Fixed Income Report:

Term Structures and Element Financial Instruments

Twitter : @Quasi quant20101

October 9, 2013

1Quasi Science

Contents

Preface i

1 Quick Mathematical Introduction 1

1.1 Prepare for Risk Neutral Measure and Numeraire Change . . . . . . . . . . 1

1.2 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Martingale Representation Theorem . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Risk Neutral Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Risk Neutral Pricin Formula by Using Numeraire as Banking Accout 6

1.4.2 Risk Neutral Pricing Formula by Using Numeraire as An Asset . . . 6

2 Stochastic Approach 9

2.1 Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Continuously-Compounded Spot Rate . . . . . . . . . . . . . . . . 10

2.1.2 Simply-Compouned Spot Interest Rate . . . . . . . . . . . . . . . . 10

2.1.3 Short Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 Simply-Compouned Forward Interest Rate . . . . . . . . . . . . . . 11

2.1.5 Instantaneous Forward Rate . . . . . . . . . . . . . . . . . . . . . . 11

2.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Stochastic Discout Factor and Zero Coupon Bond . . . . . . . . . . . . . . 12

2.3 Financial Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Forward Rate Agreement . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 CAP and FLOOR . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Expectation Hypothesis of Interest Rate . . . . . . . . . . . . . . . . . . . 20

2.5 Heath-Jarrow-Morton Frame work . . . . . . . . . . . . . . . . . . . . . . . 22

@Quasi quant2010

CONTENTS 2

2.5.1 The relation of short rate, instantaneous forward rate, and zero

coupon bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2 Arbitrage Free Model: HJM-framework . . . . . . . . . . . . . . . . 27

2.5.3 How to Use HJM Framework . . . . . . . . . . . . . . . . . . . . . 28

Bibliography 30

@Quasi quant2010

Preface

Mathematical finance is too beauty to replicate the real world. However, that we un-

derstand the theory and its fault is very important, since we do not or never recognize

our position to voyage financial markets without measures, namely mathematical finance,

statistics and so on. Therefore, understanding the theory and its fault is equal to get the

better map and compass to voyage the ocean safely or aggressively.

My object writing this paper is to introduce my knowledge since 04/01/2010. This

report focuses on the term structure on interest rate. Then, in my opinion, there are

three approaches to construct the term structures.

One is ”stochastic approach”. There are many models based on this approach. The

famous models are HJM framework, CIR process, and Vasicek process. The common

thing of those models is arbitrage free under the term. For example, when you sell a

treasury whose maturity is two years to one, you may try to take a free lunch. Then,

you consecutively buy the treasury with one maturity. However, in mathematical finance,

there is no arbitrage opportunity. In other words, Stochastic approach assumes that the

financial market is under the neutral world. Secondly, it is ”interpolating approach”.

This is very simple. If you observe zero coupon bond yield for each maturity, of course we

can not directly observe the zero coupon bond yield, then you take a curve fitting. The

famous methodologies are linear interpolation, spline interpolation, and so on. Namely,

this approach is under the actual world. The final is ”functional approach”. The famous

models are Nelson-Siegel model and Nelson-Siegel-Sevenson Mdoel. The concept is also

simple. Assuwming that yield curves have a analytic functional form, we fit the curves.

This method is similar with the second, interpolation. The difference is whether we as-

sume an analytic functional form or not.

This paper consist of two chapters. In chapter 1, we prepare the element mathemat-

ii

ics, especially the risk neutral pricing model. In chapter 2, we focus on the stochastic

approach. In section 2.1, we arrange each interest rates, and in section 2.2 we derive

lots of formulas on interest rates. Additively, we discuss the difference between discount

factor and zero coupon bond. In 2.3, we consider the financial instruments, Forward Rate

Agreement(FRA), Swap Rate, Cap, and Floor. In 2.4, we derive that forward rate is

the conditional expectation of the future spot rate under forward martingale measure. In

section 2.5, we mention Heath-Jarrow-Morton framework.

@Quasi quant2010

Chapter 1

Quick Mathematical Introduction

1.1 Prepare for Risk Neutral Measure and Nu-

meraire Change

Firstly, we mention the Radon-Nikodym derivative process, Z(t) , and the way how we

calculate the expectation and the conditional expectation in another measure, which is

determined by Z(t).

Suppose we have a probability space (Ω,F , P ) and a filtration Ft, defined for 0 ≤ t ≤ T ,

where T is a fixed final time. And suppose that Z is an almost surely positive random

variable satisfying EPZ = 1 under measure P, and we define the another measure Q;

Q(A) =

∫A

Z(ω)dP (ω) for all A ∈ F . (1.1)

For simpricity, we assume that the measure P dominate the measure Q, Q ≪ P .

Then, we can define Radon-Nikodym derivative process

Z(t) = EP [Z|Ft], 0 ≤ t ≤ T. (1.2)

This process is martingale under P. In fact, for 0 ≤ s ≤ t,

EP [Z(t)|Fs] = EP [EP [Z|Ft]|Fs] = EP [Z|Fs] = Z(s). (1.3)

Suddenly, we define Radon-Nikodym derivative process. Why we define the process? The

answer is that Z(t) is the key in changing a measure from the actual world to the risk

neutral world. We mention it in the section 1.2. In the following, we omit P if there is no

confused.

1.1 Prepare for Risk Neutral Measure and Numeraire Change 2

Lemma 1.1.1 Let t satisfying 0 ≤ t ≤ T be given and let Y be an measurable random

variable. Then

EQY = E [Y Z(t)] . (1.4)

Proof : We use the definition of the conditional expectation, the property Y is Ft-

measurable.

EQY = E[Y Z] = E[E[Y Z|Ft]] = E[Y E[Z|Ft]] = E[Y Z(t)]//. (1.5)

Using this lemma, we can calculate the unconditional expectation. However, we do not

know the way to calculate the conditional expectation in changing a measure. In fact, we

use the following lemma to derive the risk neutral pricing formula.

Lemma 1.1.2 Let s and t satisfying 0 ≤ t ≤ T be given and let Y be an Ft-measurable

random variable. Then

EQ[Y |Fs] =1

Z(s)EP [Y Z(t)|Fs]. (1.6)

Proof Taking into that EQ[Y |Fs] is the conditional expectation of Y under the Q-

measure, if we can show the following equation, for all A ∈ Fs∫A

1

Z(s)EP [Y Z(t)|Fs]dQ =

∫A

Y dQ, (1.7)

we can complete.

EQ

[1A

1

Z(s)EP [Y Z(t)|Fs]

]= EP

[1AE

P [Y Z(t)|Fs]]

= EP[EP [1AY Z(t)|Fs]

]= EP [1AY Z(t)]

= EQ [1AY ]// .

In proofing the above equation, We use the fact that lemma 1.1.1, 1A is Fs-measurable,

and the definition of the conditional expectation.

We can formally calculate the conditional expectation in changing the measure from P to

Q. However, we do not know the relation between P and Q. Girsanov theorem tell us the

relation!

@Quasi quant2010

1.2 Girsanov’s Theorem 3

1.2 Girsanov’s Theorem

Theorem 1.2.1 (Girsanov, one dimension)

Let W(t), 0 ≤ t ≤ T , be a Brownian motion on a probability space (Ω,F , P ), and let Ft,

be a brownian filtration. Let Θ(t), 0 ≤ t ≤ T , be an adapted process. Define

Z(t) = exp

−∫ t

0

Θ(u)dW P (u)− 1

2

∫ t

0

Θ(u)2(u)du

, (1.8)

WQ(t) = W P (t) +

∫ t

0

Θ(u)du, (1.9)

and Θ(t) is called Girsanov kernel. Assuming that

E

∫ T

0

Θ2(u)Z2(u)du < ∞. (1.10)

Set Z = Z(T ). Then EPZ = 1 and under the probability measure Q given by (1), the

process WQ(t), 0 ≤ t ≤ T , is a Brownian motion.

Proof According to Levy theorem, if M(t), 0 ≤ t, be a martingale relative to a filtration

Ft, 0 ≤ t, M(0) = 0, M(t) has continuous path, and [M,M ](t) = t, for all t ≥ 0, then

M(t) is a Brownian motion. Therefore, we must proof three things;

1. WQ(t) is a martingale under Q-measure,

2. Quadratic variation is t,

3. WQ(t) has a continuous path.

Since (9), the process WQ(t) starts at zero at time zero and is continuous. We calculate

the quadratic variation. From (9),

dWQ(t)dWQ(t) =(dW P (t) + Θ(t)dt

)2= dt. (1.11)

Before showing that WQ(t) is a martingale under Q-measure, firstly, we observe that Z(t)

is martingale under P-measure.

From the definition of exponential martingale, Z(t) is a martingale under P-measure. In

other ward,

dZ(t) = −ΘZ(t)dW P (t). (1.12)

@Quasi quant2010

1.2 Girsanov’s Theorem 4

We show next that WQ(t)Z(t) is a martingale under P-measure.

d(WQ(t)Z(t)) = d(WQ(t))Z(t) +WQ(t)dZ(t) + d(WQ(t))d(Z(t))

= (−WQ(t)Θ(t) + 1)Z(t)dW P (t). (1.13)

This shows that WQ(t)Z(t) is a martingale under P-measure. Finally, we show WQ(t) is

a martingale under Q-measure.

EQ[WQ(t)|Fs] =1

Z(t)EP [WQ(t)Z(t)|Fs] (Lemma2.2)

=1

Z(t)WQ(s)Z(s)

(WQ(t)Z(t) is a martingale

)= WQ(s)//. (1.14)

We know the way to analysis a Ft-measurable random variable. One is to use P-

measure. The other is Q-measure. This means that , in analyzing, P-measure is equal

to the real world and Q-measure is equal to the risk neutral world. In the following, we

confirm that the discount price process is a martingale under the risk neutral measure,

namely Q-measure.

Let W (t), 0 ≤ t ≤ T , be a Brownian motion on a probability space (Ω,F , P ), and let Ft,

0 ≤ t ≤ T , be a Brownian filtration. Consider a stock price process and a discout process;

dS(t) = α(t)S(t)dt+ σ(t)S(t)dW P (t), 0 ≤ t ≤ T, (1.15)

dD(t) = −R(t)D(t)dt. (1.16)

From the Ito-formula,

d(D(t)S(t)) = (α(t)−R(t))D(t)S(t)dt+ σ(t)D(t)S(t)dW P (t)

= σ(t)D(t)S(t)[Θ(t)dt+ dW P (t)

]. (1.17)

If we can find such Θ(t) 1, Girsanov kernel, there exsists a risk neutral measure. Take

cares that we never say the risk neutral measure is unique. From (1.17),

1In section 1.3, we mention about Martingale Representation Theorem. By this theorem, we can

derive the market price of risk.

@Quasi quant2010

1.3 Martingale Representation Theorem 5

Θ(t) =α(t)−R(t)

σ(t)(1.18)

, and we call this equation the market price of risk. If we can determine the unique

solution for the equation, the risk neutral measure is unique. In multi-dimension case,

it’s the same.

Assuming that we can find the unique solution, from (1.17),

d (D(t)S(t)) = σ(t)D(t)S(t)dWQ(t), (1.19)

dWQ(t) = Θ(t)dt+ dW P (t). (1.20)

Note that D(t)S(t)S(0)

is Radom-Nikodym derivative process because D(0) = 1. Therefore, we

can use Lemma 1.1.1 and Lemma 1.1.2. This shows that we can easily analysis the market

under Q-measure, since D(t)S(t) is a martingale under Q-measure, not martingale under

P-measure.

Next section, we mention why we can find single a Girsanov kernel.

1.3 Martingale Representation Theorem

Theorem 1.3.1 (Martingale Representation Theorem)

Let W(t), 0 ≤ t ≤ T , be a Brownian motion on a probability space (Ω,F , P ), and let Ft

be the brownian filtration. Let M(t), 0 ≤ t ≤ T , be P-martingale, E [M(t)|Fs] = E[M(s)].

Then there is an adapted process Γ(u), 0 ≤ t ≤ T , such that

M(t) = M(0) +

∫ u

0

Γ(u)dW(u), 0 ≤ t ≤ T. (1.21)

We do not proof this theorem. However, we can easily accept the theorem. Firstly,

we mention about an adapted process. This means that the process is generated by all

the information source, namely the filtration Ft.

For simplicity, we consider the discrete model; , for t1 = 0, t2, ..., tN = t ,

M(t) = M(0) +

∫ u

0

1u∈[ti≤u≤ti+1]Γ(u)dW(u) (1.22)

= M(0) +N∑k=1

Γ(tk)dW (tk) (1.23)

@Quasi quant2010

1.4 Risk Neutral Pricing Formula 6

If M(0) = 0, we can interpret (1.23) as a expansion with the base functions, dW (tk)

because dW (tk) are mutually independent. And the weight for each dW (tk) is Γ(tk).

This means that a P-martingale process is the sum of the P-martingale measure with the

weight Γ(tk).

Now, we can complete the mathematical tool for risk neutral pricing formula.

1.4 Risk Neutral Pricing Formula

1.4.1 Risk Neutral Pricin Formula by Using Numeraire as Bank-

ing Accout

In the following, we assume that the solution for the market price of risk is unique.

Therefore, if we can find risk neutral measure, it is unique.

From (1.15)・(1.16),

dV (t) = α(t)V (t)dt+ σ(t)V (t)dW P (t), 0 ≤ t ≤ T, (1.24)

dD(t) = −R(t)D(t)dt. (1.25)

Moreover,

d(D(t)V (t)) = (α(t)−R(t))D(t)V (t)dt+ σ(t)D(t)V (t)dW P (t)

= σ(t)D(t)V (t)[Θ(t)dt+ dW P (t)

],

= σ(t)D(t)V (t)dWQ(t). (1.26)

Since D(t)V (t) is a martingale under Q-measure, then, for all s ∈ 0 ≤ s ≤ t,

D(s)V (s) = EQ [D(t)V (t)|Fs] ,

V (s) =1

D(s)EQ [D(t)V (t)|Fs] . (1.27)

(1.27) is called the risk neutral priceing formula.

1.4.2 Risk Neutral Pricing Formula by Using Numeraire as An

Asset

In section 1.4.1, we select a banking account as the numeraire. In this section, we select

an general tradable asset as a numeraire. Therefore, we are allowed to use any tradable

@Quasi quant2010

1.4 Risk Neutral Pricing Formula 7

assets as numeraire.

Now we set a tradable asset, N(t), as numeraire. And the set

dV (t) = R(t)V (t)dt+ σ(t)V (t)dW P (t), (1.28)

dN(t) = R(t)N(t)dt+ ν(t)N(t)dW P (t), (1.29)

dD(t) = −R(t)D(t)dt. (1.30)

From (1.26), we can rewrite the equations;

d (D(t)V (t)) = σ(t)V (t)dWQ(t), (1.31)

d (D(t)N(t)) = ν(t)D(t)N(t)dWQ(t), (1.32)

dD(t) = −R(t)D(t)dt. (1.33)

From (1.32), D(t)N(t)N(0)

is exponetial martingale under Q-measure. And , from the Girsanov

theorem, we can find such the measure, called QN measure;

dWQN

(t) = −ν(t)dt+ dWQ(t). (1.34)

Therefore, for an random variable X,

EQN

[X|Fs] =1

Z(s)EQ[XZ(t)|Fs], (1.35)

Z(t) ≡ D(t)N(t)

N(0). (1.36)

In X = V (T ),

EQN

[V (T )|Ft] =1

Z(t)EQ[V (T )Z(T )|Ft],

⇔ EQN

[V (T )|Ft] =1

D(t)N(t)EQ[V (T )D(T )N(T )|Ft]. (1.37)

In paticular, we select the numeraire as a zero-coupund bond with T-maturity and unit

face value;

B(t, T ) =1

D(t)EQ [D(T )|Ft] (1.38)

, then

EQT

[V (T )|Ft] =1

D(t)B(t, T )EQ[V (T )D(T )B(T, T )|Ft],

⇔ EQT

[V (T )|Ft] =1

D(t)B(t, T )EQ[V (T )D(T )|Ft],

⇔ EQT

[V (T )|Ft] =1

B(t, T )V (t),

⇔ V (t) = B(t, T )EQT

[V (T )|Ft] (1.39)

@Quasi quant2010

1.4 Risk Neutral Pricing Formula 8

We call this measure T-forward (martingale) measure. Comparing (1.27) with (1.39),

we note that we need not to know the joint distribution, D(T )V (T ), under Q-measure.

To take a price for V (t), we need to know the distribution, V (T ) , under QT -measure.

Therefore, we can get very simple to take a price for the product. This is the reason why

we chage a mesure in the risk neural world.

@Quasi quant2010

Chapter 2

Stochastic Approach

The goal in this section is to derive each financial instruments, the conditional expectation

of the future spot rate under forward martingale measure, and HJM framework. In this

report, we focus on stochastic approach to consider the term structure in theoretical

formula. Then, we keep in our minds to not always understand financial markets even if

we completely know the theoretical formula, since mathematical finance is too beauty to

replicate the real world. Section 2.1, we introduce each interest rates. At the same time,

it is very important to intuitively understand those interest rates because it have you be

easy to know why forward martingale measure needs. First of all, see the following figure;

)(tr );( Stf

Short Rate Instantaneous Forward Rate

t S T

);( StL );:( TStF

Spot Rate Forward Rate

tS δ+tt δ+

Figure 2.1: Strunctue of each interest rates

2.1 Interest Rate 10

If you understand what we send to you, you skip section 2.1. In section 2.2, we arrange

the relation between stochastic discount factor and zero coupon bond. In section 2.3, we

derive the pricing formula of financial instruments. In section 2.4, we mention Expectation

Hypothesis of Interest Rate. Finally, in section 2.5, we derive Heath-Jarrow-Morton(HJM)

framework and briefly introduce how to use the framework into financial data.

2.1 Interest Rate

This section is quick introduction for each interest rates. So, we only do their definition.

If you do not understand what they suggest, you must read [2] in biography.

2.1.1 Continuously-Compounded Spot Rate

We define R(t;T ) as continuously compounded spot rate; for t ≤ T ,

exp R(t;T )(T − t)B(t;T ) ≡ 1 (2.1)

, where B(t;T ) is a zero coupon bond price at time t with T-maturity and B(T ;T ) = 1.

Therefore,

R(t;T ) =−logB(t;T )

T − t. (2.2)

2.1.2 Simply-Compouned Spot Interest Rate

We define L(t;T ) as simply-compouned spot interest rate; for t ≤ T ,

1 + L(t;T )(T − t)B(t;T ) ≡ 1. (2.3)

Therefore,

L(t;T ) =B(t;T )−B(T ;T )

(T − t)B(t;T ),

⇔ L(t;T ) =B(t;T )− 1

(T − t)B(t;T ). (2.4)

@Quasi quant2010

2.1 Interest Rate 11

2.1.3 Short Rate

We define r(t) as short rate; for 0 ≤ t,

r(t) ≡ limT→t+0

R(t;T ), (2.5)

≡ limT→t+0

L(t;T ), (2.6)

≡ limT→t+0

f(t;T ) (2.7)

, where f(t;T ) is instantaneous forward rate. This is defined in 2.1.5. From (2.5) and

(2.2), we derive the relation between short rate and zero coupon bond.

R(t; t+∆t) =−logB(t; t+∆t)

∆t,

=−(logB(t; t+∆t)− logB(t; t))

∆t(...B(t; t) = 1). (2.8)

Then,

r(t) = lim∆t→0+0

R(t; t+∆t)

= lim∆t→0+0

−(logB(t; t+∆t)− logB(t; t))

∆t

= − ∂

∂T(logB(t;T )) |T=t. (2.9)

2.1.4 Simply-Compouned Forward Interest Rate

We define F (t : S;T ) as simply-compouned forward interest rate; for t ≤ S ≤ T ,

1 + F (t : S;T )(T − S) B(t;T )

B(t;S)≡ 1. (2.10)

Therefore,

F (t : S;T ) =B(t;S)−B(t;T )

(T − S)B(t;T ). (2.11)

2.1.5 Instantaneous Forward Rate

We define f(t;T ) as instantaneous forward rate; for t ≤ S ≤ T ,

f(t;S) ≡ limT→S+0

F (t : S;T ), (2.12)

= limT→S+0

B(t;S)−B(t;T )

(T − S)B(t;T ),

@Quasi quant2010

2.2 Stochastic Discout Factor and Zero Coupon Bond 12

= limT→S+0

1

B(t;T )

B(t;S)−B(t;T )

T − S,

= − ∂

∂s(logB(t; s)) |s=S. (2.13)

This implies that r(t) = f(t; t);

f(t;S)|S=t = f(t; t),

= − ∂

∂s(logB(t; s)) |s=t (

...(2.9)),

s = r(t). (2.14)

2.1.6 Summary

From section 2.1.1 to section 2.1.5, we introduce the interest rate definitions. Then, you

notice that in equation(2.1),(2.3) and (2.8) the right hand side of those equations is unit.

Why? If you know the reason, you already the time concept of the interest rate.

Moreover, there is a very important thing. From (2.1) to (2.14), we express the interest

rate with the zero coupon bond. Namely, in interest rate world, other word is time

value world, the zero coupon bond is fundamental measure tool for interest financial

instruments.

2.2 Stochastic Discout Factor and Zero Coupon

Bond

Can you image the difference between stochastic discount factor, D(t;T ), and zero

coupon bond, B(t;T )? The important point is the difference between short rate and

instantaneous forward rate.

Firstly, we define the two;

D(t) ≡ exp

−∫ t

0

r(u)du

. (2.15)

Then,

D(t;T ) = exp

−∫ T

t

r(s)ds

. (2.16)

@Quasi quant2010

2.3 Financial Instruments 13

Here, we consider the meanings of the discount factor. The following is often question.

Comparing today’s 100yen with tomorrow’s 100yen, which is more valuable? If the interest

is positive, then the answer is the former. How do we compare the two? The unity is

today’s value, present value. So, we must know the spot rate from today to tomorrow. It

suggests that we need the time axis with different two time. This time value corresponds

to the spot rate, the discount rate.

Next, we derive the zero coupon bond price;

−∫ T

t

f(t; s)ds =

∫ T

t

∂logB(t;u)

∂u|u=sds (

...(2.13)),

= logB(t;T )− logB(t; t),

= logB(t;T ) (...B(t; t) = 1). (2.17)

Therefore,

B(t;T ) = exp

−∫ T

t

f(t; s)ds

. (2.18)

What is the difference between (2.16) and (2.18)? The single is weather the integral

function is spot rate or forward rate. Like the above, on continuous time model, short

rate and instantaneous forward rate, the former is with one different times due to (2.9).

The latter is with two different times by (2.13). Similarity, in dicrete time model, on spot

rate, we need the time axis with two different times by (2.4) and on forward rate, we need

the time axis with three different times by (2.11).

2.3 Financial Instruments

In this section, we derive some Financial Instruments in martingale pricing theory. Of

course, we need not know financial instruments to consider the term structure. However,

in terms of interpreting the financial instrument data ans analyzing the financial markets,

it is very important to understand the theoretical pricing formula. Additively, we do

take care of the limit of mathematical finance. Usually, they assume that in equivalent

martingale measure financial products obey brownian montion. Then, we doubt that is it

true? In my experience, analyzing foreign exchange data, it is not true. Rather, we can

not exactly judge whether it is true or not.

@Quasi quant2010

2.3 Financial Instruments 14

In each sections, we derive the values of Forward Rate Agreement(FRA), Interest Rate

Swap(IRS), CAP, and FLOORS.

2.3.1 Forward Rate Agreement

FRA is the contract that , at time t, sellers and buyers agree the interest rate which is

applied to a future time from S to T . In figure[], we show a trading strategy. At time

T, FRA buyers borrow unit yen with the floating rate, L(S;T ) and , at the same time,

they lend others with the fixed rate, K. From the figure[], The payoff, V (T ), is following

at time T ;

V (T ) = (S −K) (K − L(S;T )) . (2.19)

Then, what’s the value at time t? To answer the question, we use the martingale pricing

Time

Cash In

)(1 STK −+

1

Cash Out

))(;(1 STTSL −+

1

t S T

Figure 2.2: Payoff oF Forward Rate Agreement

theory. Directly, from (1.27) and (1.39), we can take a price;

V (t) =1

D(t)EQ [D(T )V (T )|Ft] ,

= EQ [D(t;T )V (T )|Ft](.

..(2.16)

),

= B(t;T )EQT

[V (T )|Ft] ,

@Quasi quant2010

2.3 Financial Instruments 15

= B(t;T )EQT

[(S −K) (K − L(S;T )) |Ft] ,

= B(t;T )(S −K)K − EQT

[L(S;T )|Ft],

⇔ FRA(t : S;T,K) = B(t;T )(S −K) K − F (t : S;T ) . (2.20)

2.3.2 Interest Rate Swap

Interest Rate Swap(IRS) is very similar with FRA. Then, what’s the difference? It is the

contract time. In a FRA, under the contract, the payoff is once time. However, in a IRS,

under the contract, the payoff is several times. See the payoff in figure[]; Therefore, the

Time

)(11 αα TTK −+ + )(1 1−−+ ββ TTK

・・・

11

))(;(111 αααα TTTTL −+ ++

αT1+αT βT

1−βT

))(;(1 11 −− −+ ββββ TTTTL

t

・・・

11

Figure 2.3: Payoff oF Interest Rate Swap, especially Receiver Swap

value at time t whose payoff is under the contract is

V (t : T , K) =

β−1∑j=α

B(t;Tj)EQTj

[V (Tj)|Ft]

V (t : T , K) =

β−1∑j=α

FRA(t : Tj;Tj+1, K) (2.21)

=

β−1∑j=α

B(t;Tj+1)(Tj+1 − Tj)(K − F (t : Tj;Tj+1)) (...(2.20))

=

β−1∑j=α

B(t;Tj+1)(Tj+1 − Tj)

@Quasi quant2010

2.3 Financial Instruments 16

×(K − F (t : S;T )

B(t;Tj)−B(t;Tj+1)

(Tj+1 − Tj)B(t;Tj+1)

)(...(2.11))

=

β−1∑j=α

B(t;Tj+1)K(Tj+1 − Tj)− (B(t;Tj)−B(t;Tj+1))

IRS(t : T , K) =

β−1∑j=α

B(t;Tj+1)K(Tj+1 − Tj)−B(t;Tα) +B(t;Tβ). (2.22)

From this, we interpret IRS as the swap contract exchaging coupon bearing bonds,

K(K(Tj+1 − Tj)), for zero coupon bond, B(t;Tα) − B(t;Tβ). This contract also express

by some zero coupon bonds.

Well, the derived (2.22) is call as Receiver Swap(RS). This contract is that the buyers

of RS receive the fixed interest rate, K, and pay the floating interest rate, L(Tj;Tj+1) at

each time Tj, : j = α, ..., β − 1. And the inverse contract is called as Payer Swap(PS).

Namely, the contract is that the buyers of PS receive the floating interest rate, L(Tj;Tj+1)

at each time Tj, : j = α, ..., β − 1, and pay the floating interest rate, K.

Next, we derive the forward swap rate(FSR), Sα,β(t). Swap rate is such interest rate

that , in (2.22), IRS(t : T , K) is zero at time t. Therefore,

Sα,β(t) =B(t;Tα)−B(t;Tβ)∑β−1

j=α(Tj+1 − Tj)B(t;Tj+1). (2.23)

Whatever we use RS or PS, the derived forward swap rate is same.

Do you think of the relation between forward swap rate and forward rate? Of course, it

does because both is the future(time is T) interest rate at today(time t). Moreover, both

is priced by non-arbitrage theory. This implies both have a relation ship. Now, we derive

it.

First of all, we consider the following equation; for α ≤ k ≤ β

B(t;Tk)

B(t;Tα)=

B(t;Tα+1)

B(t;Tα)× B(t;Tα+2)

B(t;Tα+1)× ...× B(t;Tk)

B(t;Tk−1)(2.24)

You will remember that all interest rates or financial instruments can express by zero

coupon bond. So, the inverse is also possible. From (2.11)

F (t : Tj;Tj+1) =B(t;Tj)−B(t;Tj+1)

(Tj+1 − Tj)B(t;Tj+1),

⇔ (Tj+1 − Tj)F (t : Tj;Tj+1) =B(t;Tj)−B(t;Tj+1)

B(t;Tj+1)(2.25)

@Quasi quant2010

2.3 Financial Instruments 17

Therefore,

The left hand side of (2.24) =k−1∏j=α

1 + (Tj+1 − Tj)F (t : Tj;Tj+1)−1 . (2.26)

Arranging (2.23) and substitute (2.26) into (2.23),

Sα,β(t) =1−B(t;Tβ)/B(t;Tα)∑β−1

j=α(Tj+1 − Tj)B(t;Tj+1)/B(t;Tα),

⇔ Sα,β(t) =1−

∏β−1j=α 1 + (Tj+1 − Tj)F (t : Tj;Tj+1)−1∑j−1

i=α(Ti+1 − Ti)∏j−1

i=α 1 + (Ti+1 − Ti)F (t : Ti;Ti+1)−1 . (2.27)

2.3.3 CAP and FLOOR

Time

1

))(;(111 iiii

TTTTL −+ ++

)(11 ii

TTK −+ +

1

ti

T1+iT

Figure 2.4: Payoff of Caplet

Firstly, we define a caplet whose value at time t under the above payoff is

Caplet(t : S;T,K) = B(t;T )EQT

[V (S;T )|Ft]

, where V (S;T ) = max L(S;T )−K, 0 ≡ (L(S;T )−K)+. Then, we can define CAP

as the following;

CAP (t : T , K) =

β−1∑j=α

Caplet(t : Tj;Tj+1, K) (2.28)

@Quasi quant2010

2.3 Financial Instruments 18

, where T = Tj : j = α, α + 1, ..., β. Namely, CAP is the collection of Caplet when the

realized payoff at time Tj+1 for each j is positive.

Then, why do we call it CAP? In first step, we consider the capital from markets

with the interest rate L(Tj, Tj+1) : j = α, α + 1, ..., β. In the below figure, we describe

the money flow to pay for lenders; As the above figure, we must pay the interest rate

Time

αT

1

1+αT

βT

1−βT

・・・

1:Face

1 1 1

2+αT

))(;(111 αααα TTTTL −+ ++ ))(;(1 11 −− −+ ββββ TTTTL

1:Face2+α

Figure 2.5: Money flow to pay for lenders

L(Tj, Tj+1)(Tj+1 − Tj) for each j. Moreover, at time Tβ, we must also principle value, 1.

What is the risk for this contract? It is just a rise in the interest rate! So, we want to

hedge the interest risk. How? In the situation, we propose to tie the CAP contract.

If

L(Tj, Tj+1)−K ≤ 0

, then ,from figure 2.6,

we must pay the floating rate, −L(Tj, Tj+1)(Tj+1 − Tj), for each j.

If

L(Tj, Tj+1)−K ≥ 0

, then ,from figure 2.6,

we must pay the fixed rate, −K(Tj+1 − Tj), for each j.

@Quasi quant2010

2.3 Financial Instruments 19

Time1 1

TT

))(;(111 αααα TTTTL −+ ++

1+iTiT

Time

))(;(111 iiii

TTTTL −+ ++

+TT

1 11Cash In

1+iTi

T

))(;(111 iiii

TTTTL −+ ++

1

))(;(111 iiii

TTTTK −+ ++

Cash Out

Figure 2.6: Right:Caplet cash flow under L(Ti;Ti;1) > K, Left:Caplet cash flow under

L(Ti;Ti;1) > K

Therefore, we can determine the maximum interest pay, K, whatever the market interest

rate get rises. This implied that the maximum pay is capped. So, we call it CAP .

By the way, in section 2.1.6, we mention that ”, in interest rate world, the zero coupon

bond is fundamental measure tool for interest financial instruments.” CAP is also the

interest financial instruments, so we can express it as the sum of zero coupon bond.

The payoff at time Tj+1 is (L(Tj;Tj+1) −K)+(Tj+1 − Tj). If we know the zero coupon

bond price at time Tj, then we can approximate it as the payoff at time Tj;

(L(Tj;Tj+1)−K)+(Tj+1 − Tj) ≃ (B(Tj, Tj+1))(L(Tj;Tj+1)−K)+(Tj+1 − Tj). (2.29)

You must notice that now time is just t. Therefore, we do not know B(Tj, Tj+1). However,

we can calculate it and it is called forward zero coupon bond;

B(Tj, Tj+1) =B(t, Tj+1)

B(t, Tj). (2.30)

As the result, we can regard the contract with the payoff, namely Caplet, as Tj contingent

claim. Then, from (1.27), V (Tj+1) = (B(Tj, Tj+1))(L(Tj;Tj+1)−K)+(Tj+1 − Tj)

Caplet(t : Tj;Tj+1, K) = EQ[D(t;Tj)B(Tj, Tj+1)(L(Tj;Tj+1)−K)+(Tj+1 − Tj)|Ft

],

= EQ[D(t;Tj) (1−B(Tj;Tj+1)−B(Tj;Tj+1)K(Tj+1 − Tj))

+ |Ft

],

= EQ[D(t;Tj) [1−B(Tj;Tj+1) 1 +K(Tj+1 − Tj)]+ |Ft

]= 1 +K(Tj+1 − Tj) ,

@Quasi quant2010

2.4 Expectation Hypothesis of Interest Rate 20

× EQ

[D(t;Tj)

(1

1 +K(Tj+1 − Tj)−B(Tj;Tj+1)

)+

|Ft

],

⇔ Caplet(t : Tj;Tj+1, K) ≃ 1 +K(Tj+1 − Tj)ZBP

(t : Tj;Tj+1,

1

1 +K(Tj+1 − Tj)

),

(2.31)

, where ZBP (t : S;T,K) is the european put option price, with strike price K at time t,

whose underlying is the forward zero coupon bond, B(S;T ).

Again, CAP is the sum of Caplet;

CAP (t : T , K) =

β−1∑j=α

Caplet(t : Tj;Tj+1, K).

Therefore, expressing it by zero coupon bonds,

CAP (t : T , K) =

β−1∑j=α

1 +K(Tj+1 − Tj) sZBP

(t : Tj;Tj+1,

1

1 +K(Tj+1 − Tj)

).

FLOOR is the inverse contract for CAP . Here, we describe the equation;

Floorlet(t : S;T,K) = B(t;T )EQT

[V (S;T )|Ft]

, where V (S;T ) = max K − L(S;T ), 0 ≡ (K − L(S;T ))+. Then, we can define

FLOOR as the following;

FLOOR(t : T , K) =

β−1∑j=α

Floorlet(t : Tj;Tj+1, K), (2.32)

=

β−1∑j=α

1 +K(Tj+1 − Tj)ZBC

(t : Tj;Tj+1,

1

1 +K(Tj+1 − Tj)

).

, where ZBP (t : S;T,K) is the european call option price, with strike price K at time t,

whose underlying is the forward zero coupon bond, B(S;T ).

2.4 Expectation Hypothesis of Interest Rate

In this section, we derive Expectation Hypothesis of Interest Rate and point out the

theory. In mathematical finance, it is following;

EQT

[L(S;T )|Ft] = F (t : S;T ). (2.33)

@Quasi quant2010

2.4 Expectation Hypothesis of Interest Rate 21

This means the forward rate is a good estimator for the future spot interest rate. In

other word, the forward rate is the estimator which makes the squared distance between

the future spot rate and the present forward rate least. Perhaps, it is just hypothesis,

not reality. However, we intuitively accept the hypothesis and have no choice to assume

such the hypothesis. To proof the hypothesis (2.33), to begin with, we proof the below

proposition;

Proposition 2.4.1 The simply-compounded forward rate on the interval [S, T ], 0 < t <

S < T is a QT martingale, or equivalent martingale whose numeraire is zero coupon bond,

B(t;T ), for Q;

∀0<u<t EQT

[F (t : S;T )|Fu] = F (u : S;T ) (2.34)

Proof From the definition of the simply-compounded forward rate, (2.11),

F (t : S;T ) =B(t;S)−B(t;T )

(T − S)B(t;T ),

⇒ B(t;T )F (t : S;T ) =B(t;S)−B(t;T )

(T − S). (2.35)

The right hand side of (2.31) is tradable asset. So, we can also regard the left hand side

of (2.31) as tradable assets.

By the way, QT is a equivalent martingale whose numeraire is B(t;T ). Therefore,

B(t;T )F (t:S;T )B(t;T )

is a martingale under QT . This implies F (t : S;T ) is a martingale under

QT -measure.

Using proposition 2.4.1, we can show the following

Proposition 2.4.2

EQT

[L(S;T )|Ft] = F (t : S;T ) (2.36)

Proof From the definition of the simply-compounded spot interest rate, (2.4),

L(S;T ) =B(S;T )−B(T ;T )

(T − S)B(S;T ),

= F (S : S;T ). (2.37)

From proposition 2.4.2, F (S : S;T ) is a martingale under QT -measure and its numeraire

is B(S;T ).

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 22

Intuitively, we can understand the proposition of instantaneous forward rate if proposi-

tion 2.4.2 is approved;

Proposition 2.4.3

EQT

[r(T )|Ft] = f(t;T ) (2.38)

Proof From (1.39),

V (t) ≡ B(t, T )EQT

[r(T )|Ft]

= EQ[D(t, T )r(T )|Ft]

Therefore,

B(t, T )EQT

[r(T )|Ft] = EQ[D(t, T )r(T )|Ft]

⇔ EQT

[r(T )|Ft] =1

B(t, T )EQ

[exp

−∫ T

t

r(s)ds

r(T )|Ft

]=

1

B(t, T )EQ

[− ∂

∂Texp

−∫ T

t

r(s)ds

|Ft

]=

−1

B(t, T )

∂

∂TEQ

[exp

−∫ T

t

r(s)ds

|Ft

]=

−1

B(t, T )

∂

∂TB(t;T ) (

...(1.38))

= − ∂

∂TlogB(t;T )

⇔ EQT

[r(T )|Ft] = f(t;T ) (...(2.9)) (2.39)

2.5 Heath-Jarrow-Morton Frame work

Why Heath-Jarrow-Morton(HJM) Frame work is famous? It has the two reasons. Firstly,

we require the arbitrage free condition on the term structure. For example, when you sell

a treasury whose maturity is two years to one, you may try to take a free lunch. Then,

you consecutively buy the treasury with one maturity. However, in mathematical finance,

there is no arbitrage opportunity. Secondly, we can extend the dimension of independent

variables for dependent variables. For example, if you examine a interest rate by linear

regression, then it is useful without the condition for regression dimensions. In section,

we derive some theorems and propositions in HJM frame work.

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 23

2.5.1 The relation of short rate, instantaneous forward rate, and

zero coupon bond

Firstly, We examine each relations when we model r(t), f(t;T ), and B(t;T ) by N-

dimension brownian motion. Assuming the followings;

r(t) = r(0) +

∫ t

0

a(s)ds+N∑j=1

∫ t

0

bj(s)dWj(t) (2.40)

f(t : T ) = f(0 : T ) +

∫ t

0

α(s;T )ds+N∑j=1

∫ t

0

σj(s;T )dWj(t) (2.41)

B(t : T ) = B(0 : T ) +

∫ t

0

m(s;T )B(s : T )ds

+N∑j=1

∫ t

0

vj(s;T )B(s : T )dWj(t) (2.42)

, where

b(t) = (b1(t), ..., bN(t))t, v(t;T ) = (v1(t;T ), ..., vN(t;T ))

t

, σ(t;T ) = (σ1(t;T ), ..., σN(t;T ))t

and m(t;T ), v(t;T ), α(t;T ), and σ(t;T ) are C1 class with respect to T .

Proposition 2.5.1 (Instantaneous Forward Rate and Zero Coupon Bond)

α(t;T ) = vT (t;T ) · v(t;T )−mT (t;T ) (2.43)

σ(t;T ) + vT (t;T ) = 0 (2.44)

Proof From (2.40),

dB(t;T ) = m(t;T )B(t : T )ds+ v(t;T )B(t : T ) · dW (t). (2.45)

Now, we define f(t, x) = logx(t). By the Ito’s lemma,

df(t, x) = ftdt+ fxdx+1

2fttdt

2 + ftxdtdx+1

2fxxdx

2

= ftdt+ fxdx+1

2fxxdt

Applying f(t, x) to B(t;T ),

dlogB(t, t) =1

B(t;T )+

1

2

(− 1

B2(t;T )dB(t;T )dB(t;T )

)@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 24

= m(t;T )B(t : T )dt+ v(t;T )B(t : T ) · dW (t)

− 1

2B2(t : T )B2(t : T ) v(t;T ) ·B(t;T )2

⇔ dlogB(t, t) =(m(t;T )− ||v(t;T )||2

)dt+ v(t;T ) · dB(t;T ) (2.46)

, where ||v(t;T )|| =√∑N

j=1 v2j (t;T ). By the way, from the definition of the instantaneous

forward rate,

f(t;T ) = − ∂

∂TlogB(t;T )

= − ∂

∂T

[∫ t

0

(m(s;T )− ||v(s;T )||2

)ds+

∫ t

o

v(s;T ) · dB(s;T )

]= −

∫ t

0

mT (s;T )−

1

2||v(s;T )||2

ds−

∫ t

0

vT (s;T ) · dW (s)

= the right hand side of (2.40)

⇔

α = −mT (t;T ) +

12||v(t;T )||2

σ(t;T ) + σT (t;T ) = 0

(2.40)

(2.47)

Next, we derive the relation between short rate and instantaneous forward rate.

Proposition 2.5.2 (Short Rate and Instantaneous Forward Rate)

a(t) = fT (t;T ) + α(t;T ) (2.48)

b(t) = σ(t;T ) (2.49)

Proof From (2.7) and (2.40)

r(t) ≡ limT→t+0

f(t;T ),

⇔ r(t) = f(0; t) +

∫ t

0

α(s; t)ds+

∫ t

0

σ(s; t) · dW (s). (2.50)

Using the definition of partial integration,

α(s; t) = α(s; s) +

∫ t

s

αT (s;T )|T=udu, (2.51)

σ(s; t) = σ(s; s) +

∫ t

s

σT (s;T )|T=udu, (2.52)

for simplicity, we write HT (s;T )|T=u as HT (s;u). Therefore,

The right hand side of (2.49) = f(0; t) +

∫ t

0

α(s; s)ds+

∫ t

0

∫ t

s

αT (s;u)duds

+

∫ t

0

σ(s; s)ds+

∫ t

0

∫ t

s

σT (s;u)du · dW (s). (2.53)

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 25

From stochastic fubini’s theorem,∫ t

0

∫ t

s

αT (s;u)dsdu =

∫ t

0

∫ u

o

αT (s;u)dsdu. (2.54)

So,

The right hand side of (2.52) = f(0; t) +

∫ t

0

α(s; s)ds+

∫ t

0

∫ u

0

αT (s;u)dsdu

+

∫ t

0

σ(s; s)ds+

∫ t

0

∫ u

0

σT (s;u) · dW (s)du,

= f(0; t)

+

∫ t

0

[α(u;u) +

∫ u

0

αT (s;u)ds+

∫ u

0

σT (s;u) · dW (s)

]du

+

∫ t

0

σ(s; s)ds · dW (s). (2.55)

Again, we use the definition of partial integration;

f(0; t) = f(0; 0) +

∫ t

0

fT (0;u)du,

= r(0) +

∫ t

0

fT (0;u)du. (...(2.14)) (2.56)

Substituting (2.55) into (2.54),

(2.54) = f(0)

+

∫ t

0

[fT (0;u) + α(u;u) +

∫ u

0

αT (s;u)ds+

∫ u

0

σT (s;u) · dW (s)

]du

+

∫ t

0

σ(s; s)ds · dW (s). (2.57)

Notice the following equation;

fT (0;u) +

∫ u

0

α(s;u)ds+

∫ u

0

σ(s;u)ds · dW (s) =∂

∂Tf(u;T )|T=u

= fT (u;u) (2.58)

Therefore,

The right hand side of (2.56) = r(0)

+

∫ t

0

[α(u;u) + fT (u;u)] du+

∫ t

0

σ(s; s)ds · dW (s),

= r(0) +

∫ t

0

a(u)dc+

∫ t

0

b(u) · dW (u) (...(2.39))

⇔

a(t) = fT (t;T ) + α(t;T )

b(t) = σ(t;T )

(2.39) ∪ (2.40)

(2.59)

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 26

Finally, we derive the relation of the three contents.

Proposition 2.5.3 (Short Rate, Instantaneous Forward Rate, and Zero Coupon Bond)

B(t;T ) = B(0;T ) +

∫ t

0

B(s;T )

r(s) + A(s;T ) +

1

2||S(s, T )||2

ds

+

∫ t

0

S(s;T )B(s;T ) · dW (s) (2.60)

, where

A(t;T ) = −∫ T

t

α(t; s)ds, (2.61)

S(s;T ) = −∫ T

t

σ(t; s)ds. (2.62)

Proof Let b(x) be B(x; x). Moreover, setting

λ : R → R2,

we can write b(x) in the following;

b(x) = (B λ)(x). (2.63)

From this,

db

dx

∣∣∣∣x

=∂B(t;T )

∂t

∣∣∣∣(t;T )

+∂B(t;T )

∂T

∣∣∣∣(t;T )

. (2.64)

And

b(t) = B(t; t) = 1,

⇒ db(t) = 0,

⇔ 0 = dB(t; t) +∂B(t;T )

∂T

∣∣∣∣(t;T )

dt,

= m(t; t)B(t : t)dt+ v(t; t)B(t : t) · dW (t) + BT (t; t)dt,

⇔ 0 = m(t; t) + PT (t; t) dt+ v(t; t) · dW (t),

⇒

m(t; t) + PT (t; t) = 0.

v(t; t) = 0.(2.65)

From the definition of the instantaneous forward rate and short rate,

f(t;T ) ≡ − ∂

∂TlogB(t;T ),

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 27

=−BT (t;T )

B(t;T ). (2.66)

r(t) = limT→t+0

f(t;T ),

=−BT (t; t)

B(t; t),

= −BT (t; t) (...B(t; t) = 1). (2.67)

Substituting (2.66) into (2.64), r(t) = m(t; t) = −PT (t; t).

v(t; t) = 0.(2.68)

From proposition 2.5.1,

σ(t;T ) + vT (t;T ) = 0

⇒∫ T

t

σ(t;u)du+ v(t;T )− v(t; t) = 0

⇔ v(t;T ) = −∫ T

t

σ(t;u)du = S(t;T )

α(t;T ) = vT (t;T ) · v(t;T )−mT (t;T )

⇔ mT (t;u) =∂

∂T

1

2||v(t;u)||2

− α(t;u)

⇒∫ T

t

mT (t;u)du =

∫ T

t

∂

∂T

1

2||v(t;u)||2

−∫ T

t

α(t;u)du

=1

2||v(t;T )||2 − 1

2||v(t; t)||2 −

∫ T

t

α(t;u)du

⇔ m(t;T )−m(t; t) =1

2||v(t;T )||2 −

∫ T

t

α(t;u)du (...(2.67))

⇔ m(t;T ) = r(t) + A(t;T ) +1

2||v(t;T )||2 (

...(2.60), (2.67)) (2.69)

Finally, we have done the relation for three factors, short rate, instantaneous forward

rate, and zero coupon bond by N-dimension brownian motion. Then, we derive what a

condition guarantees to make HJM-framework arbitrage free.

2.5.2 Arbitrage Free Model: HJM-framework

Modeling interest rates, there are two important conditions. The one is to make the model

arbitrage free on the term structure. The other is that we can extend the dimension of

independent variables for dependent variables. In section, we derive a important theorem

in HJM-framework.

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 28

Theorem 2.5.4 (HJM drift condition)

Let instantaneous forward rate f(t;T ) be modeled under Q-Equivalent Martingale Mea-

sure(EMM);

f(t : T ) = f(0 : T ) +

∫ t

0

α(s;T )ds+

∫ t

0

σ(s;T ) · dW (t).

Then, the following equation is approved;

α(t;T ) = σ(t;T ) ·∫ T

t

σ(t; s)ds. (2.70)

This is call ”HJM drift condition”.

Proof Assuming B()s is a standard N-brownian motion under Q-Equivalent Martingale

Measure, from proposition 2.5.3

B(t;T ) = B(0;T ) +

∫ t

0

B(s;T )

r(s) + A(s;T ) +

1

2||S(s, T )||2

ds

+

∫ t

0

S(s;T )B(s;T ) · dW (s) (2.71)

, where

A(t;T ) = −∫ T

t

α(t; s)ds, (2.72)

S(s;T ) = −∫ T

t

σ(t; s)ds. (2.73)

Under the EMM, the following equation is the necessary condition;

B(t;T )

r(t) + A(t;T ) +

1

2||S(t, T )||2

= B(t;T )r(t)

⇒∫ T

t

α(t; s)ds =1

2

∣∣∣∣∣∣∣∣∫ T

t

σ(t; s)

∣∣∣∣∣∣∣∣2⇒ α(t;T ) =

∂

∂T

∫ T

t

σ(t; s)ds

(∫ T

t

σ(t; s)ds

)⇔ α(t;T ) = σ(t;T ) ·

(∫ T

t

σ(t; s)ds

)(2.74)

2.5.3 How to Use HJM Framework

We derive HJM framework in section 2.5. And, we understand the arbitrage free condition

on the term structure. Then, how do we make a use of the framework? Moreover, how

@Quasi quant2010

2.5 Heath-Jarrow-Morton Frame work 29

do we extract the market information from financial products? In this section, briefly, we

mention how to use HJM framework into the market data.

Before doing that, notice the following important thing;

If

we determine the volatility function(the diffusion term) σ(t;T ) of instantaneous forward

rate f(t;T )

, then

we have already determined the drift term of instantaneous forward rate f(t;T ).

Rather, it is correct that HJM drift condition(theorem 2.5.4) have the drift term deter-

mined.

As the final content, we briefly introduce how to use HJM framework into financial data;

1. We determine the volatility function of instantaneous forward rate f(t;T ) from

maket’s option price(CAP/FLOOR, SWAPTION)

2. By theorem 2.5.4(HJM drift condition), we can determine the drift term. Therefore,

we make a decision of the instantaneous forward rate f(t;T ). From this step, we

know the second term,∫ t

0α(s;T )ds, and the third term,

∫ t

0σ(s;T ) ·dW (t), of (2.41)

3. If we know the initial forward rate, f(0;T ), in (2.41), then we can complete the

instantaneous forward rate model in (2.41), HJM framework.

From the today’s zero coupon bond price, B(0;T ), with maturity T, we can know

the initial f(0;T );

f(0;T ) = − ∂

∂T

(logB(0;T )

)(2.75)

4. We can construct the following model;

f(t : T ) = f(0 : T ) +

∫ t

0

α(s;T )ds+

∫ t

0

σ(s;T ) · dW (t) (2.76)

5. From B(t;T ) = exp−∫ t

0f(s;T )ds

, we can know zero coupon bond prices. There-

fore, we can use B(t;T) in price several derivatives and restructuring yield curve.

@Quasi quant2010

Bibliography

[1] 二宮祥一, 応用ファイナンス講義ノート, 東京工業大学大学院, 2009前期,

[2] B.Tuckman, Fixed Income Securities:Tools for Today’s Market, second edit , Wiley

& Sons,

[3] D.Brigo, F.Mercurio, Interest Rate Models-Theory and Practice, second edit,

Springer Finance, 2006,

[4] M.Musiela, M.Rutkowski, Maritingale Methods in Financial Modeling, second edit,

Springer, 2005,

[5] M.Choudhry, Analysing and Interpreting the Yield Curve, Wiley & Sons, 2004,

@Quasi quant2010