Fixed Income Report:
Term Structures and Element Financial Instruments
Twitter : @Quasi quant20101
October 9, 2013
1Quasi ...
Contents
Preface i
1 Quick Mathematical Introduction 1
1.1 Prepare for Risk Neutral Measure and Numeraire Change . . . . ....
CONTENTS 2
2.5.1 The relation of short rate, instantaneous forward rate, and zero
coupon bond . . . . . . . . . . . . . . ...
Preface
Mathematical finance is too beauty to replicate the real world. However, that we un-
derstand the theory and its fa...
ii
ics, especially the risk neutral pricing model. In chapter 2, we focus on the stochastic
approach. In section 2.1, we a...
Chapter 1
Quick Mathematical Introduction
1.1 Prepare for Risk Neutral Measure and Nu-
meraire Change
Firstly, we mention ...
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 a...
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...
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
...
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 ris...
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)}
b...
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...
1.4 Risk Neutral Pricing Formula 8
We call this measure T-forward (martingale) measure. Comparing (1.27) with (1.39),
we n...
Chapter 2
Stochastic Approach
The goal in this section is to derive each financial instruments, the conditional expectation...
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 ...
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
...
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)...
2.3 Financial Instruments 13
Here, we consider the meanings of the discount factor. The following is often question.
Compa...
2.3 Financial Instruments 14
In each sections, we derive the values of Forward Rate Agreement(FRA), Interest Rate
Swap(IRS...
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(...
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...
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...
2.3 Financial Instruments 18
, where T = Tj : j = α, α + 1, ..., β. Namely, CAP is the collection of Caplet when the
reali...
2.3 Financial Instruments 19
Time
1 1
TT
))(;(1 11 αααα TTTTL −+ ++
1+iTiT
Time
))(;(1 11 iiii TTTTL −+ ++
+TT
1 11Cash In...
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 ...
2.4 Expectation Hypothesis of Interest Rate 21
This means the forward rate is a good estimator for the future spot interes...
2.5 Heath-Jarrow-Morton Frame work 22
Intuitively, we can understand the proposition of instantaneous forward rate if prop...
2.5 Heath-Jarrow-Morton Frame work 23
2.5.1 The relation of short rate, instantaneous forward rate, and
zero coupon bond
F...
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...
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)d...
2.5 Heath-Jarrow-Morton Frame work 26
Finally, we derive the relation of the three contents.
Proposition 2.5.3 (Short Rate...
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...
2.5 Heath-Jarrow-Morton Frame work 28
Theorem 2.5.4 (HJM drift condition)
Let instantaneous forward rate f(t; T) be modele...
2.5 Heath-Jarrow-Morton Frame work 29
do we extract the market information from financial products? In this section, briefly...
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,...
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金利期間構造について:Forward Martingale Measureの導出

  1. 1. Fixed Income Report: Term Structures and Element Financial Instruments Twitter : @Quasi quant20101 October 9, 2013 1Quasi Science
  2. 2. 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
  3. 3. 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
  4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. @Quasi quant2010
  33. 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. 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. 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, @Quasi quant2010

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