Download A Libor Market Model Approach for Measuring

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
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 derivatives, 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, [email protected], 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
Contents
1 Introduction
3
2 Interest Rates
2.1 Interest Rate Compounding . . . . . .
2.2 Zero Coupon Bonds and Interest Rates
2.3 Interest Rate Swap . . . . . . . . . . .
2.4 Caps and Floors . . . . . . . . . . . . .
2.5 Day Count Convention . . . . . . . . .
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3 Interest Rate Modeling
11
3.1 One Factor Short Rate Models . . . . . . . . . . . . . . . . . . 11
3.2 Multi-factor Models . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Libor Market Model
14
4.1 Construction of the Libor Market Model . . . . . . . . . . . . 14
4.2 Change of measure . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Rank Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.1 Reformulation of the Dynamics of the Forward Rates . 20
4.3.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . 22
4.4 Instantaneous Volatility of Forward Rates . . . . . . . . . . . 23
4.4.1 Piecewise Constant Instantaneous Volatility Depending Only on Maturity . . . . . . . . . . . . . . . . . . . 25
4.4.2 Piecewise Constant Instantaneous Volatility Depending Only on Time to Maturity . . . . . . . . . . . . . . 26
4.4.3 Two Parameter Piecewise Constant Instantaneous Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4.4 Rebonato’s Functional Form . . . . . . . . . . . . . . . 28
4.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5 Correlation between Forward Rates . . . . . . . . . . . . . . . 30
4.6 Simulating Forward Rates . . . . . . . . . . . . . . . . . . . . 32
1
5 Application of Libor Market Model
5.1 Terminologies . . . . . . . . . . . . . . . . . .
5.2 Mark-to-Market Paths of Interest Rate Swaps
5.3 Quality of the Results . . . . . . . . . . . . .
5.4 Backtesting . . . . . . . . . . . . . . . . . . .
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6 Conclusions
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A Theorems
A.1 Radon-Nikodym Theorem . . . . . . . . . .
A.2 Change of Numeraire . . . . . . . . . . . . .
A.3 Girsanov Theorem . . . . . . . . . . . . . .
A.4 Martingale Representation Theorem . . . . .
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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 between 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.
3
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 particularly 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 Libor 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.
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.
3
4
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 continuously. 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 investment 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
R m
compounded m times per year, the balance will be (1 + m
) euro. Taking m
to infinity would mean that interest rate is continuously compounded, and
the balance will be
m
R
lim 1 +
= eR
m→∞
m
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 impor1
Note that the market is not necessary arbitrage free in every model, this is a property
that needs to be proved in a model.
5
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 construction 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
1
F (t; T, T + δ) =
δ
6
B(t, T )
−1 .
B(t, T + δ)
An important type of simple forward rate is the Libor forward rate. Libor forward 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 R(t; T, T + δ) = −
δ→0
∂
log B(t, 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 R(t, T ).
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
Z T
B(t, T ) = exp −
r(u)du .
t
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.
7
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 structured 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
δi = Ti − Ti−1
δi N R,
and pay
δi N F (Ti−1 ; Ti−1 , Ti ).
The net cash flow at time Ti is thus
δi N (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
B(t, Ti )δi N (R − F (Ti−1 ; Ti−1 , Ti )).
i=1
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)
n
X
0=
B(t, Ti )δi N (Rswap (t) − F (Ti−1 ; Ti−1 , Ti ))
i=1
n
X
B(t, Ti )δi Rswap (t) = B(t, Ti )δi F (Ti−1 ; Ti−1 , Ti )
i=1
Rswap (t) =
B(t, Ti )δi F (Ti−1 ; Ti−1 , Ti )
Pn
.
i=1 B(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
wi (t)F (t; Ti−1 , Ti ),
i=1
8
where the weights wi (t) are defined as
B(t, Ti )
wi (t) = Pn
.
j=1 B(t, Tj )
Note that we have replaced F (Ti−1 ; Ti−1 , Ti ) by F (t; Ti−1 , Ti ). This is because 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 = δi N max(0, F (Ti−1 ; Ti−1 , Ti ) − K).
Each of these payments is called a caplet with reset date Ti−1 and settlement
date Ti . The time t value of a cap Cp(t) is simply the sum of the discounted
values of its caplets.
n
X
Cp(t) =
B(t, Ti )ci .
i=1
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
B(t, Ti )δi N max(0, K − F (Ti−1 ; Ti−1 , Ti )).
i=1
Many interest rate models have explicit formula for caps and floors. They
are often used to calibrate the model parameters.
9
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 Days in leap year
+
.
365
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 important to realize for which day count convention a specific interest rate is
quoted.
10
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 developed 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, market 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 interest rate derivatives can often be explicitly expressed in analytical formulas.
Short rate models often specify the short rate r as the solution of a stochastic 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.
11
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. Unlike 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 + σt dWt ,
where θt and σt are deterministic processes. The flexibility of the parameters 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
p
drt = (µXt + θYt )dt + σt Yt dW3t ,
where
dXt = (at − bXt )dt +
p
dYt = (dt − eYt )dt +
p
Yt ft dW2t .
Xt ct dW1t ,
and
12
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 economic 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 chapters 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.
1
See Shreve [3], chapter 10.
13
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 extension 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 flexibility 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].
14
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).
1
B(t, T )
L(t, T ) := F (t; T, T + δ) =
−1 ,
δ B(t, T + δ)
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 W̃ (t) under a riskneutral probability measure P̃ 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)dW̃ (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)dW̃ (t),
(4.1)
R
T +δ
where σ ∗ (t) = t
σ(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)
Z
=B(0, T + δ) exp
0
t
1
−σ (t, T + δ) · dW̃ (u) −
2
∗
If we define
W̃
T +δ
Z
(t) = W̃ (t) +
Z
t
kσ (t, T + δ)k du .
∗
2
0
t
σ ∗ (u, T + δ)du,
0
+δ)
then we notice that D(u)B(u,T
is the Radon-Nikodim derivative process
B(0,T +δ)
Z(t) appearing in the Girsanov Theorem A.3. It follows that, under the
assumption
2
Z T +δ
D(u)B(u, T + δ)
∗
2
E
kσ (t, T + δ)k
du < ∞,
B(0, T + δ)
0
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.
2
See [3], chapter 9 for a proof.
15
the process W̃ T +δ (t) is a Brownian motion under the probability measure
P̃T +δ given by
Z
D(T + δ)
T +δ
P̃ (A) =
dP̃ for allA ∈ F.
A B(0, T + δ)
The measure P̃T +δ is called the T +δ forward measure. Furthermore, Theorem
S(t)
A.2 implies that B(t,T
is a martingale under P̃T +δ . By the Martingale
+δ)
Representation Theorem A.4, there exists an adapted process3 σ(t, T ) such
that
dL(t, T ) = σ(t, T )L(t, T )dW̃ T +δ (t),
(4.2)
where W̃ T +δ is a Brownian motion under the (T + δ) forward measure. Note
that dL has no dt term under the forward measure P T +δ . It follows immediately 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 distribution, and
Z
1 T 2
1
log(L(0, T )/K) +
σ (t, T )dt ,
d1 = qR
t 2
2
0
σ (t, T )dt
0
Z
1 T 2
d2 = q R
log(L(0, T )/K) −
σ (t, T )dt .
t 2
2
0
σ (t, T )dt
1
0
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 riskneutral measure P̃
Ẽ D(T + δ)(L(T, T ) − K)+ .
3
A stochastic process (Xi )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
filtration generated by the process (L(t, T ))t∈[0,T ] .
16
Since
have
D(T +δ)
B(0,T +δ)
is the Radon-Nikodim derivative of P̃T +δ with respect to P̃, we
Ẽ D(T + δ)(L(T, T ) − K)+
D(T + δ)
+
=B(0, T + δ)Ẽ
(L(T, T ) − K)
B(0, T + δ)
=B(0, T + δ)ẼT +δ (L(T, T ) − K)+ .
Moreover, the solution to (4.2) is
Z
Z T
1 T 2
T +δ
L(T, T ) = L(0, T ) exp −
σ (t, T )dt +
σ(t, T )dW̃
(t) .
2 0
0
It follows that log L(T, T ) is normally distributed under P̃T +δ with mean
1
log L(0, T ) −
2
and variance
Z
Z
T
σ 2 (t, T )dt
0
T
σ 2 (t, T )dt.
0
By the Black-Scholes formula, we have
ẼT +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 P̃, and the forward measure as P T instead of P̃T .
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 forward 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
17
dynamics under the different forward measures.
T
Let dPdPT +δ(t)(t) be the Radon-Nikodim derivative of P T (t) with respect to P T +δ (t).
From Theorem A.2, we have
B(t, T ) B(0, T + δ)
dP T (t)
=
.
dP T +δ (t)
B(0, T ) B(t, T + δ)
Recall from the previous section that
1
B(t, T )
L(t, T ) =
−1 .
δ B(t, T + δ)
It follows that
dP T (t)
B(0, T + δ)
=
(1 + δL(t, T )).
T
+δ
dP
(t)
B(0, T )
(4.3)
Define the process η(t) by
dP T (t)
,
η(t) :=
dP T +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)dW T +δ (t)
1
dη(t) = η(t)
cδL(t, T )σ(t)dW T +δ (t)
η(t)
1
dη(t) = η(t)
cδL(t, T )σ(t)dW T +δ (t)
c(1 + δL(t, T ))
δL(t, T )σ(t)
dW T +δ (t).
dη(t) = η(t)
1 + δL(t, T )
It follows that the Girsanov kernel4 of the measure transform from P T to
P T +δ is
δL(t, T )σ(t)
.
1 + δL(t, T )
4
See Theorem A.3
18
By the Girsanov Theorem A.3, we have
dW T +δ (t) =
δL(t, T )σ(t)
dt + dW T (t).
1 + δL(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 dynamics under any forward measure. In particular, we have
N
X
δL(t, Tk )σk (t)
dt + dW TN (t)
dW (t) = −
1
+
δL(t,
T
)
k
k=i+1
Ti
(4.5)
for i = 1, .., N . Equation (4.5) can be used to write all forward rate dynamics
under the forward measure P TN . 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 simplification 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 confirmed 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 approach 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.
5
See for example [8] and [9]
19
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 P T +δ is given by the formula
dL(t, T ) = σ(t, T )L(t, T )dW T +δ (t).
Moreover, we have seen in (4.5) that
N
X
δLk (t, T )σk (t)
dW (t) = −
dt + dW TN (t)
1
+
δL
(t,
T
)
k
k=i+1
Ti
for i = 1, .., N . If we write Li (t) := L(t, Ti ) and σi (t) := σ(t, Ti ), then it
follows that
dLi (t)
= µi (t)dt + σi (t)dW TN (t),
Li (t)
where
µi (t) := −
i = 1, ..., N
(4.6)
N
X
δσk (t)
.
1 + δLk (t)
k=i+1
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 dW TN 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 necessary 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
m
X
dLi
= µi dt +
σik dWk ,
(4.7)
Li
k=1
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
20
forward rates to reduce the number of random factors in the model. In practice, 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 W i be the Brownian
motion of the Brownian motion driving forward rate Li in equation (4.6).
i
Comparing
Pm (4.7) with equation (4.6), we see that we have replaced σi dW
with k=1 σik dWk for i = 1, ..., N . Since we do not intend to change the
dynamics of the forward rates, we must have
m
X
2
σik
= σi2 .
(4.8)
k=1
Using this equation, we have
dLi
dLi
=
Li
Li
= µi dt + σi
= µi dt + σi
m
X
σik
k=1
m
X
σi
σ
pPmik
2
k=1 σik
k=1
If we define
dWk
σik
bik := pPm
2
k=1 σik
dWk .
(4.9)
.
Then condition (4.8) becomes
m
X
b2ik = 1,
i = 1, ..., n.
(4.10)
k=1
Moreover, equation (4.9) can be rewritten as
m
X
dLi
= µi dt + σi
bik dWk .
Li
k=1
Note that we have taken the volatilities of the forward rates σi out of σik .
Moreover, let ρij be the correlation between the Brownian motions W i and
21
W j , 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 decomposition 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 number 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
b2ik = 1,
(4.11)
k=1
and the distance between bbT and ρ is minimized. We define the distance as
the χ2 measure
X X
2
(bbT )ik − ρik .
(4.12)
i=1:h k=1:m
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.
6
See Rebonato (2004) [2]
22
This equation can be generalize to m factors
X
(sin2 (θj ) + cos2 (θk ))
k−1
Y
!
sin2 (θj )
=1
j=1
k=1:m−1
for all θj , j = 1, , k − 1. If we define
bik = cos θik
k−1
Y
sin θij ,
k = 1, ..., m − 1
j=1
bim =
m−1
Y
sin θij ,
j=1
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 Brownian 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 volatilities of the forward rates. Before we begin with the calibration strategies, we
will show the analytical expressions that relate instantaneous volatilities and
23
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 consistent 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
s
Z
1 T 2
σ (t, T )dt.
(4.13)
σimplied (T ) =
T 0
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 volatilities. 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 discussed in the next chapter. Throughout this section, we will assume that the
at-the-money cap implied volatilities are known8 , furthermore we do not consider 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:
F1 (t)
F2 (t)
..
.
FN (t)
7
8
t ∈ (T0 , T1 ] t ∈ (T1 , T2 ] · · ·
σ1,1
σ2,1
σ2,2
..
..
.
.
σN,1
σN,2
···
t ∈ (TN −1 , TN ]
σN,N
See for example [5]
In this thesis, we will read the cap implied volatilities from Bloomberg.
24
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
2
σimplied
(T ) =
k
k
1X 2
σ (Ti − Ti−1 ).
T i=1 k,i
(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 Depending 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 ]
for all any fixed T . The resulting volatility structure is as follows:
25
F1 (t)
F2 (t)
..
.
FN (t)
t ∈ (T0 , T1 ] t ∈ (T1 , T2 ] · · ·
σ1
σ2
σ2
..
...
.
σN
σN
···
t ∈ (TN −1 , TN ]
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.
F1 (t)
F2 (t)
F3 (t)
F4 (t)
F5 (t)
F6 (t)
F7 (t)
F8 (t)
F9 (t)
F10 (t)
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]
1.1260
1.0293
1.0293
0.9425
0.9425
0.9425
0.8670
0.8670
0.8670
0.8670
0.7984
0.7984
0.7984
0.7984
0.7984
0.7394
0.7394
0.7394
0.7394
0.7394
0.7394
0.6872
0.6872
0.6872
0.6872
0.6872
0.6872
0.6872
0.6410
0.6410
0.6410
0.6410
0.6410
0.6410
0.6410
0.6410
0.5999
0.5999
0.5999
0.5999
0.5999
0.5999
0.5999
0.5999
0.5999
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 Depending 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 ] · · ·
σ1
σ2
σ1
..
...
.
FN (t)
σN
σN −1
···
t ∈ (TN −1 , TN ]
F1 (t)
F2 (t)
..
.
9
σ1
The Nelson-Siegel function is a function in the form of a + (b + cT ) exp(−dT ).
26
Note that under this assumption, the model volatilities will be uniquely determined in the following way: by equation (4.14), at k = 1, we have
σimplied1 (T ) = σ12 .
at k = 2, we have
1 2
σ1 (T2 − T1 ) + σ22 (T1 − T0 ) .
T
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
2
σimplied
(T ) =
2
σk2
=
2
T σimplied
k
−
k
X
σi2 .
(4.15)
i=2
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:
F1 (t)
F2 (t)
F3 (t)
F4 (t)
F5 (t)
F6 (t)
F7 (t)
F8 (t)
F9 (t)
F10 (t)
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]
1.1260
0.9213
1.1260
0.7412
0.9213
1.1260
0.5814
0.7412
0.9213
1.1260
0.4363
0.5814
0.7412
0.9213
1.1260
0.2986
0.4363
0.5814
0.7412
0.9213
1.1260
0.1501
0.2986
0.4363
0.5814
0.7412
0.9213
1.1260
i
0.1501
0.2986
0.4363
0.5814
0.7412
0.9213
1.1260
i
i
0.1501
0.2986
0.4363
0.5814
0.7412
0.9213
1.1260
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
numbers, we assume that the model volatilities have the following structure:
27
F1 (t)
F2 (t)
..
.
FN (t)
t ∈ (T0 , T1 ] t ∈ (T1 , T2 ] · · ·
φ1 ψ1
φ2 ψ1
φ2 ψ2
..
...
.
φN ψ1
φN ψ2
···
t ∈ (TN −1 , TN ]
φN ψN
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
(σimpliedk − σk )2 ,
k=1
where σimpliedk is the k-th implied volatility and
k
1X
φk ψi .
σk =
T i=1
An example of the model volatilities is shown in the following table with the
same settings as the previous approaches:
F1 (t)
F2 (t)
F3 (t)
F4 (t)
F5 (t)
F6 (t)
F7 (t)
F8 (t)
F9 (t)
F10 (t)
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]
1.1260
0.9910
1.0659
1.0188
.08679
0.9334
1.0056
0.8821
0.7514
0.8082
0.9556
0.8736
0.7663
0.6528
0.7021
0.9346
0.8307
0.7595
0.6662
0.5675
0.6103
0.8756
0.8227
0.7312
0.6685
0.5864
0.4995
0.5372
0.7757
0.7892
0.7415
0.6591
0.6025
0.5285
0.4502
0.4842
0.7983
0.6900
0.7020
0.6596
0.5863
0.5360
0.4701
0.4005
0.4307
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 function.
4.4.4
Rebonato’s Functional Form
In [2], Rebonato suggested a functional form for the instantaneous volatility
function as follows:
σ(t, T ) = (a + b(T − t)) exp(−c(T − t)) + d,
28
(4.16)
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
Z
1 Ti 2
2
σimplied (T2 ) =
σ (t, T )dt,
T Ti−1
Hence we can optimize the parameters a, b, c and d such that
Z
1 Ti 2
2
σ (t, T )dt
σimplied (T ) −
T Ti−1
is minimized. An example of the resulting model volatilities is shown in the
following table with the same settings as the previous approaches:
F1 (t)
F2 (t)
F3 (t)
F4 (t)
F5 (t)
F6 (t)
F7 (t)
F8 (t)
F9 (t)
F10 (t)
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]
1.1966
1.0730
1.1966
0.4906
1.0730
1.1966
0.1840
0.4906
1.0730
1.1966
0.0629
0.1840
0.4906
1.0730
1.1966
0.0202
0.0629
0.1840
0.4906
1.0730
1.1966
0.0062
0.0202
0.0629
0.1840
0.4906
1.0730
1.1966
0.0019
0.0062
0.0202
0.0629
0.1840
0.4906
1.0730
1.1966
0.0006
0.0019
0.0062
0.0202
0.0629
0.1840
0.4906
1.0730
1.1966
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
Z
1 Ti
σ(t, T )dt,
i = 1, ..., N.
T Ti−1
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
Z
1 Ti
2
σi (τ ) =
σ(τ )dt
T Ti−1
uniquely, but the solution contains negative model volatilities. With the
extra restrictions that guarantees positive model volatilities, the solution
29
to the above equation no long exists. In this case, instead of solving the
equation, we try to optimize the parameters such that
Z
1 Ti
2
σi (τ ) ≈
σ(τ )dt.
(4.17)
T Ti−1
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 volatilities, 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 structure between the forward rates. Interest rates of the same currency usually
have very high correlation between each other. In general, a correlation matrix {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 correlation matrix to have the following properties:
• i → ρij is decreasing
• i → ρi+p,i is increasing for any fix p in {1, ..., N − 1}
30
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 generate 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
between the forward rates are assumed to converge as their distances increase.
31
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 )dW Ti+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
m
X
dLi (t)
= µi (t)dt + σi (t)
bik dWk (t).
Li (t)
k=1
The solution of this equation is
!
Z t
Z t
m
X
1 2 Li (t) = Li (0) exp
µi (u) − σi (u) du +
σi (u)
bik dWk (u) .
2
0
0
k=1
This is equivalent to
Z
log Li (t) = log Li (0) +
0
t
1
µi (u) − σi2 (u) du +
2
Z
t
σi (u)
0
m
X
bik dWk (u).
k=1
Discretizing the above equation gives
m
X
√
1
2
log Li (t + dt) = log Li (t) + µi − σi (t) dt + dtσi (t)
bik Zk .
2
k=1
11
(4.18)
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
32
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.
33
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 calculated 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.
34
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 exposure that would be exceeded with a probability of no more than 5%.
Banks are particularly interested in measuring EE and PFE of their portfolio’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 measure 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.
35
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 candidates 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.
36
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 functions that have been introduced in the previous chapter. The quality of the
volatility functions will be determined by the differences between the generated 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.
37
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 observed 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 generated 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:
38
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
39
40
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 profiles: 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 fluctuate 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 assumption for the volatility function 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
41
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 extensively. 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 volatilities 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.
42
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 dramatically 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
43
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
Figure 5.9: EE and PFE at 6 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
44
Figure 5.10: EE and PFE at 9 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
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
45
Figure 5.12: EE and PFE at 15 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
Figure 5.13: EE and PFE at 18 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
46
Figure 5.14: EE and PFE at 21 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
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 subsequent 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.
47
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 structures, 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.
48
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
Z
Q(E) =
f dP
E
for all measurable sets E ∈ Ω. f is usually denoted by
Radon-Nikodym derivative of Q with respect to P .
A.2
dQ
dP
and is called the
Change of Numeraire
Assume that Q0 and Q1 are martingale measures for the numeraire S0 and
dQ1
S1 respectively. Then the Radon-Nikodym derivative dQ
0 is given by
dQ1
S1 (t) S0 (0)
(t) =
.
0
dQ
S1 (0) S0 (t)
Moreover, the process
A.3
S0 (t)
S1 (t)
is a martingale under the measure Q1 .
Girsanov Theorem
Let T be a fixed positive number, and let
W (t) = (W1 (t), ..., Wd (t)),
49
0 ≤ t ≤ T,
be a d-dimensional Brownian motion on a probability space (Ω, F, P). Let
Θ(t) = (Θ1 (t), ..., Θd (t))
be a d-dimensional adapted process. Define
Z t
Z
1 t
2
Z(t) = exp
Θ(u) · dW (u) −
kΘ(u)k du
2 0
0
and
Z
t
Θ(u)du.
W̃ (t) = W (t) +
0
Assume that
Z
E
T
kΘ(u)k2 Z 2 (u)du < ∞.
0
Then EZ(T ) = 1, and the process W̃ (t) is a d-dimensional Brownian motion
under the probability measure P̃ given by
Z
Z(ω)dP(ω)
P̃(A) =
A
for all A ∈ F. The function Θ is call the Girsanov kernel of the measure
transform from P to Q.
Let W P be a standard P -Wiener process on a measure space (Ω, F, P ), let
ψ be a adapted process. Define a process η on [0, T ] by
η(0) = 1
dη(t) = ψ(t)η(t)dW P (t).
Assume that E P (η(T )) = 1 and define a new probability measure Q on FT
by
dQ
.
η(T ) =
dP
Then
Z t
W P (t) =
ψ(s)ds + W Q (t),
0
where W
Q
is a standard Q-Wiener process.
50
A.4
Martingale Representation Theorem
Let W (t), 0 ≤ t ≤ T , be a Wiener process defined on a probability space
(Ω, F, P). Let F(t), 0 ≤ t ≤ T , be the filtration generated by this Wiener
process. Let M (t), 0 ≤ t ≤ T , be a martingale with respect to this filtration.
Furthermore, assume that EΓ(t)2 < ∞. Then there is an adapted process
Γ(u), 0 ≤ u ≤ T such that
Z t
Γ(u)dW (u), 0 ≤ t ≤ T.
M (t) = M (0) +
0
51
Bibliography
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Continuing Challenge for Global Financial Markets (2012)
[2] Riccardo Rebonato, Volatility and Correlation (2004)
[3] R. E. Shreve, Stochastic Calculus for Finance II (2004)
[4] Tomas Bjoerk, Arbitrage Theory in Continuous Time (2009)
[5] Damiano Brigo, Fabio Mercurio, Interest Rate Models - Theory and Practice
[6] Damir Filipovic, Term-Structure Models (2009)
[7] Alan Brace, Dariusz Gatarek and Marek Musiela, The Market Model of
Interest Rate Dynamics, Mathematical Finance, Vol. 7, No. 2, 127-154
[8] Jakob Sidenius, LIBOR market models in practice, Journal of Computational Finance, 2000 April
[9] Damiano Brigo, A Note on Correlation and Rank Reduction (2002),
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52