A Libor Market Model Approach for Measuring Counterparty Credit Risk Exposure

(1)

A Libor Market Model Approach for

Measuring Counterparty Credit Risk Exposure

Junsheng Huang

July 11, 2014

Master’s Thesis

Supervisors:

dr. P.J.C. Spreij (University of Amsterdam) Dimitar Mechev, MSc. (NIBC Bank N.V.)

KdV Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

(2)

Abstract

This thesis studies the Libor market model and its application for measuring the counterparty credit risk exposure of interest rate derivatives. We will study the forward rate dynamics of the model and perform Monte Carlo simulations to simulate the future forward curves. Using the simulated forward curves, the future mark to market curves and the exposure profiles of the interest rate derivatives can be obtained. Results are reasonable and are comparable to those obtained by Bloomberg and other interest rate models.

Calibration of the model volatilities is done using cap implied volatilities. The correlation structure between the forward rates is calibrated using Rebonato’s method. Different assumptions on the volatility structure will be discussed and comparisons of the resulting exposure profiles will be shown.

Keywords: Counterparty credit risk, interest rate deriva-tives, valuation, credit valuation adjustment (CVA), expected exposure (EE), potential future exposure (PFE), Libor Market Model (LMM).

Title: A Libor Market Model Approach for Measuring Counterparty Credit Risk Exposure

Author: Junsheng Huang, jh15926@gmail.com, 5789338 Supervisor: dr. P.J.C. Spreij (University of Amsterdam) Dimitar Mechev, MSc. (NIBC Bank N.V.)

Date: July 11, 2014

Korteweg de Vries Instituut voor Wiskunde Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

(3)

Chapter 1 Introduction

Risk is the exposure to the possibility of loss. It can come from uncertainty in financial markets. Financial risk management is an important part of any business. Of all the different areas of financial risks, counterparty credit risk is arguably one of the most complex risk to deal with. Counterparty credit risk is a specific form of credit risk between derivatives counterparties. It is the risk that a counterparty will not be able to pay what it is obligated to on any trade or transaction when it is supposed to. Ever since the credit crisis of 2007 and the failure of many large financial institutions, counterparty risk has been considered by most market participants to be the key financial risk. One type of the financial derivatives that can lead to significant counterparty risk is over-the-counter (OTC) derivatives. OTC trading is done directly be-tween two parties, without any supervision of an exchange. OTC derivatives are an important part of the financial industry. Its markets are large and have grown dramatically in the last decade1.

One particular interesting concept related to the counterparty credit risk is Credit Valuation Adjustment (CVA). CVA is defined as the difference between the risk free portfolio and the true portfolio value that takes into account the possibility of a counterparty’s default. Intuitively, CVA is the market value of counterparty credit risk. The CVA of a financial instrument is usually calculated as follows:

CV A = P D ∗ LGD ∗ EE.

Where PD is the probability of default, LGD is the loss given default, and

1_{According to the Year-End 2012 market analysis published by ISDA on June 20,}

2013, the notional amount outstnading of OTC derivatives at December 31, 2012 was $565.2 trillion.

(6)

EE is the expected exposure at the time of default. In this thesis, we are interested in building internal models to measuring the amount of exposure in their portfolio of interest rate derivatives. Hence we will focus on measure the expected exposure component of CVA in this thesis.

One of the possible approaches to measure interest rate risk exposure is to implement an interest rate model to simulate future interest rates. One par-ticularly important interest rate is the Libor2 _{rate. It is widely used as a}

reference rate for many financial instruments in both financial markets and commercial fields.

In this thesis, we study and implement an interest rate model called the Li-bor Market Model to measure interest rate risk exposure. We will mainly focus on measuring exposures of interest rate swaps in this thesis. However, since the Libor Market Model simulates the underlying Libor rate and not the value process of any particular financial products, the model can be used to measure the exposure of any interest rate derivatives.

The structure of this thesis is as follows: Chapter 2 is a summary of the basic introduction of the interest rate market3_{. We will introduce the basis of}

interest rates and different types of interest rate derivatives. In chapter 3, we will introduce various types of short rate models and explain their advantages and disadvantages compared to the Libor Market Model4_{. Chapter 4 will}

introduce the Libor Market Model explicitly. Furthermore, we will also show the calibration of important parameters in the Libor Market Model. Chapter 5 will analyze some of the results produced by the Libor Market Model. We have also designed a simple backtest to test the robustness of the model. Finally, Chapter 6 is a summary of this thesis and will sum up important findings of this project.

2_{London Interbank Offer Rate.}

3_{See for example [3] or [4] for extensive explanation of interest rates and its derivatives.} 4_{See for example [6] for an extensive introduction of various interest rate models.}

(7)

Chapter 2 Interest Rates

Money has time value. A euro today is worth more than a euro tomorrow due to the existence of interest rates. In this chapter, we introduce the basic construction of interest rates and its derivatives. Throughout this thesis, we will assume that the market is frictionless and trades can be made continu-ously. We also assume that the market is free of arbitrage1_{. In an arbitrage}

free market, investment strategies that replicate each other must have the same rate of return. This property will be used throughout this thesis.

2.1 Interest Rate Compounding

An important concept about interest rates is compounded interest. Briefly speaking, it is the interest of interest. For example, let us consider an invest-ment of one euro for one year at an annual interest rate of R. If the interest rate is paid once per year, then we will receive 1 + R euro after a year. On the other hand, if the interest rate is paid once per half year, the balance will be (1 +R₂)2 _{euro in the end of the year. In general, if the interest rate is}

compounded m times per year, the balance will be (1 +_mR)m euro. Taking m to infinity would mean that interest rate is continuously compounded, and the balance will be

lim m→∞ 1 + R m m = eR

euro after a year. Since the exponential function has nice analytic properties, we often study continuously compounded interest rates. However, interest rates are not always continuously compounded in practice. It is very

impor-1_{Note that the market is not necessary arbitrage free in every model, this is a property}

(8)

tant to remember that the differences do matter when the nominal investment is large enough.

2.2 Zero Coupon Bonds and Interest Rates

In this section, we will introduce zero coupon bonds and show the construc-tion of interest rates from zero coupon bonds. Zero coupon bonds are the primary building blocks of interest rates.

Definition 2.1. A zero coupon bond with face value N and maturity T (also known as a T -bond) is a contract that pays N units of money at time T . The price of a T -bond with face value 1 at time t is denoted by B(t, T ). Remark. The value B(t, T ) is also known as the discount factor. Meaning that 1 euro at time T is worth B(t, T ) at time t. Note that we must have B(T, T ) = 1 in an arbitrage free market.

To construct interest rates from zero coupon bonds, we fix three time points t ≤ T < T + δ. Consider a contract at time t that allows us to invest 1 euro at time T to receive a deterministic rate of return at time T + δ. We observe that this contract can be replicated in the following way:

• At time t, sell a T -bond to receive B(t, T ) euro. Use the income to buy B(t, T )/B(t, T + δ) amount of T + δ bonds. Note that T + δ bonds cost B(t, T + δ) each, and thus our net investment at time t is 0.

• At time T , the T bond matures and we will pay 1 euro.

• At time T +δ, the T +δ bonds mature and we will receive B(t, T )/B(t, T + δ) euro.

Note that all of the above cash flows are deterministic. It follows that, the investment of 1 euro at time T yields B(t, T )/B(t, T + δ) euro at time T + δ. This leads to the following definition.

Definition 2.2. The simple forward rate for the period [T, T + δ] contracted at time t, denoted by F (t; T, T + δ), is defined as the solution to the equation

1 + δF (t; T, T + δ) = B(t, T ) B(t, T + δ). The solution of this equation is

F (t; T, T + δ) = 1 δ B(t, T ) B(t, T + δ)− 1 .

(9)

An important type of simple forward rate is the Libor forward rate. Libor for-ward rates are interest rates at which deposits between banks are exchanged. They are quoted for a series of possible maturities ranging from overnight to 12 months.

Another important concept in many interest rate models is the instantaneous short rate. It is being modeled by many interest rate models because the short rate has some nice analytic properties. For instance, all forward rates and discount factors can be uniquely determined by the short rate in an arbitrage free model. To define the short rate, we need to walk through a serie of different interest rate definitions first.

Definition 2.3. The continuously compounded forward rate for the period [T, T + δ] contracted at time t, denoted by R(t; T, T + δ), is defined as the solution to the equation

exp(δR(t; T, T + δ)) = B(t, T ) B(t, T + δ). It follows that

R(t; T, T + δ) = −log B(t, T + δ) − log B(t, T )

δ .

As δ goes to 0 and assuming differentiability of log B(t, T ), we obtain the instantaneous forward rate with maturity T contracted at time t

f (t, T ) = lim

δ→0R(t; T, T + δ) = −

∂

∂T log B(t, T ).

The function T → f (t, T ) is called the forward curve at time t. Finally, we define the instantaneous short rate at time t by

r(t) = f (t, t) = lim

T ↓tR(t, T ).

Note if the short rate process r(t) is a deterministic function of time, the discount factor B(t, T ) can be written as

B(t, T ) = exp − Z T t r(u)du .

Remark. Although the the discount factors can be uniquely determined by the short rate, it has some undesirable properties. For instance, short rates cannot be directly observed in the market. They must be estimated using market observable interest rates such as the forward Libor rates. More details about short rate models will be discussed in Chapter 3.

(10)

2.3 Interest Rate Swap

An interest rate swap is an instrument in which two parties agree to exchange interest rate cash flows. It has a huge number of varieties that can be struc-tured to meet the specific needs of the counterparties. In this section, we will introduce the floating-for-fixed interest rate swap settled in arrears. It is specified by a nominal value N , a fixed rate R and a number of future dates T0 < T1 < ... < Tn. At time Ti, i = 1, ..., n, we will receive

δiN R, δi = Ti − Ti−1

and pay

δiN F (Ti−1; Ti−1, Ti).

The net cash flow at time Ti is thus

δiN (R − F (Ti−1; Ti−1, Ti)).

Let the Ti bond prices B(t, Ti), t ≤ T0, be the discount factors, the value of

the swap at time t is thus Π(t) =

n

X

i=1

B(t, Ti)δiN (R − F (Ti−1; Ti−1, Ti)).

The fixed rate R = Rswap that gives Π(t) = 0 is called the swap rate. Hence

the swap rate is calculated as follows 0 = Π(t) 0 =

n

X

i=1

B(t, Ti)δiN (Rswap(t) − F (Ti−1; Ti−1, Ti)) n

X

i=1

B(t, Ti)δiRswap(t) = B(t, Ti)δiF (Ti−1; Ti−1, Ti)

Rswap(t) =

B(t, Ti)δiF (Ti−1; Ti−1, Ti)

Pn

i=1B(t, Ti)δi

.

Furthermore, if we assume that δi are the same for all i, then the swap rate

can be written as the weighted average of the simple forward rates Rswap(t) =

n

X

i=1

(11)

where the weights wi(t) are defined as wi(t) = B(t, Ti) Pn j=1B(t, Tj) .

Note that we have replaced F (Ti−1; Ti−1, Ti) by F (t; Ti−1, Ti). This is

be-cause the market usually assumes that F (t; Ti−1, Ti) is the expected value

F (Ti−1; Ti−1, Ti) at time t.

Interest rate swaps are extremely liquid. They are a popular instrument to hedge interest rate risk. Swaps can also be used by investors who expect changes in interest rates to make profit. For instance, investors who expect interest rates to fall could enter a floating-for-fixed interest rate swap. As the interest rates fall, they will make profit by paying a low floating rate in exchange for the same fixed rate.

2.4 Caps and Floors

In this section, we will introduce two common interest rate derivatives, caps and floors. An interest rate cap is a contract that protects the holder against high interest rates. Similar to interest rate swaps, it is specified by a nominal value N , a strike rate K and a number of future dates T0 < T1 < ... < Tn

with Ti− Ti−1= δi. Its net cash flow at time Ti, i = 1, ..., n, is

ci = δiN max(0, F (Ti−1; Ti−1, Ti) − K).

Each of these payments is called a caplet with reset date Ti−1and settlement

date Ti. The time t value of a cap Cp(t) is simply the sum of the discounted

values of its caplets.

Cp(t) =

n

X

i=1

B(t, Ti)ci.

A floor is the converse to a cap. It is a contract that protects the holder against low interest rates. With similar arguments, the time t value of a floor is

F l(t) =

n

X

i=1

B(t, Ti)δiN max(0, K − F (Ti−1; Ti−1, Ti)).

Many interest rate models have explicit formula for caps and floors. They are often used to calibrate the model parameters.

(12)

2.5 Day Count Convention

The market uses different day count conventions to determine the number of days between two dates in different ways. It determines how interest accrues over time for a variety of investments. It is also used to quantify periods of time when discounting a cash flow to its present value. In this section, we will introduce some frequently used day count conventions and show how to calculate the year fraction between two dates in those different day count conventions. By convention, the unit of time is years. The two dates are expressed as d1/m1/y1 and d2/m2/y2, representing their dates as day/month/year.

• Actual/Actual: This convention counts the actual days in the period. The number of years is calculated based on the portion in a leap year and the portion in a non-leap year. The day count convention is given by

Days not in leap year

365 +

Days in leap year

366 .

• Actual/365: This convention also counts the actual days in the period. Every year counts as 365 days. The day count convention is given by

Actual number of days in the period

365 .

• Actual/360: This convention is the same as Actual/365 except that every year counts as 360 days.

• 30/360: Every month counts as 30 days and every year counts as 360 days. The day count convention is given by

360(y2 − y1) + 30(m2 − m1) + d2 − d1

360 .

When extracting information on interest rates from market data, it is im-portant to realize for which day count convention a specific interest rate is quoted.

(13)

Chapter 3 Interest Rate Modeling

Since the changing of interest rates constitutes one of the major risk sources for financial institutions, a large number of interest rate models have been de-veloped attempting to model the dynamics of the interest rates and correctly price interest rate derivatives. From a theoretical point of view, the short rate is a convenient object to be modeled because the future evolution of all discounting factors can be written in terms of the short rate. However, mar-ket observations suggest that the dynamics of interest rates are not driven by only one stochastic process. This, together with other considerations, has led various researchers to consider models that are driven by multiple stochastic processes.

In this chapter, we will introduce various types of interest rate models and discuss their differences. The model of choice for this thesis is the Libor Market Model, which will be explicitly discussed in the next chapter.

3.1 One Factor Short Rate Models

The earliest stochastic interest rate models were one factor short rate models. The main advantage of short rate models is that the prices of bonds and in-terest rate derivatives can often be explicitly expressed in analytical formulas. Short rate models often specify the short rate r as the solution of a stochas-tic differential equation. Below we discuss some of the popular short rate models that have been used over the last few decades. All of the parameters are real-valued.

(14)

The Vasicek model models the short rate as

drt= (θ − αrt)dt + σdWt.

Assuming that α > 0, the Vasicek model is the first interest rate model to capture mean reversion, an essential characteristic of the interest rate. Un-like some other financial instruments such as stocks, interest rates cannot rise indefinitely. Similarly, interest rates usually do not decrease below 0. As a result, interest rates move in a limited range, showing a tendency to revert to a long run value.

One disadvantage of the Vasicek model is that the short rate in this model will not move in a limited rage and may decrease below 0. Moreover, it has only a finite number of free parameters. Because of this, it is impossible to calibrate these parameters in such a way that the model produces observed market prices of certain interest rate derivatives. The natural solution to this problem is to extend the Vasicek model by allowing some of the parameters to be deterministic functions of time. This extension of the Vasicek model is called the Hull-White model, which models the short rate as

drt= (θt− αrt)dt + σtdWt,

where θt and σt are deterministic processes. The flexibility of the

parame-ters of the Hull-White model allows it to produce correct market prices of certain interest rate derivatives. However, the dynamics of all interest rates in this model are still driven by a single stochastic process, which may not be consistent with actual market movements.

3.2 Multi-factor Models

Besides one-factor models, there are also multi-factor models of the short rate. An example of multi-factor short rate model is the Longstaff-Schwartz model. It models the short rate as

drt= (µXt+ θYt)dt + σt p YtdW3t, where dXt = (at− bXt)dt + p XtctdW1t, and dYt= (dt− eYt)dt + p YtftdW2t.

(15)

Although multi-factor short rate models are able to capture many of the observed properties of the term structure, the entire money market is still driven by the short rate in multi-factor short rate models. From an eco-nomic point of view, it is unlikely that the entire money market is driven by a single explanatory variable. Moreover, from a numerical point of view, it is very complicated to calibrate a short rate model to cap or swaption data. Furthermore, the market practice has been using a formal extension of the Black-Scholes model to value caps, floors and swaptions. It can be shown1

that short rate models are inconsistent with Black’s formula. Thus there has been a natural demand for arbitrage free models with the property that the theoretical values for caps, floors and swaptions are consistent with Black’s formula.

In 1997, Brace, Gatarek and Musiela developed an interest rate model that models the discrete market rates instead of the instantaneous interest rates. Under suitable choice of numeraires, these market rates can be modeled log normally and the resulting valuation of caps and floors are thus consistent with Black’s formula. This model is called the Libor Market Model because it models discrete market rates such as the Libor rates. In the following chap-ters of the thesis, we will introduce the Libor Market Model and show how to implement it to measure credit risk exposures of interest rate derivatives.

(16)

Chapter 4 Libor Market Model

In the previous chapter, we have seen that short rate models have limitations when it comes to calibration on cap or swaption prices. Furthermore, the market practice has been pricing caps, floors and swaptions by using an ex-tension of the Black-Scholes model. Hence there has been a natural demand for interest rate models that produce cap, floor and swaption prices of the Black-Scholes form. This can be done by modeling the simply compounded interest rate instead of the short rate. Suitable candidates for this framework are discrete market rates like forward Libor rates or swap rates, hence the name Libor/Swap Market Model. It has been proven that under a suitable choice of measures, these market rates can indeed be modeled log normally and they are thus consistent with the market practice.

One of the most interesting properties of the Libor Market Model is its flexi-bility because we have the freedom to make assumptions on some of the model parameters. The goal of this thesis is to find a set of model parameters such that the Libor Market Model produces reasonable expected exposure and potential future exposure curves for interest rate derivatives. The first half of this chapter will introduce the theory behind the modeling process. The second half of this chapter will study the assumptions of previous researchers on the model parameters and apply them to current market data’s. Modeling results under different assumptions will be shown in the end of this chapter and the next chapter.

4.1 Construction of the Libor Market Model

In this section, we will introduce the basics of the Libor Market Model. This is a summary based on chapter 9 and 10 of [3].

(17)

Let δ be a fixed positive number, for example 3 months. The forward Libor rate with tenor δ for the future date T contracted at time t, denoted by L(t, T ), is defined as the simple forward rate F (t; T, T + δ).

L(t, T ) := F (t; T, T + δ) = 1 δ B(t, T ) B(t, T + δ) − 1 ,

where B(t, T ) is a stochastic process that represents the price of a T -bond with face value 1 at time t. Moreover, we define

S(t) = B(t, T + δ)L(t, T ), 0 ≤ t ≤ T.

If we build a term-structure model driven by a single Brownian motion under the actual probability measure P and satisfying the Heath-Jarrow-Morton no-arbitrage condition1_{, then there is a Brownian motion ˜}_{W (t) under a}

risk-neutral probability measure ˜_{P such that forward rates f (t, T + δ) are given} by

df (t, T + δ) = σ(t, T + δ)σ∗(t, T + δ)dt + σ(t, T + δ)d ˜W (t), and bond prices B(t, T ) by

dB(t, T ) = R(t)B(t, T + δ)dt − σ∗(t, T + δ)B(t, T + δ)d ˜W (t), (4.1) where σ∗(t) =R_tT +δσ(t, v)dv and R(t) = f (t, t). Note that (4.1) is equivalent to the following equation2

D(t)B(t, T + δ) =B(0, T + δ) exp Z t 0 −σ∗(t, T + δ) · d ˜W (u) −1 2 Z t 0 kσ∗(t, T + δ)k2du . If we define ˜ WT +δ(t) = ˜W (t) + Z t 0 σ∗(u, T + δ)du,

then we notice that D(u)B(u,T +δ)_{B(0,T +δ)} is the Radon-Nikodim derivative process Z(t) appearing in the Girsanov Theorem A.3. It follows that, under the assumption E Z T +δ 0 kσ∗(t, T + δ)k2 D(u)B(u, T + δ) B(0, T + δ) 2 du < ∞,

1_{Since the Libor Market Model is a term-structure model for the forward Libor rates,}

it falls under the Heath-Jarrow-Morton (HJM) framework. See [3], chapter 10 for more detail information about the HJM framwork.

(18)

the process ˜WT +δ(t) is a Brownian motion under the probability measure ˜ PT +δ given by ˜ PT +δ(A) = Z A D(T + δ) B(0, T + δ)d˜P for allA ∈ F .

The measure ˜_PT +δis called the T +δ forward measure. Furthermore, Theorem A.2 implies that _{B(t,T +δ)}S(t) is a martingale under ˜_PT +δ_{. By the Martingale}

Representation Theorem A.4, there exists an adapted process3 _{σ(t, T ) such}

that

dL(t, T ) = σ(t, T )L(t, T )d ˜WT +δ(t), (4.2) where ˜WT +δ is a Brownian motion under the (T + δ) forward measure. Note that dL has no dt term under the forward measure PT +δ. It follows imme-diately that L is log normally distributed under PT +δ _{if σ is deterministic.}

This leads to the following pricing formula.

Theorem 4.1. Let K be a nonnegative constant. Consider a caplet that pays (L(T, T ) − K)+ _{at time T + δ. If we assume that the forward Libor rate is}

given by (4.2) and σ(t, T ) is deterministic. Then the price of the caplet at time zero is

B(0, T + δ)(L(0, T )N (d1) − KN (d2)),

where N is the cumulative distribution function of a standard normal distri-bution, and d1 = 1 q Rt 0 σ2(t, T )dt log(L(0, T )/K) + 1 2 Z T 0 σ2(t, T )dt , d2 = 1 q Rt 0 σ 2_{(t, T )dt} log(L(0, T )/K) − 1 2 Z T 0 σ2(t, T )dt .

Proof. According to the risk-neutral pricing principle, the price of the caplet at time zero is the discounted expected value of the payoff under the risk-neutral measure ˜P

˜

ED(T + δ)(L(T, T ) − K)+ .

3_{A stochastic process (X}

i)i∈I is said to be adapted to the filtration (Fi)i∈I if Xi is

Fi-measurable for all i ∈ I. In this case, the process (σ(t, T ))t∈[0,T ] is adapted to the

(19)

Since _{B(0,T +δ)}D(T +δ) is the Radon-Nikodim derivative of ˜_PT +δ with respect to ˜_{P, we} have ˜ ED(T + δ)(L(T, T ) − K)+ =B(0, T + δ)˜_E D(T + δ) B(0, T + δ)(L(T, T ) − K) + =B(0, T + δ)˜_ET +δ(L(T, T ) − K)+. Moreover, the solution to (4.2) is

L(T, T ) = L(0, T ) exp −1 2 Z T 0 σ2(t, T )dt + Z T 0 σ(t, T )d ˜WT +δ(t) .

It follows that log L(T, T ) is normally distributed under ˜_PT +δ _{with mean}

log L(0, T ) − 1 2 Z T 0 σ2(t, T )dt and variance Z T 0 σ2(t, T )dt. By the Black-Scholes formula, we have

˜

ET +δ(L(T, T ) − K)+= L(0, T )N (d1) − KN (d2).

Hence the risk-neutral price of the caplet at time zero is B(0, T + δ)(L(0, T )N (d1) − KN (d2)).

In the rest of this thesis, we will not work with the real world measure P anymore. For the sake of simplicity, we will write the risk neutral measure as P instead of ˜_{P, and the forward measure as P}T _{instead of ˜}

PT.

4.2 Change of measure

In the previous section, the dynamics of each Libor forward rate is defined under its own forward measures depending on the effective period of the for-ward rate. For analytical purposes, it is more convenient to work under one single measure. In this section, we will show how to write the forward rate

(20)

dynamics under the different forward measures.

Let _dPdPT +δT(t)_(t) be the Radon-Nikodim derivative of PT(t) with respect to PT +δ(t).

From Theorem A.2, we have dPT(t) dPT +δ_(t) = B(t, T ) B(0, T ) B(0, T + δ) B(t, T + δ). Recall from the previous section that

L(t, T ) = 1 δ B(t, T ) B(t, T + δ)− 1 . It follows that dPT(t) dPT +δ_(t) = B(0, T + δ) B(0, T ) (1 + δL(t, T )). (4.3)

Define the process η(t) by

η(t) := dP T_(t) dPT +δ_(t), and write c := B(0, T + δ) B(0, T ) . Then by (4.3), we have η(t) = c(1 + δL(t, T )) dη(t) = cδdL(t, T ) dη(t) = cδL(t, T )σ(t)dWT +δ(t) dη(t) = η(t) 1 η(t)cδL(t, T )σ(t)dW T +δ_(t) dη(t) = η(t) 1 c(1 + δL(t, T ))cδL(t, T )σ(t)dW T +δ_(t) dη(t) = η(t)δL(t, T )σ(t) 1 + δL(t, T )dW T +δ (t).

It follows that the Girsanov kernel4 _{of the measure transform from P}T _to

PT +δ is

δL(t, T )σ(t) 1 + δL(t, T ).

(21)

By the Girsanov Theorem A.3, we have dWT +δ(t) = δL(t, T )σ(t)

1 + δL(t, T )dt + dW

T_(t). _(4.4)

Note that (4.4) is true for any fixed T . Let

Ti = T + iδ, i = 1, ..., N

and applying equation (4.4) inductively, we can write any forward rate dy-namics under any forward measure. In particular, we have

dWTi_{(t) = −} N X k=i+1 δL(t, Tk)σk(t) 1 + δL(t, Tk) dt + dWTN_(t) _(4.5)

for i = 1, .., N . Equation (4.5) can be used to write all forward rate dynamics under the forward measure PTN_{. P}TN _{is called the terminal measure.}

4.3 Rank Reduction

One of the major problems in carrying out Monte Carlo simulations for the Libor market model is the lack of computational speed. Without any sim-plification techniques, each simulated forward curve will require as many Brownian motions as the number of forward rates to generate. However, practical observations suggest that the number of factors driving a forward curve is much smaller than the number of forward rates. This has been con-firmed by various research papers5_{. It has been demonstrated that, when}

applying principal components analysis (PCA) on the covariance matrix of the forward rates, only 3 or less factors are needed to recover more than 95% of the information. However, the covariance matrix modified by the PCA ap-proach does not generate the correct market cap prices in this model anymore because the PCA technique does not preserve all information of the original covariance matrix. An alternative approach to reduce the number of factors in the model and recover the correct market cap prices has been proposed by Rebonato [2]. In this section, we will discuss Rebonato’s approach and show how to apply it to calculate the model correlations between the forward rates.

(22)

4.3.1 Reformulation of the Dynamics of the Forward

Rates

Recall from (4.2) that, for every fixed T , the dynamics of the forward rates under its forward measure PT +δ _{is given by the formula}

dL(t, T ) = σ(t, T )L(t, T )dWT +δ(t). Moreover, we have seen in (4.5) that

dWTi_{(t) = −} N X k=i+1 δLk(t, T )σk(t) 1 + δLk(t, T ) dt + dWTN_(t)

for i = 1, .., N . If we write Li(t) := L(t, Ti) and σi(t) := σ(t, Ti), then it

follows that dLi(t) Li(t) = µi(t)dt + σi(t)dWTN(t), i = 1, ..., N (4.6) where µi(t) := − N X k=i+1 δσk(t) 1 + δLk(t) .

In the rest of this chapter, we will omit the dependency on t for the sake of simplicity. Furthermore, we will only work with Brownian motions under the terminal measure dWTN _{for the rest of this thesis.}

Note that even though we have written all dynamics of the forward rates under identical Brownian motions, these Brownian motions are not neces-sary independent of each other. From a modeling point of view, it may be profitable to have identical and independent Brownian motions as drivers for the model. Rebonato [2] suggested a method to do this as follows.

Instead of having one Brownian motion driving each forward rate process, we assume that all forward rate processes are driven by the same set of m independent Brownian motions under the terminal measure, we can rewrite equation (4.6) as dLi Li = µidt + m X k=1 σikdWk, (4.7)

where σik is the volatility of the k-th Brownian motion on the i-th forward

rate. Note that we have the freedom to choose m in the above formulation. The intention is to choose m much smaller than the number of underlying

(23)

forward rates to reduce the number of random factors in the model. In prac-tice, m will usually be equal to 2 or 3.

Note that we know very little about σik at this moment. We only know

that it might be related to the volatilities and correlations of the forward rates. From a modeling point of view, it is not very convenient to have a parameter that depends on both the volatility and correlation structure of the underlying product. Hence we would like to reformulate (4.7) in such a way that each parameter depends only on the volatility or the correlation of the forward rates. We can do this as follows: Let Wi be the Brownian motion of the Brownian motion driving forward rate Li in equation (4.6).

Comparing (4.7) with equation (4.6), we see that we have replaced σidWi

with Pm

k=1σikdWk for i = 1, ..., N . Since we do not intend to change the

dynamics of the forward rates, we must have

m

X

k=1

σ_ik2 = σ2_i. (4.8)

Using this equation, we have dLi Li = dLi Li = µidt + σi m X k=1 σik σi dWk = µidt + σi m X k=1 σik pPm k=1σ 2 ik dWk. (4.9) If we define bik := σik pPm k=1σ 2 ik . Then condition (4.8) becomes

m

X

k=1

b2_ik = 1, i = 1, ..., n. (4.10)

Moreover, equation (4.9) can be rewritten as dLi Li = µidt + σi m X k=1 bikdWk.

Note that we have taken the volatilities of the forward rates σi out of σik.

(24)

Wj, it can be shown6 that b is related to ρ, the correlation matrix of the original Brownian motions, by

bbT = ρ.

Intuitively, the quantities bik can be interpreted as the sensitivities of the

i-th forward rate to the k-th Brownian motion. For instance, if bik = 0.8,

then the k-th Brownian motion will determine 80% of the change of the i-th forward rate.

4.3.2 Polar coordinates

In the previous section, we have achieved the task of separating the volatility and correlation components. Moreover, the correlation matrix ρ is completely determined by b. Suppose that ρ is known and b has the same rank as ρ. Since correlation matrices are always symmetric, b is just the cholesky decom-position of ρ. However, this would also imply that the number of Brownian motions needed to generate each forward curve will be the same as the num-ber of underlying forward rates. This is not always affordable because the amount of observations of Brownian motions needed to perform monte carlo simulations in this model would be huge. Moreover, practical observations suggest that 2 or 3 Brownian motions are sufficient to model most of the market movements of forward curves. In this section, we will discuss a rank reduction technique suggested by Rebonato [2]

Suppose that the rank of the correlation matrix ρ is h and the number of Brownian motions driving the model is m, m < h. The task is to determine the elements of the matrix b such that

m

X

k=1

b2_ik = 1, (4.11)

and the distance between bbT and ρ is minimized. We define the distance as the χ2 _measure X i=1:h X k=1:m (bbT)ik− ρik 2 . (4.12)

Optimizing the coefficients of b seems to be a complicated task. However, we observe that for any θ, we have

sin2(θ) + cos2(θ) = 1.

(25)

This equation can be generalize to m factors X k=1:m−1 (sin2(θj) + cos2(θk)) k−1 Y j=1 sin2(θj) ! = 1

for all θj, j = 1, , k − 1. If we define

bik = cos θik k−1 Y j=1 sin θij, k = 1, ..., m − 1 bim= m−1 Y j=1 sin θij,

then condition (4.11) will always be satisfied. Therefore, any sets of angles {θij} specifies a possible set of coefficients {bik}. The optimization is thus

translated into finding a set of angles such that (4.12) is minimized. Note that after the translation, we only need to optimize over (m − 1)h variables, while initially we needed to optimize over mh variables. Since we intend to choose a small m, such as m = 2 or m = 3, this translation will reduce the amount of variables by 1/3 to 1/2, which will lead to a significant speed improvement for the model simulation process.

4.4 Instantaneous Volatility of Forward Rates

In the previous section, we have shown how to rewrite the dynamics of the forward rates such that all forward rates are driven by m independent Brow-nian motions. Note that the number m can be chosen freely. This means that we can choose the number of factors in the model, which will, in most of the cases, reduce the complexity of the model and we have thus completed the task of increasing computational speed. The only task that is left to do is the calibration of the model parameters. One of the properties of the Libor market model is its flexibility to allow the user to make assumptions on the structures of its parameters. During this project we have studied and implemented several parameter structures proposed by different authors. In the rest of this chapter, we will discuss these structures and their respective results to decide which structure to use for our implementation.

The most important parameters in the Libor market model are the volatili-ties of the forward rates. Before we begin with the calibration strategies, we will show the analytical expressions that relate instantaneous volatilities and

(26)

cap implied volatilities. Cap implied volatility is the volatility that gives the correct market price when plugged into the Black Scholes formula for caps and caplets.

In practice, volatility implied by caplets and swaptions are not necessary con-sistent with each other. Hence we must not use both caplets volatilities and swaption volatilities for the calibration. In this chapter, we will investigate several calibration strategies that use either caplet volatilities or swaption volatilities.

Recall from Theorem 4.1 that the Black caplet price is given by B(0, T + δ)(L(0, T )N (d1) − KN (d2)),

where d1, d2 are as specified in Theorem 4.1. The implied volatility is defined

as σimplied(T ) = s 1 T Z T 0 σ2_{(t, T )dt.} _(4.13)

Note that the implied volatilities depend only on T and do not uniquely determine the instantaneous volatility function σ(t, T ). Hence we have the freedom to make assumptions about the structure of the forward rate volatil-ities. In this section, we will discuss some of the assumptions that were proposed in a number of books 7. Comparison of the results will be dis-cussed in the next chapter. Throughout this section, we will assume that the at-the-money cap implied volatilities are known8_{, furthermore we do not}

con-sider volatility smiles due to its complexity and the time limit of this project. In this project, we model forward rates by discretizing the model dynamics and taking small steps to the future every time. The volatilities between every small step are assumed to be constant. Hence the model volatility structure is as follows: t ∈ (T0, T1] t ∈ (T1, T2] · · · t ∈ (TN −1, TN] F1(t) σ1,1 F2(t) σ2,1 σ2,2 .. . ... . .. FN(t) σN,1 σN,2 · · · σN,N

7_{See for example [5]}

(27)

Where Fk(t) := F (t; Tk−1, Tk) and σi,j is the model volatility of Fj(t) for the

period t ∈ (Ti−1, Ti]. Note that equation (4.13) becomes

σ_implied2 k(T ) = 1 T k X i=1 σ_k,i2 (Ti− Ti−1). (4.14)

Where σimpliedk is the implied volatility of the k-th forward rate.

We expect that the volatility structure should have certain properties. For instance, all volatilities should be real and positive. In the rest of this section, we will consider different assumptions on the volatility function and observe the resulting model volatility structure. The implied volatility that is used for calibration is the cap implied volatility on May 10 2013, which is stripped from Bloomberg and shown in the following table.

Cap maturity Implied volatility

1 Year 113.84% 2 Year 70.48% 3 Year 70.55% 4 Year 65.99% 5 Year 59.11% 6 Year 53.40% 7 Year 48.13% 8 Year 43.84% 9 Year 40.15% 10 Year 37.30% 12 Year 33.10% 15 Year 29.51% 20 Year 27.03%

4.4.1 Piecewise Constant Instantaneous Volatility

De-pending Only on Maturity

One of the simplest assumptions that uniquely determines the instantaneous volatility structure is to assume that the instantaneous volatility function σ(t, T ) depends only on T . In this case, we have

σ(t, T ) = σimplied(T ), t ∈ [0, T ]

(28)

t ∈ (T0, T1] t ∈ (T1, T2] · · · t ∈ (TN −1, TN] F1(t) σ1 F2(t) σ2 σ2 .. . ... . .. FN(t) σN σN · · · σN

An example of the model volatilities will be shown in the next table. In this example, we will only show volatilities of forward rates with 6 month tenor, up to 5 years forward in time. Furthermore, only a few points of the implied volatility curve are known. The whole curve is obtained by Nelson-Siegel interpolation9 of the known implied volatilities.

t ∈ (0, 0.5] t ∈ (0.5, 1] t ∈ (1, 1.5] t ∈ (1.5, 2] t ∈ (2, 2.5] t ∈ (2.5, 3] t ∈ (3, 3.5] t ∈ (3.5, 4] t ∈ (4, 4.5] t ∈ (4.5, 5] F1(t) 1.1260 F2(t) 1.0293 1.0293 F3(t) 0.9425 0.9425 0.9425 F4(t) 0.8670 0.8670 0.8670 0.8670 F5(t) 0.7984 0.7984 0.7984 0.7984 0.7984 F6(t) 0.7394 0.7394 0.7394 0.7394 0.7394 0.7394 F7(t) 0.6872 0.6872 0.6872 0.6872 0.6872 0.6872 0.6872 F8(t) 0.6410 0.6410 0.6410 0.6410 0.6410 0.6410 0.6410 0.6410 F9(t) 0.5999 0.5999 0.5999 0.5999 0.5999 0.5999 0.5999 0.5999 0.5999 F10(t) 0.5640 0.5640 0.5640 0.5640 0.5640 0.5640 0.5640 0.5640 0.5640 0.5640

The implied volatilities will uniquely determine the model volatilities in this approach. Moreover, the volatility structure is very stable in the sense that it does not produce negative or imaginary volatilities. However, it might not be very realistic because forward rate volatilities could change over time in practice.

4.4.2 Piecewise Constant Instantaneous Volatility

De-pending Only on Time to Maturity

The second approach is to assume that the instantaneous volatilities are piecewise constant and depend only on the time to maturities T − t. The resulting volatility structure is as follows:

t ∈ (T0, T1] t ∈ (T1, T2] · · · t ∈ (TN −1, TN] F1(t) σ1 F2(t) σ2 σ1 .. . ... . .. FN(t) σN σN −1 · · · σ1

(29)

Note that under this assumption, the model volatilities will be uniquely de-termined in the following way: by equation (4.14), at k = 1, we have

σimplied1(T ) = σ 2 1. at k = 2, we have σ_implied2 2(T ) = 1 T σ 2 1(T2− T1) + σ22(T1− T0) .

Note that all variables except σ2 is known in this equation. Hence we can

uniquely solve σ2 as well. In general, σimpliedk depends on σ1, ..., σk. Since

σimpliedi is known for all i = 1, ..., N , we can solve σ3, ..., σN uniquely using

equation (4.14) by rewriting it as σ_k2 = T σ_implied2 k− k X i=2 σ_i2. (4.15)

Although this equation uniquely determines the volatility structure, it does not guarantee that the left hand side will be positive. It will be negative when there is a large difference between short term and long term implied volatilities, which leads to imaginary model volatilities. An example of the model volatilities is shown in the following table with the same settings as the previous approach. The letter i in this table indicates that the result is a complex number: t ∈ (0, 0.5] t ∈ (0.5, 1] t ∈ (1, 1.5] t ∈ (1.5, 2] t ∈ (2, 2.5] t ∈ (2.5, 3] t ∈ (3, 3.5] t ∈ (3.5, 4] t ∈ (4, 4.5] t ∈ (4.5, 5] F1(t) 1.1260 F2(t) 0.9213 1.1260 F3(t) 0.7412 0.9213 1.1260 F4(t) 0.5814 0.7412 0.9213 1.1260 F5(t) 0.4363 0.5814 0.7412 0.9213 1.1260 F6(t) 0.2986 0.4363 0.5814 0.7412 0.9213 1.1260 F7(t) 0.1501 0.2986 0.4363 0.5814 0.7412 0.9213 1.1260 F8(t) i 0.1501 0.2986 0.4363 0.5814 0.7412 0.9213 1.1260 F9(t) i i 0.1501 0.2986 0.4363 0.5814 0.7412 0.9213 1.1260 F10(t) i i i 0.1501 0.2986 0.4363 0.5814 0.7412 0.9213 1.1260

We see that the model volatilities can be imaginary. Hence we conclude that this approach for the model volatility is not suitable for practical applications.

4.4.3 Two Parameter Piecewise Constant Instantaneous

Volatility

Next, we assume that the instantaneous volatilities are piecewise constant and follow a separable structure. Let φ1, ...φN and ψ1, ...ψN be constant

(30)

t ∈ (T0, T1] t ∈ (T1, T2] · · · t ∈ (TN −1, TN] F1(t) φ1ψ1 F2(t) φ2ψ1 φ2ψ2 .. . ... . .. FN(t) φNψ1 φNψ2 · · · φNψN

Note that this is a generalization of the first two cases. We hope that the dependence of T can be modeled by the parameter φ, and the dependence of T −t can be modeled by the parameter ψ. Note also that the model volatilities will not be uniquely determined by this structure. However, equation (4.14) should still be valid here. Hence, to obtain the model volatilities, we simply find the constants φ1, ..., φN and ψ1, ..., ψN such that the error function is

minimized. The error function is defined as

N X k=1 (σimpliedk − σk) 2 , where σimpliedk is the k-th implied volatility and

σk = 1 T k X i=1 φkψi.

An example of the model volatilities is shown in the following table with the same settings as the previous approaches:

t ∈ (0, 0.5] t ∈ (0.5, 1] t ∈ (1, 1.5] t ∈ (1.5, 2] t ∈ (2, 2.5] t ∈ (2.5, 3] t ∈ (3, 3.5] t ∈ (3.5, 4] t ∈ (4, 4.5] t ∈ (4.5, 5] F1(t) 1.1260 F2(t) 0.9910 1.0659 F3(t) 1.0188 .08679 0.9334 F4(t) 1.0056 0.8821 0.7514 0.8082 F5(t) 0.9556 0.8736 0.7663 0.6528 0.7021 F6(t) 0.9346 0.8307 0.7595 0.6662 0.5675 0.6103 F7(t) 0.8756 0.8227 0.7312 0.6685 0.5864 0.4995 0.5372 F8(t) 0.7757 0.7892 0.7415 0.6591 0.6025 0.5285 0.4502 0.4842 F9(t) 0.7983 0.6900 0.7020 0.6596 0.5863 0.5360 0.4701 0.4005 0.4307 F10(t) 0.5507 0.7524 0.6503 0.6617 0.6217 0.5526 0.5052 0.4431 0.3775 0.4060

The model volatilities obtained by this approach fluctuates at a seemingly random way, which is not one of the desired properties for the volatility func-tion.

4.4.4 Rebonato’s Functional Form

In [2], Rebonato suggested a functional form for the instantaneous volatility function as follows:

(31)

where a, b, c and d are positive constants. Note that this functional form resembles the Nelson-Siegel function. It is relatively simple very flexible function. Moreover, it also guarantees positivity of the model volatilities. Note that this functional form is continuous and it does not directly specify the discretized model volatilities. However, by following the principles as formula (4.13), the following equation should hold

σ2_implied(T2) = 1 T Z Ti Ti−1 σ2(t, T )dt,

Hence we can optimize the parameters a, b, c and d such that σ_implied2 (T ) − 1

T Z Ti

Ti−1

σ2(t, T )dt

is minimized. An example of the resulting model volatilities is shown in the following table with the same settings as the previous approaches:

t ∈ (0, 0.5] t ∈ (0.5, 1] t ∈ (1, 1.5] t ∈ (1.5, 2] t ∈ (2, 2.5] t ∈ (2.5, 3] t ∈ (3, 3.5] t ∈ (3.5, 4] t ∈ (4, 4.5] t ∈ (4.5, 5] F1(t) 1.1966 F2(t) 1.0730 1.1966 F3(t) 0.4906 1.0730 1.1966 F4(t) 0.1840 0.4906 1.0730 1.1966 F5(t) 0.0629 0.1840 0.4906 1.0730 1.1966 F6(t) 0.0202 0.0629 0.1840 0.4906 1.0730 1.1966 F7(t) 0.0062 0.0202 0.0629 0.1840 0.4906 1.0730 1.1966 F8(t) 0.0019 0.0062 0.0202 0.0629 0.1840 0.4906 1.0730 1.1966 F9(t) 0.0006 0.0019 0.0062 0.0202 0.0629 0.1840 0.4906 1.0730 1.1966 F10(t) 0.0002 0.0006 0.0019 0.0062 0.0202 0.0629 0.1840 0.4906 1.0730 1.1966

We found that it is pushing model volatilities to zero. A possible explanation is as follows. The model volatilities between interval [Ti−1, Ti] is given by

1 T

Z Ti

Ti−1

σ(t, T )dt, i = 1, ..., N.

Note that the model volatilities are constant within the interval [Ti−1, Ti].

Moreover, recall from 4.16 that σ(t, T ) depends only on T − t. This means that we are in a similar situation as in section 4.4.2, but with more restrictions on the instantaneous volatilities. One of these restrictions forces the model volatilities to be positive. Recall that in section 4.4.2, we were able to solve

σ_i2(τ ) = 1 T

Z Ti

Ti−1

σ(τ )dt

uniquely, but the solution contains negative model volatilities. With the extra restrictions that guarantees positive model volatilities, the solution

(32)

to the above equation no long exists. In this case, instead of solving the equation, we try to optimize the parameters such that

σ2_i(τ ) ≈ 1 T

Z Ti

Ti−1

σ(τ )dt. (4.17)

Since we already knew that the solution contains negative model volatilities, we could expect that the optimized paramters to produce close to zero model volatilities.

Since volatilities are almost never close to zero in practice, we conclude that Rebonato’s approach is also not suitable for practical applications.

4.4.5 Conclusion

Having seen the results of different assumptions on the instantaneous volatili-ties, we conclude that none of the above assumptions can perfectly model the dynamics of the instantaneous volatilities. However, some of the assumptions are still clearly more impractical than the others. In chapter 5, we will show how the choice of assumption affects some of the practical applications of the Libor Market Model to determine the best assumption on the instantaneous volatilities among the choices that we have discussed in this chapter.

4.5 Correlation between Forward Rates

Another important input of the Libor market model is the correlation struc-ture between the forward rates. Interest rates of the same currency usually have very high correlation between each other. In general, a correlation ma-trix {ρij} must satisfy the following properties:

• Real and symmetric • 1 on the diagonal

• It is positive semi-definite

Furthermore, practical observations suggest that we could expect the corre-lation matrix to have the following properties:

• i → ρij is decreasing

(33)

There are two main approaches to estimate the correlation structures: using historical data to estimate the future correlations empirically, or assume that the correlation matrix follows a certain parametric form. A natural choice to estimate the correlations would be to use historical data because it is widely used in other statistical applications. However, correlation matrices produced by this approach do not always satisfy all of the conditions above. On the other hand, the parametric form approach gives us a lot of flexibility and we can choose to work with parametric forms that satisfy some of the conditions above by definition.

One of the simplest functional form of correlation function that satisfies most of the conditions is

ρij = exp(−β|Ti− Tj|),

where Ti and Tj are the expiries of the i-th and j-th forward rates, and β is a

positive constant. Although the exponential function is convenient to work with, it contains only small amount of correlation structures. Moreover, the correlation among forward rates goes asymptotically to zero as their distance increases, which is not consistent with what we often observe in the market. Based on these criticisms, Rebonato [2] suggested a very useful generalization of the exponential form as follows.

ρij = ρ∞+ (1 − ρ∞) exp(−βij|Ti− Tj|),

where

ρ∞ = lim |Ti−Tj|→∞

ρij.10

Moreover, β is no longer a constant but a function of the forward rates. This extended exponential form is much more flexible than the simple exponential form. However, the parameters ρ∞ and βij must be carefully chosen to

produce reasonable results. In chapter 20 of [2], Rebonato suggested the following set of parameters:

ρ∞= 0.3

βij = 0.12 − 0.005 max(Ti, Tj).

By empirical experiments, we found that this set of parameters is able to gen-erate reasonable correlation matrices, with only one drawback: if max(Ti, Tj) >

24, then βij < 0. In that case, the exponent will be positive and the function

10_{A practical implication of ρ}

∞is that it represents the asymptotic limit of correlation

among forward rates as their distance, |Ti−Tj|, goes to infinity. In practice, the correlations

(34)

will generate correlations greater than 1. During this project, we found a possible solution to this problem is to adjust the parameters to avoid this problem. For instance, we can define

β = 0.12 − 0.001 max(Ti, Tj).

Then the problem above will only appear when max(Ti, Tj) > 120. It means

we are simulating the forward rates for 120 years, which is very uncommon and it is usually unnecessary to simulate interest rates for such a long period. We will keep using this correlation form for the rest of this thesis.

4.6 Simulating Forward Rates

In this section, we will implement theories from previous sections to formulate Monte Carlo simulations for the forward euribor11_{. Recall from (4.2) that}

the dynamics of the forward rates are given by

dLi(t, T ) = σi(t, T )Li(t, T )dWTi+1(t), i = 1, ..., n

Where Li(t, T ) = F (t, Ti, Ti+1), Ti+1 = Ti+ δ, is the forward euribor rate for

the period [Ti, Ti+1] at time t. After a few modifications and simplifications

shown in section 4.3, we can write dLi(t) Li(t) = µi(t)dt + σi(t) m X k=1 bikdWk(t).

The solution of this equation is Li(t) = Li(0) exp Z t 0 µi(u) − 1 2σ 2 i(u)du + Z t 0 σi(u) m X k=1 bikdWk(u) ! . This is equivalent to log Li(t) = log Li(0) + Z t 0 µi(u) − 1 2σ 2 i(u)du + Z t 0 σi(u) m X k=1 bikdWk(u).

Discretizing the above equation gives log Li(t + dt) = log Li(t) + µi − 1 2σi(t) 2_{dt +}√_dtσ i(t) m X k=1 bikZk. (4.18)

11_{Although the model is called the Libor Market Model, it is a model for all simply}

compounded interest rates. Hence euribor is also a suitable candidate to be modeled by the Libor Market Model

(35)

Where dt is a small positive real number and {Zk} are independent identically

distributed standard normal random variables. Since the formula above is valid for any t between 0 and T , we can use the forward rates at time 0 as initial values and simulate the future forward rates step by step according to (4.18). The simulated forward rates can then be used to solve path dependent problems such as derivative pricing. In the next chapter, we will show how to use Monte Carlo simulation to measure the credit risk exposures of interest rate derivatives.

(36)

Chapter 5 Application of Libor Market

Model

In this chapter, we will apply the LMM to measure the counterparty credit risk exposure of interest rate swaps. We will present the results and access their quality by measuring the calibration errors and comparison to other similar results obtained by other sources.

5.1 Terminologies

Before we start, we will introduce some terminologies that will be used throughout this chapter first.

Mark-to-Market Value (MtM)

The mark to market value of a financial product is its fair value. It is usually calculated as the discounted value of the cash flows under a risk neutral measure.

Exposure

Exposure is defined as the amount an investor has at risk. It is calcu-lated as the maximum of 0 and the MtM, i.e.

exposure = max(0, M tM ). Expected exposure (EE)

Expected exposure is the amount expected to be lost if the counterparty defaults before the end of the contract. Note that EE will always be greater or equal to the expected MtM by definition.

(37)

Potential future exposure (PFE)

Intuitively, potential future exposure is the worst exposure an investor could have at a certain time in the future. It has similar definition as the value-at-risk (VAR). Let 1 − α be a confidence level and E be the exposure, the PFE is defined as the solution of

P (E > P F E) = α.

For example, the PFE at a confidence level of 95% will define an expo-sure that would be exceeded with a probability of no more than 5%. Banks are particularly interested in measuring EE and PFE of their port-folio’s in order to estimate their counterparty credit risks. Since MtM is closely related to EE and PFE by definition, it could be helpful to simulate the future MtM paths of some of the frequently traded financial products.

5.2 Mark-to-Market Paths of Interest Rate

Swaps

In this section, we will show how to use Monte Carlo simulations to mea-sure the EE and PFE of interest rate swaps. We design our Monte Carlo simulation method as follows:

• Fix a tenor (for example 3 months), obtain the initial underlying 3 month forward curve from market data and make 10000 copies of them. • For each of the initial curves, simulate forward interest rate curves for

3 months later according to 4.18.

• Calculate the future MtM of the interest rate swap according to each curve.

• Calculate the exposure for every path.

• Calculate EE and PFE by taking the mean and 95% quantile of the exposures respectively.

• Stop if this is the last simulation point. Otherwise remove the first point of the current curves (they are now spot rates), take the rest as new initial curves and go to step 2.

For convenience of modeling, all simulations are done under the terminal forward measure. Note that different volatility and correlation choices may lead to completely different EE and PFE profiles. In the next section, we will present some of the results obtained by different volatility function choices.

(38)

Figure 5.1: Bloomberg EE curve

5.3 Quality of the Results

In this section, we will show the results obtained by our simulations. The results will help us to determine which volatility functions are suitable candi-dates for interest rate modeling according to the Libor market model. First of all, we will examine the shapes of the EE curves. To have an idea what an EE curve of an IRS should look like, we have referred to the EE curve given by Bloomberg. We will compare the EE/PFE curves obtained by different assumptions on the volatility function shown in the previous chapter. All other parameters will remain the same. The parameters are as follows:

• 10 year floating for fixed interest rate swap • Notional value: 1.

• Starting date: May 10 2013. • Tenor: 3 months.

• Swap rate is chosen such that the present value of the swap is 0. • Volatilities are calibrated from cap implied volatilities on May 10 2013. The EE profile given by Bloomberg is shown in figure 5.1.

We see that The EE of this swap quickly increases to its peak in the first few years, and then decrease gradually back to 0 during the rest of the years.

(39)

Figure 5.2: Piecewise constant instantaneous volatility depending only on maturity. The blue line show the generated EE and the red line show the generated PFE

We have tried several different sets of parameters, and we found that this is a typical shape of the EE curve of a interest rate swap.

Since Bloomberg is one of the most popular pricing systems, we will use its results as benchmark. In the rest of this section, we will observe the EE curve generated by the same parameters with different volatility func-tions that have been introduced in the previous chapter. The quality of the volatility functions will be determined by the differences between the gener-ated EE curves and the Bloomberg EE curve.

First we observe the EE curve generated by a volatility function that only depends on maturity, as introduced in section 4.4.1. The EE/PFE profiles are shown in figure 5.2.

We see that the shape of this EE profile is very close to the profile provided by Bloomberg. However, the magnitudes of the two profiles are not similar to each other.

(40)

Figure 5.3: Piecewise constant instantaneous volatility depending only on time to maturity. The blue line show the generated EE and the red line show the generated PFE

Secondly, we observe the EE curve generated by a volatility function that only depends on time to maturity, as introduced in section 4.4.2. The EE/PFE profiles are shown in figure 5.3.

We observe that both the EE and the PFE profile are negative during the lifetime of the interest rate swap. This is not consistent with what we ob-served from Bloomberg. Hence we conclude that this volatility function does not generate realistic EE and PFE profiles.

The next EE curve that we observe is generated by a volatility function that follows a parametric structure, as introduced in section 4.4.3. The EE/PFE profiles are shown in figure 5.4.

We see that the resulting EE and PFE profiles are very similar to those gen-erated by the first volatility function.

The last EE curve that we observe is generated by Rebonato’s functional form, as introduced in section 4.4.4. The EE/PFE profiles are shown in figure 5.5:

(41)

Figure 5.4: Two parameter piecewise constant instantaneous volatility. The blue line show the generated EE and the red line show the generated PFE

Figure 5.5: Rebonato’s parametric form. The blue line show the generated EE and the red line show the generated PFE

(42)

(43)

We see that the EE and PFE profiles are flat during most of the interest rate swap’s life time. This is not a desirable property and we conclude that this volatility assumption is not realistic for modeling.

After comparing the different EE and PFE profiles, we observed that only two out of the four volatility functions generate reasonable EE and PFE pro-files: the volatility function that depends only on T and the two parameter form approach. Moreover, in the previous chapter, we have concluded that model volatilities generated by the two parameter approach seem to fluctu-ate in a random way, which is not a desirable property of the volatility. On the other hand, despite the fact that we completely ignore the dependence on t, the volatility function that depends only on T generates very practical EE and PFE profiles. Hence we will conclude that the simplest assump-tion for the volatility funcassump-tion is the most reasonable to use for modeling. This is an unexpected result. A possible explanation of this observation is as follows: Since the instantaneous volatility function cannot be observed, the structure of this function is unknown to anyone. Making complicated assumptions about this function can lead to unreasonable results because the correct dynamics of the volatility function are very difficult to describe. On the other hand, if we assume only that the volatility function depends on T , then the volatility function will be constant for every fixed T , which is consistent with the Black caplet formula. Furthermore, the dependence of the volatility function on T can be easily observed from the cap implied volatilities of different maturities. This explains why the EE/PFE profiles generated by volatility function depending only on T are the most practical. Hence we decide to use the simplest volatility function for our model.

5.4 Backtesting

In this section, we will perform a simple backtest to the Libor Market Model. Backtesting is the process of testing a model on prior time periods. It is used to estimate the performance of a model if it had been employed during a past period. Most quantitative analysis strategies are tested with this approach. Backtesting usually requires simulating past conditions with sufficient detail. In this case, it means that we have to apply the Libor Market Model to generate the exposure profiles of a large portfolio consisting of many interest rate derivatives continuously over a few years in the history. It is very time consuming to read and process large amount of detailed historical data. Due

(44)

Figure 5.6: Actual exposure of the interest rate swap we use for backtesting to the time constraint of this project, we will not backtest the model exten-sively. Instead, we will consider the exposure profile of a single interest rate swap that was active during the credit crisis in 2008. Since a credit crisis is one of the most extreme scenarios, we expect that the exposure of this interest rate swap will be around the PFE given by the model. We will judge the model by measuring the differences between the PFE given by the model and the actual exposure during the crisis.

The interest rate swap that we use for the backtesting has the following properties:

• Notional value: 1.

• Active period: November 2 2007 to November 2 2013. • Tenor: 3 months.

• Swap rate: 4.495%.

Furthermore, the model volatilities are calibrated from cap implied volatili-ties available at the calculation time under the assumption that the volatility function depends only on the maturity times of the caps. The actual exposure of this interest rate swap over its active period is shown in figure 5.6.

(45)

Figure 5.7: EE and PFE at the starting time of the interest rate swap. The blue line show the generated EE and the red line show the generated PFE We will calculate the EE and PFE every 3 months (up to 2 years) after the starting date of the swap according to the data available at those time points. The resulting EE and PFE are shown in figure 5.7 to 5.14

Note that, the actual exposure of this interest rate swap has increased dra-matically between the 9th month and the 12th month of its active period. The exposure jumped from 0 to roughly 0.75 between these 3 months due to

(46)

Figure 5.8: EE and PFE at 3 month after the starting time of the interest rate swap. The blue line show the generated EE and the red line show the generated PFE

(47)

Figure 5.11: EE and PFE at 1 year after the starting time of the interest rate swap. The blue line show the generated EE and the red line show the generated PFE

(48)

(49)

the decrease of Libor rates during the credit crisis. If we look at figure 5.10, we see that the model’s PFE reaches 0.65 after 1 year. The measurement of PFE in this case seems to be inaccurate. However, if we look at the subse-quent figures, figure 5.11 to 5.14, we see the model was able to capture the market movement very quickly. It’s PFE reaches 0.8, 0.9 and then 1.0 very quickly, which was close to the actual exposure during that time. It appears that the PFE’s generated by our model based on the data available during the crisis were reasonable estimates of the actual exposure during the crisis, which means that the PFE’s given by this model were good indications of the worst case scenario’s. Hence we conclude that the Libor Market Model has passed the simplified version of a backtest.

(50)

Chapter 6 Conclusions

In this thesis, we have studied and implemented the Libor Market Model to measure the counterparty risk exposure of interest rate derivatives. In contrast to short rate models, the LMM models the forward rate directly. This model is more practical because it is built around market observable parameters. We have also shown that the LMM is very flexible and has a lot of different parameter configurations. The main challenge of this project was to find the best configuration of volatility and correlation structure that produces accurate exposure profiles for the interest rate derivatives.

We have studied and implemented different assumptions on the volatility proposed by several different literatures. We have concluded that the most reasonable results are obtained by using a volatility function that depends only on the effective time of the forward rates. As for the correlation struc-tures, we found Rebonatos approach very useful. This approach is able to perform rank reduction techniques on the correlation matrix and recover the correct market cap prices at the same time.

(51)

Appendix A

Theorems

A.1 Radon-Nikodym Theorem

Let P and Q be probability measures on a measurable space (Ω, F ), if Q is absolutely continuous with respect to P , then there exists a almost surely unique non-negative measurable function f on Ω such that

Q(E) = Z

E

f dP

for all measurable sets E ∈ Ω. f is usually denoted by dQ_dP and is called the Radon-Nikodym derivative of Q with respect to P .

A.2 Change of Numeraire

Assume that Q0 _{and Q}1 _{are martingale measures for the numeraire S} 0 and

S1 respectively. Then the Radon-Nikodym derivative dQ

1 dQ0 is given by dQ1 dQ0(t) = S1(t) S1(0) S0(0) S0(t) .

Moreover, the process S0(t)

S1(t) is a martingale under the measure Q

1_.

A.3 Girsanov Theorem

Let T be a fixed positive number, and let

A Libor Market Model Approach for Measuring Counterparty Credit Risk Exposure