Modelling Stochastic Interest Rate With spot rate and term structure simulation

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Modelling Stochastic Interest Rate

With spot rate and term structure

simulation

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List of Figures

A.1 EUR001M Interest Rate Index data . . . II A.2 Simulation of spot rates using Wiener process . . . II A.3 Simulation of spot rates using vasicek model . . . III A.4 Vasicek simulation for 1000 data series . . . III A.5 CIR simulation for 1000 date series . . . IV A.6 Vasicek’s Upward Sloping Term Structure Model . . . IV A.7 Vasicek’s Downward Sloping Term Structure Model . . . V A.8 Vasicek’s Humped Sloping Term Structure Model . . . V A.9 Approach 1 Modelling the Force of Interest Accumulation

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List of Tables

B.1 Resume Statistics for The EUR001M Interest Rate Index data VIII B.2 Table expected value of annuity, Modelling the force of

inter-est accumulation function . . . VIII B.3 Table expected value of annuity, Modelling the force of interest IX B.4 Table standard deviation of annuity, Modelling the force of

interest accumulation function . . . IX B.5 Table standard deviation of annuity, Modelling the force of

interest . . . X B.6 Table coefficient of skewness of annuity, Modelling the force

of interest accumulation function . . . X B.7 Table coefficient of skewness of annuity, Modelling the force

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Abstract

Gary Parker presents two approaches for modelling interest random-ness in his paper, ”Two Stochastic Approaches For Discounting Actuarial

Functions”. They are the modelling of the force of interest accumulation

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Chapter 1

Introduction

The goal of this paper is to investigate the two stochastic approaches for discounting actuarial function proposed by Gary Parker in his paper, ”Two

Stochastic Approaches For Discounting Actuarial Functions”. The main

question of this paper is: ”Are these two approaches good when they are applied in real data?” To answer this question I will simulate the two ap-proaches, discount them and then compare the results with the real data in the form of discount factor.

Why is modelling interest rate as random variable preferable than mod-elling it as a constant? I would try to give the reason in the following. Interest rates and the process of discounting future cash flow play a funda-mental role in the financial world. One of the usefulness of interest rate is in pricing a financial instrument like bonds, interest rate swaps, mortgage backed securities, insurance reserves, duration and convexity of a portfolio and many other examples. Interest rates are used in time discounting. A good, accurate and reliable model of interest rate is then becoming very important. A lot of companies use constant interest rate in their calculation to price and value financial transactions. But it’s not really accurate and reliable to do that because interest rates fluctuate every second. Its fluctu-ation is influenced by macro economics factors such as inflfluctu-ation, oil prices, price of dollars or euros, hot financial and political issues and many other factors. In other words, assuming interest rate as random variable capture better reality than assuming it to be a constant. To model interest rate as a random variable is more natural than to assume it as a constant.

I would like to introduce some important things from the paper. There are two approaches presented in the paper to model the interest randomness. First is by modelling the force of interest accumulation function and second is by modelling the force of interest. Later on in the paper, the expected value, standard deviations and coefficient of skewness of the present value of annuities immediate are also presented.

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Force of interest accumulation function; White Noise process; Wiener pro-cess; Ornstein-Uhlenbeck propro-cess; Present value function; Annuity-Immediate

I would like to begin with explaining the terms that are going to be used a lot in this paper. I would begin with the concept of the force of interest. The force of interest is a measure of interest that operates at a very small intervals of time. It is denoted by δ_s, that is the force of interest at time s. The force of interest accumulation function is the integration of the force of interest over some interval of time. It is denoted by y(t), which

y(t) =R₀tδsds

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Chapter 2

Stochastic Process

2.1 Wiener Process

In the theory of stochastic process and its applications, a fundamental role is played by the Wiener process. Among other applications, the Wiener process provides a model for Brownian motion, an irregular movement of a particle in a liquid or gas. It is named after the English botanist Robert Brown, who discover the phenomenon in 1827.

A stochastic process {X(t), t ≥ 0} is said to be a Wiener process if: 1. {X(t), t ≥ 0} has stationary independent increments,

2. For every t > 0, X(t) is normally distributed, 3. For all t > 0, E[X(t)] = 0,

4. X(0) = 0

A stochastic process {X(t), 0 ≤ t < ∞} is said to have independent increments if X(0) = 0 and for all choices of indices t0 < t1 < t2. . . < tn,

the n random variables X(t1) − X(t2), X(t2) − X(t1), . . . , X(tn) − X(tn−1)

are independent.

The process is said to have a stationary independent increments if X(t₂+

h) − X(t1+ h) has the same distribution as X(t2) − X(t1) for all choices of

indices t1 and t2 and every h > 0.

2.2 White Noise

{X(t), t ≥ 0} is said to be a white noise process if it possesses a constant

spectral density function:

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White noise is a stationary Gaussian stochastic process with mean zero and a constant spectral density. Such a process does not exist in the conven-tional sense since it would have an infinite variance. Nonetheless, the white noise is a very useful mathematical idealization for describing influences that fluctuate rapidly. White noise is the derivatives of the Wiener process.

2.3 Ornstein-Uhlenbeck process

Models for Brownian motion which are somewhat more realistic than the Wiener process can be constructed. One such model is the Ornstein-Uhlenbeck process. The so-called Langevin equation:

˙

X(t) = −αX(t) + σξt, α > 0, σ constant

The corresponding stochastic differential equation is:

dX(t) = −αX(t)dt + σdWt

2.4 Annuity-immediate

An annuity is defined as a series of payments made at equal intervals of time. Examples of annuity is house rents, mortgage payment, installment payment on car, etc. Annuity immediate is an annuity under which payments of 1 are made at the end of each period for n periods. The formula of annuity immediate:

a_ne = e−y(1)_{+ e}−y(2)_{+ . . . + e}−y(n)

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Chapter 3

Modelling Interest

Randomness

According to the paper of Gary [6], there are two ways of modelling interest randomness. A first approach is to model y(t), the force of interest accumu-lation function. A second approach is to model δs, the force of interest.

3.1 Modelling the force of interest accumulation

function

In this section two process will be described as a model for the y(t), the force of interest accumulation function. The two processes are Wiener pro-cess with deterministic drift δ and an Ornstein-Uhlenbeck propro-cess also with deterministic drift δ.

3.1.1 Modelling the force of interest accumulation function

by Wiener Process

Let y(t) be the sum of a deterministic drift of slope δ and a perturba-tion/disturbance modelled by a Wiener process.

y(t) = δt + σWt, σ ≥ 0, Wt: Wiener process

The expected value of y(t) is

E[y(t)] = E[δt + σWt] = E[δt] + E[σWt] = δt + σE[Wt] = δt (3.1)

The autocovariance function of y(t) is

cov[y(s), y(t)] = σ2min(s, t) (3.2) The conditional expected value of y(t) given y(s) and δs for s < t when

y(t) follows a Wiener process is:

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E[y(t)|y(s) = x, δs= ²] = δt + σE[Wt|Ws= x − δs_σ , δs= ²] (3.4)

because Wt|Ws is independent of δs, then

E[y(t)|y(s) = x, δs = ²] = δt +σ_σ(x − δ) (3.5)

E[y(t)|y(s) = x, δs= ²] = x + δ(t − s) (3.6)

3.1.2 Modelling the force of interest accumulation function

by Ornstein-Uhlenbeck Process

Let y(t) be the sum of a deterministic drift of slope δ and a perturba-tion/disturbance modelled by Ornstein-Uhlenbeck Process.

y(t) = δt + Xt, Xt: Ornstein-Uhlenbeck process

˙

X(t) = −αX(t) + σξ(t); α ≥ 0, σ is constant ξ(t) is a scalar white noise

The expected value of y(t) is:

E[y(t)] = E[δt + Xt] = E[δt] + E[Xt] = δt + E[Xt] = δt (3.7)

The covariance of y(t) is:

cov[y(s), y(t)] = σ2 2α(e −α(t−s)_{− e}−α(t+s)₎ _(3.8) or cov[y(s), y(t)] = ρ2(e−α(t−s)− e−α(t+s)) (3.9) where ρ2 ₌ σ2 2α If s=t we have:

V [y(t)] = cov[y(t), y(t)] = ρ2(e0− e−2αt) = ρ2(1 − e−2αt) (3.10) The conditional expected value of y(t) given y(s) and δs for s < t when

y(t) follows an Ornstein-Uhlenbeck process is:

E[y(t)|y(s) = x, δs = ²] = E[δt + X(t)|δs + X(s) = x, δs= ²] (3.11)

E[y(t)|y(s) = x, δs = ²] = δt + E[X(t)|X(s) = x − δs, δs= ²] (3.12)

because X(t)|X(s) is independent of δs for s < t, then

E[y(t)|y(s) = x, δs= ²] = δt + E[X(t)|X(s) = x − δs], (3.13)

and because E[X(t)|X(s) = x] = xe−α(t−s) [see [7] (1990, Section 2)], then

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3.2 Modelling the force of interest

A second approach to model interest randomness is to model δs, the force

of interest at time s. The paper of Gary [6] modelled δs by White-Noise

process, Wiener process and Ornstein-Uhlenbeck process. The modelling of the force of interest by White-Noise is the same as modelling the force of interest accumulation function by Wiener process.

3.2.1 Modelling the force of interest by Wiener process

Let the force of interest be defined by

δt= δ + σWz, σ ≥ 0, Wt: Wiener process

We can obtain the expected value of δ_t by

E[δt] = E[δ + σWt]

= E[δ] + E[σWt]

= δ + σE[W_t]

= δ (3.15)

In this case we can obtain the expected value of y(t), that is

E[y(t)] = E[ Z _t 0 δ_tds] = Z _t 0 E[δt]ds = Z _t 0 δds = δt (3.16)

The covariance of y(t) is:

cov[y(s), y(t)] = σ2(s2t 2 −

s3

6 ) (3.17)

The variance of y(t) is:

V [y(t)] = σ2(t3 2 − t3 6) = 2σ2t3 6 = σ2t3 3 (3.18)

The conditional expected value of y(t) given y(s) and δs for s < t when

δs follows a Wiener process is:

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3.2.2 Modelling the force of interest by Ornstein-Uhlenbeck process

Let the force of interest be defined by the following equation:

dδt= −α(δt− δ)dt + σdWt, α ≥ 0, σ ≥ 0 (3.20)

with initial value of δ0 = δ

The expected value of δt is

E[δ_t] = δ (3.21) and its autocovariance function is

cov[δs, δt] = σ

2

2α(e

−α(t−s)_{− e}−α(t+s)_{), s ≤ t} _(3.22)

This makes the force of interest accumulation function y(t) follow a Gaus-sian process with expected value

E[y(t)] = δt (3.23)

and autocovariance function

cov[y(s), y(t)] = σ2

α2 min(s, t) +

σ2

2α3[−2 + 2e

−αt_{− e}−α|t−s|_{− e}−α(t+s)_{] (3.24)}

which makes its variance

V [y(t)] = 2ρ 3_t α + ρ2 2α(−3 + 4e −αt_{− e}−2αt₎ _(3.25)

The conditional expected value of y(t) given y(s) and δ_s for s < t when

δs follows a Wiener process is:

E[y(t)|y(s) = x, δs= ²] = x + δ(t − s) + (² − δ) Ã 1 − e−α(t−s) α ! (3.26) All the models have all been defined such that their expected value of

y(t) is always δt. What varies over the model is the variance of y(t). Table

Summary of Results about y(t) summarize the results that we have obtained above.

Table Summary of Results about y(t)

Process E[y(t)] V [y(t)] E[y(t)|y(s) = x, δs= ²]

Modelling the force of interest accumulation function

Wiener δt σ2.t x + δ(t − s)

O-U δt ρ2.(1 − e−2αt) δ.t + (x − δ.s).e−α(t−s)

Modelling the force of interest

Wiener δ_t σ2_.t3_/3 _{x + ²(t − s)}

O-U δ_t 2ρ_α2t+_2αρ2(−3 + 4e−αt_{− e}−2αt₎ _{x + δ(t − s) + (² − δ).}³1−e−α(t−s)

α

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3.3 n-Year Annuity-Immediate Contract

We now consider an n-year annuity-immediate contract. Let a_ne be the present value of n equal payments of 1 made at the end of the next n years.

a_ne = n X t=1 e−y(t) (3.27) then E[a_ne] = E[ n X t=1 e−y(t)] = n X t=1 E[e−y(t)] (3.28) Assuming that y(t) is Gaussian, then the present value function is log-normally distributed with parameter E[−y(t)] and V [y(t)].

y(t) ≈ Gaussian (3.29)

e−y(t) ≈ Log − normal(E[−y(t)], V [y(t)]) (3.30) Its mth moment about the origin is :

E[(e−y(t))m] = E[e−m.y(t)] = e−mE[y(t)]+12m2V [y(t)] (3.31)

Then equation 3.28 can be written as:

E[a_ne] =

n

X

t=1

e−E[y(t)]+12V [y(t)] (3.32)

If we apply the above method to find the expectation of a_ne we will get the following results:

W iener : E[a_ne] = n X t=1 e−δ.t+12σ2.t (3.33) O − U : E[a_ne] = n X t=1 e−δ.t+12ρ2(1−e−2αt) (3.34)

Modelling the force of interest

W iener : E[a_ne] = n X t=1 e−δ.t+12σ2.t3/3 (3.35) O − U : E[a_ne] = n X t=1 e−δ.t+12[ 2ρ2t α + ρ2 2α(−3+4e−αt−e−2αt)] (3.36)

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standard deviation of a_ne in table 2 and the coefficient of skewness of a_ne in table 3. I reproduce the results Gary’s paper using software package excel and show that I get the same results. I show my results using excel in table expected value of a_ne, table standard deviation of a_ne and table coefficient of skewness of a_ne. My choices of parameters in these tables are made to be similar with the Gary’s choice of parameters. The parameter δ is set at 0.06 and 0.10. For Wiener process the parameter σ is 0.01 and 0.02. For O-U process, the parameter α is chosen to be 0.17. Parameter ρ is set at 0.01 and 0.02.

From table expected value of a_ne we can see that the expected value of

a_ne does not depend on the modelling approach. Whether modelling the force of interest accumulation function or modelling the force of interest we will get more or less the same number. The expected value of a_ne also does not depend on the parameters of the process, except parameter δ. Take for example modelling the force of interest accumulation function with Wiener process (δ = 0.06 and σ = 0.01) with modelling the force of interest with also Wiener process, we will get the result of 4.1920 and 4.1943. Both numbers are quite similar. Another example would be to take different parameter value, take σ = 0.01 and σ = 0.02, we get the result of 4.1920 and 4.1938. These two numbers are also quite similar.

The Wiener process, for n larger than 20, when used to model the force of interest, is another exception.

On table standard deviation of a_ne, standard deviation differs between models. Moreover we can not use multiplication by factors to get similar results.

On table coefficient of skewness of a_ne, coefficient of skewness also differs between two approaches. This fact supports that the two models are not equivalent. This means that modeling the force of interest accumulation function has different implications on the random present value function and other actuarial functions than modeling the force of interest. When modeling the force of interest, it is δ_s that varies according to the chosen stochastic process. When modeling y(t), then δs varies so that y(t) follows

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Chapter 4

Simulation

In this section I will simulate the Ornstein-Uhlenbeck process to model the interest rate. I will use the EUR001M Index of interest rate as a proxy for the spot rate. These data series are obtain from Bloomberg. The data consists of 80 monthly observations. It ranges from 31 Dec 1998 to 29 July 2005. The monthly interest rate has an average of 3.1% and standard deviation of 0.945%. The minimum and maximum rate from the whole series are 2.0430% and 4.9480%. The summary statistics of EUR001M Index of interest rates are presented in table B.1. The plot of the EUR001M Index can be seen on figure A.1.

First I am going to simulate the force of interest, in practice used to be called spot rate, using Wiener process. After simulating Wiener process I will simulate Ornstein-Uhlenbeck process. Using Ornstein-Uhlenbeck pro-cess to simulate the force of interest has been done by Vasicek (1977) in [8]. Vasicek model has a property that interest rate can become negative which in practice most likely will not happened. Cox, Ingersoll and Ross (CIR) in [2] solve the problem that Vasicek model has. There will be no negative in-terest rates in CIR model. I will also simulate term structure using Vasicek model.

One important property in one-factor interest rate model using Ornstein-Uhlenbeck process is that interest rates appear to be pulled back to some long-run equilibrium level over time. This property is known as mean

rever-sion. Mean reversion will pull back interest rate down when interest rate is

high and mean reversion will pull up the interest rate when it is low.

4.1 Simulating the Wiener Process

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dx = adt + bdz (4.1) where a and b are constants and a variable z follows a basic Wiener process that has a drift rate zero and a variance rate of 1. I simulate the generalized Wiener process x and the basic Wiener process z with initial value of 3.256, a = 3.1 and b = 0.945 . We can see the results in figure A.2. The script of the Matlab program can be seen in the Appendix.

4.2 Simulating the Vasicek Model

Ornstein-Uhlenbeck process is used by Vasicek (1977) in modelling the force of interest, in practice used to be called spot rate, δ_t:

dδt= α(δ − δt)dt + σdWt, α ≥ 0, σ ≥ 0 (4.2)

In 4.2, the parameters have the following interpretation:

δ : the long-term mean spot interest rate,

α : the pressure to revert to the mean α > 0, and σ : the instantaneous standard deviation.

The Vasicek model is a one-factor model. It means that all rates depend on the shortest-term interest rate, which we call the spot rate. To be able to make simulation for this rate we have to make a little manipulation to the equation 4.2 by incorporating the changes of the interest rate over a short period ∆t into the equation. Equation 4.2 becomes:

∆δt= α(δ − δt)∆t + σ²

√

∆t (4.3)

where ² is standard normal distribution.

The characteristic of the term structure model is additive. This is in contrast with the characteristic to stock price model, which is multiplicative. This additive characteristic of term structure make the calculation of the spot rate δtat time t + ∆t is:

δt+∆t= δt+ ∆δt= δt+ α(δ − δt)∆t + σ²

√

∆t (4.4) Set α = 3.256%, δ = 3.1%,σ = 0.945, ∆t = 0.0001. Program script in Matlab can be seen in Appendix.

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the bank instead of receiving the interest from the bank. They will rather choose to put their money under their pillow than depositing their money in the bank, at least they don’t have to pay interest when they put their money under their pillow.

4.3 Simulating the CIR model

As I mentioned in earlier section, Vasicek model has a problem that interest rate can become negative. Cox, Ingersoll and Ross have an alternative model where interest rate can never be negative. CIR proposed a model for interest rate in equation 4.5:

dδ_t= α(δ − δ_t)dt +pδ_tσdW_t (4.5) CIR model has the same property like Vasicek model, that is mean rever-sion. The only difference with Vasicek model is that the standard deviation of the change in the spot rate is proportional to√δt. This makes sure that

the interest rates can never be negative. We can see the results of CIR simulation in figure A.5.

4.4 Vasicek’s Term Structure Model

The whole term structure can be constructed using Vasicek model for the spot rate. The price of the zero-coupon bond is given in equation 4.6:

P (t, T ) = A(t, T )eB(t,T )r(t) (4.6) where r(t) is the value of the spot rate r at time t.

B(t, T ) = 1 − e −α(t−t) α (4.7) A(t, T ) = exp µ (B(t, T ) − T + t)(α2_{δ − σ}2_/2) α2 − σ2_{B(t, T )}2 4α ¶ (4.8) In the special case when α = 0, equation 4.7 and 4.8 become

B(t, T ) = T − t and A(t, T ) = eσ2_{(T −t)}3_/6

Equation 4.9 shows that the whole term structure can be constructed from a function of δtonce α, δ and σ are chosen or estimated from historical

data.

R(t, T ) = − 1

T − tln A(t, T ) +

1

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The three parameters (α, δ and σ) are arbitrarily chosen. The initial short rate is 3% and assumed that the long term rate is 5%. The model is set up to produce interest rate from year zero until year 100. You can see the result of Vasicek’s term structure model in figure A.6. This figure shows the upward sloping term structure. Vasicek model can produce the downward sloping term structure if we start with interest rate above the long-run equilibrium rate. The downward sloping Vasicek term structure can be seen in figure A.7. The Vasicek model can also produce humped term structure. We can see it in figure A.8. The program script in Matlab can be seen in the Appendix.

4.5 Fitting Discount Factor Using Wiener Process

and Ornstein-Uhlenbeck Process

The goal of this paper as written on the Introduction is to investigate whether the two stochastic approaches for discounting actuarial function proposed by Gary Parker are good when applied to real data. The investi-gation works like this. We take a EUR001M as a proxy for the spot rate. We already have the data simulation of the force of interest accumulation function (approach 1) and also the data simulation of the force of interest (approach 2).

We will compare the discount factor resulting from different sources, discount factor from EUR001M, discount factor from the first approach and discount factor from the second approach. If they fit each other we can say that the two approaches are good and if they don’t fit each other we can say that the two approaches are not good.

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Chapter 5

Remarks and Summary

As a conclusion we can say that both approach, modelling the force of in-terest accumulation function and modelling the force of inin-terest are good when applied to real data.

Furthermore the two stochastic approaches used to model interest ran-domness are not equivalent. Modelling the force of interest accumulation function has different consequences on the random present values and annuity-immediate than modelling the force of interest. The expected value of a_ne does not depend on the modelling approach, but the standard deviation and coefficient of skewness does.

Interest rates can be modelled by Wiener process, Ornstein-Uhlenbeck process as in Vasicek model and CIR model. Vasicek model has one problem, that is interest rates can become negative. In reality, nominal interest rate will never become negative because people has to pay interest when they put their money in the bank. This problem is solved by Cox, Ingersoll and Ross by making the standard deviation of the change proportional to √δt.

CIR model is an improvement of Vasicek model.

Mean reversion is an important property in Vasicek and CIR model. Interest rates appear to be pulled back to some long-run equilibrium level over time. When interest rate is high, mean reversion will pull back interest rate down and when interest rate is low mean reversion will pull it up. The Vasicek and the CIR model are both one factor interest rate model. It means that the process depend on the spot rate.

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Appendix A

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0 10 20 30 40 50 60 70 80 0.02 0.025 0.03 0.035 0.04 0.045 0.05

EUR001M Monthly Interest Rate Index

Time

Value

Figure A.1: EUR001M Interest Rate Index data

0 100 200 300 400 500 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Simulating Wiener Process

Value of variable, x Time dx dx=adt dz Generalized Wiener Process dx = a dt + b dz Wiener Process, dz

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0 10 20 30 40 50 60 70 80 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Vasicek Simulation Time Value

Figure A.3: Simulation of spot rates using vasicek model

0 200 400 600 800 1000 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

Vasicek Simulation for 1000 data series

Time

Value

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0 200 400 600 800 1000 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 Time Value

Simulating CIR Model

Figure A.5: CIR simulation for 1000 date series

0 20 40 60 80 100 120 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048

Vasicek’s Term Structure Model

Time

Value

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0 20 40 60 80 100 120 0.045 0.05 0.055 0.06 0.065 0.07 0.075

Vasicek’s Term Structure Model

Time

Value

Figure A.7: Vasicek’s Downward Sloping Term Structure Model

0 20 40 60 80 100 120 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032

Vasicek’s Humped Term Structure Model

Time

Value

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0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Approach 1 (Modelling the Force of Interest Accumulation Function)

Time Present Value Real Data Wiener Process Ornstein−Uhlenbeck Process EUR001M discount factor Wiener Process discount factor O−U Process discount factor

Figure A.9: Approach 1 Modelling the Force of Interest Accumulation Func-tion 0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Present Value

Approach 2 (Modelling the Force of Interest) Real Data Wiener Process O−U Process EUR001M discount factor Wiener Process discount factor O−U Process discount factor

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Appendix B

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Statistics EUR001M Index Minimum 2.0430 1st Quantile 2.1192 Mean 3.0995 Median 3.0580 3rd Quantile 3.5290 Maximum 4.9480 Total Observations 80 Standard Deviations 0.9450

Table B.1: Resume Statistics for The EUR001M Interest Rate Index data

Table expected value of a_ne

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Table expected value of a_ne Modelling the force of interest

n 5 10 20 30 40 Wiener δ σ 0.06 0.01 4.1943 7.3273 11.5925 14.4863 17.0285 0.06 0.02 4.2030 7.4217 12.6140 19.5880 48.6888 0.1 0.01 3.7437 6.0327 8.3788 9.4388 10.0567 0.1 0.02 3.7510 6.1008 8.9232 11.3948 18.0414 O-U δ α ρ 0.06 0.17 0.01 4.1920 7.3007 11.3221 13.5410 14.7658 0.06 0.17 0.02 4.1938 7.3135 11.3862 13.6702 14.9531 0.1 0.17 0.01 3.7417 6.0135 8.2336 9.0548 9.3586 0.1 0.17 0.02 3.7432 6.0229 8.2703 9.1151 9.4331 Table B.3: Table expected value of annuity, Modelling the force of interest

Table standard deviation of a_ne

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Table standard deviation of a_ne Modelling the force of interest

n 5 10 20 30 40 Wiener δ σ 0.06 0.01 0.1251 0.5171 0.1960 4.2762 8.6273 0.06 0.02 0.2515 1.0710 5.1457 27.4239 1111.8356 0.1 0.01 0.1073 0.3880 1.1483 1.9504 2.9114 0.1 0.02 0.2157 0.8019 2.8968 10.1266 240.2379 O-U δ α ρ 0.06 0.17 0.01 0.0576 0.1968 0.5294 0.7975 0.9767 0.06 0.17 0.02 0.1152 0.3952 1.0736 1.6334 2.0169 0.1 0.17 0.01 0.0495 0.1495 0.3263 0.4202 0.4610 0.1 0.17 0.02 0.0991 0.3001 0.6604 0.8563 0.9433 Table B.5: Table standard deviation of annuity, Modelling the force of in-terest

Table coefficient of skewness of a_ne

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Table coefficient of skewness of a_ne Modelling the force of interest

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Appendix C

Matlab Script

%Simulating a basic Wiener proces dz and

%a Generalized Wiener Process dx = a dt + b dz rep=500; e=normrnd(0,1,1,rep); dt=1/rep; z=zeros(size(e)); x=zeros(size(e)); z(1)=3.256; x(1)=3.256; w(1)=3.256; a=3.1; b=0.945; for t=2:rep; z(t)=z(t-1)+e(t)*sqrt(dt); w(t)=w(t-1)+b*(z(t)-z(t-1)); x(t)=x(t-1)+ a*dt + b*e(t)*sqrt(dt); end t=1:rep; plot(w) plot(t,x,t,z);

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for i = 2:rep; r(i)=r(i-1)+deltaR(i-1); end; X(1)=0; X=0; deltaX=-alpha*deltaT*X+sigma*nor*sqrt(deltaT); y(1)=0.03256; for i = 2:rep; y(i)=y(i-1)+gamma*deltaT+deltaX(i-1); end; plot(y);

%CIR simulation for the spot rate alpha = 0.03256; gamma=0.031; sigma=0.945; deltaT=0.0001; rep=1000; r=0.03256; r(1)=0.03256; nor = normrnd(0,1,rep); deltaR = alpha*(gamma-r)*deltaT+sigma*sqrt(r)*nor*sqrt(deltaT); for i = 2:rep; r(i)=r(i-1)+deltaR(i-1); end; plot(r);

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Bibliography

[1] Beninnga, Ben and Wiener, Zvi (1998) Term Structure of Interest Rate,

Mathematica in Education and Research, Vol. 7 No. 2.

[2] Cox,J.C., J.E. Ingersoll, and S.A. Ross (1985) A theory of the Term Structure of Interest Rate, Econometrica 53, 385-402

[3] Das, Satyajit (1998) Risk Management and Financial Derivatives,

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[4] Hull, J.C. (2003) Options, Futures, and Other Derivatives, Prentice

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Modelling Stochastic Interest Rate With spot rate and term structure simulation