4. 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-
5. 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
6. 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 EP
Z = 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.
7. 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
EQ
Y = E [Y Z(t)] . (1.4)
Proof : We use the definition of the conditional expectation, the property Y is Ft-
measurable.
EQ
Y = 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
[
1AEP
[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
8. 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)dWP
(u) −
1
2
∫ t
0
Θ(u)2
(u)du
}
, (1.8)
WQ
(t) = WP
(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 EP
Z = 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) =
(
dWP
(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)dWP
(t). (1.12)
@Quasi quant2010
9. 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)dWP
(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)dWP
(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)dWP
(t)
= σ(t)D(t)S(t)
[
Θ(t)dt + dWP
(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),
1
In section 1.3, we mention about Martingale Representation Theorem. By this theorem, we can
derive the market price of risk.
@Quasi quant2010
10. 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 + dWP
(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
11. 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)dWP
(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)dWP
(t)
= σ(t)D(t)V (t)
[
Θ(t)dt + dWP
(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
12. 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)dWP
(t), (1.28)
dN(t) = R(t)N(t)dt + ν(t)N(t)dWP
(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
13. 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
14. 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
15. 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
16. 2.1 Interest Rate 11
2.1.3 Short Rate
We define r(t) as short rate; for 0 ≤ t,
r(t) ≡ lim
T→t+0
R(t; T), (2.5)
≡ lim
T→t+0
L(t; T), (2.6)
≡ lim
T→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) ≡ lim
T→S+0
F(t : S; T), (2.12)
= lim
T→S+0
B(t; S) − B(t; T)
(T − S)B(t; T)
,
@Quasi quant2010
17. 2.2 Stochastic Discout Factor and Zero Coupon Bond 12
= lim
T→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
18. 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
19. 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
20. 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
)(1 1 αα TTK −+ + )(1 1−−+ ββ TTK
・・・
11
))(;(1 11 αααα TTTTL −+ ++
αT 1+αT βT1−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
21. 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
22. 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−1
j=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
))(;(1 11 iiii TTTTL −+ ++
)(1 1 ii TTK −+ +
1
t iT 1+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
23. 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
))(;(1 11 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
24. 2.3 Financial Instruments 19
Time
1 1
TT
))(;(1 11 αααα TTTTL −+ ++
1+iTiT
Time
))(;(1 11 iiii TTTTL −+ ++
+TT
1 11Cash In
1+iTiT
))(;(1 11 iiii TTTTL −+ ++
1
))(;(1 11 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
25. 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
26. 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
27. 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
[
−
∂
∂T
exp
{
−
∫ T
t
r(s)ds
}
|Ft
]
=
−1
B(t, T)
∂
∂T
EQ
[
exp
{
−
∫ T
t
r(s)ds
}
|Ft
]
=
−1
B(t, T)
∂
∂T
B(t; T) (
.
.
.
(1.38))
= −
∂
∂T
logB(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
28. 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
2
fttdt2
+ ftxdtdx +
1
2
fxxdx2
= ftdt + fxdx +
1
2
fxxdt
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
29. 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 v2
j (t; T). By the way, from the definition of the instantaneous
forward rate,
f(t; T) = −
∂
∂T
logB(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) + 1
2
||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) ≡ lim
T→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
30. 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) =
∂
∂T
f(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
31. 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) ≡ −
∂
∂T
logB(t; T),
@Quasi quant2010
32. 2.5 Heath-Jarrow-Morton Frame work 27
=
−BT (t; T)
B(t; T)
. (2.66)
r(t) = lim
T→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.
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33. 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
34. 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
0
α(s; T)ds, and the third term,
∫ t
0
σ(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, ˆB(0; T), with maturity T, we can know
the initial f(0; T);
ˆf(0; T) = −
∂
∂T
(
log ˆB(0; T)
)
(2.75)
4. We can construct the following model;
ˆf(t : T) = ˆf(0 : T) +
∫ t
0
ˆα(s; T)ds +
∫ t
0
ˆσ(s; T) · dW(t) (2.76)
5. From B(t; T) = exp
{
−
∫ t
0
ˆf(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
35. 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,
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