rajkumar misha rap df

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Study of Positivity Preserving Numerical Methods

for Cox Ingersoll Ross Interest Rate Model

A Project Report

submitted in partial fulfillment of the

requirements for the degree of

Master of Technology

in

Computational Science

by

Raju Kumar Mishra

Supercomputer Education Research Centre

Indian Institute of Science

Bangalore - 560012

June 2010


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Acknowledgement

Being a believer, I pay my respect to god for whatever I have got in my life. This

project report originated from my interest in numerical finance and support, guidanceand encouragement of my project guide, Dr. Soumyendu Raha to whom I express my

sincere gratitude.

I wish to thank my professors Dr. Atanu Mohanty, Dr. Virendra Singh, Dr. R.

Govindarajan, Dr. Sathish Vadhiyar, Dr. Mrinal K. Ghosh, Dr. P. S. Sastry, Dr.

Chiranjit Mukhopadhyay for their insightful teaching through the courses which I

undertook.

I would like to thank Prof. R. Govindarajan, Chairman, Supercomputer Education

and Reserch Centere, for letting me use the computer facilities.

I would like to thank my batchmates, my freinds Pawan, Rajnish, Manoj Kr. Ma-

hala, Avinash Dash, Vinayak, Ravi, Abhishek Sahi, Hari Gupta, Jitendra Singh,

Rajesh , Devendra Mani Tripathi, Gaurav Sharma and many more for their encour-

agement and supports throughout my stay at Indian Institute of Science. I would

also like to thank my juniors.

I am very glad to be grateful to office staff of SERC for helping me with various

issues during their stay at SERC.

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Finally, we have developed a new discretization sceme named as, the New Method,

to simulate the CIR interest rate model. This method was found to be the most timeefficient one amongst the existing scheme in most of the simulations.

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Contents

1 Introduction 1

1.1 Properties of the CIR process: . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Simulating evolution of single path of the CIR process. . . . . . . . . 6

1.3 Organization of the report . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Related Work 9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 The Euler-Maruyama Method . . . . . . . . . . . . . . . . . . . . . . 10

2.3 The Milstein Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 The Balanced Implicit Method (BIM). . . . . . . . . . . . . . . . . . 11

2.5 The Moment Matched Log-Normal Approximation . . . . . . . . . . 11

2.6 The Logarithmic Transformed CIR Process . . . . . . . . . . . . . . . 12

2.7 The Balanced Milstein method (BMM) . . . . . . . . . . . . . . . . . 12

2.8 The New Scheme(News) . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.9 The Pathwise Adapted Linearisation Quadratic . . . . . . . . . . . . 13

2.10 The Pathwise Adapted Linearisation Quartic . . . . . . . . . . . . . . 13

2.11 Simulation of evolution of the CIR process: . . . . . . . . . . . . . . . 14

3 Implementation of FIS- on the CIR process 16

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 FISnumerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Computation ofP . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Discretisation of the CIR process with the FIS method . . . . . . . 19

3.3.1 CIR process with alpha scheme . . . . . . . . . . . . . . . . . 19

3.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 The CIR process is strictly positive (2 > 2). . . . . . . . . 223.4.2 The CIR process is nonnegative (2 2). . . . . . . . . . . 27

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3.5 Problem with the Existing Schemes . . . . . . . . . . . . . . . . . . . 34

3.6 Mixed Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6.1 Mixed Method Algorithm . . . . . . . . . . . . . . . . . . . . 36

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 New Method 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Splitting-step algorithm: . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Development of New Method: . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Simulation of evolution of the CIR process:. . . . . . . . . . . 41

4.4 Numerical Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 The CIR process is strictly positive (2 > 2): . . . . . . . . . . . . 44

4.5.1 Simulation for Interval [0 1]: . . . . . . . . . . . . . . . . . . . 45


4.6 The CIR process is nonnegative (2 2): . . . . . . . . . . . . . . 484.6.1 Simulation for Interval [0 1]: . . . . . . . . . . . . . . . . . . . 49


4.7 Order of convergence comparison . . . . . . . . . . . . . . . . . . . . 534.8 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusion 55

A 56

A.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.2 Ito-Doeblin formula for an Ito process. . . . . . . . . . . . . . . . . . 57

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List of Tables

1.1 Selected Interest-Rate Models . . . . . . . . . . . . . . . . . . . . . . 2

3.1 Simulation results for interval [0 1] of the CIR parameter = 1, =

0.0625 and = 0.8, FIS-(N) parameter and , are 0.3 and 2 re-

spectively, FIS-(log) parameter and , are 0.6 and 5 respectively

by Pathwise Adapted Linearization Quartic method . . . . . . . . . . 32

4.1 CPU time comparison of simulation interval [0 1] for the CIR parameter

= 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3

and 1.1 respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 CPU time comparison of simulation interval [0 4] for the CIR parameter = 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3




and 2 respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49



and 2 respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5 Comparison of order of convergence() for the CIR process parameters

=1, =0.5 and varying . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Comparison of order of convergencefor the CIR process parameters

=1, =0.0625 and varying . . . . . . . . . . . . . . . . . . . . . 54

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List of Figures

1.1 CIR simulation for strict positive condition . . . . . . . . . . . . . . . 7

1.2 CIR simulation for non-negative condition . . . . . . . . . . . . . . . 7

2.1 CIR simulation for strict positive condition . . . . . . . . . . . . . . . 14

2.2 CIR simulation for non-negative condition . . . . . . . . . . . . . . . 15

3.1 CIR simulation for strict positive condition with FIS-method. . . . 20

3.2 CIR simulation for non-negative condition with FIS-method . . . . 21



and 1.1 respectively, FIS-(log) parameter and , are 0.5 and 1.1respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Error comparison of simulation interval [0 1] for the CIR parameter


and 1.1 respectively, FIS-(log) parameter and , are 0.5 and 1.1

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


= 1, = 0.0625 and = 0.2, the FIS-(N) parameter and , are

0.3 and 1.1 respectively, the FIS-(log) parameter and , are 0.9 and9.1 respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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= 1, = 0.0625 and = 0.5, FIS-(N) parameter and , are 0.3and 1.1 respectively, FIS-(log) parameter and , are 0.3 and 4.1

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


= 1, = 0.0625 and = 0.5, FIS-(N) parameter and , are0.3 and 1.6 respectively, FIS-(log) parameter and , are 0.3 and

4.6respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


= 1, = 0.0625 and = 0.8, FIS-(N) parameter and , are

0.3 and 2 respectively, FIS-(log) parameter and , are 0.6 and 5

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.14 Interest rate evalution in simulation interval [0 1] for the CIR parameter



respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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= 1, = 0.0625 and = 1.2, FIS-(N) parameter and , are0.3 and 2 respectively, FIS-(log) parameter and , are 0.9 and 12

respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37




respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Comparison of evolution of one pathe of the CIR process in strictly

positive condition with different methods . . . . . . . . . . . . . . . 42

4.2 Comparison of evolution of one pathe of the CIR process in non-

negative condition with different methods. . . . . . . . . . . . . . . . 43













A.1 One path of the Brownian Motion . . . . . . . . . . . . . . . . . . . 57

x


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Chapter 1

Introduction

An investor knows that the financial market is uncertain and this introduces risk

in any financial activity. Due to the complex nature of a given financial system

and the large number of variables involved, the present day investors are still unable

to predict the market with reasonable certainty. Several financial instruments exist

in the financial market, which are traded actively at many exchanges throughout

the world. The existing financial models have been developed to aide investors in

managing their risk. In this context, the interest rate models are quite popular inthe financial market. The present work deals with these models in particular.

The dynamics of interest rates, plays an important role in the decisions related to

an investment, its risk management and also the transactions based on lending and

borrowing. The uncertainty pertaining to the future evolution of stochastic interest

rates, thus becomes an important part of the financial decision making. Interest rate

models are typically used in the pricing of the interest rate derivatives. Hence, their

study has gained importance in the developing economics literature.

An interest rate model is a probabilistic description of the future evolution of inter-

est rates. The one factor model of interest rates, are popularly used for the pricing

of interest rate derivatives. They are represented using the following stochastic dif-

ferential equation:

dR(t) =(R(t), t)dt + (R(t), t)dW(t) (1.1)

1


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Chapter 1. Introduction 2

Where :

(R(t), t) is drift term of the process(R(t),t) is diffusion term of the process

dW(t) is Brownian Motion with initial condition R(0) > 0.

Interest rate models have two properties:

1) Mean Reversion: The stochastic process is said to mean reverting, if the pro-

cess once deviates from the mean, it is brought back to the stationary mean value

again.Interest rate tends to return to an average level.

2)Positivity : In real condition negative interest rate does not exist. Therefor R(t)

should be non-negative.

Several interest rate models have been developed with different combinations of

drift and diffusion coefficients. Selected interest rate models are given below in the

following Table1.1:

Interest Rate Models Drift Term Diffusion TermVasicek(1977) (- R(t)) Dothan(1978) R(t)

Rendleman-Barter(1980) R(t) R(t)Courtadon(1982) (-R(t)) R(t)

Cox-Ingersoll-Ross(1985) (-R(t))

R(t)Exponential Vasicek R(t)(-a ln(R(t))) R(t)

Table 1.1: Selected Interest-Rate Models

The Cox-Ingersoll-Ross (CIR) model [2] is a significant one-factor model, which

is used to model the dynamics of interest rates. This model is the main focus of the

present work. The CIR interest rate model is given by:

dR(t) = ( R(t))dt +

R(t)dW(t) (1.2)

Here ,and are positive constants.

R(t) is interest rate W(t) is Brownian motion The CIR model is also parameterized

as:

dR(t) =( R(t))dt +

R(t)dW(t) (1.3)

Where :

is reversion rate or drift factor, > 0


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is mean level or long term rate constant, >0

is volatility, >0R(0) is positive with probability 1.

The CIR process has also been used in the Heston Stochastic Volatility Model,

which is given by:

dSt =Stdt +

VtStdW1t (1.4a)

dVt =( Vt)dt + VtdW2t (1.4b)WhereSt are Vt are the price and volatility process respectively and W1t and W

2t are

co-releated Brownian Motion with co-relation parameter . Here it can be observed

that,Vt is nothing but the CIR process. is the risk neutral drift of the asset price.

1.1 Properties of the CIR process:

The following are the properties of the CIR process.

1) If 2 > 2, the CIR process is strictly positive, otherwise non-negetive. Hence,

the CIR interest rate model depicts the actual condition of the market where interest

rate is non negetive.

2)The CIR process is mean reverting in nature. If the process deviates from the

stationary mean level , it is brought back to at the rate . If the process reaches

the zero state, the term multplying dW(t) vanishes and the positive drift the dt inequation (1.3) drives the interest rate back into positive territory.

3)The CIR process has no general explicit solution. However, its mean and variance

can be calculated explicitly

Proof:


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Let us consider the function f(t,x)=etx. Using the Ito-Debolin formula to compute.

The Ito-Debolin formula is as follows:

df(t, R(t)) =ft(t, R(t)dt + fx(t, R(t))dR(t) +1

2fxx(t, R(t))dR(t)dR(t) (1.5)

Therefore,

d(etR(t)) = etR(t)dt + et( R(t))dt + et

R(t)dW(t) (1.6)

d(etR(t)) =etdt + et

R(t)dW(t) (1.7)

Integration of both sides of equation (1.7)

etR(t) =R(0) +

t0

eudu+

t0

eu

R(u)dW(u)

=R(0) +

(et 1) +

t0

eu

R(u)dW(u)

(1.8)

Equivalently for equation (1.3)

etR(t) =R(0) +

t0

etdu+

t0

eu

R(u)dW(u)

=R(0) + (et 1) + t0

eu

R(u)dW(u)

(1.9)

It is known that the expectation of an Ito integral is zero. Therefore

etER(t) =R(0) +

(et 1) (1.10)

ER(t) =etR(0) +

(1 et) (1.11)

Equivalent formulation of expection for equation (1.3) is

ER(t) =etR(0) + (1 et) (1.12)

Now it is interesting to note that limxER(t) = which is its mean level.


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Calculation of the Variance To compute the variance of R(t), setX(t) =etR(t),

for which we have already computed

dX(t) =etdt + et

R(t)dW(t))

=etdt + et

2

X(t)dW(t)

(1.13)

and

EX(t) =R(0) +

(et 1) (1.14)

According to the Ito-Debolin formula (with f(x) =x2, f

(x) = 2x, andf

(x) = 2)

d(X2(t)) = 2X(t)dX(t) + dX(t)dX(t) (1.15)

d(X2(t)) = 2X(t)dX(t) + dX(t)dX(t)

= 2etX(t)dt + 2et

2 X32 (t)dW(t) + 2etX(t)dt

(1.16)

Integeration of equation (1.17) yields

X2(t) =X2(0) + (2+ 2)

t0

euX(u)du+ 2

t0

eu2 X

32(udW(u) (1.17)

taking expections on both side of equation (1.17), using the fact that the expection

of an Ito integeral is zero and the formula already derived for EX(t), we obtained

EX2(t) =X2(0) + (2+ 2)

t0

euEX(u)du

=R2(0) + (2+ 2)

t0

eu(R(0) +

(eu 1))du

=R2(0) +(2+ 2)

(R(0)

)(et 1) +(2+

2)

2

(e2t 1)

(1.18)

Therefore,

ER2(t) =e2tEX2(t)

=e2t

R

2

(0) +

(2+ 2)

(R(0)

)(et

e2t

) +

(2+ 2)

22 (1 e2t

)(1.19)


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Finally, the variance can be calculated as

V ar(R(t)) = ER2(t) (ER(t))2

=e2tR2(0) +(2+ 2)

(R(0)

)(et e2t) + (2+

2)

22

(1 e2t) e2tR2(0) 2

R(0)(et e2t) 2

2(1 et)2

=2

R(0)(et e2t) +

2

22(1 2et + e2t)

(1.20)

Hence, the variance of CIR process will be given by equation(1.20) Similarly for

equation (1.3) the variance of cir process is given as

V ar(R(t)) =2

R(0)(et e2t) +

2

2(1 2et + e2t) (1.21)

1.2 Simulating evolution of single path of the CIR

processThe following section will show the simulation of equation(1.9) for evolution of

one path of the CIR process. Simulation is done in interval [0 1].For the simulation

the interval of simulation [0 1] is divided into 216 parts. For the simulation result

depicted in Figure(1.1), the CIR process parameters are = 1, = 0.8 and = 1.

Here: 2 2=2 0.8 1=0.6

It can be noted that the strict positivity of the solution of the CIR process is

ensured analytically.

For the simulation depicted in Figure(1.2), the CIR process parameters are =

1, = 0.45 and = 1. Here:

2 2=2 0.45 1=-0.1

It can be noted that the non-negative condition of the solution of the CIR process

is ensured analytically.


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Figure 1.1: CIR simulation for strict positive condition

Figure 1.2: CIR simulation for non-negative condition


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1.3 Organization of the report

The rest of the report is organised as follows:

Chapter 2 explains the existing numerical methods to discretize the CIR process.

This chapter also deals with the merits and demerits of the existing methods.

Chapter 3 deals with implementation of the FIS- numerical scheme on the CIR

and the logrithmically transformed CIR process. This chapter also discusses the

motivation of new method requirment. In this chapter, we have developed a new

method which ensures numerical positivity in simulation of the CIR process.

Chapter 4 discusses the development of a new method to discretize the CIR process

and the properties of this new method are discussed.

Chapter 5 concludes the present work by elaborating on how the methods developed

in the present work drastically improves the state of the existing methods of simulating

the interest rate models. Further, the scope of further work is discussed.


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Chapter 2

Related Work

2.1 Introduction

Analytically, the CIR process ensures an analytical non-negative solution, which

makes the CIR interest rate model more in line with the actual behavior of inter-

est rates. This makes the CIR model popular among existing interest rate models.

However, the CIR process is not amenable to a closed form solution. Hence, the

usefulness of the CIR process requires the development of a numerical discretisation

scheme with the view of providing better approximations to the problem of interest-

rate modeling. Existing literature elaborates various numerical methods to simulate

the CIR process. The results of such simulations is required in risk management. Two

issues need to be addressed regarding discretisation schemes, namely the correctness

of scheme and the CPU time to simulate one path of scheme. Error in simulations

lead to wrong decisions which may have adverse effects. Therefore, the correctness

of simulation results is vital. In financial market, huge amount of data is analysed

within short span of time, which demands the simulation to be considerably fast to

meet the market requirement.

In this chapter we explain some of existing discretisation schemes on the CIR

process.

Let T be time span of process simulation where T>0 and N is a positive integer.

The variables t, Rn andT denote TN

, R(tn) andtf t0 respectively, wheret0 < t1 0,where W(t) is the Brownian Motion.

The following paragraphs elaborate on the existing method for discretisation of the

CIR process

2.2 The Euler-Maruyama Method

This is oldest and simplest method. The EulerMaruyama scheme has order= 12of strong convergence. The CIR process discretisation with the EulerMaruyamascheme is given by:

Rn+1=Rn+ ( Rn)t +

RnWn (2.1)

Discretisation (2.1) has been studied in [3,4] and not able to preserve positivity

2.3 The Milstein Method

The Milstein Method makes use of the Itos lemma to increase the accuracy of the

approximation by adding the second order term. This method has order = 1 of

strong convergence. The CIR process discretisation with the The Milstein Method is

given by:

Rn+1 = Rn+ ( Rn)t + RnWn+1

42

(W2n t) (2.2)

The Milstein discretisation (2.2) has been studied in [3,4]. This scheme preserves the

positivity of the CIR model but is not sufficiently accurate.


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Chapter 2. Related Work 11

2.4 The Balanced Implicit Method (BIM)

The Balanced Implicit Method was developed by G.N.Milstein et.al.in[5].The CIR

process discretisation with the The Balanced Implicit Method is given by

Rn+1=Rn+ ( Rn)t +

RnWn+ C(Rn)(Rn Rn+1) (2.3a)C(Rn) =c0(Rn)t + c1(Rn)|Wn| (2.3b)

The functions, c0 and c1, are called control functions [20]. Discretisation (2.3a) and

(2.3b) has been analysed in [3,4,10].It is not meeting requirment in context of con-

vergence and the CPU time to simulate one path.

2.5 The Moment Matched Log-Normal Approxi-

mation

This method was suggested by Andersen and Brotherton-Ratcliffe.The Moment

Matched Log-Normal Approximation of the CIR process is following

Rn+1= (+ (Rn )et)e12 2n+nz (2.4a)

n= ln(1 + 2Rn(1 et)

2(+ (Rn )et)2 ) (2.4b)

In the equation above, z is a gaussian random variable with mean zero and variance

1. The Moment Matched Log-Normal approximation has been analysed in [3]

and it is found that practically, for a small value of , this approximation has no

convergence advantage over the Euler-Maruyama Method, which is straight-forward

and explicit. However this approximation is very good for the Brennan-schwartz

model.The Brennan-schwartz model is given by:

dR= ( R)dt + RdW (2.5)


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2.6 The Logarithmic Transformed CIR Process

The CIR process can be transformed to logrithmic co-ordinates using Itos lemma as

suggested by Andersen and Brotherton-Ratcliffe [19]. The transformed equation has

been given in by Christan Kahl and Peter Jackel in [3], as given below.

d ln R=2( R) 2

2R dt +

R

dW (2.6)

The transformed stochastic differential equation (2.6) can be solved by aid of the

Euler-Maruyama Method and it is also able to preserve the positivity. But it be-

comes unstable for suitable time steps. These instabilities are basically caused by the

divergence of both the drift and the diffusion term near zero.

2.7 The Balanced Milstein method (BMM)

This method was developed by Christan Kahl and Henri Schurz in [4]. The CIR

discretisation with The BMM is given as follows:

Rn+1=Rn+ ( Rn)t + RnWn+ D(Rn)(Rn Rn+1) (2.7a)D(Rn) =d0(Rn)t + d1(Rn)(W

2n t) (2.7b)

The functions d0 and d1 are called control functions. The choice of these control

functions strongly depends upon the structure of stochastic differential equation. The

discretisation (2.7a) (2.7b) was analysed by Christan Kahl and Henri Schurz in [4]

and was found to be the best suitable method for CIR in context of accuracy. It is

also able to preserve positivity. The CIR discretisation with the BMM was furtheranalysed in [3,10] in context of accuracy and CPU time taken to simulate on path

and was found to be promising.

2.8 The New Scheme(News)

This method was developed by D. Ding and C. I. Chao in [10]. The CIR process

discretisation with News is as follows:

Rn+1=et(

Rn+

1

2Wn)

2 +1

( 1

42)(1 et) (2.8)


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News method of the CIR discretisation (2.8) was analysed in [10]. This method took

less time to simulate one path in comparison to BIM and BMM.

2.9 The Pathwise Adapted Linearisation Quadratic

The Pathwise Adapted Linearisation Quadratic numerical integeration scheme was

given by Christan Kahl and Peter Jackel in [3].This scheme is as follows:

Rn+1=Rn+ ((( Rn) + bnRn)t)[1 +(bn 2

Rnt)

4

Rn] (2.9)

where

= 2

4 (2.10)

and Wn=bnt

The method (2.9) was discussed in [3]. This scheme is good for The CIR model

simulation. The Pathwise Adapted Linearisation Quadratic is remarkably effective

for small value of but this scheme is inappropriate for larger value of, as disscused

in [3].

2.10 The Pathwise Adapted Linearisation Quartic

The Pathwise Adapted Linearisation Quartic numerical integeration scheme was

given by Christan Kahl and Peter Jackel in [3]. This scheme is as follows:

Rn+1=Rn+ (((

Rn) + bnRn)t)

[1 +((bn 2Rn)t)

4

Rn+

((Rn(4Rn 3bn) bn))24

Rn

3 t2

+(3bn

2 + R2n(7bn 8

Rn) + 2bn

Rn(bn+

Rn))

192

Rn5 ]

(2.11)

where

= 2

4 (2.12)

and Wn=bnt


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The method in equation (2.11) was discussed in [3] and was able to preserve pos-

itivity. This method is promising in context of error. However, the CPU time tosimulate one path is lager than BMM.

2.11 Simulation of evolution of the CIR process:

In this part we are showing the simulation of one path of the CIR process with some

selected method discussed above and its comparision with the actual simulation which

is done with equation(1.9).The simulation is done in simulation interval [0 1]. For

actual simulation the simulation interval is divided in 216 parts whereas, the schemes

to followes are done with time step size 0.25. For the simulation result depicted in

Figure(2.1), the CIR process parameters are = 1, = 0.0625 and = 0.2, and R(0)

is 1. Here: 2 2=2 0.0625 .04=0.085


ensured analytically.

Figure 2.1: CIR simulation for strict positive condition

For the simulation depicted in Figure(2.2), the CIR process parameters are =

1, = 0.06255 and = 0.8, and R(0)is 1. Here:


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2 2=2 0.0625 0.8=-0.515.

It can be noted that the non-negative condition of the solution of the CIR process

is ensured analytically.

Figure 2.2: CIR simulation for non-negative condition

It is observed from Figure(2.2) that in non-negative condition the numerical posi-

tivity is maintained by each concerned method.


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Chapter 3

Implementation of FIS- on the

CIR process

3.1 Introduction

The Fully Implicit Stochastic (FIS) method was developed by Sk. Safiqueet. al. in [1].The FIS

numerical integeration scheme for stochastic differential

equations, was constructed to handle stiffness by providing numerical dissipation, in

both the drift and the diffusion terms.

In this chapter we describe the FISnumerical scheme, the CIR process discreti-sation with the FISmethod. Results of simulation of the CIR process discretisationwith the FISmethod and its comparision with other existing method with respectto the CPU time taken for the simulation of one path and accuracy. The special

attention of this chapter is numerical positivity. One new method Mixed Method

has been developed, which ensures positive numerical solution of the CIR process in

every condition.

Let t :=[t0, tf] is time on which vector valued stochastic process x(t) evolves.Along a given sample path x(t)is denoted by xt. Initial value, Ito Stochastic differ-

ential equation with multiplicative noise is given as follows:

dx= f(x, t)dt +m

j=1

b(j)(x, t)dW(j)t (3.1)

16


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Chapter 3. Implementation of FIS- on the CIR process 17

Where x Rl with probability 1

f : Rl Rl is deterministic drift coefficient and function in x and t

bj : Rl Rl for j=1,....,m is diffusion coefficient (corresponding to jth compo-nent of the Wiener process) function in x and t

W(j)t is the jth component of the componentwise independent mdimentional vector

Wiener process W at time t

P(x(t0)) =x0) = 1

3.2 FIS numerical schemeThe present section eleborates the Fully Implicit Stochastic method to solve

the initial value stochastic differential equations given in equation (3.1) on a uniform

mesh t0 < t1 < .... < tn < ... < tN=tfof time step size t= tn+1 tn on the simu-

lation interval [t0, tf].Numerically computed value of x(tn) on a given sample path isdenoted as xn.Let

Wn=tn+1tn

dWs=Wn+1 Wn; I(j1, j2)n=tn+1tn

stn

dW(j1)u dW

(j2)s

B(x, t) = {b(1),...,b(m)} : Rl Rll

(i) =

x(i) is the ith component of the gradient operator

L(j) =b(j)T

is a scalar operator.

c(j1,j2)(x, t) =L(j1)b(j2)(x, t) : Rl Rl

fn= f(xn, tn); fn = f(xn, tn); Bn = B(xn, tn); c(j1,j2)n =c(j1,j2)(xn, tn)


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fn=fn+1 fn; Bn = Bn+1 Bn; c(j1,j2)n =c(j1,j2)n+1 c(j1,j2)n

f(xn, tn) =f(xn, tn)h + B(xn, tn)Wn+

mj1=1

mj2=1

c(jn,j2)n(xn, tn)I(j1, j2) (3.2a)

n = fnh +

mj1=1

mj2=1

c(jn,j2)nI(j1, j2) (3.2b)

Then the F IS numerical integeration is given as follows:

xn+1=xn+ fn+

P\n+ (1

2

)an (3.3a)

an+1= 1

n+ (1 1

)an (3.3b)

where

is a user selectable real parameter

(1,

)

is a real user selectable parameter [0,1)

= 2+1

12

= 1(+1)2

a0=h2

ft

+ fx

ft=t0,x=x0 =h2k0 2)

In this section we explain simulation results for strictily positive condition, when

the CIR process solution will not be zero. The time intervals for the simulations has

been taken to be [0 1] and [0 4]. The simulation corresponding to the latter is done

with larger time steps. For the simulation discussed this section, the CIR process

parameters are =1, =0.0625 and = 0.2. Here:


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2 2=2 0.0625 0.2 0.2=0.085


ensured analytically

Simulation for Interval [0 1]

In this section the FIS-(N) method parameter, and , are 0.3 and 1.1 respec-

tively. In order to simulate FIS-(log) method, the method parameters and are

0.5 and 1.1 respectively.

Figure 3.3: CPU time comparison of simulation interval [0 1] for the CIR parameter

= 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3 and 1.1respectively, FIS-(log) parameter and , are 0.5 and 1.1 respectively

The left sub-figure in Figure(3.3) shows the comparison of the average time taken

by the CPU for one path of the simulation as performed using the selected schemes

mentioned in chapter 2 and the FIS-method. It can be seen that the use of the FIS-

method after the logarithmic trasnformation of the CIR method consumes more

CPU time per path, throughout the interval of the simulation, at all the time-steps

taken. Further, it is evident from the left figure that, the difference between the

CPU time taken for this method and the remaining ones, is of the order 103. In


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Figure 3.4: Error comparison of simulation interval [0 1] for the CIR parameter= 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3 and 1.1

respectively, FIS-(log) parameter and , are 0.5 and 1.1 respectively

order to study the CPU times for the other methods, the figure has been magnified

as seen on the right sub-figure in Figure(3.3). It is observed that, the CPU time for

the Pathwise Adapted Linearization Quartic method is more than the rest of the

methods throughout the interval for all time-steps. The remaining methods are at

par with each other, with the present scheme (FIS-) being between the BMM and

the Pathwise Adapted Linearization Quadratic schemes at time-step of about 0.125 .

With increase in the time-step, it is seen that the present scheme converges with the

above two schemes to a satisfactory level.

The comparison of accuracy amongst the methods of concern is shown in Fig-

ure(3.4). The FIS-(log) scheme shows accuracy at par with the other schemes. It

is however interesting to note, that the FIS-(N) method is as accurate as the Path-

wise Adapted Linearization Quartic method and is hence the most accurate of all the

methods shown. Further, as it was seen in the CPU-time comparison, the FIS- (N)

method also took significantly less time in comparison to the Pathwise Adapted Lin-

earization Quartic method. This clearly shows that the FIS-(N) method is, on one

hand, time-efficient, and on the other hand, also significantly accurate.


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In this section the FIS-(N) method parameter, and , are 0.3 and 1.1 respec-

tively. In order to simulate FIS-(log) method, the method parameters and are

0.9 and 9.1 respectively.


= 1, = 0.0625 and = 0.2, the FIS-(N) parameter and , are 0.3 and 1.1

respectively, the FIS-(log) parameter and , are 0.9 and 9.1 respectively


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Figure 3.6: Error comparison of simulation interval [0 4] for the CIR parameter

= 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3 and 1.1




mentioned in chapter 2 and the FIS-(N) method. It can be seen that the use of

the FIS-method after the logarithmic trasnformation of the CIR method consumes

more CPU time per path, throughout the interval of the simulation, at all the time-

steps taken. Further, it is clear from the left figure that, the difference between the

CPU time taken for this method and the remaining ones, is of the order 103. In

order to study the CPU times for the other methods, the figure has been enlarged

as seen on the right sub-figure in Figure(3.5). It is observed that, the CPU time

for the Pathwise Adapted Linearization Quartic method is more than the rest of

the methods throughout the interval for all time-steps. The averge time consumed

to simulate one path of The FIS-(N) method is less then thePathwise Adapted

Linearization Quartic ,but more than rest methods at time step size 1.However, the

FIS-(N) method is consuming approximately equal time as the rest of methods for

time step size 2 and 4.


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ure(3.6).The The FIS- scheme used on logarithmically transformed coordinatesshows accuracy at par with the other schemes.The Pathwise Adapted Lineariza-

tion Quadratic error is less than the Pathwise Adapted Linearization Quartic and

the logrithmically transformed CIR with the FIS-method ,but more than rest meth-

ods with time step 1 ,there after it is diverging for time step 2 and 4.The Pathwise

Adapted Linearization Quartic is also not at par with rest of methods. It is however

interesting to note, that the FIS-(N) method is more accurate than the BMM there-

fore it is the most accurate of all the methods shown. It is evident Figure(3.5) that the

CPU time consumed to simulate one path is comparable with the BMM.Therefore,the FIS-(N) method is time efficient as well as significantly accurate.

3.4.2 The CIR process is nonnegative (2 2)In this section we explain simulation results for nonnegative condition, when the

CIR process solution can reach the state zero analytically. The time intervals for

the simulations has been taken to be [0 1] and [0 4]. In this section of work, the

CIR process is simulated with some existing method ,disscused in chapter 2 and the

FIS-(N) method.


The simulation of this section is done in two parts. First part analyses effect on

accuracy of the simulation results by varying the parameters of the FIS- method

namely and . Whereas, the second part discusses the effect on accuracy by varia-

tion in parameters of the CIR process namely , and . Furthermore, by changing

the parameters of the CIR method the behaviour of other methods has been depicted.

Simulation result by varying FIS-(N) parameters The simulation is done

for the CIR process parameters are = 1, = 0.0625 and = 0.5. Here:

2 2=2 0.0625 0.52=-0.125.It can be noted that the non negativity of the solution of the CIR process is ensured

analytically.

In this part of work the FIS-(N) method parameters , namely and, are 0.3

and 1.1 for the simulation results shown in Figure(3.8), respectvely , whereas these


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values are 0.3 and 1.6 for Figure(3.9).The FIS-(log) parameters are 0.3 and 4.1 for

Figure(3.8) and 0.3 and 4.6 for Figure(3.9)

Figure 3.7: CPU time comparison of simulation interval [0 1] for the CIR parameter= 1, = 0.0625 and = 0.5, FIS-(N) parameter and , are 0.3 and 1.1




mentioned in chapter 2 and the FIS-(N) method. It can be seen that the use of

the FIS-method after the logarithmic trasnformation of the CIR method consumes

more CPU time per path, throughout the interval of the simulation, at all the time-

steps taken. Further, it is evident from the left figure that, the difference between

the CPU time taken for this method and the remaining ones, is of the order 103. In

order to study the CPU times for the other methods, the figure has been magnified as

seen on the right sub-figure in Figure(3.7). It is observed that, the CPU time for the

the FIS-(N) method is more than the rest of the methods throughout the interval

for all time-steps. The averge time consumed to simulate one path of the Pathwise

Adapted Linearization Quartic method is less then the FIS-(N) ,but more than rest

methods throughout the interval for all time-steps. Least average time is consumed


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respectively, FIS-(log) parameter and , are 0.3 and 4.6respectively

by the BMM to simulate one path of the CIR process.Rest of the methods are in

between the FIS-(N) method and the BMM method.


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Figures Figure(3.8)and Figure(3.9) show the comparison of the error through vari-

ations in the parameter and for the FIS-(N) method as well as the FIS-(log). It is to be noted that the CIR parameters have been kept unvaried. It can be

observed that, for the second parametric variation Figure(3.8) the the FIS-(log)

shows less error in comparison to the other cases. The FIS- method, on the other

hand, is between with the Pathwise Adapted Linearization Quadratic method and

the BMM method. Interestingly, the second variation of and Figure(3.9) shows

adverse effect on the accuracy of the logarithmically transformed case, while showing

significant improvement in the FIS-(N) case.In Figure(3.9) ,it is however interesting

to note that the FIS-(N) method is as accurate as the Pathwise Adapted Lineariza-tion Quartic method.The News diverges with the increment of time step however ,it

is more accurate than the FIS-(log) but less accurate than rest of the methods.the

Pathwise Adapted Linearization Quadratic depicts more accuracy than the News and

the FIS-(log) but, less accurate than rest of methods.

Simulation result by varying the FIS-(N) parameters and CIR parametrs

The simulation is done for the CIR process parameters are = 1, = 0.0625 and

= 0.8. Here:2 2=2 0.0625 0.82=-0.515It can be noted that the non negetivity of the solution of the CIR process is ensured

analytically. In this part of work the FIS-(N) method parameters , namely and ,

are 0.3 and 2 respectively and same parameters for the FIS-(log) are 0.6 and 5.



mentioned in chapter 2 and the FIS-(N) method. It is evident from Figure(3.10),that the use of the FIS- method after the logarithmic trasnformation of the CIR

method consumes more CPU time per path, throughout the interval of the simulation,

at all the time-steps taken. Further, it is observed from the left figure that, the

difference between the CPU time taken for this method and the remaining ones, is of

the order 103. In order to study the CPU times for the other methods, the figure has

been enlarged as depicted on the right sub-figure in Figure(3.10). The FIS-(N) takes

less average time to simulate one path of the CIR process than the Pathwise Adapted

Linearization Quartic method but more than the rest of the methods throughout theinterval for all time-steps. Least average time is consumed by the News to simulate


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Figure 3.10: CPU time comparison of simulation interval [0 1] for the CIRparameter = 1, = 0.0625 and = 0.8, FIS-(N) parameter and , are 0.3 and

2 respectively, FIS-(log) parameter and , are 0.6 and 5 respectively

Figure 3.11: Error comparison of simulation interval [0 1] for the CIR parameter= 1, = 0.0625 and = 0.8, FIS-(N) parameter and , are 0.3 and 2

respectively, FIS-(log) parameter and , are 0.6 and 5 respectively


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one path of the CIR process.Rest of the methods are in between the News method

and the Pathwise Adapted Linearization Quartic method.

The accuracy comparison amongst the concerned schemes is shown in Figure(3.11).

The FIS- scheme used on logarithmically transformed coordinates having rather

more error with respect to other schemes.The Pathwise Adapted Linearization Quadratic

error is less than the FIS- method and the BMM,but more than the News for

timestep size 0.0625 and 0.125.However, thePathwise Adapted Linearization Quadratic

has minimum error for timestep size 0.25 and afterwards it is coming between the

News and the BMM. It is evident from Figure(3.11), that for time step size 0.5 and 1the FIS-method is having least error among all schemes and followd by the BMM.It

is interesting to note that in given condition of the FIS-(log) parameters and the

CIR process parameters for timestep size 0.25 and greater its error is not increas-

ing.Therefore, the FIS-(N) method is time efficient as well as significantly accurate

for larger time step as well as at par with other schemes in context of time consump-

tion to simulate one path.The Pathwise Adapted Linearization Quartic is diverging

in this given condition and it is also mentioned by Christian Kahl and Peter Jackel

in [3].The following table depicts the timestep size and coresponding error and theaverage time taken to simulate one path.

Timestep size Error The average CPU time0.0625 2.076418101 1.4017131040.1250 1.273897103 6.0441741050.2500 2.163967102 3.5205911050.5000 5.290098102 2.3030371051.0000 6.806635102 1.734766105

Table 3.1: Simulation results for interval [0 1] of the CIR parameter= 1, = 0.0625 and = 0.8, FIS-(N) parameter and , are 0.3 and 2

respectively, FIS-(log) parameter and , are 0.6 and 5 respectively by PathwiseAdapted Linearization Quartic method

It is clear from Table (3.1) that, the Pathwise Adapted Linearization Quartic is

diverging for time step size 0.0625, 0.1250, 0.2500, and 0.5000.However, it is converg-

ing for time step 1. In order to get more accuracy, smaller time steps are required.

However, the Pathwise Adapted Linearization Quartic does not meet this require-

ment .


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The FIS-(N) method parameter,and , are 0.3 and 2 respectively . Whereas, for

the FIS-(log) method, the method parameters and are 0.9 and 10 respectively.

Figure 3.12: CPU time comparison of simulation interval [0 4] for the CIRparameter = 1, = 0.0625 and = 0.5, FIS-(N) parameter and , are 0.3 and




mentioned in chapter 2 and the FIS-(N) method. From left subfigure, it can be seen

that the use of the FIS- method after the logarithmic trasnformation of the CIR

method consumes more CPU time per path, throughout the interval of the simulation,

at all the time-steps taken. Further, it is clear from the left figure that, the difference

between the CPU time taken for this method and the remaining ones, is of the order

103. It is evident from concerned figure that, the BMM has taken least average

time to simulate one path, at at all time steps. It can be observed easily that, at

time step size 1 the FIS-(N) method has taken largest time to simulate one path

but, there is significant decrese for time step size 2 and 4. It is observed that, theCPU time for the Pathwise Adapted Linearization Quartic method is more than


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Figure 3.13: Error comparison of simulation interval [0 4] for the CIR parameter= 1, = 0.0625 and = 0.5, FIS-(N) parameter and , are 0.3 and 2

respectively, FIS-(log) parameter and , are 0.9 and 10 respectively

the rest of the methods throughout the interval for all time-steps. The averge time

consumed to simulate one path of the FIS-(N) method is less then thePathwise

Adapted Linearization Quartic at all timestep sizes but 1.


ure(3.13). At time step size 1 and 2 the Pathwise Adapted Linearization Quartic

has least error but at time step size 4 it is highest. The FIS-scheme used on logarith-

mically transformed coordinates shows accuracy at par with the other schemes and

its error is decreasing with the increment in time step size. The error of News,the

Pathwise Adapted Linearization Quartic ,the BMM and the FIS- is positively

coreleated with time step size.The FIS-(N) accuracy is similar to the BMM at time

step size 1 and 2 but, it is least at time step size 4.

3.5 Problem with the Existing Schemes

In the sections above, we have analysed the accuracy and the average time con-

sumed by selected schemes to simulate one path of the CIR process. It has been


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observed that, the FIS-(N) is best in accuracy in most of the conditions of the pa-

rameters of the CIR process; or atleast, it is at par with the other schemes. Usually,it was observed to perform best with larger time step such (0.25 and above). On the

other hand, the FIS-(log) was observed to be less accurate and more time-consuming

with respect to the other scheme while simulating one path in maximum conditions.

This section discusses the effect of decrement of value of 2 2 on the preserva-tion of positivity. The following simulation has been done with the CIR parameters,

namely , and as 1, 0.0625 and 1.2 respectively. The FIS-(N) parameters and

has been taken as 0.3 and 2, and that for the FIS-(log) are 0.9 and 12 respectively.

The time step size of simulation is 0.0625. Here:2 2=2 0.0625 1.22=-1.315

Figure 3.14: Interest rate evalution in simulation interval [0 1] for the CIR

parameter = 1, = 0.0625 and = 1.2, FIS-(N) parameter and , are 0.3 and2 respectively, FIS-(log) parameter and , are 0.9 and 12 respectively

It is evident from Figure(3.14), that in given the CIR parameters condition, no

method apart from the FIS-(log) method is able to numerically preserve the posi-

tivity. It is observed in previous discussions that the FIS-(log) is less accurate than

other methods. However, it is mentioned in [3] that the logrithmicaly transformed

CIR process can be simulated using the Euler-Maruyama scheme. However,it be-comes unstable because of divergence of both the drift and the diffusion terms near


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the zero state of interest rate. It is interesting to note that, the FIS- sceme is able

to cope with the unstability near the zero state of the numerically simulated interestrate.

3.6 Mixed Method

From the discussions above, it is clear that the FIS-(N) scheme provides maximum

accuracy and time efficency in maximum number of conditions and the FIS-(log)

ensures positivity in every condition but not the efficency in context of error and time

cunsiomption to simulate one path. In this section we will develop a new method

which is able to preserve positivity in any condition as well as most accurate and

time efficient amongst all the existing methods. The newly developed method will be

referred to as the Mixed Method. The algorithm of mixed method is as follows:

3.6.1 Mixed Method Algorithm

Let Rn is the simulated value of interest rate at time tn

Step1 : Calculate Rn using the FIS-(N) scheme

Step2 : IfRn 0 exit

Else

Step3 :CalculateRn using the FIS-(log) scheme

The following simulation has been done with the CIR parameters, namely , and

as 1, 0.0625 and 1.2 respectively as in the previous section (3.5) for one-to-one

comparision. The FIS-(N) parameters and have been taken as 0.3 and 2 and

that for the FIS-(log) are 0.9 and 12 respectively. The time step size of simulation

is 0.0625

Results Figure(3.15) show that, the Mixed Method progresses through the time

interval with significant similarity to all the other existing methods. Interestingly,

the Mixed Method is capable of preserving the numerical positivity even in non-

negative conditions of the CIR process for any value of the CIR process parameters.


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Figure 3.15: Interest rate evalution in simulation interval [0 1] for the CIRparameter = 1, = 0.0625 and = 1.2, FIS-(N) parameter and , are 0.3 and


This observation can be clearly verified in Figure(3.15) between time-steps 0.5 to

0.812500 .

The following simulation has been done with the CIR parameters, namely , and

as 1, 0.0625 and 1.5 respectively here section. The FIS-(N) parameters and

has been taken as 0.3 and 2 and similar parameters for the FIS-(log) are 0.9 and 12

respectively. The time step size of simulation is 0.0625.Here:

2 2=2 0.0625 1.52=-2.125.

It is evident from Figure(3.16) that even in the new condition, where the value of

2 2 is further decreased the Mixed Method is preserves the numerical positivityof the CIR process.


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Figure 3.16: Interest rate evalution in simulation interval [0 1] for the CIRparameter = 1, = 0.0625 and = 1.5, FIS-(N) parameter and , are 0.3 and


3.7 Summary

In this chapter we implemented the FIS- scheme on the CIR process and logrit-

mically transformed the CIR process. The FIS-(N) is approprite in maximum con-

ditions and the FIS-(log) is able to preserve positivity. With the combination of

both we developed a new scheme to simulate the CIR process namely the Mixed

Method. The Mixed method is able to preserve positivity in any condition of the

CIR process parameters keeping the accuracy and the efficiency intact. This was not

available in any of the previously existing schemes.


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Chapter 4

New Method

4.1 Introduction

In this chapter we are developing a new method namely New Method.This

method is based on Splitting-step algorithm discussed by Esteben Moro and Henri

Schurz in [9].

4.2 Splitting-step algorithm:

The Splitting-step algorithm is as followes. Let the stochastic differential equation

which is to be integrated is of the following form:

dX(t) = [a(X(t), t) + b(X(t), t)]dt + (X(t), t)dW(t) (4.1)

Where a(X(t),t), b(X(t),t) and (X(t), t) are function of X(t) and t and W(t) is

Brownian motion.It can be decomposed into two following equations:

dX1(t) =b(X(t), t)dt + (X(t), t)dW(t) (4.2)

dX2(t) =a(X(t), t)dt (4.3)

It is assumed while splitting is done that ,exact solution of X1 is known or the

conditional probability P[X1(t)/ X1(0)] is known.Thus, the solution of equition(4.1)

can be approximated by a stochastic process Yt along time intervals [t,t+t] usingthe following algorithm. Which is known as splitting-step algorithm t

39


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Chapter 4. New Method 40

Step1: Knowing the value ofYt we obtain an intermediate value Yt which is obtained

through the exact integration of (4.2) and Yt = X1(t + t) and with initial conditionX1(t)=Yt.

Step 2: Then Yt is used as the initial condition for equation (4.3) which is now

integrated using any converging deterministic numerical algorithm to get X2. Then

Yt+t = X2(t + t).

4.3 Development of New Method:

In this section of the present work , we develop a new method to simulate the CIR

method. The New Method development is based on the Splitting-step algorithm

mentioned above.The logarithmic transformed CIR process is given as follows:

d ln Rt=2( Rt) 2

2Rtdt +

Rt

dWt (4.4)

Equation (4.4) is discussed in chapter 2 and in [3]. To make easy in understanding

set rt=ln Rt then equation is transformed as follows:

drt =2( ert) 2

2ertdt +

ert

dWt (4.5)

The New Method method is to solve the stochastic differential equations given in

equation (4.5) on a uniform mesh t0 < t1 < .... < tn < ... < tN=tfof time step size

t = tn+1 tn on the simulation interval [t0, tf]. Equation (4.5) can be written asfollows:

drt = (2( ert)

2ert

2

4ert)dt + (

2

4ertdt +

ert

dWt) (4.6)

drt = dpt+ dqt (4.7)

Where

dpt = (2( ert)

2ert

2

4ert)dt (4.8)

and

dqt= ( 2

4ertdt +

ert

dWt) (4.9)


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It is interesting to note that, there is a closed form solution of equation (4.9) , and it

is given by as follows:

qt = ln(

2Wt+ e

rn2 )2, t [tntn+1] (4.10)

Where qt is approximation ofqt and rn is initial condition in time interval[tntn+1] It

is intresting to note that, equation (4.8) is also having closed form solution. Let rt

is an approximation ofrt ,then using the Splitting-step algorithm the rt is calculated

as follows

rt = ln((4 2) et(4 4eqt 2)

4 ) t [tntn+1] (4.11)

Again setting ln Rn+1=rn+1 we get following discretisation

Rn+1 =(4 2) et(4 4eqt 2)

4 (4.12)

4.3.1 Simulation of evolution of the CIR process:

In this part we are showing evolution of equation (4.12) through time along with itscomparison with actual simulation, selected methods of chapter 2 and the FIS-(N).

The actual simulation is done with equation(1.9). The actual simulation is done by

partitioning the simulation interval [0 1] in 216 parts whereas, the schemes to followes

are done with time step size 0.25. The initial value of simulation is 0.0625.

The CIR process is strictly positive (2 > 2):

This simulation is done for the CIR parameter namely, =1, = 0.0625 and =.2.

Whereas, the FIS-(N) parameters and are 0.3 and 1.1 respectevely.Here:

2 2=2 0.0625 0.2 0.2=0.085It can be noted that the strict positivity of the solution of the CIR process is ensured

analytically


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Figure 4.1: Comparison of evolution of one pathe of the CIR process in strictly

positive condition with different methods

It can be observed from Figure(4.1) that the BMM is moving apart from other

methods.

The CIR process is nonnegative (2 2):

This simulation is done for the CIR parameter namely, =1, = 0.0625 and =.8.

Whereas, the FIS-(N) parameters and are 0.3 and 2 respectevely.Here:

2 2=2 0.0625 0.8 0.8=-0.515It can be noted that the non-negativity of the solution of the CIR process is ensured

analytically


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Figure 4.2: Comparison of evolution of one pathe of the CIR process in

non-negative condition with different methods

In both condition the New Method is able to preserve the numerical positivity .

4.4 Numerical Results:

This section eleborates the simulation results to compare the results obtained fromselected existing methods discussed in chapter 2, the CIR process discretisation with

the FIS(N) method and the New Method. The comparison thus done in the present

work is in terms of the error and the CPU time taken to simulate one path of the

CIR process.

The strong convergence of different the CIR process discretisation schemes are

measured byL2 norm of the difference between the simulated terminal value, and the


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terminal value of the reference calculation, averaged over all M path, i.e.

error=

1M

Mi=1

(Rnstepi (T) Rreferencei (T))2 (4.13)

Comparison between the average CPU time per path for different schemes, is calcu-

lated using the following expression.

average CPU time per path =

Mi=1CPU time to simulate path i

M (4.14)

The total number of paths (M) considered for the present simulation are 35000. The

reference terminal simulation has been done with discretisation (give here chapter

one reference).

For simulation time interval [0 1], the reference simulation is done with equidistant

time step t= 1216

and, the simulations to follow, are done with equidistant time step

t { 129

, 128

, 127

, 126

, 125

, 124

, 123

, 122

, 12

, 1}. Whereas,for simulation time interval [0 4], thereference simulation is done with equidistant time step t= 1

2

18 and, the simulations

to follow, are done with equidistant time step t {1, 2, 4}. The BMM controlfunctions ared0=and d1=0 as in equation(2.7). In each simulation the initial value

of interest rate is 0.0625.

4.5 The CIR process is strictly positive (2 > 2):

In this section we explain simulation results for strictily positive condition, when the

CIR process solution will not reach zero state. The time intervals for the simulationshas been taken to be [0 1] and [0 4].The simulation corresponding to the latter is done

with larger time steps. In this section of work, the CIR process is simulated with some

selected existing method ,disscused in chapter 2, the FIS-(N) method and the New

Method. For the simulation discussed this section, the CIR process parameters are

= 1, = 0.0625 and = 0.2. Here:

2 2=2 0.0625 0.2 0.2=0.085It can be noted that the strict positivity of the solution of the CIR process is ensured

analytically


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4.5.1 Simulation for Interval [0 1]:

In this part of work the simulation is done for simulation interval [0 1].

Timestep size PALQar FIS (N) PALQad BMM New Method

0.001953 4.665345 5.226097 8.817580 4.587186 4.189826

e-04 e-05 e-05 e-05 e-05

0.003906 2.322434 2.436683 4.501594 2.309400 2.108031

e-04 e-05 e-05 e-05 e-05

0.007812 1.174236 1.337740 2.366223 1.286749 1.170014

e-04 e-05 e-05 e-05 e-05

0.015625 5.995517 7.801057 1.296949 7.525914 7.032829

e-05 e-06 e-05 e-06 e-06

0.031250 3.120174 5.036057 7.626771 4.854771 4.727000

e-05 e-06 e-06 e-06 e-06

0.062500 1.701846 3.825286 5.222314 3.751571 3.728457

e-05 e-06 e-06 e-06 e-06

0.125000 9.801486 3.104571 3.698743 2.979629 3.051514

e-06 e-06 e-06 e-06 e-06

0.250000 6.074229 2.662229 2.994000 2.574029 2.669571

e-06 e-06 e-06 e-06 e-06

0.500000 4.319086 2.467229 2.659486 2.494086 2.576629

e-06 e-06 e-06 e-06 e-06

1.000000 3.320943 2.367429 2.416743 2.265943 2.440057

e-06 e-06 e-06 e-06 e-06

Table 4.1: CPU time comparison of simulation interval [0 1] for the CIR parameter= 1, = 0.0625 and = 0.2, FIS-(N) parameter and , are 0.3 and 1.1

respectively

In tables, PALQar and PALQad are the Pathwise Adapted Linearization Quar-

tic and the Pathwise Adapted Linearization Quadratic respectively.

The Table(4.1) shows the comparison of the average time taken by the CPU for one

path of the simulation as performed using the selected schemes mentioned in chapter 2

and the FIS-(N) method ,and the New Method. It is evident from Table(4.1) that,


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the CPU time for the Pathwise Adapted Linearization Quartic method is more

than the rest of the methods throughout the interval for all time-steps. It is observedthat, the CPU time for the Pathwise Adapted Linearization Quadratic method is

more than the rest of the methods throughout the interval for all time-steps, but less

than the Pathwise Adapted Linearization Quartic. This sequence is followed by

the FIS-(N) method. From time step size 0.001953 to 0.0625 the New Method has

taken least average CPU time to simulate one path of the CIR process. However, for

time step size from 0.125 to 1 the BMM has taken least average CPU time to simulate

one path. Hence for small time step the New method is very efficient in context of

average CPU time to simulate one path.



respectively


ure(4.3). It is clear from figure, that error of each method co-related is positively


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concerned with time. It is evident from figure that upto time step size 0.007812 the

New Method is at par in context of error but for larger time step its error is increasingmore with respect to other methods of discussion, hence it is least accurate than all

method of concern. The Pathwise Adapted Linearization Quartic has least error

amongst all the methods throughout the interval for all time-steps. It is interesting to

note that,the FIS- (N) scheme is as accurate as the Pathwise Adapted Lineariza-

tion Quartic throughout the interval for all time-steps. The Pathwise Adapted

Linearization Quadratic is more accurate than the BMM and the New Method upto

time step size 0.5 but becomes less accurate than the BMM at time step size 1.


In this part of work the simulation results are from simulation time interval [0 4]


1 1.263256 1.296617 5.174686 4.462029 4.927971

e-04 e-05 e-06 e-06 e-06

2 5.087371 2.587371 2.591800 2.410914 2.552429

e-05 e-06 e-06 e-06 e-06

4 3.023300 2.409314 2.390514 2.277029 2.499514

e-05 e-06 e-06 e-06 e-06

Table 4.2: CPU time comparison of simulation interval [0 4] for the CIR parameter


respectively

It is observed from Table(4.2) that, the BMM is least average time consuming to

simulate one path throughout the interval for all time-steps amongst all the concerned

methods. It is evident from Table(4.2) that, for time step size 1 and 2, the New

Method is less average time consuming than the Pathwise Adapted Linearization

Quadratic but more than the BMM. The Pathwise Adapted Linearization Quartic

is taking highest the average time to simulate on path amongst all throughout the

interval for all time-steps. The FIS-(N) shows mixed effects.


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respectively

The Figure(4.4) depicts accuracy comparision amongst the concerned methods.

It is clear from concerned figure that, the FIS-(N) and the BMM are most accurate

and show equal accuracy. The New Method is less accurate than the BMM and the

FIS-(N) but more accurate than the Pathwise Adapted Linearization Quadraticfor all time step size.

4.6 The CIR process is nonnegative (2 2):In this section we explain simulation results for non-negative condition, when the

CIR process solution can reach the state zero analytically. The time intervals for the

simulations has been taken to be [0 1] and [0 4]. In this section of work, the CIR

process is simulated with some selected existing method ,disscused in chapter 2, theFIS-(N) method and the New Method.


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. In this section, simulation is done for the CIR process parameters are = 1, =

0.0625 and = 0.8. Here:

2 2=2 0.0625 0.82=-0.515It can be noted that the non negetivity of the solution of the CIR process is ensured

analytically. In this part of work the FIS-(N) method parameters , namely and ,

are 0.3 and 2 respectively


0.001953 3.033672 4.541425 4.005685 3.783377 2.880292e-03 e-04 e-04 e-04 e-04

0.003906 1.510831 2.243651 2.006872 1.901531 1.448534

e-03 e-05 e-04 e-04 e-04

0.007812 7.603546 1.160553 1.034111 9.800189 7.569097

e-04 e-04 e-04 e-05 e-05

0.015625 3.857018 6.188934 5.478331 5.195171 4.099163

e-04 e-05 e-05 e-05 e-05

0.031250 1.981492 3.464974 3.046966 2.888306 2.364769

e-04 e-05 e-05 e-05 e-05

0.062500 1.046358 2.135726 1.835020 1.738589 1.511831

e-04 e-05 e-05 e-05 e-05

0.125000 5.769806 1.454909 1.216217 1.163566 1.073046

e-05 e-05 e-05 e-05 e-05

0.250000 3.421420 1.100869 8.993914 8.692000 8.474686

e-05 e-05 e-06 e-06 e-06

0.500000 2.255977 9.290400 7.451971 7.258000 7.374829

e-05 e-06 e-06 e-06 e-06

1.000000 1.685334 9.210657 6.851857 6.698629 6.991143

e-05 e-06 e-06 e-06 e-06


= 1, = 0.0625 and = 0.8, FIS-(N) parameter and , are 0.3 and 2

respectively


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The Table(4.3) shows the comparison of the average time taken by CPU for one path

of simulation using the selected schemes mentioned in chapter 2, the FIS- (N) and theNew Method. It can be seen that, the New Method consumes least average time per

path to simulate the CIR process from time step size 0.001913 to 0.25. However, it is

taking more average CPU time per path for time step size 0.5 and 1 than the BMM but

less than other schemes of concern. The Pathwise Adapted Linearization Quartic

is taking the highest average CPU time to simulate on path amongst all methods

throughout the interval for all time-steps.The FIS-(N) consumes less average time to

simulate per path than the Pathwise Adapted Linearization Quartic but more than

the rest of methods,throughout the interval for all time-steps. The BMM consumesthe least average CPU time amongst rest of the methodes for time step size 0.5 and

1.



respectively


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The accuracy comparision amongst the concerned methods is shown in Figure(4.5).

It is observed from figure Figure(4.5) that, the New Method is most accurate amongstremaining methods upto time step size 0.25, after that it is least accurate. It is

interesting to note that, in one hand the FIS-(N) is least accurate upto time step

size 0.25, on the other hand, it is most accurate for time step size 0.5 and 1.it is

evident from concerned figure that, The BMM is second most accurate at time step

size 1. The Pathwise Adapted Linearization Quadratic is at par with others in context

of error upto time step size 0.5 but, it is second to the New Method at time step size

1. The Pathwise Adapted Linearization Quartic has not converged as discussed in

chapter 3.



1 1.235846 1.271706 5.180257 4.441686 4.815829

e-05 e-05 e-06 e-06 e-06

2 4.096486 2.631029 2.657314 2.396057 2.503571

e-06 e-06 e-06 e-06 e-06

4 3.149171 2.417429 2.437343 2.408886 2.435943

e-06 e-06 e-06 e-06 e-06



respectively

The comparison of the average CPU time to simulate one path of the CIR processis shown in Table(4.4). It is evident from Table(4.4) that, in one hand the BMM

takes least average CPU time to simulate one path of the CIR process, on the other

hand the Pathwise Adapted Linearization Quartic takes highest average time to

simulate the same. Rest of the methods are between the BMM and the Pathwise

Adapted Linearization Quartic in context of average time to simulate one path of the

CIR process. The FIS-(N) consumes more average CPU time than the Pathwise

Adapted Linearization Quadratic at time step size 1 but less for others. The New

method is at second position in context of least the average CPU time to simulate

one path of the CIR process.


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respectively

The Figure(4.6) depicts accuracy comparision amongst the concerned methods. It

is evident from Figure(4.6)that, the New Method is more accurate than the Pathwise

Adapted Linearization Quadratic. However, it is less accurate than rest of the meth-

ods throughout the interval for all time-steps. The Pathwise Adapted LinearizationQuartic is most accurate for time step size 1 and 2 but it is diverging at time step

size 4. The Pathwise Adapted Linearization Quadratic is least accurate amongst

all the concerned methods throughout the interval for all time-steps.


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4.7 Order of convergence comparison

In this section of work compare the the convergence rate of methods of concerned by

estimating the in following equation:

e=

1M

Mi=1

(Rnstepi (T) Rreferencei (T))2 Ct (4.15)

Taking log on both side of equation((4.15)) we get

log(e) log(C) + log(t) (4.16)

Let is least squre estimation of, which is calculate using MATLAB functionpolyfit.

Table(4.5) shows the estimated value offor the CIR parameters=1,=0.5 keeping

constant while varying. Whereas, Table(4.6) shows the estimated value offor the

CIR parameters =1, =0.065 keeping constant while varying

Scheme = 0.1 = 0.2 = 0.3

BMM 0.775413 0.893105 0.924123PALQad 0.821518 0.919095 0.939535

PALQar 0.938893 1.230824 1.484425

FIS (N) 0.771027 0.877806 0.919602

New Method 0.846300 0.913514 0.937681

Table 4.5: Comparison of order of convergence() for the CIR process parameters

=1, =0.5 and varying

The comparison of estimated order of convergence () amongst the methods of

concern is shown in Table(4.5). It is done by keeping the cir process parameters=1

and =0.5 constant while varying . It is evident from concerned table that, the

Pathwise Adapted Linearization Quartic shows highest order of convergence. The

Pathwise Adapted Linearization Quadratic convergence order is second highest for

0.2 and 0.3 but less than New Method for value 0.1. The BMM has value less

than the Pathwise Adapted Linearization Quadratic. The New Methods value is

more than the FIS (N) and the BMM.


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Scheme = 0.4 = 0.5 = 0.6

BMM 0.770434 0.688399 0.628421PALQad 0.837337 0.757582 0.695612

PALQar 0.777813 0.650235 0.731274

FIS (N) 0.762968 0.680018 0.636744

New Method 0.852433 0.811057 0.752117

Table 4.6: Comparison of order of convergence for the CIR process parameters

=1, =0.0625 and varying

The comparison of estimated order of convergence() amongst the methods of

concern is shown in Table(4.6). It is done by keeping the cir process parameters

=1 and=0.065 constant while varying . It can be observed from Table(4.6) that,

the New Method has highest order of convergence followed by the Pathwise Adapted

Linearization Quadratic. The BMM convergence order is greater than the FIS (N)

at 0.4 and 0.5 but less than at value 0.6.

4.8 Summary:

From above discussion, it is observed that the New Method behaves in a mixed way.

It is time efficient in many condition and in some it is efficient in context of error.

The mean squre order of convergence of the New Method is very good in comparison

of methods concerned


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Chapter 5

Conclusion

In this chapter we conclude our work highlighting its potential and the results

obtained. This is followed by a brief overview of future scope and challenges ahead.

Conclusions

We have shown that, the FIS- method on the CIR process is time effecient as

well as accurate than all methods in variying parameters of the CIR process. Hence,

it is useful in market condition where the accuracy and time efficiency is the most

important parameter. The newly developed the Mixed Method is able to preserve

numerical positivity at conditions where all the existing methods failed to do the

same.

The New Method is time efficient and as accurate as the existing methods for

small time step size of simulation. Since the predictions in the real market require the

time-efficiency of interest rate model simulations, this method fairs well in comparison

to all the existing methods in this aspect. The mean squre order of convergence of

the New Method is more in comparison of most of existing method.

Scope for future work

Implementation of the FIS-scheme on the Heston model can be studied in order

to improve upon the accuracy of the model. Further, it can be generalised for the

mean reverting constant elasticity of variance (CEV) volatility process.

55


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Appendix A

A.1 Brownian Motion

Defnition A stochastic process{X(t), t 0} called a Brownian Motion if

(1) X(0)=0

(2) X(t) has stationary and independent increments.

(3) For every t >0 X(t) is normal with mean zero and the variance 2t

Brownian Motion is continuous process. When =1, the process is called standerd

Brownian Motion

56


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Appendix A. 57

Figure A.1: One path of the Brownian Motion

A.2 Ito-Doeblin formula for an Ito process

Let X(t) t

0, be an Ito process which is given by as follows:

dX(t) = (t)dW(t) + (t)dt

Let f(t,x) be a function for which the partial derivatives ft(t, x),fx(t, x) and fxx(t, x)

are defined and continuous. Then, for every T 0

df(t, X(t)) =ft(t, X(t))dt+fx(t, X(t))(t)dW(t)+fx(t, X(t))(t)dt+1

2fxx(t, X(t))

2(t)dt

(A.1)


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Bibliography

[1] Sk. Safic Ahmad, Nigam Chandra Parida and Soumyendu Raha, The Fully Im-

plicit Stochastic-alpha Method For Stiff Stochastic Differential Equations, jour-nal of Computational Physics, 228(22), 1 December 2009, pp. 8263-8282.

[2] John C. Cox, Jonathan E. Ingersoll, Jr., Stephen A. Ross, A Theory of The Term

Structure of Interest Rates, Econometrica, 53(2), Mar., 1985, pp. 385-408.

[3] C. Kahl and Peter Jackel, Fast Strong App.roximation Monte-Carlo Schemes for

Stochastic Volatility Models, Quantitive Finance 6, 2006, pp. 513-536.

[4] C. Kahl and H. Schurz, Balanced Milstein Methodes for Ordinary SDEs, Monte

Carlo Methodes and App.lications 12, (2006), pp. 143-170.

[5] G. N. Milstein, E. Platen and H. Schurz, Balanced Implicit Methodes for Stiff

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rajkumar misha rap df

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