0.5% 0.6%2007-3-31 0.3% 0.2% 1.0% 0.4% 2007-3-31 0.4% 0.0% 1.0% 1.1%2007-4-30 0.3% 0.3% 1.0% 1.1% 2007-4-30 0.4% 0.5% 1.0% 0.8%2007-5-31 0.3% 0.3% 0.7% 1.4% 2007-5-31 0.7% 0.8% 0.7% 0.7%2007-6-30 0.3% 0.3% 0.4% 1.1% 2007-6-30 0.2% 0.2% 0.4% 0.9%2007-7-29 0.4% 0.4% 1.0% 0.9% 2007-7-29 0.3% 0.3% 1.0% 0.7%2007-8-31 0.7% 0.7% 0.3% 0.8% 2007-8-31 0.2% 0.3% 0.3% 0.3%2007-9-28 0.6% 0.6% 1.1% 1.3% 2007-9-28 0.5% 0.6% 1.1% 1.2%

Total 8.2% 8.3% 15.8% 21.0% Total 10.6% 11.1% 15.8% 14.3%

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMM stands for Swap Market Model and LMMstands for Libor Market Model) by date. '11Y' denotes calibration results for 11Y co-terminal European swaptoins from Feb 2005 toSep 2006 ; '10Y' denotes calibration results for 10Y co-terminal European swaptoins from Feb 2006 to Sep 2007. 'EUR' denotesEuro market and 'USD' denotes US-dollar market.

Date a b c d Date a b c d2005-2-28 0.0100 0.0062 0.0062 0.0072 2005-2-28 -0.0208 -0.0082 0.1031 0.17532005-3-31 0.0100 0.0065 0.0061 0.0067 2005-3-31 0.0188 0.0215 0.6595 0.13512005-4-29 0.0100 0.0061 0.0060 0.0064 2005-4-29 0.0197 0.0143 0.5514 0.13432005-5-31 0.0100 0.0064 0.0064 0.0059 2005-5-31 0.0441 -0.0016 0.0919 0.12882005-6-30 0.0100 0.0069 0.0063 0.0064 2005-6-30 0.0465 0.0158 0.4431 0.14802005-7-29 0.0100 0.0063 0.0064 0.0066 2005-7-29 0.0196 0.0136 0.4466 0.14662005-8-31 0.0100 0.0063 0.0066 0.0065 2005-8-31 0.0190 0.0164 0.4144 0.15332005-9-30 0.0100 0.0063 0.0066 0.0064 2005-9-30 0.0117 0.0196 0.4089 0.1542

2005-10-31 0.0100 0.0065 0.0067 0.0071 2005-10-31 0.0019 0.0164 0.5374 0.15802005-11-30 0.0100 0.0066 0.0068 0.0065 2005-11-30 0.0032 0.0197 0.4492 0.15522005-12-30 0.0100 0.0064 0.0068 0.0064 2005-12-30 0.0060 0.0153 0.3217 0.15852006-1-31 0.0100 0.0061 0.0067 0.0058 2006-1-31 0.0300 0.0114 0.1652 0.12212006-2-28 0.0100 0.0059 0.0065 0.0059 2006-2-28 0.0181 0.0096 0.1807 0.12962006-3-31 0.0100 0.0061 0.0063 0.0060 2006-3-31 0.0030 0.0075 0.3124 0.13602006-4-28 0.0100 0.0063 0.0061 0.0065 2006-4-28 0.0045 0.0089 0.5624 0.13322006-5-31 0.0100 0.0063 0.0061 0.0062 2006-5-31 0.0152 0.0049 0.3026 0.12592006-6-30 0.0100 0.0063 0.0062 0.0062 2006-6-30 0.0171 0.0030 0.1787 0.12182006-7-31 0.0100 0.0062 0.0062 0.0060 2006-7-31 0.0094 0.0075 0.3131 0.12742006-8-31 0.0100 0.0063 0.0063 0.0060 2006-8-31 0.0041 0.0118 0.4010 0.13712006-9-29 0.0100 0.0064 0.0063 0.0060 2006-9-30 0.0041 0.0118 0.4010 0.1371

Date a b c d Date a b c d2005-2-28 0.0087 0.1642 0.1328 0.1521 2005-2-28 -0.0558 0.2099 1.1154 0.13732005-3-31 0.0087 0.1740 0.1312 0.1405 2005-3-31 -0.0169 0.1665 0.8245 0.13522005-4-29 0.0087 0.1716 0.1337 0.1351 2005-4-29 0.0203 0.0997 0.6972 0.13382005-5-31 0.0087 0.1860 0.1465 0.1229 2005-5-31 0.1163 0.0052 0.2577 0.12712005-6-30 0.0087 0.2123 0.1488 0.1412 2005-6-30 0.1555 0.0856 0.5681 0.14682005-7-29 0.0087 0.1867 0.1530 0.1401 2005-7-29 -0.0089 0.0891 0.5949 0.14592005-8-31 0.0087 0.1966 0.1654 0.1406 2005-8-31 -0.0537 0.1083 0.5871 0.15422005-9-30 0.0087 0.1933 0.1719 0.1348 2005-9-30 -0.0958 0.1051 0.5195 0.1521

2005-10-31 0.0087 0.1856 0.1657 0.1562 2005-10-31 -0.1023 0.1017 0.6665 0.15752005-11-30 0.0087 0.1864 0.1708 0.1397 2005-11-30 -0.1175 0.1020 0.5525 0.15372005-12-30 0.0087 0.1882 0.1850 0.1425 2005-12-30 -0.0836 0.0634 0.4002 0.15472006-1-31 0.0087 0.1677 0.1705 0.1257 2006-1-31 -0.0031 0.0362 0.2119 0.10622006-2-28 0.0087 0.1627 0.1659 0.1310 2006-2-28 -0.0217 0.0319 0.2372 0.12022006-3-31 0.0087 0.1539 0.1503 0.1300 2006-3-31 -0.0429 0.0320 0.4028 0.13422006-4-28 0.0087 0.1523 0.1366 0.1354 2006-4-28 -0.0298 0.0510 0.6778 0.13292006-5-31 0.0087 0.1523 0.1350 0.1213 2006-5-31 0.0143 0.0240 0.4130 0.12472006-6-30 0.0087 0.1484 0.1373 0.1231 2006-6-30 0.0256 0.0086 0.2143 0.11652006-7-31 0.0087 0.1505 0.1406 0.1205 2006-7-31 -0.0182 0.0304 0.3983 0.12582006-8-31 0.0087 0.1597 0.1500 0.1266 2006-8-31 -0.0575 0.0536 0.4868 0.13542006-9-29 0.0087 0.1638 0.1552 0.1283 2006-9-29 0.0148 0.0207 0.2445 0.1228

Note: HW stands for Hull-White modle,a in HW model is the mean reversion rate, b ,c and d is sigma (0),sigma(3) and sigma(11)respectively; BK stands for Black-Karasinski model, a in HW model is the mean reversion rate, b ,c and d is sigma0,sigma 3 and sigma 11respectively; SMM stands for Swap Market Model and LMM stands for Libor Market Model.a,b,c,d in SMM and LMM are the parametersof the volatility functional form as suggested by Rebonato 1999. '11Y'('10Y') denotes calibration is made to co-terminal 11-year (10-year)European swaptoins. All calibrations are performed with both EUR and USD markets.

Table 2(a): Model Calibrated Parameter Values for EUR Market fromFebruary 2005 to September 2006

HW(EUR) SMM(EUR)

BK(EUR) LMM(EUR)

Table 2(b): Model Calibrated Parameter Valuesfor USD Market from February 2005 to September 2006

Date a b c d Date a b c d2005-2-28 0.0060 0.0092 0.0089 0.0062 2005-2-28 0.0166 0.0488 0.2342 0.13172005-3-31 0.0060 0.0102 0.0087 0.0062 2005-3-31 0.0149 0.0524 0.2897 0.13542005-4-29 0.0060 0.0098 0.0094 0.0064 2005-4-29 0.0211 0.0470 0.2687 0.14402005-5-31 0.0060 0.0094 0.0094 0.0064 2005-5-31 0.0211 0.0461 0.2394 0.14382005-6-30 0.0060 0.0097 0.0096 0.0065 2005-6-30 0.0284 0.0326 0.3098 0.16102005-7-29 0.0060 0.0096 0.0090 0.0066 2005-7-29 0.0228 0.0286 0.3125 0.15212005-8-31 0.0060 0.0097 0.0094 0.0070 2005-8-31 0.0219 0.0306 0.2954 0.16592005-9-30 0.0060 0.0097 0.0094 0.0069 2005-9-30 0.0223 0.0277 0.2931 0.1537

2005-10-31 0.0060 0.0099 0.0095 0.0067 2005-10-31 0.0174 0.0340 0.2581 0.14422005-11-30 0.0060 0.0101 0.0101 0.0076 2005-11-30 0.0158 0.0278 0.2398 0.15632005-12-30 0.0060 0.0097 0.0098 0.0083 2005-12-30 0.0169 0.0039 0.3350 0.17402006-1-31 0.0060 0.0092 0.0099 0.0079 2006-1-31 0.0151 0.0007 0.2796 0.16542006-2-28 0.0060 0.0081 0.0097 0.0079 2006-2-28 0.0152 -0.0372 0.4159 0.17162006-3-31 0.0060 0.0083 0.0097 0.0066 2006-3-31 0.0206 -0.0077 0.2870 0.14502006-4-28 0.0060 0.0081 0.0088 0.0069 2006-4-28 0.0166 -0.0082 0.3910 0.13742006-5-31 0.0060 0.0084 0.0095 0.0071 2006-5-31 0.0199 -0.0154 0.3859 0.14492006-6-30 0.0060 0.0080 0.0093 0.0077 2006-6-30 0.0137 -0.0230 0.3872 0.14632006-7-31 0.0060 0.0080 0.0094 0.0076 2006-7-31 0.0160 -0.0220 0.3703 0.14792006-8-31 0.0060 0.0077 0.0096 0.0077 2006-8-31 0.0171 -0.0322 0.3567 0.15642006-9-29 0.0060 0.0078 0.0096 0.0077 2006-9-30 0.0171 -0.0322 0.3567 0.1564

Date a b c d Date a b c d2005-2-28 0.0060 0.1985 0.1737 0.1004 2005-2-28 0.0255 0.0606 0.3062 0.12232005-3-31 0.0060 0.2028 0.1576 0.1058 2005-3-31 0.0674 0.0612 0.3763 0.13032005-4-29 0.0060 0.2133 0.1854 0.1044 2005-4-29 0.0253 0.0697 0.3149 0.13002005-5-31 0.0060 0.2123 0.1957 0.1076 2005-5-31 -0.0080 0.0756 0.3045 0.13102005-6-30 0.0060 0.2239 0.2044 0.1138 2005-6-30 0.0082 0.0757 0.2864 0.13002005-7-29 0.0060 0.2045 0.1858 0.1144 2005-7-29 -0.0484 0.0885 0.3866 0.14542005-8-31 0.0060 0.2214 0.2037 0.1316 2005-8-31 -0.0370 0.0814 0.3598 0.15702005-9-30 0.0060 0.2053 0.1923 0.1170 2005-9-30 -0.0454 0.0802 0.3516 0.14412005-10-31 0.0060 0.1969 0.1819 0.1150 2005-10-31 -0.0001 0.0577 0.3029 0.13142005-11-30 0.0060 0.2034 0.1958 0.1371 2005-11-30 -0.0213 0.0536 0.3024 0.14782005-12-30 0.0060 0.2013 0.1995 0.1552 2005-12-30 -0.0858 0.0648 0.4062 0.17012006-1-31 0.0060 0.1872 0.1959 0.1502 2006-1-31 -0.0801 0.0512 0.3185 0.15632006-2-28 0.0060 0.1628 0.1940 0.1586 2006-2-28 -0.2150 0.0609 0.4785 0.17002006-3-31 0.0060 0.1613 0.1827 0.1182 2006-3-31 -0.1376 0.0707 0.3364 0.13612006-4-28 0.0060 0.1513 0.1548 0.1175 2006-4-28 -0.1236 0.0674 0.4668 0.13522006-5-31 0.0060 0.1559 0.1677 0.1202 2006-5-31 -0.1638 0.0786 0.4562 0.14202006-6-30 0.0060 0.1455 0.1645 0.1316 2006-6-30 -0.1527 0.0527 0.4492 0.14432006-7-31 0.0060 0.1501 0.1696 0.1297 2006-7-31 -0.1650 0.0617 0.4344 0.14532006-8-31 0.0060 0.1513 0.1824 0.1378 2006-8-31 -0.2088 0.0661 0.4190 0.15332006-9-29 0.0060 0.1574 0.1869 0.1411 2006-9-29 -0.2322 0.0762 0.4611 0.1591

Note: HW stands for Hull-White modle,a in HW model is the mean reversion rate, b ,c and d is sigma (0),sigma(3) and sigma(11)respectively; BK stands for Black-Karasinski model, a in HW model is the mean reversion rate, b ,c and d is sigma0,sigma 3 and sigma 11respectively; SMM stands for Swap Market Model and LMM stands for Libor Market Model.a,b,c,d in SMM and LMM are the parametersof the volatility functional form as suggested by Rebonato 1999. '11Y'('10Y') denotes calibration is made to co-terminal 11-year (10-year)European swaptoins. All calibrations are performed with both EUR and USD markets.

HW(USD) SMM(USD)

BK(USD) LMM(USD)

Table 3(a): 11Y and 10Y Bermudan Swaption Pricesin EUR Market from February 2005 to September 2007

DATE(11Y) HW BK SMM LMM2005-2-28 0.0465 0.0441 0.0463 0.04602005-3-31 0.0467 0.0441 0.0479 0.04802005-4-29 0.0476 0.0447 0.0481 0.04852005-5-31 0.0491 0.0460 0.0499 0.05012005-6-30 0.0508 0.0474 0.0534 0.05492005-7-29 0.0485 0.0455 0.0484 0.04782005-8-31 0.0486 0.0455 0.0483 0.04752005-9-30 0.0464 0.0438 0.0458 0.04642005-10-31 0.0459 0.0437 0.0450 0.04532005-11-30 0.0452 0.0432 0.0444 0.04482005-12-30 0.0429 0.0414 0.0415 0.04152006-1-31 0.0411 0.0398 0.0395 0.03932006-2-28 0.0396 0.0384 0.0379 0.03762006-3-31 0.0386 0.0376 0.0375 0.03742006-4-28 0.0387 0.0377 0.0386 0.03902006-5-31 0.0396 0.0384 0.0397 0.04042006-6-30 0.0386 0.0376 0.0384 0.03872006-7-31 0.0387 0.0376 0.0382 0.03852006-8-31 0.0389 0.0379 0.0384 0.03862006-9-29 0.0381 0.0373 0.0379 0.0382

DATE(10Y) HW BK SMM LMM Strike2006-2-28 0.0271 0.0272 0.0273 0.0271 0.04022006-3-31 0.0379 0.0367 0.0365 0.0364 0.03952006-4-28 0.0534 0.0510 0.0519 0.0521 0.03772006-5-31 0.0610 0.0582 0.0591 0.0592 0.03632006-6-30 0.0744 0.0720 0.0722 0.0723 0.03442006-7-31 0.0610 0.0584 0.0585 0.0586 0.03562006-8-31 0.0600 0.0573 0.0575 0.0576 0.03392006-9-29 0.0545 0.0522 0.0519 0.0522 0.03442006-10-31 0.0417 0.0404 0.0399 0.0397 0.03692006-11-30 0.0390 0.0378 0.0371 0.0369 0.03712006-12-29 0.0563 0.0543 0.0540 0.0537 0.03572007-1-31 0.0526 0.0508 0.0504 0.0502 0.03772007-2-28 0.0443 0.0427 0.0423 0.0422 0.03782007-3-30 0.0379 0.0332 0.0328 0.0329 0.04092007-4-30 0.0308 0.0301 0.0293 0.0290 0.04312007-5-31 0.0391 0.0380 0.0370 0.0369 0.04332007-6-29 0.0476 0.0461 0.0464 0.0466 0.04422007-7-31 0.0487 0.0471 0.0473 0.0474 0.04282007-8-31 0.0510 0.0492 0.0502 0.0504 0.04082007-9-28 0.0578 0.0557 0.0573 0.0576 0.0401

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMM stands for Swap Market Model andLMM stands for Libor Market Model.11Y Bermudan swaption are priceed as at-the-money from February 2005 toSeptember 2006; 10 year Bermudan swaptions are priced with the corresponding 11Y ATM strikes one year ago.Forexample, on February 2006, the strike rate is 0.0402 which is ATM strike rate on February 2005. 10 year Bermudanswaption are priced from February 2006 to September 2007.

Table 3(b): 11Y and 10Y Bermudan Swaption Pricesin USD Market from February 2005 to September 2007

DATE(11Y) HW BK SMM LMM2005-2-28 0.0543 0.0528 0.0562 0.05762005-3-31 0.0526 0.0516 0.0568 0.05972005-4-29 0.0552 0.0539 0.0574 0.05922005-5-31 0.0545 0.0532 0.0555 0.05662005-6-30 0.0557 0.0546 0.0570 0.05822005-7-29 0.0512 0.0508 0.0529 0.05392005-8-31 0.0537 0.0531 0.0547 0.05582005-9-30 0.0522 0.0518 0.0532 0.05402005-10-31 0.0517 0.0515 0.0531 0.05422005-11-30 0.0543 0.0541 0.0545 0.05522005-12-30 0.0525 0.0526 0.0520 0.05212006-1-31 0.0517 0.0517 0.0500 0.04952006-2-28 0.0470 0.0475 0.0430 0.04152006-3-31 0.0487 0.0486 0.0459 0.04482006-4-28 0.0470 0.0465 0.0455 0.04502006-5-31 0.0485 0.0483 0.0463 0.04542006-6-30 0.0465 0.0464 0.0432 0.04202006-7-31 0.0480 0.0477 0.0446 0.04342006-8-31 0.0488 0.0484 0.0441 0.04242006-9-29 0.0493 0.0489 0.0448 0.0431

DATE(10Y) HW BK SMM LMM Strike2006-2-28 0.0472 0.0468 0.0430 0.0418 0.04962006-3-31 0.0507 0.0497 0.0471 0.0464 0.05222006-4-28 0.0741 0.0713 0.0701 0.0698 0.04822006-5-31 0.0848 0.0817 0.0802 0.0798 0.04642006-6-30 0.0926 0.0899 0.0883 0.0877 0.04542006-7-31 0.0696 0.0670 0.0643 0.0635 0.04862006-8-31 0.0716 0.0688 0.0653 0.0640 0.04562006-9-29 0.0534 0.0519 0.0485 0.0473 0.04922006-10-31 0.0422 0.0419 0.0383 0.0368 0.05202006-11-30 0.0369 0.0372 0.0349 0.0338 0.05172006-12-29 0.0464 0.0453 0.0428 0.0418 0.05032007-1-31 0.0467 0.0455 0.0424 0.0429 0.05122007-2-28 0.0393 0.0388 0.0372 0.0366 0.05142007-3-30 0.0337 0.0336 0.0327 0.0319 0.05482007-4-30 0.0271 0.0277 0.0270 0.0264 0.05732007-5-31 0.0308 0.0311 0.0289 0.0276 0.05772007-6-29 0.0385 0.0381 0.0371 0.0368 0.05852007-7-31 0.0460 0.0453 0.0453 0.0451 0.05662007-8-31 0.0447 0.0437 0.0458 0.0467 0.05382007-9-28 0.0506 0.0491 0.0526 0.0547 0.0525

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMM stands for Swap Market Model andLMM stands for Libor Market Model. 11Y Bermudan swaptions are priced as at-the-money from February 2005 toSeptember 2006; 10-year Bermudan swaptions are priced with the corresponding 11-year ATM strikes one year ago. Forexample, on February 2006, the strike rate is 0.0402 which is ATM strike rate for the 11-year Bermudan swaption onFebruary 2005. 10-year Bermudan swaptions are priced from February 2006 to September 2007.

Table 4: PCA Factor Loadings of Libor Rates from January 2000 to September 2007

EUR USDRates PCA1(EURO) PCA2(EURO) PCA3(EURO) PCA1(USD) PCA2(USD) PCA3(USD)

1 0.16 -0.41 0.16 0.15 -0.25 0.452 0.37 -0.44 0.12 0.34 -0.30 0.423 0.40 -0.34 0.02 0.37 -0.20 0.224 0.35 -0.17 -0.02 0.34 -0.09 0.035 0.33 -0.04 -0.12 0.32 -0.55 -0.436 0.30 0.11 -0.09 0.32 0.58 0.407 0.23 0.14 -0.76 0.30 0.10 -0.158 0.33 0.44 0.60 0.29 0.16 -0.199 0.27 0.33 -0.02 0.29 0.19 -0.2610 0.24 0.30 -0.06 0.28 0.25 -0.2111 0.25 0.27 -0.05 0.26 0.14 -0.22

EUR Percentagevariance

Cumulativevariance USD Percentage

vararianceCumulativevarariance

PCA1 69.2% 69.2% PCA1 79.6% 79.6%PCA2 16.2% 85.4% PCA2 8.7% 88.3%PCA3 7.6% 93.0% PCA3 5.9% 94.2%

Table 5: Explainary Power of the First Three Pinciple Comp

Table 6: Hedging Profit and Loss in EUR and USD Market

EUR HW BK SMM LMM2006-02-28 0.0064 0.0024 0.0037 0.00472006-03-31 0.0081 0.0069 0.0117 0.01372006-04-28 0.0064 0.0065 0.0080 0.00802006-05-31 0.0063 0.0074 0.0118 0.01332006-06-30 0.0018 0.0042 0.0137 0.00652006-07-31 0.0050 0.0075 0.0127 0.01262006-08-31 0.0054 0.0069 0.0109 0.00992006-09-29 0.0057 0.0076 0.0078 0.00682006-10-31 0.0109 0.0112 0.0115 0.01412006-11-30 0.0109 0.0104 0.0130 0.00942006-12-29 0.0053 0.0077 0.0059 0.00572007-01-31 0.0060 0.0088 0.0080 0.00392007-02-28 0.0162 0.0198 0.0184 0.01552007-03-30 0.0194 0.0260 0.0259 0.02482007-04-30 0.0258 0.0314 0.0381 0.03882007-05-31 0.0361 0.0399 0.0385 0.03572007-06-29 0.0266 0.0317 0.0334 0.02962007-07-31 0.0211 0.0261 0.0246 0.02092007-08-31 0.0061 0.0090 0.0118 0.00402007-09-28 -0.0065 -0.0037 -0.0058 -0.0069

RMSS 0.0149 0.0175 0.0189 0.0175

USD HW BK SMM LMM2006-02-28 0.0102 0.0099 0.0159 0.01682006-03-31 0.0009 0.0010 0.0081 0.02372006-04-28 -0.0024 0.0027 0.0034 0.01852006-05-31 -0.0083 -0.0017 -0.0010 0.00572006-06-30 -0.0109 -0.0035 -0.0040 0.00452006-07-31 -0.0031 0.0028 0.0052 0.01482006-08-31 -0.0007 0.0052 0.0084 0.01592006-09-29 -0.0003 0.0012 0.0056 0.01562006-10-31 0.0104 0.0107 0.0157 0.02792006-11-30 0.0193 0.0193 0.0216 0.03072006-12-29 0.0041 0.0052 0.0064 0.01722007-01-31 0.0043 0.0057 0.0073 0.01402007-02-28 0.0094 0.0103 0.0072 0.00702007-03-30 0.0197 0.0200 0.0180 0.01722007-04-30 0.0274 0.0268 0.0265 0.02602007-05-31 0.0239 0.0238 0.0239 0.02422007-06-29 0.0149 0.0156 0.0132 0.00842007-07-31 0.0096 0.0106 0.0071 0.00132007-08-31 0.0115 0.0127 0.0055 -0.00272007-09-28 -0.0010 -0.0005 -0.0081 -0.0126

RMSS 0.0554 0.0548 0.0570 0.0775

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMMstands for Swap Market Model and LMM stands for Libor Market Model. All numbersare in real value. RMSS is the root mean sum of sqaure of all P&L in each period

Figure 1: Root Mean Square Prices Error by Date from Calibration to 11-year Co-terminal European Swaptions over the period of February 2005 to September 2006

Note: HW stands for Hull-White model; BK stands for Black-Karasinski model;SMM stands forSwap Market Model and LMM stands for Libor market Model.

EUR

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Date

RM

SE

HWBKSMMLMM

USD

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Date

RM

SE

HWBKSMMLMM

Figure 2: Root Mean Square Prices Error by Contractin EUR and USD Market from February 2005 to September 2006

Note: HW stands for Hull-White model; BK stands for Black-Karasinski model;SMM stands forSwap Market Model and LMM stands for Libor market Model. x*y means x-year option on y-yearswap. Calibration is made simultaneously to 11-year (x+y) co-terminal European swapition.

EUR

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

1.0%

1*10 2*9 3*8 4*7 5*6 6*5 7*4 8*3 9*2 10*1

Contract

RM

SE

HW

BK

SMM

LMM

USD

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1*10 2*9 3*8 4*7 5*6 6*5 7*4 8*3 9*2 10*1

Contract

RM

SE

HW

BK

SMM

LMM

Figure 3: Calibrated Parameter Values of Four Term Structure Modelsin EUR and USD Markets from February 2005 to September 2006

Note: HW stands for Hull-White modle,a in HW model is the mean reversion rate, b ,c and d is sigma0, sigma 3 and sigma 11 respectively; BK standsfor Black-Karasinski model, a in HW model is the mean reversion rate, b ,c and d is sigma(0),sigma(3) and sigma(11) respectively; SMM stands forSwap Market Model and LMM stands for Libor Market Model.a,b,c,d in SMM and LMM are the parameters of the volatility functional form assuggested by Rebonato 1999. Wecalibrated to co-terminal 11Y European swaptoins from Feb 2005 to Sep 2006. All calibration are performed withboth EUR and USD market.

HW Parameters(EUR)

-

0.002

0.004

0.006

0.008

0.010

0.012

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

BK Parameters(EUR)

-

0.05

0.10

0.15

0.20

0.25

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

HW Parameters(USD)

-

0.002

0.004

0.006

0.008

0.010

0.012

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

BK Parameters(USD)

-

0.05

0.10

0.15

0.20

0.25

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

SMM Parameters(USD)

-0.10-0.05

-0.050.100.150.200.250.300.350.400.45

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

LMM Parameters(USD)

-0.3

-0.2

-0.1

-

0.1

0.2

0.3

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0.5

0.6

Feb-05

Apr-05

Jun-05

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Dates

value

a

b

c

d

SMM Parameters(EUR)

-0.1

-

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

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Aug-06

Dates

value

a

b

c

d

LMM Parameters(EUR)

-0.2

-

0.2

0.4

0.6

0.8

1.0

1.2

Feb-05

Apr-05

Jun-05

Aug-05

Oct-05

Dec-05

Feb-06

Apr-06

Jun-06

Aug-06

Dates

value

a

b

c

d

Figure 4: 11Y and 10 Year Bermudan swaption Prices in EUR and USD Markets from February 2005 to September 2007

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMM stands for Swap Market Model and LMM stands for Libor Market Model.11YBermudan swaption is price as at-the-money from Feb 2005 to Sep 2006; 10 year Bermudan swaptions are priced with the corresponding 11Y ATM strikes one year ago. 10year Bermudan swaption are priced from Feb 2006 to Sep 2007.

11Y Bermudan Swaption Prices(EUR)

0%

1%

2%

3%

4%

5%

6%

Feb-05

Apr-05

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Dates

Pric

esHWBKSMMLMM

10Y Bermudann Swaption Prices (EUR)

0%

1%

2%

3%

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6%

7%

8%

Feb-06

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es

HWBKSMMLMM

11Y Bermudan Swaption Prices (USD)

0%

1%

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Feb-05

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es

HWBKSMMLMM

10Y Bermudan Swaption Prices (USD)

0%1%2%3%4%5%6%7%8%9%

10%

Feb-06

Apr-06

Jun-06

Aug-06

Oct-06

Dec-06

Feb-07

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Date

Pric

es

HWBKSMMLMM

Figure 5: PCA factor loading for the first three principle component in EUR and USD Market

P1 P2 P3EUR 0.005216 0.002344 0.0010106USD 0.0082656 0.002051 0.0020943

Mean of first Three Principal Components

PCA Loadong for Factor 1

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

1 2 3 4 5 6 7 8 9 10 11

Rates

Val

ue P1(Euro)P1(USD)

PCA Loadong for Factor 2

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

1 2 3 4 5 6 7 8 9 10 11

Rates

Val

ue P2(Euro)P2(USD)

PCA Loadong for Factor 3

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

1 2 3 4 5 6 7 8 9 10 11

Rates

Val

ue P3(Euro)P3(USD)

Figure 6: Factor Loadings for the first three principle components of forwardrates term structure in the EUR and USD markets

PCA 1 Facotr Loading

0.0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 10 11

Rates

Loadings

PCA1(EURO)

PCA1(USD)

PCA 2 Facotr Loading

-0.8-0.6-0.4-0.20.00.20.40.60.8

1 2 3 4 5 6 7 8 9 10 11

Rates

Loadings

PCA2(EURO)

PCA2(USD)

PCA 3 Facotr Loading

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 10 11

Rates

Loadings

PCA3(EURO)

PCA3(USD)

Figure 7: Hedging Profit and Loss in EUR and USD market from Feb 2006 to Sep 2007

Note: HW stands for Hull-White modle, BK stands for Black-Karasinski model, SMM stands forSwap Market Model and LMM stands for Libor Market Model. All numbers are in real monetaryvalue.

EUR

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

Feb-06

Apr-06

Jun-06

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Date

Val

ue

HWBKSMMLMM

USD

-0.02

-0.01

-0.01

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

Feb-06

Apr-06

Jun-06

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Date

Val

ue

HWBKSMMLMM