a market model approach for measuring counterparty credit...

Master Thesis

A Market Model Approach for MeasuringCounterparty Credit Risk of Interest Rate Derivatives

April 2010

Tobias Beck

.

Master of Science in Business Mathematics

Submission date: April 30, 2010

Tobias Beck

Weberstraße 3260318 Frankfurt am [email protected]

Supervised by:

Prof. Dr. Marcus R. W. Martin (University of Applied Sciences Darmstadt)Prof. Dr. Wilfried Hausmann (University of Applied Sciences Friedberg)

Abstract

This thesis presents a LIBOR market model to simulate interest rates and to mea-sure the counterparty credit risk of single currency, over-the-counter interest ratederivatives. The model is calibrated under empirical measure and therefore presentsa method for estimating the model’s volatility and correlation structure from his-torical forward rates.

This work analyzes the behavior of the simulated forward rates under different mea-sures and the effects of a reduction in the number of model factors. The model isused to simulate based on different time grids, which leads to a recommendation forthe placement of portfolio valuation in the simulation model. Moreover, the model’ssensitivity to changes in its parameters is analyzed and the importance of the ini-tial forward rate curve is observed. The expected exposure distribution of differentover-the-counter derivatives is presented and compared to the market quotations ob-served in and after the financial crises of 2008. This leads to the discovery of somedisadvantages with the model. Finally, the behaviors of the estimated EAD, RWA,exposure profiles, and potential exposure profiles of swaps are studied over timesof both moderate and strong market movements. For this purpose, the model isre-calibrated in ten day steps. This analysis shows that the particular market modelapproach presented in this thesis is highly depend on the initial forward curve. Forthis reason the estimated quantities change strongly over time.

i

Acknowledgements

I would like to express my gratitude to my supervisor Prof. Dr. Marcus R.W.Martin for providing the topic of this thesis, and for the guidance and assistanceduring its preparation. I am also indebted to my cosupervisor Prof. Dr. WilfriedHausmann.

I am particularly thankful for the invaluable feedback from my parents, my sister,colleagues and friends which have greatly improved and clarified this work. Espe-cially I would like to thank Rahel Franziska Finckh, Marlies Beck, Rainer Beck, andKristopher K. Kruger for their support during this thesis.

I dedicate this thesis to my parents who unremittingly supported me during myyears of study. They made this work possible.

Frankfurt am Main, April 30, 2010

ii

Contents

1 Introduction 1

2 Counterparty Credit Risk 32.1 Introduction to Counterparty Credit Risk . . . . . . . . . . . . . . . 32.2 Quantification of Counterparty Credit Risk . . . . . . . . . . . . . . 4

3 Interest Rates and Interest Rate Products 103.1 Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Interest Rate Products and Derivatives . . . . . . . . . . . . . . . . 14

3.2.1 Forward Rate Agreement (FRA) . . . . . . . . . . . . . . . 153.2.2 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Caps/Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 The Term Structure of Interest Rates . . . . . . . . . . . . . . . . . 183.4 Svensson’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 The Interest Rate Simulation Model 224.1 Interest Rate Model Selection . . . . . . . . . . . . . . . . . . . . . 224.2 The LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . 264.3 Reduced-Rank Formulations of the Correlation Matrix . . . . . . . 314.4 Simulating Forward Rates using the LIBOR Market Model . . . . . 33

4.4.1 Simulating under the T -Forward Measure . . . . . . . . . . . 334.4.2 Simulating under the Spot Measure . . . . . . . . . . . . . . 35

4.5 Estimation of the Model Parameters . . . . . . . . . . . . . . . . . 364.5.1 Interest Rate Data Structure and Forward Rates . . . . . . . 364.5.2 Estimating the Instantaneous Correlation of Forward Rates . 374.5.3 Estimating the Instantaneous Volatility of Forward Rates . . 38

4.6 Pricing of Interest Rate Products and Derivatives in the LMM Sim-ulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6.1 Pricing Bonds, FRAs and Swaps within the LMM Simulation

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6.2 Pricing Caps/Floors within the LMM Simulation Model . . 42

5 Data Description and Model Calibration 445.1 Historical Interest Rate Curves . . . . . . . . . . . . . . . . . . . . 445.2 Parameterized Functional Form of the Swap Rates . . . . . . . . . . 455.3 Estimated LMM Parameters . . . . . . . . . . . . . . . . . . . . . . 47

iii

Contents iv

5.3.1 Estimation of the Correlation Structure . . . . . . . . . . . . 475.3.2 Estimation of the Volatility Structure . . . . . . . . . . . . . 48

5.4 Reducing the Number of LMM Factors . . . . . . . . . . . . . . . . 515.5 The Validation Sample . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 Implied Cap Volatilities for Pricing IR-Options . . . . . . . . . . . . 54

6 Results 576.1 Forward Rate Distribution . . . . . . . . . . . . . . . . . . . . . . . 576.2 Exposure Profiles of Interest Rate Products . . . . . . . . . . . . . 65

6.2.1 Exposure Profiles of Interest Rate Swaps . . . . . . . . . . . 656.2.2 Exposure Profiles of Caps and Floors . . . . . . . . . . . . . 71

6.3 Reducing the Number of Factors . . . . . . . . . . . . . . . . . . . . 736.4 Reducing the Simulation’s Time Grid . . . . . . . . . . . . . . . . . 82

6.4.1 Reduction of Portfolio Valuations . . . . . . . . . . . . . . . 826.4.2 Different Reductions of the Simulation’s Time Grid . . . . . 85

6.5 Sensitivity of the Simulation Results due to changes in the ModelParameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.5.1 Impact of the Initial Forward Rate Curve on the Simulation

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.5.2 Impact of the Forward Rate Volatility on the Simulation Results 936.5.3 Impact of the Correlation Structure on the Simulation Results 966.5.4 Pricing using Different Cap Volatility Surfaces . . . . . . . . 101

6.6 Change of Estimation Results over Time . . . . . . . . . . . . . . . 1036.6.1 Estimation Results in a Period of High Market Stress . . . . 1036.6.2 Estimation Results in a Period of Moderate Market Movements 1086.6.3 Closing Conclusion of the Estimation Results over Time . . 113

7 Concluding Remarks 1167.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A Results from Stochastic Calculus 123

List of Figures

2.1 Exposure measures applied on a swap contract. . . . . . . . . . . . 72.2 Capital requirement function for fixed maturity . . . . . . . . . . . 9

3.1 Examples of the Svensson Model fitted to term structures of interestrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Critical examples of the Svensson Model fitted to term structures ofinterest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1 The historical term structure of interest rates . . . . . . . . . . . . 455.2 Svensson calibration results . . . . . . . . . . . . . . . . . . . . . . 465.3 Estimated historical forward rates for calibration . . . . . . . . . . 485.4 Estimated correlation structure . . . . . . . . . . . . . . . . . . . . 495.5 Volatility estimation results . . . . . . . . . . . . . . . . . . . . . . 505.6 Estimated volatility coefficients of the LMM . . . . . . . . . . . . . 515.7 Estimated historical forward rates for validation . . . . . . . . . . . 535.8 Example of estimated historical forward rates curves used for validation 545.9 Historical evolution of ATM implied cap/floor volatilities . . . . . . 555.10 Examples of cap volatility surface . . . . . . . . . . . . . . . . . . . 56

6.1 Evolution and distribution of F1(t) . . . . . . . . . . . . . . . . . . 606.2 Evolution and distribution of F5(t) . . . . . . . . . . . . . . . . . . 616.3 Evolution and distribution of F10(t) . . . . . . . . . . . . . . . . . . 626.4 Evolution and distribution of F20(t) . . . . . . . . . . . . . . . . . . 636.5 Distribution of forward rates at expiry . . . . . . . . . . . . . . . . 646.6 Simulated exposure profiles for a one year swap . . . . . . . . . . . 676.7 Simulated exposure profiles for a two and a half year swap . . . . . 686.8 Simulated exposure profiles for a five year swap . . . . . . . . . . . 696.9 Simulated exposure profiles for a ten year swap . . . . . . . . . . . 706.10 Simulated exposure profiles for a ten and five year cap and floor . . 726.11 Reducing model factors - example path of a swap’s value . . . . . . 746.12 Reducing model factors - EE(t) and PFE(t) of a payer swap . . . . 756.13 Reducing model factors - EE(t) and PFE(t) of a receiver swap . . . 766.14 Reducing model factors - example path of F10(t) and F20(t) . . . . . 776.15 Reducing model factors - mean of F10(t) and F20(t) . . . . . . . . . 786.16 Reducing model factors - spread F20(t) minus F10(t) . . . . . . . . . 806.17 EAD and RWA of caps and floors depending on simulation’s time grid. 84

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List of Figures vi

6.18 EE(t) of caps and floors depending on simulation grids . . . . . . . 866.19 PFE(t) of caps and floors depending on simulation grids . . . . . . 876.20 Applied shifts to the forward rate curve . . . . . . . . . . . . . . . . 906.21 F20(t) simulation using different forward rate curves . . . . . . . . . 916.22 EE(t) profiles of a cap and swaps using different initial forward rate

curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.23 Applied shifts to the forward rate volatilities . . . . . . . . . . . . . 936.24 F20(t) simulation using different forward rate volatility functions . . 946.25 EE(t) profiles of a cap and swaps using different forward rate volatilities 956.26 Correlations of F10 and F20 to other forward rates . . . . . . . . . . 976.27 F20(t) simulation using different forward rate correlations . . . . . . 996.28 EE(t) profiles of a cap and swaps using different forward rate corre-

lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.29 EE(t) and PFE(t) of caps and floors using two volatility surfaces . . 1026.30 Initial yields and forward rates over a period of high market stress . 1046.31 EE(t) of swaps over a period of high market stress . . . . . . . . . . 1056.32 Estimated EAD and RWA over a period of high market stress . . . 1066.33 PFE(t) of swaps over a period of high market stress . . . . . . . . . 1076.34 New model calibration for a period of moderate market movements 1096.35 EE(t) of swaps over a period of moderate market movements . . . . 1106.36 Estimated EAD and RWA over a period of moderate market move-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.37 PFE(t) of swaps over a period of moderate market movements . . . 1126.38 Estimated EAD using CEM and LMM simulation over time . . . . 114

List of Tables

4.1 Piecewise-constant volatility structure of the LMM . . . . . . . . . 284.2 LMM volatility structure only depending on the time to maturity . 40

5.1 Results on reducing the rank of the correlation matrix . . . . . . . . 525.2 Results on reducing the rank of the default correlation matrix . . . 52

6.1 Statistics on simulated forward rates at there time of expiry . . . . 596.2 Parameters of simulated interest rate swaps . . . . . . . . . . . . . 656.3 Parameters of simulated caps and floors . . . . . . . . . . . . . . . . 716.4 Reducing model factors - risk-weighted assets (example A) . . . . . 816.5 Reducing model factors - risk-weighted assets (example B) . . . . . 816.6 Time grids used for simulation . . . . . . . . . . . . . . . . . . . . . 856.7 Estimated RWA simulated on different time grids . . . . . . . . . . 886.8 Mean absolute bias in the PFE(t) simulated on different time grids 88

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1 Introduction

The financial crisis of 2008 highlighted the huge risks coming from over-the-counter(OTC) derivatives. When large financial players such as Lehmann Brothers, BearnStearns or AIG struggled during that time, their possible failure to fulfill their obli-gations from OTC derivatives was one of the issues which destabilized the financialsystem. The risk that a counterparty to a OTC transaction could default, is denotedas counterparty credit risk (CCR).

The largest class of OTC derivatives are single currency interest rate derivatives,which account for 69% of the OTC derivative market, with an outstanding notionalof 653,397 billion US dollars in June 2009.1

Therefore, a deep understanding and adequate assessment of the CCR coming fromsingle currency interest rates OTC derivatives is extremely important. MeasuringCCR leads to the question of how to model underlying risk factors. These can beclassified in counterparty specific risk factors like the counterparty’s default proba-bilities and recoveries at default and in contract specific risk factors. The latter areinterest rates and volatilities of interest rates.

This thesis presents a market model approach to model and simulate the contractspecific risk factors, the interest rates. For this, a LIBOR Market Model (LMM) cal-ibrated to the evolution of the historical yield curve is used. Based on the simulatedinterest rates we valuate interest rate derivatives and hence the expected exposuresto a counterparty and the CCR.

Furthermore, interest rates are typically somewhat of a cornerstone for measuringthe CCR of any other class of OTC derivatives, since for that purpose, potentialfuture interest rate scenarios are crucial.

1The amounts of notional amount were published by Bank for International Settlements (BIS)(2009), relating to the size and structure of derivatives markets in the G10 countries andSwitzerland.

1

1 Introduction 2

The thesis is organized as follows: In the second chapter the measurement of coun-terparty risk is discussed in more detail. Chapter 3 presents basic definitions oninterest rates and interest rate derivatives, which are used throughout the thesis.Furthermore, Chapter 4 illustrates the LIBOR Market Model and its calibration formodeling and simulating interest rates. Chapter 5 presents the model calibrationand calibration results. Finally, the simulation results are presented in Chapter 6and Chapter 7 concludes this thesis and highlights further points of research.

2 Counterparty Credit Risk

This chapter introduces Counterparty Credit Risk (CCR) and presents a possibleapproach to measure this particular credit risk. The description follows the publica-tions Basel Committee on Banking Supervision (2005b), Basel Committee on Bank-ing Supervision (2005c) as well as a collection of articles written by various authorspublished in the market standard monographically edited by Pykhtin (2005).

2.1 Introduction to Counterparty Credit Risk

Counterparty Credit Risk denotes the risk that the counterparty to a transactioncould default, i.e. fail to fulfill its obligations, before the final settlement of a trans-action’s cash flows. Contracts privately negotiated between counterparties, such asover-the-counter (OTC) derivatives, are subject to counterparty credit risk. In caseof a default a loss occurs, if the transaction’s economic value is positive. The eco-nomic value can be quantified as the price of a transaction reproducing all open cashflows. These costs are denoted as positive replacement costs. When the economicalvalue is negative, the transaction is again replaced, but counterpart is entitled to thisvalue. CCR creates a bilateral risk of loss, since the market value of a transactioncan be positive or negative to either counterparty.

Following the classical approaches for quantifying CCR, one can distinguish threecomponents. There is the counterparty’s probability of default (PD), which could beestimated based on the counterparty’s credit-worthiness. Additionally the exposureat default (EAD) has to be estimated, which is the transaction’s replacement costat time of default. Finally, in case a counterparty defaults, there is a possibility ofpartial recovery of the outstanding exposure. The percentage of the exposure, whichis lost due to default is denoted as loss given default (LGD).

3

2 Counterparty Credit Risk 4

In practice, these three components are not stochastically independent. To ease thepresentation, their independence is assumed following the general Basel II Frame-work, considering a special type of correlation risk later.

The exposure of OTC derivatives depends on underlying market factors such asinterest rates, foreign exchange rates, credit spreads, commodity yields, equity pricesor other. The counterparty’s probability of default can be effected by the samemarket factors. Hence, PD of and EAD to a counterparty can be either positivelyor negatively correlated. In such cases the terms wrong-way and right-way risk areused.

Future transactions with a counterparty are usually uncertain, but can be expectedto occur. The resulting risk is denoted as rollover risk and should be included intothe exposure estimation under some circumstances.

Typically there are several ways to reduce CCR, which are individually negotiatedbetween counterparties. This could be the arrangement of netting agreements whichallows to offset positive against negative transaction’s values. A portfolio affected bysuch an agreement is denoted as netting set. Additionally collateral agreements couldbe in place, which obligate to balance positive and negative exposure by collateral.

Furthermore, contract specific agreements such as early termination clauses couldbe part of a transaction. In a simple case, this could be an early cash settlementwhich allows one or both parties to determine the transaction at a specific date or aset of dates. The difference between the transaction’s value and the market value atthe determination date is balanced between the parties by cash. Therefore, it doesnot effect the transaction’s market value. In case the termination clause does notinclude the cash payment, it is comparable to an option and therefore a contractitself.

2.2 Quantification of Counterparty Credit Risk

The most referred measures of CCR can be found in the publications of the BaselCommittee on Banking Supervision, especially Basel Committee on Banking Super-vision (2005b) and Basel Committee on Banking Supervision (2005c), which were


developed in cooperation with the International Organization of Securities Com-missions (IOSCO). The Basel Committee on Banking Supervision presented threeapproaches to estimate the EAD for transactions exposed to CCR. In this thesis,only take a closer look on the Internal Model Method is taken, which defines, ac-cording to Pykhtin and Zhu (2006), the most risk-sensitive approach within thisframework. After presenting this EAD estimation method the description proceedswith the definition of the minimum capital requirements due to CCR according tothe current capital amount.1

Assume a set of transactions negotiated with a counterparty and affected by a nettingagreement. Let V (t) denote the value of this netting set at time t. The ExpectedExposure EE(t) is defined as the expectation of the exposures distribution of V (t)

at any particular future date t, floored by zero, hence

EE(t) = E[max(V (t), 0)].

This could be estimated by the mean exposure over some future market scenariosgenerated by adequate simulations. The Effective EE(t) at a specific date t is spec-ified as the maximum expected exposure that occurs at this or any prior date, i.e inmathematical terms

Effective EE(t) = maxu≤t

EE(u).

Furthermore, the Expected Positive Exposure EPE(t) denotes the average over aspecific time interval of the EE(t), i.e.

EPE(t) =1

t

∫ t

0

EE(u) du.

The Effective Expected Positive Exposure (Effective EPE) denotes the average ofthe Effective EE given by

Effective EPE(t) =1

t

∫ t

0

Effective EE(u) du,

For calculating the positions EAD, t is set to the lesser of the longest remaining

1While writing, further amendments to the current IMM frameworks were on the way, whichdo not influence our results here, but which do influence other areas of IMM. For more detailsee for example Basel Committee on Banking Supervision (2009a) and Basel Committee onBanking Supervision (2009b)


time to maturity of contracts within the netting set and one. This is due to the factthat the calculation of the minimum capital requirements, hence the EAD, is basedon a capital horizon of one year.2

Besides this expectation-measures, the Potential Future Exposure (PFE) is set to themaximum exposure estimated to occur on a future date at a high level of statisticalconfidence. According to (Basel Committee on Banking Supervision, 2005c, Page 4)banks often use PFE when measuring an exposure against the counterparty’s creditlimits and for decision making.

Coming back to the estimation of the EAD following the notation of Basel Commit-tee on Banking Supervision (2005b), which is computed according to

EAD = α× Effective EPE(t∗),

where t∗ is set to the minimum of one and the longest remaining time to maturitywithin the netting set. The scaling factor α is set to be 1.4 or higher if supervisorsrequire. Banks may also apply for a regulatory approval to use own estimates ofα which, by regulation, will necessarily be floored at 1.2. Such estimates of αbasically equal the ratio of economic capital from an extensive simulation acrossall counterparties to the economic capital based on EPE. An extensive simulationwould cover all risk factors, all securities and default probabilities and especiallythe correlations among these components. In other words, the Basel Committeeon Banking Supervision proposed this offset to capture wrong-way risk and thepotential lack of granularity of the simulation model.

Since wrong-way risk is captured by the offset α and right-way risk is reducing CCR,roll-over risk is left from the list of CCR-related risks. This risk is mainly dependenton the bank’s investment strategy. However, the Basel Committee on Banking Su-pervision placed the offset for this risk by including the running maximum (EPE(t))in the calculation of the EAD (at least on the one year capital horizon).

Figure 2.1 shows examples of the capital measures as described above, based ona swap contract maturing in ten years. The figure shows in particular possibledifferences between exposure measures and measures on a high level of statisticalconfidence.

2Note that some authors define the Effective EPE(t) and Effective EE(t) just for t ≤ 1, since itis designed for the calculation of EAD.


0 2 4 6 8 10t

0.5

1.0

1.5

2.0

2.5

3.0

(a) The measures EE(t) andEffective EE(t) in blue. EPE(t)and Effective EPE(1) in purple.

0 2 4 6 8 10t

10

20

30

40

50

(b) EE(t) and the Max(t) are shown inblue, the quantiles at a level of .95, .975,and .99 in purple.

Figure 2.1: Chart (a) shows the described CCR, i.e. exposure measures based on asingle payer swap example. The Effective EPE is calculated for a oneyear period as used for regulatory minimum capital requirements. Theswap is paying floating and receiving fixed both semi-annualy over aperiod of 10 years. Figure (b) shows possible PFE(t)-measures in con-trast to the expected exposure and Max(t), which denotes the maximumobserved exposure over all simulations.

However, the EAD denotes just one component of the calculation of capital re-quirements due to exposures to counterparties. Other components such as PD andLGD were already introduced. We will not consider possible estimation methodsfor these probabilities. Again we follow Basel Committee on Banking Supervision(2005b) and consider the counterparty to be a bank. Furthermore, no securities,such as collateral, are taken into account.3

To calculate the minimum capital requirements CB, some definitions are required.At first the effective maturity (M), which is used in case the maturity of the longest-dated contract Tmax is greater than one year. M is defined by

M = max

(∫ 1

0Effective EE(u)D(0, u) du+

∫ Tmax

1EE(u)D(0, u) du∫ 1

0Effective EE(u)D(0, u) du

, 5

),

where D(t, T ) denotes a risk-free discount factor4. Additionally a correlation factor

3The Basel II Framework distinguishes various approaches to calculate risk-weigthed assets andminimum capital requirements dependet on the asset class of the counterparty. For more detailsee Basel Committee on Banking Supervision (2005b). For this description it is restricted tothe treatment of corporate, sovereign, and bank exposures.

4An introduction to interest rates is given in Chapter 3. Definition 3.2 presents the discountfactor D(t, T ).


ρB is calculated according to

ρPD = 0.24− 0.121− e−50 PD

1− e−50.

The capital requirement CB, expressed as a percentage of the exposure, is definedby

CB(PD,LGD,M) =(LGDΦ

(1√

1− ρPDΦ−1(PD) +

√ρPD

1− ρPDΦ−1(.999)

)−PD · LGD

)1 + (M − 2.5) (0.11852− 0.05478 lnPD)2

1− 1.5 (0.11852− 0.05478 lnPD)2

but floored at zero in case CB returns negative results. Φ denotes the cumulativedistribution function of a standard normal random variable and Φ−1 its inverse.

Since CB is expressed as percentage of exposure, the total minimum capital re-quirement is given by the product of CB and EAD. Finally, the risk-weighted assets(RWA) are given by RWA = 12.5 · CB · EAD.5

Recall the swap example given by Figure 2.1. One can calculate the effective matu-rity, which is given by 5 years due to the floor, where the risk free rate flat is set to5%. Figure 2.2 shows the function CB dependent on PD and LGD.

5Amore detailed discussion and interpretation on these formulas can be found in Basel Committeeon Banking Supervision (2005a).


0.1

0.2

PD 0.6

0.8

1.0

LGD

0.4

0.6

0.8

Figure 2.2: The capital requirement CB expressed as a percentage of the exposuredependent on PD and LGD for fixed effective maturity M = 5.

3 Interest Rates and Interest Rate

Products

In this chapter basic definitions on interest rates and interest rate derivatives areintroduced. This definitions can be found in fundamental textbooks on interest ratederivatives such as Musiela and Rutkowski (2005), Brigo and Mercurio (2005), andRebonato (2002).

This chapter ends with the description of an approach to model interest rate curvesin a functional form.

3.1 Interest Rates

Expressing interest rates in mathematical terms requires a large number of defi-nitions to develop a consistent theoretical apparatus. This also includes generalassumptions and conditions. A comprehensive composition of this aspect can befound in Musiela and Rutkowski (2005), which is used as given.

Given this general assumption, the introduction starts with the definition of a bankaccount.

Definition 3.1 (Bank account). Let B(t) define the price of a security with avalue of 1 at time t = 0 following the differential equation:

dB(t) = rtB(t) dt, B(0) = 1, (3.1)

where rt denotes a risk free and positive interest rate over time. The process B(t)can also be considered as the value of a bank account. The instantaneous rate rt isalso denoted as instantaneous spot interest rate, or briefly as short rate.

10

3 Interest Rates and Interest Rate Products 11

Consequently, one can rewrite the differential Equation (3.1) as follows

B(t) = exp

(∫ t

0

rs ds

)This expression can also be used to calculate the present value of a future payoffusing a so-called discount factor.

Definition 3.2 (Discount factor). The (stochastic) discount factor D(t, T ) rep-resents the value at time t < T of one unit of currency payable at time T , and isgiven by

D(t, T ) =B(t)

B(T )= exp

(−∫ T

t

rs ds

)(3.2)

Futhermore, we need the notation of a zero-coupon bond, also referred to as purediscount bond.

Definition 3.3 (Zero-coupon bond). A zero-coupon bond denotes a securitywhich pays a fixed amount of one unit of currency at its maturity T and has nointermediate payments. The value of this contract at a time t < T is referred to asP (t, T ), where P (T, T ) = 1.

In financial markets, there are different conventions for counting the time to matu-rity, or more precisely a year fraction.

Definition 3.4 (Time to maturity). The time to maturity T − t is the amountof time (in years) from the present time t to the maturity time T > t.

Definition 3.5 (Year fraction). Let τ(s, t) denote the time difference between thedates s and t, which is given in a year fraction under a particular time measurement,where τ(s, t) is chosen to reflect day-count conventions.

There are a number of different day-count conventions in the financial markets.Some common conventions are:

• Actual/365: The year fraction between the two dates D1 and D2 is measuredas the actual number of days D2 −D1 divided by 365, hence

τ(D1, D2) =D2 −D1

365.


• Actual/360: The year fraction under this day-count convention is calculatedas

τ(D1, D2) =D2 −D1

360.

• 30/360: Under this convention, months are assumed to consist of 30 days andyears to consist of 360 days. To express this method in mathematical termsthe dates D1 and D2 are separated into the components day d, month m, andyear y to define

τ(D1, D2) =max(30− d1, 0)−min(d2, 30) + 360 (y2 − y1) + 30 (m2 −m1 − 1)

360.

Beside these different day-count conventions, there are also different interest ratecompounding conventions, which are explained within the following two defini-tions.

Definition 3.6 (Continuously-compounded spot interest rate). The continuously-compounded spot interest rate prevailing at time t for maturity T is denoted byR(t, T ). Its relation to the zero-coupon bond P (t, T ) is given by

R(t, T ) := − lnP (t, T )

τ(t, T ).

This continuously-compounded interest rate is therefore a constant yield which isconsistent with the zero-coupon-bond price in that

eR(t,T )τ(t,T )P (t, T ) = 1

and the bond price can expressed by

P (t, T ) = e−R(t,T )τ(t,T ). (3.3)

An alternative to continuous compounding is the so-called simply-compounding,which is defined hereafter.

Definition 3.7 (Simply-compounded spot interest rate). The constant rateL(t, T ) is denoted as the simply-compounded spot interest rate prevailing at timet for the maturity T . It is the constant yield which an investment has to produce


for paying one unit at time T starting from P (t, T ) units currency under someday-measure τ(t, T ). In formulas and solved to L(t, T ) it is given by

L(t, T ) :=1− P (t, T )

τ(t, T )P (t, T ). (3.4)

In the following sections, L(t, T ) is also referred to as LIBOR rates, which arealso simply-compounded spot rates and which are typically linked to a day-countconvention of Actual/360 for computing τ(t, T ).

Equation (3.4) can be rewritten to

P (t, T ) =1

1 + L(t, T )τ(t, T ), (3.5)

which expresses the zero-coupon bond price in terms of L(t, T ).

To describe rates that can be locked today for investments starting in the future forsome period, forward rates will be defined.

Definition 3.8 (Simply-compounded forward rate). The simply-compoundedforward rate prevailing at time t for the expiry T > t and maturity S > T is denotedby F (t, T, S) and is defined by

F (t, T, S) :=1

τ(T, S)

(P (t, T )

P (t, S)− 1

). (3.6)

The description proceeds with the definition of the instantaneous forward rate. Forthis consider the limit

limS→T+

F (t, T, S) = − limS→T+

1

P (t, S)

(P (t, S)− P (t, T )

S − T

)= − 1

P (t, T )

∂P (t, T )

∂T

= −∂ lnP (t, T )

∂T

where the day-count convention τ(T, S) is set to S − T for simplicity.

Definition 3.9 (Instantaneous forward interest rate). The instantaneous for-ward interest rate prevailing at time t for the maturity T > t is denoted by f(t, T )


and defined asf(t, T ) := lim

S→T+F (t, T, S) = −∂ lnP (t, T )

∂T.

It shows the connection to the zero-coupon-price by

P (t, T ) = exp

(−∫ T

t

f(t, u)du

). (3.7)

In the case that money is invested over a one year period receiving a simply-compounded rate Y and it is reinvested for another period afterwards, this rateis denoted as annually-compounded spot interest rate. To extent this, one can allowto reinvest k times per year, which leads to Definition 3.10.

Definition 3.10 (k-times-per-year compounded spot interest rate). The k-times-per-year compounded spot interest rate prevailing at time t for maturity T isdenoted by Y k(t, T ) and describes a constant rate (referred to a period of one year)at which an amount P (t, T ) invested at time t accrues to 1 unit of currency at timeT , if proceeds are reinvested k times per year:

Y k(t, T ) :=k

P (t, T )1/(k τ(t,T ))− k.

For k = 1 this defines the annually-compounded spot interest rate. It can be rewrittento express the price of a zero-coupon bond in terms of the compounded spot rateas

P (t, T ) :=

(1 +

Y k(t, T )

k

)−k τ(t,T )

. (3.8)

3.2 Interest Rate Products and Derivatives

A (huge) universe of financial derivatives is based on interest rates as underling.For the purpose of this thesis just three basic derivates are considered. They aredescribed in the sections below.

In the following, we take the current textbook approach for pricing standard interestrate derivatives. Note that after the financial crisis a complete review has to be doneand current research indicates that all available approaches have to be extended by


liquidity and counterparty risk features to fit the market (see Mercurio (2009) andPiterbarg (2010) for more details).

3.2.1 Forward Rate Agreement (FRA)

Recall Definition 3.8 of simply-compounded forward rates. It can be used to define afuture investment in interest rates at time t for an expiry T and a maturity S > T . Attime S, an interest payment based on a fixed rate K is exchanged against a paymentbased on a floating rate L(T, S) fixed at time T with a maturity S, where the ratesare simply compounded. Including some notional amount N , the holder of thiscontract receives τ(T, S)KN and pays τ(T, S)L(T, S)N units of the same currencyat maturity S. Such a contract is defined as a forward rate agreement (FRA). Thecontract value in terms of zero-coupon bonds and the simply-compounded forwardrate is given by

FRA(t, T, S,N,K) := N(P (t, S)τ(T, S)K − P (t, T ) + P (t, S)) (3.9)

= NP (t, S)τ(T, S)(K − F (t, T, S)).

3.2.2 Interest Rate Swaps

The economical principle underlying a FRA can be extended to a contract includingmore but similar payments at different payment-dates based on a number of floatingrates. This generalization leads to the definition of an (Forward-start) Interest RateSwap (IRS), which is a contract that exchanges payments between two differentlyindexed legs, settled today or in a future time instant. At every instant Ti in aprespecified set of days T := {Tα+1, . . . , Tβ} the so-called fixed leg pays out theamount

N τiK

corresponding to a fixed interest rate K, a nominal N and a year fraction τi =

τ(Ti−1, Ti), whereas the contract’s so-called floating leg pays an amount of

N τi L(Ti−1, Ti)


at time Ti, where L(Ti−1, Ti) is set at time Ti−1. Note the simplification of simulta-neous payments of the fixed- and floating-rate. If the fixed rate is paid, the IRS istermed Payer IRS (from the payer’s view), if a fixed rate is received it is denotedas Receiver IRS.

The discounted payoff at time t < Tα of a payer IRS can be expressed as

β∑i=α+1

D(t, Ti)N τi (L(Ti−1, Ti)−K).

A receiver IRS can also be considered as a contract of FRAs, where the value ofeach FRA is given by formula (3.9). Hence, the value of a payer IRS is given by

PIRS(t, T , N,K) = −β∑

i=α+1

FRA(t, Ti−1, Ti, N,K)

= −Nβ∑

i=α+1

P (t, Ti−1) τi (K − F (t, Ti, Ti−1)) (3.10)

= N

(P (t, Tα)− P (t, Tβ)−

β∑i=α+1

τiKP (t, Ti)

).

One can now think about a particular rateK, where the contract above has a presentvalue of zero at its settlement date t. This leads to the following definition.

Definition 3.11 (Par swap rate). The forward swap rate Sα,β(t) at time t fora set of times T is defined as the IRS fixed leg rate K that makes the contract’svalue equal to zero, i.e. PIRS(t, T , N,K) = 0. This par swap rate (obtained fromEquation (3.10)) is given by

K := Sα,β(t) =P (t, Tα)− P (t, Tβ)∑βi=α+1 P (t, Ti)τ(Ti−1, Ti)

. (3.11)


3.2.3 Caps/Floors

This section derivative products concludes by introducing two main derivative prod-ucts of interest rate market, which are caps and floors.

A cap can be simply viewed as a payer IRS where each exchange payment is executedonly if it has positive value. The discounted payoff is therefore given by

β∑i=α+1

D(t, Ti)N τi (L(Ti−1, Ti)−K)+,

where t < Tα. A contract just consisting of one element of the summation above isdenoted as caplet, hence the cap is also a sum of caplets.

Analogously, a floor is equivalent to a receiver IRS where each exchange paymentis only executed, if it has positive value. Furthermore, one can define a floorlet as acontract on one payment of a floor, as seen from today’s perspective.

According to (Brigo and Mercurio, 2005, Page 17) and (Glasserman, 2003, Page 566)it is market practice, i.e. convention, to price caplets and floorlets using Black’sformula. Hence the price of caps and floors is given by a sum of Black’s formulas.

The value of a cap at time zero is given by

CAPBlack(0, T , N,K, σα,β) = N

β∑i=α+1

P (0, Ti) τi BL(K,F (0, Ti−1, Ti), σα,β√Ti−1, 1),

(3.12)where

BL(K,F, v, w) = FwΦ

(w

ln(F/K) + v2/2

v

)−KwΦ

(w

ln(F/K)− v2/2

v

)(3.13)

and with a common volatility parameter σα,β received from market quotations ofcaps and floors, generally depends on the strike K and the time to maturity T . Thevaluation of floors follows the same approach and results in

FloorBlack(0, T , N,K, σα,β) = N

β∑i=α+1

P (0, Ti) τi BL(K,F (0, Ti−1, Ti), σα,β√Ti−1,−1).


3.3 The Term Structure of Interest Rates

A basic input for the valuation of interest rate products is the so-called term structureof interest rates. It shows interest rates as a continuous function of their maturityand can be constructed from market data. It is common to use simply-compoundedmoney market rates for maturities of less than one year. For longer maturities, it iscommon to use either annually-compounded yield rates, zero-coupon rates, forwardrates, or swap rates, depending on the context.1 The definition of the term structurein mathematical terms is given below.

Definition 3.12 (Term structure of interest rates). The term structure (alsoreferred to as yield curve or zero-coupon curve) at a time t is a graph of the function

T 7→

L(t, T ) t < T < t+ 1

Y (t, T ) T > t+ 1, (3.14)

where t and T are expressed in years and Y (t, T ) refers to Definition 3.10 with k = 1.

In this thesis, annually-compounded swap rates are used for maturities longer thanone year. Thus, in Equation (3.14) the rate Y (t, T ) is replaced by the par swap rateSα,β(t) from Equation (3.11).

3.4 Svensson’s Model

For many purposes it is necessary to interpolate between the yield curves. Thereare various methods proposed in the literature. On one hand, there are classicalnumerical techniques such as linear, exponential, or spline interpolation. On theother hand, one can use models especially adapted to the term structure of interestrates.

One of the later models proposed by Svensson (1994) in principle can be used toestimate the instantaneous forward rates, where the description follows Packham(2005) among others. One advantage of the model is that it produces a parameter-ized functional form of the forward rate curve. This compact specification has many

1As mentioned in the introduction of Section 3.2, this topic is reviewed by current research oninterest rate markets.


advantages for following estimation procedures. However, this model cannot repro-duce the given yield curves in a perfect way like a spline interpolation would do.The model also creates a smoothness effect on the yield curve. The disadvantagesare in context of the missing abstinence of arbitrage opportunities due to the limitedreproducing of market quotations, and fitting problems in situations of high marketstress.2 But for the purposes this model as applied in this thesis, the approach seemsto be acceptable.

The Svensson’s Model describes the instantaneous forward rate f(τ) = f(t, t+ τ) ina functional way in terms of the parameter vector b = (β0, β1, β2, β3, t1, t2) through

f(τ, b) = β0 + β1 exp

(− τt1

)+ β2

τ

t1exp

(− τt1

)+ β3

τ

t2exp

(− τt2

), (3.15)

where β0, t1 and t2 are strictly positive. The second term of Equation (3.15) is amonotonically decreasing function to the settlement day, if β1 > 0 and monotonicallyincreasing otherwise. The third and forth term generate a hump-shaped, if theparameters β2 and β3 are positive or U-shaped if the parameters are negative.

0 10 20 30 40Maturity

0.02

0.03

0.04

0.05Yield

Figure 3.1: The term structures of interest rates captured by the Svensson Model atdifferent points in time over the recent years.

Using the Definition 3.6, including Equation (3.7) one can express the continuously-compounded spot rates in terms of instantaneous forward rates using

R(t, T ) =1

τ(t, T )

∫ T

t

f(t, u) du.

2See Brousseau and Sahel (2001) for a detailed discussion


Setting τ = τ(t, T ) and combining this expression with Equation (3.15), one findsthat the spot rate in the Sevensson Model is given by

R(t, T ) = β0 + β1

1− exp(− τt1

)τt1

+ β2

(1− exp(− τ

t1)

τt1

− e−τt1

)

+β3

(1− exp(− τ

t2)

τt2

− e−τt2

). (3.16)

This expression of the spot rate R(t, T ) in terms of the parameters gives the firststarting point for the model calibration, since it now can be based on yield curvesas explained in Definition 3.12. Before using Equation (3.16), yields have to beconverted into continuously-compounded rates. As mentioned in Section 3.3, rateswith maturities of less than one year are simply-compounded rates, whereas rateswith longer maturities are annually-compounded interest rates. This leads to twocases requiring conversion, which is done using

rc(t, T ) =

1τ(t,T )

ln(1 + r(t, T )τ(t, T )) t ≤ 1 year

1τ(t,T )

ln((1 + r(t, T ))τ(t,T )) else,(3.17)

where the day-count convention τ(t, T ) could differ in the two cases.

For fitting the model parameters, an intuitive least square method can be used(as applyid in this thesis), minimizing the squared difference between the obtainedmarket data rc(0, ti) and R(0, ti) for every tenor ti in the yield curve. This leads tothe model parameters b. Figure 3.1 shows some fits of the Svensson Model to yieldcurves in terms of the spot rate R(0, t) given by Equation 3.16. One should alsomention that the model fails to fit the interest rate curve if it has too many humps,as we will observe in times of high market stress later on. Figure 3.2 shows suchcritical fits. We will not use the model to capture the yield curve for such times.


0 10 20 30 40Maturity

0.02

0.03

0.04

0.05

Yield

Figure 3.2: The term structures of interest rates captured by the Svensson Model attwo points in time, observed in a period of high market stress.

4 The Interest Rate Simulation

Model

Since it is the aim of this thesis to estimate exposure profiles of portfolios, whichcontain interest rate derivatives, we need to choose an interest rate simulation model.This chapter presents a specific simulation model and its calibration to observedmarket data. First, a short description of different methods to simulate interestrates is given. Furthermore the decision of using a pricing interest rate model, moreprecisely a market model, to simulate interest rates, is discussed.

4.1 Interest Rate Model Selection

There are a high number of approaches to simulate interest rates. In the following wewant to distinguish two classes of models, econometric and pricing models, knowingthere are possible intersections.

In general, from a purely theoretical point of view, no special model or model classis principally favorable or advantageous. Any model selection bears a certain modelrisk as discussed in Martin, M. R.W. (2010b) and Martin, M. R.W. (2010a). Theaim of this paper is to provide insights in the modeling of CCR using a specialpricing model (in terms of our broad classification). However, a rough overview ofpotential approaches to measure CCR of interest rate derivatives is given, beginningwith the class of econometric models.

In the scope of econometric models one can highlight the publications of Nelson andSiegel (1987) and Svensson (1994), which present analytical expressions to be fittedto the term structure of interest rates. The analysis of Bürger (2008) has shownthat this type of model is not flexible enough to capture all possible types of interest

22

4 The Interest Rate Simulation Model 23

rate term structures. This work supports this finding for times of high market stressas observed in the recent years. Another approach, which delivered good resultsaccording to Bürger (2008), was presented by Rebonato et al. (2005) and Nyholmand Rebonato (2008). The basic idea is to use observed yield curves to create aforecast using a bootstrap method. The model produces good interest rate curvestructures in terms of realistic scenarios in comparison to the observed structures.

The first disadvantage of such models is that they just reflect observations of thepast. When we think of the financial crisis of 2008 and assume a calibration to databefore strong market movements, this seems to be highly critical. A second maindisadvantage of empirical models is that the step towards a valuation of interest ratederivatives is difficult, even if a risk neutral pricing is unclaimed. Assume the pricingof OTC options as caps and swaptions. In respect to the usually used Black-likepricing approaches, volatility surfaces are the key input parameter. Such volatilitysurfaces have to be simulated or estimated simultaneously to interest rates, wherethe level of stochastic flexibly should not be to high, with respect to the possiblesimulation effort. Some authors propose an approximative simulation approach forthe pricing of options following Longstaff and Schwartz (2001), which extends thesimulation effort by a significant amount.

Such arguments lead to the second class of models, which is refer to as pricing models.These models were developed to price derivatives under risk neutral aspects, i.e. tofit current market quotations. They can be found in basic textbooks on (interestrate) derivatives, like Brigo and Mercurio (2005), Rebonato (2002), and Musielaand Rutkowski (2005). They are defined by a stochastic process (Definition A.1)mostly driven by a number of Brownian motions (Definition A.4), which describe theevolution of interest rates. Clearly, these models are not created for the simulationof interest rates, but this does not necessarily tell about the possibility of providinggood results, if they are calibrated accurately. When talking about such models,there are two different classes of models distinguished in the literature, which areshort-rate models and market models.

As the name implies, short-rate models are modeling the short-rate as defined byDefinition 3.1. Classical models, named by their developers, are the Vasicek Model,the Cox, Ingersoll, Ross (CIR) Model, the Ho and Lee Model and the Hull-WhiteModel. The main disadvantage of those models, in case of just one stochastic el-ement, i.e. one driving Brownian motion, is that they just model the short-rate.


The whole term structure of interest rates has to be adopted from just one rate.This does not allow for enough flexibility concerning the evolution of yields andadditionally, as shown in Bürger (2008) leads to somewhat unrealistic yield curves.Moreover, the Vasicek, and the Hull-White Model cannot guarantee positive shortrates. One can extend most of the above mentioned models to multi-factor models,i.e. include more than one stochastic element. This can possibly solve the argumentof stochastic flexibility.

However, using short rate models for simulation purposes, one needs to guaranteepositive forward rates. This has a large impact on the model selection since theprice of forward rate agreements, swaps as well as caps, depends on forward rates.One-factor short rates models cannot guarantee this characteristic.1 For multi-factor short rate models, one has to pay close attention to the model calibrationto handle this point, since they model the instantaneous short interest rates andnot forward rates. However, if we find and calibrate such a model and somehowguarantee positive forward rates, high complexity and high simulation effort has tobe accepted. Therefore, one-factor models do not seem to be a good choice for thesimulation of interest rate term structures and multi-factor models do not offer theadvantage of simpleness.

The other class of pricing models, the market models, model a number of simple-compounding forward rates or forward swap rates simultaneously. According to(Brigo and Mercurio, 2005, Page 195) this family of interest-rate models is themost popular and promising, due to the fact that it agrees with well establishedmarket formulas for basic interest rate derivatives such as the Black formulas for capsor swaptions. Authors like Brigo and Mercurio (2005) distinguish the LognormalForward-LIBOR Model, which agrees with Black’s cap formula and the LognormalForward-swap Model for pricing swaptions using the Black’s swaption formula. Thisis due to the fact that forward LIBOR rates and forward swap rates cannot belognormal at the same time, i.e. under the same measure. However, like someauthors do, we will denote the Lognormal Forward-LIBOR Model as LIBOR MarketModel (LMM). At this point it should be mentioned that all presented pricing modelsare special cases of the Heath-Jarrow-Morton (HJM) Model, which can be seen asa generalization of the models above.

Most negative facts, which were mentioned in the context of short-rate models, do1See (Choudhry, 2001, Page 797).


not hold in the context of the LMM. Another positive fact is the models perfectcompatibility with Black’s cap formula, if used for pricing purposes. This leadsto easy extendibility for pricing purposes and the pricing of counterparty creditrisk adjustments (also called counterparty credit valuation adjustments (CVA)), ifcalibrated under risk neutral instead of empirical (historical) measure as presentedin Cesari et al. (2010). Additionally, well tested analytical expressions for pricingswaptions are available. However, the simulation effort may be as high as for multi-factor short-rate models. In the context of CCR, the LMM has been applied byHegre (2006) and Øvergaard (2006), where some criticism on the model calibrationshould be placed.2

As said before, there is no special model or model class principally favorable froma purely theoretical point of view. We highlighted some advantages and disadvan-tages. In this thesis we apply a market model approach to the purpose of exposureestimation, which we expect to be a promising.The basic idea is to simulate forwardrates rather than the short rate and then using the results for the valuation of OTCinterest rate derivatives up to their maturity. Assume one wants to simulate theexposure of a single payer IRS paying some fixed rate and receiving a semi-annualLIBOR rate as described in Section 3.2.2. For simplification we assume the fixed legto pay semi-annually. Let us say the contract has a time to maturity of 10 years,thus the last fixing of the floating leg is done in 9.5 years.

To estimate the value of this swap on a future date we need specific interest rates.One could first simulate the short rate, then estimate zero-coupon bonds and finallyvalue the swap on this term structure. The problems one would face have beendiscussed earlier. On the other hand, one could simulate all underlying simplecompounding spot rates. More precisely, one could simulate the simple compoundingforward rates, which will match the spot rates in the date of their fixing.

This description leads us to a market model approach. What we want to point outis the number of forward rates needed in this scenario. For the floating leg, oneneeds 19 forward rates spaced by six months. Additionally one requires a forwardrate expiring in ten years time, used for the estimation of the swap value for thetime between the last expiry and the swap’s maturity. We end up with 20 forward

2Both authors estimate the volatility and correlation structure of the LMM out of historic Nor-wegian zero-coupon yields instead of forward rates derived from those yield curves.


rates which have to be simulated. This is the basic scenario for which the followingdescription is designed and the further numerical analysis is based on.

Another issue is the valuation of caps/floors within the market model approach.Clearly, the LMM is designed to valuate caps/foors consistent with Black’s capformula, but it depends on the volatility structure to which it is calibrated. Whatwould one achieve if the LMM is calibrated to the current Black’s implied volatilityof caps, besides the matching of the current cap prices? Let us have a look onhistorical observations.

In general, the historical volatility of forward rates does not correspond with his-torically obtained implied volatilities of caps and floors. Since we want to use theLMM to simulate somehow realistic forward rates, one could calibrate the model onhistorical volatilities of forward rates. We could then expect realistic forward rates,given an accurate calibration method, but we cannot match the current cap andfloor prices. The question is, of how one can achieve both and at the same timestay within the market model framework. A basic idea could be to keep the currentimplied Black cap/floor volatility surface for pricing purposes. This is clearly a hugesimplification but this will be the applied method. A possible approach could be theidentification of a linear or non-linear connection between the stochastic modeledrisk factors, i.e. the forward rates as well as the shape of the yield curve and thelevel of volatility implied by Black’s cap/floor pricing formula. This would result in achanging (pricing) volatility surface depending on the simulated forward rates. Thisidea or other approaches are not analyzed in this thesis and are therefore addressedto further research.

4.2 The LIBOR Market Model

The LIBOR Market Model (LMM) describes the dynamics of the term structure ofinterest rates by modeling the simply-compounding forward rates. The name refersto the London Interbank Offered Rate (LIBOR) which denotes a daily interbankreference rate. The model will be introduced according to the notation of Brigo andMercurio (2005). We also follow Glasserman (2003) for descriptions of simulationapproaches. In this chapter results from continuous time stochastic calculus, whichare presented in Appendix A, are used.


The LMM models the simply-compounded LIBOR rate L(t, T ) at time t for aninterval [t, T ] as described in Definition 3.7. A contract entered at time t allows toborrow 1 unit of currency at time T and repay 1 unit of currency with cummulatedinterest rates at time T . The actual interest rate is then given by τ(t, T )L(0, T ).Typical values for the time interval [t, T ] are (approximately) three or six months.

Let t = 0 be the current time and T = {T0, T1, . . . , TM} a finite set of ascendingsorted maturities respectively expiry dates (Ti−1, Ti). Let τi = τ(Ti−1, Ti) denotethe years fraction between the settlement day Ti−1 and the expiry day Ti as set inDefinition 3.5, where dates are generally expressed in years from current time t = 0.To complete the notation, the definition of T−1 := 0 is suggested.

The forward rate Fk(t) = F (t, Tk−1, Tk) for k = 0, . . . ,M matches with the simplecompounded spot rate L(Tk−1, Tk) at time Tk−1.

Let now Qk be the forward (adjusted) measure as described in Definition A.15, i.e.the probability measure associated with the zero-coupon bond P (·, Tk) as numeraire.When considering simply-compounding discounting, it follows, by Definition 3.8that

Fk(t)P (t, Tk) =1

τk(P (t, Tk−1)− P (t, Tk)) .

Since the right hand side of the equation shows the difference of two zero bonds,Fk(t)P (t, Tk) denotes the price a tradable asset. Since Fk(t)P (t, Tk) divided by thenumeraire is Fk(t) itself, it follows that Fk has to be a martingale under the measureQk. Therefore, Fk needs to be driftless under Qk if it is modeled according to adiffusion process.

It is assumed that Fk satisfies the stochastic differential equation

dFk(t) = σk(t) Fk(t) dWQk

k (t), t ≤ Tk−1, (4.1)

where WQk

k (t) is a one-dimensional Brownian motion under Qk with some instan-taneous and constant covariance ρ = (ρi,j)i,j=1,...,M among the Brownian motionsgiven by

dWQk

i (t) dWQk

j (t)′ = d〈Wi,Wj〉t = ρi,j dt. (4.2)

Additionally σk(t) in Equation (4.1) denotes the volatility function associated withthe forward LIBOR rate Fk(t). As described in Equation (4.1) the process of dFk(t)ends at the time Tk−1 since it matches with the spot rate L(Tk−1, Tk).


Assuming that all σk(t) are bounded and deterministic functions (as we will do forall further applications), Equation (4.1) has a unique solution.3 It is obtained byapplying Itô’s formula, as given by Theorem A.1 with G(Xt) := ln(Xt), on theprocess given by Equation (4.1), to find

d lnFk(t) = −σk(t)2

2dt+ σk(t) dWk(t), t ≤ Tk−1, (4.3)

which shows the lognormal distribution of Fk(t) under the forward measure Qk givenbounded coefficients. The solution is given by

lnFk(T ) = lnFk(0)−∫ T

0

σk(t)2

2dt−

∫ T

0

σk(t) dWk(t).

There are a several approaches to model deterministic instantaneous volatility func-tions. For an overview we refer to (Brigo and Mercurio, 2005, Chapter 6.3.1) or(Rebonato, 2002, Chapter 6). In general, they can be distinguished into piecewise-constant and parametric models for the instantaneous volatility, where both ap-proaches have advantages and disadvantages. For the purposes of this thesis, apiecewise-constant modeling is suggested, leaving more flexibility and allowing anunproblematic calibration, even if the description is not that compact compared toparametric approaches. The approach considered in the forth-coming sections mod-els the instantaneous volatility piecewise-constant in time until the correspondingforward rate expires. This leads to a volatility structure as shown in Table 4.1.

(0, T0] (T0, T1] (T1, T2] . . . (TM−2, TM−1]F1(t) σ1,1 expired expired . . . expiredF2(t) σ2,1 σ2,2 expired . . . expired...

......

... . . . ...FM(t) σM,1 σM,2 σM,3 . . . σM,M

Table 4.1: A schematic description of the volatility structure assuming piecewise-constant volatility functions over time.

For further purposes, let β(t) denote a function defined on [0, TM) → {1, . . . ,M}3The constraint on the volatility function σk(t) simplifies further analysis and has some positiveaspects, even if the LMM loses flexibility. In the context of such market models, this strongsimplification is not set, especially from a HJM point of view. Here, as presented in (Musiela andRutkowski, 2005, Page 450), σk(t) for any expiry Tk ∈ T are RM -valued, bounded F−adapted(stochastic) processes.


such that β(t) = m if t ∈ (Tm−2, Tm−1]. It follows that the function β(t) gives theindex of the first not expired forward rate Fβ(t) = F (t, Tβ(t)−1, Tβ(t)) at time t.

By using one numeraire per (single) maturity, every forward rate Fk(t) is necessarilylognormal under its particular Tk-forward measure Qk. Since it is clear that notevery forward rate can be lognormal at the same time under the same numeraire,the dynamics of Fk(t) under different measures will be discussed.

Remark 4.1 (Relationship between Brownian motions under different nu-meraires). Using the Girsanov transformation, as given by Theorem A.2, for chang-ing from a measure Qi to Qi+1 one finds

dWQi+1

= dWQi +τi+1Fi+1(t)

1 + τi+1Fi+1(t)ρ σi+1(t)e

′i+1dt, (4.4)

as the relationship between forward rates under these two measures, where e′i denotesa unit vector with 1 in the ith component. Please note that we use a vector notationfor the Brownian motion WQi , i.e. WQi+1 and the matrix notation for ρ in theequation above.

Applying the Girsanov transformation using the Radon-Nikodym derivative on thedynamics of forward rates under one particular measure Qi, as done in remark above,one ends up at the following proposition.

Proposition 4.1 (Forward-measure dynamics in the LMM). Given the log-normal assumption, the dynamics of Fk under the forward-adjusted measure Qi inthe cases i < k, i = k and i > k are

i < k, t ≤ Ti :dFk(t)

Fk(t)= σk(t)

k∑j=i+1

ρk,jτjσj(t)Fj(t)

1 + τjFj(t)dt+ σk(t)dW

Qi

k (t),

i = k, t ≤ Tk−1 :dFk(t)

Fk(t)= σk(t) dW

Qi

k (t), (4.5)

i > k, t ≤ Tk−1 :dFk(t)

Fk(t)= −σk(t)

i∑j=k+1

ρk,jτjσj(t)Fj(t)

1 + τjFj(t)dt+ σk(t)dW

Qi

k (t).

All equations admit a unique solution if the coefficients σ(·) are bounded and deter-ministic functions.

Unlike to the i = k case, the forward measure dynamics given by Proposition 4.1 do


not feature any known transition densities in the cases i 6= k. Even the knowledgeof the parameters σ and ρ does not allow to write analytical expressions for thedensity of future rates in a future instant on the basis of forward rates in a pastinstant. Hence, there are no known analytical formulas for contingent claims whichare based on the joint dynamics of different forward rates.

When using the LMM for the pricing of some interest rate derivatives, it is necessaryto apply a measure to the dynamics of the LMM, which is associated with a bankaccount numeraire B(t). The main reason why we introduce this measure is thatthe resulting dynamics of the forward rates is better suited for simulation purposes.This is discussed in more detail in Section 4.4.2.

However, at this point a problem occurs since the numeraire B(t) based on theshort-rate process rt is not a natural choice for simply-compounded forward rateswith preassigned tenors and maturities. The authors of (Brigo and Mercurio, 2005,Chapter 6.3) propose new dynamics of forward LIBOR rates, which end up with thenecessity to model the instantaneous forward volatility in some way. An alternativeapproach is to apply new settings to B(t).

Definition 4.1 (Discretely replaced back account). Let Bd(t) define a bankaccount whose value is rebalanced only on a set of discrete times {T0, . . . , Tn}. It isdefined through zero-coupon bonds P (t, Tj) and the forward rates Fk(t) by

Bd(t) :=P (t, Tβ(t)−1)∏β(t)−1

j=0 P (Tj−1, Tj)=

β(t)−1∏j=0

(1 + τjFj(Tj−1))P (t, Tβ(t)−1) (4.6)

where β(t)− 1 gives the index of the last tenor date at time t. While using Bd(t) asthe numeraire, the corresponding probability measure is denoted with Qd and calledspot LIBOR measure.

The numeraire asset Bd(t) can be interpreted as a portfolio starting with Bd(0) = 1

and investing in a quantity of 1/P (0, T0) in a T0 zero-coupon bond P (0, T0). At timeT0 the payoff 1/P (0, T0)P (T0, T1) is cashed and reinvested into a T1 zero-couponbond. Continuing this procedure, the present value of the portfolio at time t isgiven by Equation (4.6).

The next step is to transfer the dynamics given by Equation (4.1) for all forwardrates from the current forward measure to the spot LIBOR measure Qd, where the


model’s tenor structure is applied to the numeraire Bd(t). This leads to the followingproposition, where the Girsanov transformation is used again.

Proposition 4.2 (Spot-LIBOR-measure dynamics in the LFM). The spotLIBOR measure dynamics of forward LIBOR rates in the LMM are given by

dFk(t) = σk(t)Fk(t)k∑

j=β(t)

τj ρj,k σj(t)Fj(t)

1 + τjFj(t)dt+ σk(t)Fk(t) dW

dj (t). (4.7)

There are some benefits in using the spot LIBOR measure instead of the forward-adjusted measure Qi, even if no single forward rate is modeled without drift terms.These are caused by the different distributions of the drift terms, which result in abetter distribution of the bias among the forward rates over time.4

4.3 Reduced-Rank Formulations of the Correlation

Matrix

When modeling M forward rates in a LIBOR Market Model, the model is driven byM Brownian motions. The correlation matrix ρ is given by

ρi,j = dWi(t) dWj(t), i, j = 1, . . . ,M.

Ideally one could think of a reduction of the number of Brownian motions, while stillsimulating all forward rates in order to reduce simulation effort. For this purpose,finding a reduced formulation of the correlation matrix ρ is a key point. For this,several approaches are discussed in the literature, as shown in Brigo and Mercurio(2005). A detailed illustration and analysis of parameterizations of the correlationmatrix can also be found in Packham (2005). This thesis will refer to only one ap-proach, which reduced the correlation matrix by zeroing the smallest eigenvalues.

Given a correlation matrix ρ, it is assumed (as usual) that ρ is a real positive definitematrix. Therefore

ρ = PHP ′,

4This topic is discussed in more detail in Section 4.4.2.


where P is a real orthogonal matrix with PP ′ = P ′P = IM , where IM denotes a M -dimensional identity matrix and H is a diagonal matrix of the eigenvalues of ρ. Thematrix P contains (column by column) the eigenvectors of ρ, to the correspondingeigenvalue in H.

In a next step, let Λ denote a matrix consisting of the square root of the elementsof H and set A := PΛ, which has the property to give

AA′ = ρ, A′A = H.

The idea of reducing the rank of ρ is now to replicate the decomposition ρ = AA′

by means of a suitable n-rank M × n matrix B with n < M such that BB′ isan M -rank correlation matrix. One can take the matrix B to replace the model’sM -dimensional random shocks dW (t) by B dW ∗(t) since

dWdW ′ = ρ dt

andB dW ∗(B dW ∗)′ = B dW ∗dW ∗′B′ = BB′dt.

Therefore, we letρB = BB′.

The main advantage of decreasing the model’s noise to n is the reduction of thesimulation effort.

We use the approach of reducing the rank of the matrix ρ by applying a principalcomponent analysis and zeroing the smallest eigenvalues. In more detail, let Λ̄(n) bethe Matrix Λ, where the smallest M − n eigenvalues are set to zero. FurthermoreB(n) := PΛ(n) is defined, where all zero columns are removed. When setting

ρ̄(n) := B(n)B(n)′,

one achieves generally a positive semidefinite matrix, having the disadvantage ofdiagonal elements unequal to one. To ensure diagonal element equal to one, all


matrix elements are rescaled by

ρ(n)i,j =

ρ̄(n)i,j√

ρ̄(n)i,i ρ̄

(n)j,j

,

analogous to the connection between correlation and covariance matrices.

4.4 Simulating Forward Rates using the LIBOR

Market Model

Simulating stochastic processes leads to the problem of choosing an accurate dis-cretization scheme. In this section, the forward rate dynamics under the T -Forwardmeasure given by Equation (4.5) and under the spot measure presented by Equa-tion (4.7) discretized. Initially, a time grid {t0, . . . , tn} is introduced. It is assumedthat it also includes the tenor dates {T1, . . . , TM} of the LMM, which gives a lowerbound of M on the dimension of the simulation grid.

Basically one has to discretize the drift terms of the dynamics, to use Gaussianshocks to capture the Brownian motion and to discretize the particular volatilityterms σk.

4.4.1 Simulating under the T -Forward Measure

Consider a LMM under the forward measure QM corresponding to the numeraireP (0, TM) and the dynamics given by Equation (4.5), where M still denotes the lastsimulated forward rate.

Furthermore applying Itô’s formula to lnFk(t) assuming deterministic and boundedfunctions σk leads to

d lnFk(t) = −σk(t)M∑

j=k+1

ρk,j τj σj(t)Fj(t)

1 + τjFj(t)dt− σk(t)

2

2dt+ σk(t) dW

Mk (t), (4.8)


which also has deterministic diffusion coefficients. Applying the Euler scheme on theequation given above and inverting the logarithm, the discretization is given by

F̂k(ti+1) = F̂k(ti) exp

(−σk(ti)(ti+1 + ti)

M∑j=k+1

ρk,j τj σj(ti) F̂j(ti)

1 + τjF̂j(ti)

)

· exp

(−(ti+1 + ti)

σk(ti)2

2+ σk(ti)(Z

(k)ti−1− Z(k)

ti )

)(4.9)

where (in vector notation) the difference (Zti+1− Zti) ∼

√(ti+1 + ti) N (0, ρ) and

time-independent.5 Note that this scheme keeps F̂k positive, whereas a scheme juston Fk(t) could produce negative rates.

Alternatively, one can rewrite the stochastic term in (4.8) to describe it in terms ofthe covariance matrix COVt. This can be done by integrating the stochastic termin between a time difference ti+1 − ti for all k. This results in

∆ζi :=

∫ ti+1

ti

σ(s) dW (s) ∼ N (0,COVti),

where σ is a vector with dimension M containing the volatility functions of theforward rates and the matrix COVt is given by

(COVt)h,k =

∫ ti+1

ti

ρh,k σh(s)σk(s)ds.

Therefore there is no need to approximate the stochastic term by

σ(t)(Zti−1− Zti) ∼ N (0, τi σ(t)ρσ(t)′)

as done in the scheme as given by Equation (4.9). Instead, one can also use ∆ζi as anapproximation, which can easily be simulated through its gaussian distribution.

Now recall and discuss the reduction of the Brownian motions as described in Sec-tion 4.3. The reduction results in a decomposition B(n)B(n)′ of ρ. One can nowreplace the vector dW (t) by B(n) dW ∗(t) in Equation (4.8), where dW ∗(t) is ann-dimensional standard Brownian motion. This result can also be applied on Equa-

5We do not discuss different discretization schemes in here. Details on properties and otherdiscretization approaches can be found in (Brigo and Mercurio, 2005, Section 6.10, i.e. C.3)and Kloeden and Platen (1992).


tion (4.9), where (in vector notation) the changes (Zti+1−Zti) ∼

√(ti+1 + ti) N (0, ρ)

can be replaced by

(Zti+1− Zti) ∼

√(ti+1 + ti)B

(n)N (0, I),

where I is the n× n identity matrix.

4.4.2 Simulating under the Spot Measure

The simulation of forward rates under the spot measure starts as before by discretiz-ing the dynamics as in Equation (4.7). Again, the discretization scheme is appliedto the dynamics of lnFk(t) (which is calculated using Itô’s formula) and results in

F̂k(ti+1) = F̂k(ti) exp

σk(ti)(ti+1 + ti)k∑

j=β(t)

ρk,j τj σj(ti) F̂j(ti)

1 + τjF̂j(ti)

· exp

(−(ti+1 + ti)

σk(ti)2

2+ σk(ti)(Z

(k)ti−1− Z(k)

ti )

), (4.10)

where, as before, the function β(t) returns the index of the first not expired forwardrate at time t. Note that only the drift term differs from Equation (4.9). Addi-tionally, the simplification of the stochastic part (for simulation) can be appliedanalogous to the simulation under the T -forward measure.

A simulation under the spot measure Qd has some advantages compared to a sim-ulation under a forward measure, especially in case of a simulation until the lastforward rate expires. Note that in this case one has to use the QM -measure in theforward measure case. The main advantage is that the number of drift terms is moreequally distributed over all forward rates. Hence, a possible bias coming from thediscretized drift is more evenly distributed among the different rates. This can beexamined by counting the number of drift terms per forward rate at different datesof the simulation horizon. Additionally, with this approach fewer values have to besaved from iteration to iteration.6

6See (Brigo and Mercurio, 2005, Page 219 f.) and (Glasserman, 2003, Page 176 ff.) for a moredetailed discussion.


4.5 Estimation of the Model Parameters

This section gives an illustration of the technique used to estimate the model’svolatility structure and the instantaneous correlation under the empirical measure.Thus, the model is calibrated to historical market data rather than current marketquotations of particular interest rate derivatives as a risk neutral calibration wouldrequire.

4.5.1 Interest Rate Data Structure and Forward Rates

The estimation of the model’s correlation and volatility structure is based on histor-ical term structures of interest rates following Definition 3.12. Such a term structureis constructed using certain number of tenors. The term structure could be rewrittento be expressed in terms of zero-coupon bonds using Equations (3.5) and (3.8).

Given historical term structures in terms of P (t, T ), one has time series of zero-coupon bond prices in the way

P (t, δ), P (t+ 1, t+ δ + 1), . . . , P (t+ n, t+ δ + n)

for a single time to maturity δ, where n denotes the time series length. The timeto maturity can be one of the tenor dates of the term structure or any other, if aninterpolation method is used.

When it comes to the estimation of forward rate correlations and volatilities, onehas to construct differently shaped time series of zero-coupon bond rates. One canthen use the relationship

Fk(t) =1

τk

(P (t, Tk−1)

P (t, Tk)− 1

), (4.11)

which we recall from Definition 3.8. For a forward rate Fk(t) with a certain matu-rity or settlement date T , one needs to create a time series of zero coupon bondsstructured like

P (t, T ), P (t+ 1, T ), . . . , P (t+ n, T ),

i.e. the maturity has to be kept fixed. An interpolation of the obtained termstructure is required to generate such time series of forward rates. One possible


interpolation method was presented in Section 3.4. This method, however, deliversforward rates directly by Equation (3.15) without expressing the yield curve in termsof zero-coupon bonds. Nonetheless, Equation (4.11) has to be used in the case alinear interpolation is applied to zero-coupon bonds in order to calculate forwardrates. Both ways lead to a construction of time series using the expression

Fk(tj) := F (tj, Tk−1, Tk), j = 1, . . . , n , (4.12)

where n denotes the length of the time series bounded by the condition tj ≤ Tk−1.

4.5.2 Estimating the Instantaneous Correlation of Forward

Rates

Assume the yield curve has been interpolated in a parametrized functional formas suggested in Section 3.4 or by any other interpolation method. Then, everyzero-coupon bond with a maturity between the shortest and the longest historicalspot rate can be produced. Thus, forward rates can be created in a very flexiblemanner using Equations (3.4), (4.11), or (4.12) respectively including some day-count convention.

As mentioned in Section 4.5.1 to estimate the model’s correlation structure thematurity of forward rates has to be fixed over time. Thus a data sample of historicalforward rates constructed according to the LMM’s structure will reduce its size afterthe first forward rate expires.

To handle this problem, different approaches could be taken. At first, one couldestimate the correlations among the forward rates based only on data sets with amaximum length equal to the first expiry. This leads to short data sets and highlybiased estimators, which could be somehow stabilized by averaging over a numberof such estimations. Secondly, one could estimate correlations over the whole timeseries of forward rates while considering empty parts of the series (due to expiredforward rates) as such and not setting them to zero. This leads to better estimationsfor the correlation and volatility parameters among the forward rates with longermaturity. On the other hand, all estimators, including any early expiring forwardrates, tend to deliver highly biased results.


Keeping this in mind, an accurate combination of both could be more adequate.This could be done by splitting the time series into a number of subsamples with asufficient length and then estimating the parameters on each data sample. All thoseestimations could then be averaged to receive final estimations.

We follow this estimation approach and construct time series of all forward rates Fkwith a particular length n using Equation (4.12), where in the case of tj > Tk−1 thevalues of Fk(tj) are signed as empty.

One can construct a number of such subsamples each starting with the full number offorward rates. Based on a single sample set the volatility and correlation parametersare estimated using

µ̂mi =1

m

m∑k=0

lnFi(tk+1)

Fi(tk)(4.13)

and

v̂mi,j =1

m− 1

m∑k=0

(lnFi(tk+1)

Fi(tk)− µ̂i

)(lnFj(tk+1)

Fj(tk)− µ̂j

), (4.14)

which implies a Gaussian distribution of the logarithmized forward rates. The cor-relation among the forward rates can now be estimated using

ρ̂i,j =v̂mi,j√v̂mi,i√v̂mj,j

,

where m denotes the minimal index of the tenors i, j and where the values are notempty (again due to the expiry of the forward rates before the end of the sample).

This procedure is applied to all subsamples and the variance v̂i,j. The correlationρ̂i,j is calculated by averaging over these estimations. This yields an estimatedcorrelation matrix of the forward rates.

4.5.3 Estimating the Instantaneous Volatility of Forward

Rates

The central question when calibrating the LMM on historical term structures is:How does the model match this structure? In other words, is the model reproducingthe historical term structures? To find an answer, let us first note some consequencesof Section 4.2.


The LMM models a LIBOR rate Fi(t) lognormal under the forward measure Qi.Hence lnFk(t) is normally distributed. The case i = k has already been treated inEquation (4.3). Applying Itô’s formula for lnFk(t), where the dynamics of Fk(t)under Qi are given in Equation (4.5) one can find the dynamics of lnFk(t) in thecase i < k given by

d lnFk(t) = σk(t)k∑

j=i+1

ρk,jτjσj(t)Fj(t)

1 + τjFj(t)dt− σk(t)

2

2dt+ σk(t) dW

Qi

k (t). (4.15)

In the case of i > k, an analogous equation to Equation (4.15) follows, differing justin the sign of the sum of drift terms, as determined by Equation (4.8).

Let us first assume a fixed and known correlation structure and also constant, de-terministic and bounded variance functions. Then, lnFk(t) is normally distributedif k = i and the distribution parameters can directly given by the equation above.Hence, when estimating σk(t) = σk using the estimators for the mean and the vari-ance given by Equations (4.13) and (4.14), one should find a mean of −σ2

k/2 and avariance of σ2

k.

Facing the more difficult cases i < k and i > k, one finds a dynamic mean forall tenors depending on the current forward rate level. Therefore we produce anestimation bias if we use the same method as in the i = k case. This, of course, isjust one of a number of points which might be criticized. For example, one couldalso scrutinize the calibration method of the correlation structure and argue aboutthe ignored direct connection between σi(t), σj(t) and ρi,j given by the covariancebetween i and j. Moreover, one could simply ask if forward rates are actuallylognormally distributed.

Despite, knowing all these issues, there is still the problem of finding a realisticvolatility structure. Let us assume the forward rates to be equally spaced over timewith some time distance δ. When thinking about the structure of forward rates,it seems to be realistic that the behavior of future forward rates in their last timeinterval before their expiry (of length δ) is the same for all forward rates, especiallyfrom a perspective at time zero. To simplify and prevent from over-fitting, one couldalso think of a constant volatility for every interval with length δ. Furthermore, onecould expect a different behavior for rates one time interval before. This leads to avolatility structure given in Table 4.2 below.


(0, T0] (T0, T1] (T1, T2] . . . (TM−2, TM−1]F1(t) σ1 expired expired . . . expiredF2(t) σ2 σ1 expired . . . expired...

......

... . . . ...FM(t) σM σM−1 σM−2 . . . σ1

Table 4.2: Deterministic and piecewise constant volatility structure only dependingon the time to maturity of the forward rate.

The matrix in Table 4.2 shows that the volatility of forward rates depends on thetime to maturity of the forward rate, rather than on the forward rate itself.

The volatility is estimated by applying Equations (4.13) and (4.14) on daily changesof forward rates. The estimations are transformed using the so-called square-root-of-time rule

σ̂k =√v̂k,k/

√dy, (4.16)

where dy denotes the number of trading dates per year y and we assume v̂k,k to bebased on daily forward rates.

4.6 Pricing of Interest Rate Products and

Derivatives in the LMM Simulation Model

In Section 3.2 different basic interest rate products and derivatives were discussed.This section shows how those derivatives can be priced within the context of aLMM simulation approach. One challenge is to interpolate modeled forward ratesto generate a larger set of information.

4.6.1 Pricing Bonds, FRAs and Swaps within the LMM

Simulation Model

Recall the definition of forward rate agreements and interest rate swaps. Theseproducts clearly depend on a number of forward rates and some zero-coupon bondsused for discounting purposes only.


From the definition of the LMM, it is clear that it is possible to model all requiredforward rates. However, in case that we want to price a huge number of productsdepending on different forward rates, the total number of rates can get very large.In this case, an interpolation between forward rates increases efficiency. We will usea simple linear interpolation between forward rates.

For any required zero-coupon bond, one runs into similar interpolation requirements.Recall that the price of a zero-coupon bond in terms of the forward rate is given byEquation (3.5). The equation shows the bounded quantity of zero-coupon bonds,which are given in LMM analytically without any bias. More precisely, a bondP (s, S) can be priced if s and S both correspond with the current simulation time tand the models tenor set T . The bond’s value is then given at any simulation dateprior s. The price of those zero-coupon bonds is given by

P (t, Tk) = P (t, Tβ(t)−1)k∏

j=β(t)

1

1 + τjFj(t).

In the case of pricing other bonds, an interpolation has to be used. More precisely,the time to maturity S − s of a zero-coupon bond P (s, S) is divided into threepieces: the time to the next fixing date, the time corresponding with the modeltenor structure, and the resulting time afterwards. Only the second time instantcan be priced without any bias. Therefore, we require an interpolation method. Thevalue of a zero-coupon bond is calculated by

P (s, S) =1

1 + τ(s, S)Fβ(t)−1(t), S ≤ Tβ(t)

and

P (s, S) =1

1 + τ(s, Tβ(s))Fβ(s)−1(t)

β(S)−1∏j=β(s)

1

1 + τjFj(t)

1

1 + τ(Tβ(S)−1, S)Fβ(S)−1(t)

in every other case, where s ≥ t. As greater the year-fractions τ(s, Tβ(s)) andτ(Tβ(S)−1, S), the larger the interpolation bias.

Note that in some cases it can be more appropriate (compared to the presentedapproach) to use a weighted average between Fβ(s)−1 and Fβ(s) for the first factorand a weighted average between Fβ(S)−1 and Fβ(S) for the last factor.


A more advanced, but also more computationally intensive approach, could be theapplication of some drift interpolation or bridge technique to receive a larger set offorward rates as presented by (Brigo and Mercurio, 2005, Section 6.21).

4.6.2 Pricing Caps/Floors within the LMM Simulation

Model

Recall the payoff of a cap (or floor) as described in Section 3.2.3. The price of a capis given by Equation (3.12), which denotes a sum of caplets. The pricing of capletsis now discussed below.

The expectation of the stochastic part of a caplet’s payoff expiring at time Ti−1

under the probability measure Qi at time 0 is given by

P (0, Ti) EQi [(Fi(Ti−1)−K)+],

since the process of Fi as given by Equation (4.1) is a martingale and Fi is lognormaldistributed under Qi. Hence, the expectation above can be easily computed similarto a Black Scholes Price for a stock option. Therefore the price of a Ti−1-caplet(named by the expiry date), Ti−1 ∈ T , corresponds with the pricing given as partof Equation (3.12), i.e.

CplLMM∗(t, Ti−1, Ti, K) = P (t, Ti) τi BL(K,Fi(t), vi, 1),

where

vi(t)2 =

∫ Ti−1

t

σi(t)2dt,

and where σi denotes the volatility function of Fi. The coefficient vi(t) is calculatedout of the square root of the integrated instantaneous variance of Fi over t ∈ [t, Ti−1).An additional notional value can be multiplied within the pricing formula.

The above conclusion implies a corresponding of the variance of forward rates andthe caplets volatility implied by market quotes. As discussed earlier, this is neitherobvious nor observable in the market data. Hence, to use this pricing function, thevolatility structure of the LMM must actually be fitted to the implied volatilities


of cap or floor prices. In this thesis the LMM fitted to historical volatilities of for-ward rates expecting more realistic simulated forward rates. Therefore, the pricingapproach above would lead to unrealistic cap and floor prices, if they are comparedto prices based on implied cap/floor volatilities.

To price caps and floors, we use the last obtained implied volatility parameters(before a simulation start date) evaluated for cap and floor quotations using Black’sformula. Here, the caps and floors on the EURIBOR have the tenor spacing of sixmonths, which equals to the simulation model. Such volatility coefficients dependon the time to maturity T and the strike K. Hence, we can describe the volatilitythrough a function v(T,K). The pricing formula used for a cap and floor is givenby

CapLMM(0, Tj, K) =

j∑i=1

P (0, Ti) τi BL(K,Fi(0), v(T,K)√Ti−1, 1)

and

FloorLMM(0, Tj, K) =

j∑i=1

P (0, Ti) τi BL(K,Fi(0), v(T,K)√Ti−1,−1),

where BL denotes Black’s formula as given by Equation (3.13). The function v(T,K)

is derived from quoted implied volatilities for combinations of K and T , where alinear interpolation is used between the observed values. Note that the reduction ofthe time to maturity within a simulation causes a change of the volatility coefficientover time.7

For volatilities of caps with a time to maturity of less than one year, no cap andfloor volatilities are provided by market makers, to the best of the master candidatesknowledge. This seems clear since, by convention, the first caplet of a cap expiresafter six months. For our simulation approach, we need volatility coefficients fortimes to maturity of less than one year. In this case we use the one year volatility toprice caps and floors. Furthermore, in the case that a maturity of the option doesnot correspond to the model’s tenor times, a linear interpolation is applied to theforward rates used for pricing purposes.

7Notice that we use an average volatility depending on the cap’s (floor’s) maturity T for all caplets(floorlets) within the option. Thus, we cannot price caplets (floorlets) at this stage because wedo not have single volatilities available. The caplet (floorlet) volatilities can be evaluated byapplying some stripping algorithm. We refer to (Brigo and Mercurio, 2005, Section 6.4.3) formore details and restrict on the pricing of caps (floors).

5 Data Description and Model

Calibration

This chapter presents the market data used and the calibration of the LIBOR marketmodel according to historical term structures of interest rates.

5.1 Historical Interest Rate Curves

The interest rate model calibration and analysis is based on a sample of the historicalEURIBOR money market and EURIBOR swap rates over a period of five and a halfyears. The period starts on July 1, 2004 and ends December 31, 2009, representing1435 trading days. For a maturity of one day the EONIA rate, and for maturities of1 week and from 1 up to 12 months, EURIBOR rates are chosen. For maturities of2 through 25 and 30, 35, 40 years, the data set contains Euro swap rates. The termstructure is set up on these 41 rates according to Definition 3.12 and is presentedin Figure5.1. It becomes noticeable that the shape of the term structure changesstrongly over time. Furthermore the short end of the term structure is sometimesvery humped, which could be difficult to capture in a functional form.

We divide the data sample into two subsets. The first set, spanning from July 1,2004 to June 30, 2008, is used for the model calibration. The second set, from July1, 2008 to December 31, 2009 is used for the model validation. The terms were splitdue to the market movements after the bankruptcy of Lehman Brothers HoldingINC, in the middle of September 2008. The extreme effects of this event on theinterest rate market are interesting to compare with the simulation results.

44

5 Data Description and Model Calibration 45

Figure 5.1: The historical term structure of interest rates shown over a 5.5 yearperiod, where every term structure is build out of 41 tenors.

5.2 Parameterized Functional Form of the Swap

Rates

As described in Section 3.4 there are several approaches to cover yield curves. Inthis thesis, the Svensson Model is chosen to find a functional representation of thehistorical term structures.

As mentioned in Section 3.4, historical yield curves have to be converted to continuously-compounded rates rc(0, T ) using Equation (3.17). Furthermore, the 30/360-day-count convention is used within the Svensson model.

Money market rates of less than one year are quoted based on the day-count con-vention actual/360 and thus have to be transformed. For all other rates, the yearfraction is given by the 30/360-convention. For converting rates with maturitiesup to one year, τ(t, T ) = (T−t)

360has to be used as the day-count function in the

numerator. The conversion function is given by

rc(t, T ) =

1τ(t,T )

ln(1 + r(t, T )τ(t, T )365360

) t ≤ 1 year1

τ(t,T )ln((1 + r(t, T ))τ(t,T )) else,

where τ(t, T ) denotes the 30/360-day-count convention.


As mentioned in Section 3.4, the model parameters are estimated for every singleterm structure using a least square method.

The accuracy of the approximation is quantified by an estimation error defined asthe square root of the sum of squared differences between the observed and theestimated continuously-compounded spot interest rates. The estimation error isshown in Figure 5.2. It is apparent that there are high mismatches at the end of2008 and in 2009. These differences are due to a humped short end of the termstructure and a drastic change in the yield level of the money market to swap rates,which is also observable in Figures 3.1 and 3.2.

2005 2006 2007 2008 2009 2010Time

0.002

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Estimation Error

(a) The estimation error of the Svensson cali-bration.

(b) The differences of the Svensson spot ratesto the observed spot rates rc(t).

Figure 5.2: The estimation error from capturing the term structure using a Svens-son model is shown in chart (a). The differences between the observedcontinuously-compounded spot rates and the spot rates estimated by theSvensson model are presented in Figure (b). Here, the absolute differ-ences are given in percent.

Along with showing the estimation error, Figure 5.2 also shows the differences be-tween the estimated and the observed continuously-compounded rates. The figureshows higher differences at the short end of the term structure, particularly in 2009.This result corresponds with the examination of the estimation error.

Large estimation errors of the Svensson model have been observed since autumn2008, which corresponds with large changes in the structure of interest rates. Themain cause could have been the bankruptcy of Lehman Brothers Holding INC andsubsequent monetary interventions by the European Central Bank during this periodof extreme market stress. However, the calibration results for this period are not


accurate, thus the Svensson model should not be used for such periods, as was alsomentioned in Section 3.4.

For this reason we use a simple linear interpolation for the time after the 30st of June2008, hence the validation data set. This method guaranties that all observed inter-est rates are captured, which is not generally guarantied by the Svensson model.

5.3 Estimated LMM Parameters

Given the parametric form of the forward rates by the presented Svensson modelone can now estimate the parameters of the LMM according to Section 4.5.

The correlation and the volatility structure will be estimated for a LMM modelling20 LIBOR rates. All rates are modeled with a time to maturity of six months andthe rates are equally spaced over a 10 year period. This results in a half year’s timebetween the expiry dates of two subsequent forward rates.

We have to estimate a correlation and a volatility matrix of dimension 20 × 20.Additionally, the initial forward rate curve is derived from the interest rate termstructure of June 30, 2008. It is shown later as a part of Figure 5.8.

As described in Section 4.5.1 data sets with the above specified shape have to becreated. To use and to analyze the four year historical data from July 2004 to June2008, four data sets spanning over a one year period are constructed. Given thetenor structure mentioned above, the first LIBOR rate expires after six months ineach data set. Furthermore a data sample over four years is constructed, which isshown in Figure 5.3.

5.3.1 Estimation of the Correlation Structure

The estimation procedure follows the methods provided in Section 4.5.2 for all fivedata sets. The estimated correlations can be found in Figure 5.4, where the meancorrelation structure of the yearly data samples is given in addition.

Due to the changing market situation, the estimated correlations are changing overtime. Since our focus is to simulate over a long term, the general setup of all furthersimulations is based on the mean correlation structure of the four data sets. To


2005 2006 2007 2008Time

0.025

0.030

0.035

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0.045

0.050

0.055

0.060Forward rate

(a) The estimated forward rates. (b) The estimated forward rates in a three di-mensional presentation.

Figure 5.3: The estimated historical forward rates over a four year period are shownin a two (a) and a three (b) dimensional graph. The forward rates wereestimated starting 1st of July 2004. The maturity of the left figure refersto this date.

quantify the influence of this choice on the model results, we use the estimatedcorrelation matrixes based on one-year samples to simulate forward rates.

5.3.2 Estimation of the Volatility Structure

Besides the correlation structure, one needs to estimate the volatility structure of theLMM to calibrate the model. The theoretical aspects are explained in Section 4.5.3,where some possible approaches to estimate a matrix of piecewise constant volatili-ties based on the presented historical forward rates are described.

We apply the approach presented in Table 4.2. Here the volatility only dependson the time to maturity. Our estimation sample has to be shaped analogically.The estimation is based on a four year data set, similar to the one described inSection 5.3.1, extended by eight forward rates (to 28 in total) spaced by six months.Since the sample reduces over time by eight forward rates (due to their expiry), thesample is extended to 28 rates to guarantee 20 tenors for every sub-sample.

As shown in Figure 5.3, the number of forward rates does reduce over time. Basedon this sample, we have eight estimations for the variance as a piecewise constantfunction of time to maturity for the time from zero to ten years. Note that the sample


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(f) Mean of (a) to (d).

Figure 5.4: The estimated correlation structure for five data sets and the mean ofthe correlation structures of (a) to (d) in Figure (f).


extension is due to ensure eight estimations also for the forward rate expiring in tenyear’s time.

As implied above, the data sample is split into eight sub-samples, all spanning overa six month period. We use the estimators as given by the Equations (4.13) and(4.14) to estimate the mean and variance of each forward rate in each sample. Theresults are shown in Figure 5.5. Analyzing the estimation results one can clearlyobserve the dependence on the data sample, hence on the time. This leads to thequestion how the model should be calibrated, if such different market situationsappear. At this stage we restrict ourselves for the general model setup to simplyaverage the estimations, as done for the correlation estimation. Nonetheless, themodels sensitivity to changes in the volatility structure will be analyzed, where wecome back to the presented estimations.

5

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(a) The estimated mean.

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(b) The estimated variance.

Figure 5.5: The estimated mean and variance of eight samples of logarithmized for-ward rates.

To estimate the yearly volatility of the forward rates we use Equation (4.16) andaverage over all eight estimations to receive the final model volatility. The resultingvolatility function is shown in Figure 5.6. In this special case, the volatility is afunction of only time to maturity.


0 2 4 6 8 10T-t

0.05

0.10

0.15

0.20

0.25

Σi

Figure 5.6: The estimated volatility coefficients as a function of time to maturity.

5.4 Reducing the Number of LMM Factors

Modeling a LMM with 20 LIBOR rates needs basically 20 Brownian motions, whichcauses an extreme simulation effort. To reduce this burden the rank-reducing meth-ods presented in Section 4.3 are used. The core problem is to reduce the correlationmatrix using a principal component analysis and to neglect a number of the matrix’seigenvalues, with the constraint to explain nearly its full variance.

The correlation matrices are just containing 20 rows. Hence, they cannot carry ahigh variance and the reduced form should explain at least 98% of the observedvariance. In some cases it might be necessary to use an additional principal com-ponent to capture the correlation matrix adequately. Table 5.1 shows the numberof components, the explained variance, and the mean bias between the full and thereduced correlation matrix, defined by the mean absolute difference (in column es-timation error). The estimation error is low compared to numbers between −1 and+1 as given within the correlation matrix.

As described in Section 5.3.1 we focus on the mean correlation structure (generaldefault setup) as presented in row (f) in Table 5.1. As a (general) simulation pa-rameter, this correlation matrix is reduced to five principal components. However,further analysis will highlight this choice of components again, by comparing simu-lation results of a full correlation matrix with reduced forms. For this illustration,reduced forms of the correlation matrix with ranks 10, 8, 5, 4, 3, 2, and 1 are used.Table 5.2 shows information about these, similarly to the information of Table 5.1.


sample sample period principal explained estimationset components variance error(a) July 2004 until June 2005 5 99.6251 0.00271444(b) July 2005 until June 2006 5 99.7950 0.00143060(c) July 2006 until June 2007 5 99.4633 0.00438014(d) July 2007 until June 2008 5 99.2902 0.00415306(e) July 2004 until June 2008 7 98.7548 0.00740668(f) Mean of yearly matrixes (a) to (d) 5 99.2334 0.00535736

Table 5.1: The necessary principal components to explain percentages of the non-reduced matrix’s variance given in column “explained variance” are showin this table. This approximation causes an estimation error between thefull and the reduced correlation matrix which is specified as the meanabsolute difference and given in the last column.

sample principal explained estimationset components variance error(a) 20 100 0(b) 10 99.9989 0.00000880(c) 8 99.9856 0.00009938(d) 5 99.2334 0.00535736(e) 4 96.8694 0.01960221(f) 3 91.1375 0.06068956(g) 2 81.6723 0.17318545(h) 1 68.8542 0.34289929

Table 5.2: The results of reducing the dimensions of the default correlation matrixby zeroing a number of its eigenvalues are shown in this table . Dependingon the remaining principal components, the explained variance as well asthe estimation error are given. The error is specified as the mean absolutedifference between the full and the reduced matrices.


5.5 The Validation Sample

As mentioned before, we use the data set from July 1, 2008 to December, 31 2009for the validation of the simulation model. As described in Section 5.2, the Svens-son model fails to capture the dynamics especially in the beginning of this period.For this reason we use a simple linear interpolation on the converted continuously-compounded rates rc(t). We use the Equations (3.3) and (3.6) to estimate forwardrates according to the simulation scheme as described in Section 4.5.1. We receive 20forward rates spaced by 6 months in the same shape i.e. structure as the simulatedforward rates will be. The results are presented in Figure 5.7.

Figure 5.8 shows forward rate curves during the validation period. Here one cansee a nearly flat initial level. The short end declines strongly, whereas the long endshows less variation. Toward the end of the observed period, the long end movesslightly upwards. This observation, which can also be seen in Figure 5.7, is necessaryto understand the movement of different interest rate products within the validationperiod evaluated on this market data. We will refer to this later.

2009 2010Time

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(a) The estimated forward rates.

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(b) The estimated forward rates in a three di-mensional presentation.

Figure 5.7: The estimated historical forward rates over a one and a half year period,which will be used for validation purposes.


0.5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 7. 7.5 8. 8.5 9. 9.510.Time0.01

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Figure 5.8: Examples of estimated historical forward rate curves during the valida-tion period are shown in this figure. Here we start with the forward ratesobserved at the first date, which is also used as initial forward rate curveof the simulation model. Additionally the figure shows observed forwardrate curves spaced by 30 trading days over the whole period, where thecolor of the curves switches between blue and purple sample by sampleordered in time. The observations can easily be distinguished, since theshort end is monotone decreasing from sample to sample, i.e over time.

5.6 Implied Cap Volatilities for Pricing

IR-Options

As described in Section 4.6.2 we use the latest observable cap/floor volatility surfacebefore the start date of the simulation for the valuation of caps and floors withinthe simulation model. Hence, the effects of a variation of the surface over timeare not modeled. Such changes can be significant, as shown in Figure 5.9. Thisfigure presents the observed end of day implied at the money (ATM) volatility fora period of 5.5 years starting at July 1, 2004 until December 31, 2009 representing1435 trading days. The market quotations were provided by ICAP. Note that thisperiod corresponds to the market data of forward rates presented in Section 5.1.

Additionally, Figure 5.10 shows the cap volatility surface used within the simulationmodel as well as the surface six months later. The function shows the impliedvolatility of caps and floors containing caplets (floorlets) spaced by six months.Every caplet pays the difference between EURIBOR and the strike rate K giventhat it is non-negative, whereas the floorlets pay the difference between K andEURIBOR if non-negative. The first caplet on the cap expires after six months,


the last caplet six months prior to final maturity. The payment exchange of everycaplet, if any, is done six months after its fixing. The implied volatility depends onthe final maturity as well as on the fixed strike K, which is equal for all caplets. Theimplied volatility is derived from market quotations of caps and floors using Black’sformula.1

Figure 5.9: The historical evolution of ATM implied cap volatilities, shown over fiveand a half years.

Figure 5.9 shows a very strong variation at the short end starting September 2009,which corresponds to the high volatility of interest rates shown in Figure 5.1. Clearly,this leads to the question of how one can model cap volatilities correlated in a waywith the term structure within the simulation model. Furthermore, Figure 5.10shows a possible change of the surface’s geometry. Hence, not only the level ofvolatility, but also the geometry should somehow be modeled. However, we use thelast observed volatility surface for pricing. We will return to this aspect at the endof this thesis.

1See Sections 3.2.3 and 4.6.2 for cap/floor pricing formulas and also (Brigo and Mercurio, 2005,section 6.4) for more details on the evaluation of volatility coefficients from market quotationsfor pricing of options in the LMM context.


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Figure 5.10: Examples of a cap/floor volatility surface, where surface (a) is used forvaluation purposes within the simulation model.

6 Results

In this section the LIBOR market model, as introduced in Chapter 4 and calibratedin Chapter 5, is studied by analyzing simulation results. In the first section, theresulting forward rate distributions are analyzed depending on the spot and forwardmeasure. Section 6.2 proceeds with a presentation of the estimated exposure distri-bution of swaps and caps. In Section 6.3, the bias due to a reduction of the modelfactors is analyzed, and Section 6.4 proceeds with a study of the effects of a reductionof the time grid. Furthermore, Section 6.5 presents a study of the model’s sensitivityto changes in the model’s parameters. Finally, Section 6.6 shows how expected ex-posures, capital requirements and potential exposures, change over time, when thesimulation model is re-calibrated and new simulations are performed periodically.

6.1 Forward Rate Distribution

The simulation model is used to estimate the forward rate distribution under differ-ent measures. Recall, the LMM is calibrated to a historical correlation and volatilitystructure, where the non-deterministic drift terms are not taken into account1. Theresulting LMM models 20 forward rates Fk(t), k = 1, . . . 20, all spaced by half ayear. For simulation purposes, we use a 30/360-day-count convention as describedin context of Definition 3.5. We simulate the evolution of forward rates using theestimation of daily changes. To be consistent with the market data used for modelcalibration only work days were simulated, which are set to be 260 each year. Thus,between any two successive forward rates 130 simulations are performed.

We simulate both, a model under spot measure Qd and a model under forwardmeasure Q20 simultaneously. Theoretically, one would expect the last forward rate

1For more details, see Section 4.5.3.

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