1. VASICECK ONE AND TWO
FACTOR
Fall 2015
Projects in Mathematical and Computational
Finance
CEDRIC L. MELHY
NJIT | New Jersey Institutes of Technology

2. 1 | P a g e
Abstract
The Vasicek Model introduced by Oldrich Vasicek in 1977 is one of the earliest stochastic
models of the term structure of interest rate. It represents one of the first and well known interest
rates models and is still in active use. This method of modeling interest rates movement
describes the movement of an interest rate as a factor of market risk, time and equilibrium value
that the rate tends to revert towards. It is also used in the valuation of interest rates futures. The
scope of this study is to show how to estimate the single factor model but we will also show how
this can be extended to the multifactor framework, using available data such as Zero coupon
Bond and then estimate the zero-coupon bond yield curve of tomorrow by using Vasicek yield
curve model with the zero-coupon bond yield data of today. We will give the results for the
estimation of the model parameters using two different ways including extensive R code for
parameters estimation and bond pricing
Finally, we will emphasize a comparative study of the Ornstein-Uhlenbeck and the Cox,
Ingersoll, and Ross (CIR, developed in 1985) model of the short rate
Keywords: one and two factors Vasicek model, CIR model, Equilibrium model, Bond Pricing,
interest rate term structure.

3. 2 | P a g e
TABLE OF CONTENTS
Chapter
Page
1 Introduction ……………………………………………………………. Page 3
1.1 Background ……..……………………………………………………. Page 3
1.2 Interest Rate Model ……………………………………………………... Page 3
1.2.1 Description…………………….……………………………… Page 3
1.2.2 Equilibrium Model…………………………………………….. Page 4
2 Vasicek Interest Rate Model………………………………………………….. Page 6
3 Parameter Estimation ………………………………………………………… Page 7
3.1 Least Square Regressions…..……………………………………………. Page 7
3.1.1 Data…………………………………………………………… Page 7
3.1.2 Parameter Estimation using Least Square Regression Technique Page 10
3.1.3 Maximum Likelihood Estimator……………………………… Page 12
4 Zero Coupon Bond Pricing……………………………………………………… Page 14
4.1 Data……………………………………………………………………… Page 14
4.2 Calibration of model parameters using bond prices…………………….. Page 14
4.3 Data: Euribor from January 18, 1999 to November 18, 2008…………… Page 16
4.4 Data: Euribor from January 2005 to March 2006…………..…………… Page 17
5 Monte Carlo Simulation of the term Structure for Vasicek Model Parameters Page 21
6 Term Structure………………………………………………………………… Page 22
7 The two-factor Vasicek Model………………………………………………… Page 23
7.1 Case Studied ……………………………………………………………... Page 26

4. 3 | P a g e
8 Comparative study of the Vasicek and the CIR Model……………………… Page 26
9 Conclusion…………………………………………………………………… Page 28
10 Appendix…………………………………………………………………….. Page 29
11 References…………………………………………………………………… Page 35

5. 4 | P a g e
1. Introduction
1.1. Background
The short rate r, at time t is the rate that applies to an infinitesimally short period of time at time
t. It sometimes referred to as the instantaneous short rate. Bond prices and other derivatives
prices depend only on the process followed by r in a risk neutral world. The process for r in the
real world is irrelevant. The value at time t of an interest rate derivative that provides a payoff of
at time T is [ ( )
) where denotes the expected value in the traditional risk-
neutral world. Then at time t, the price of zero coupon bond [Z (t, T)] that pays off $1 at time T
is Z (T, t) = [ ( )
]
= ( , )( )
where R (t, T) is the continuously compounded rate at time t for a term
of T-t
That will lead us to the relation R (t, T) = ln [ ( )
]
1.2 Interest Rates model
1.2.1 Description
Interest rate models can be used to model the dynamics of the yield curve, which is vital in
pricing and hedging of fixed-income securities and also of great importance from a macro
economical point where we can estimate the future short-term interest rates, inflation and
economic activity. Traditionally these models specify a stochastic process for the term structure
dynamics in a continuous-time setting. A stochastic process means that the outcome depends
both from a determined and a random component and is thus given the form:
dr(t) = dt + dW(t) Here the first term is deterministic and called the drift-term. The second
term describes the randomness of the process. Models can be roughly divided into equilibrium
models and no-arbitrage models. The essential difference between the two types of models is

6. 5 | P a g e
therefore as follows: In an equilibrium model, the initial term structure of interest rates is an
output from the model and doesn’t fit the current term structure automatically whereas the
arbitrage free model use the current term structure as an output and is designed to be exactly
consistent with the current term structure. Also, for no-arbitrages models , bond prices of the
model are equivalent to the bond prices. Only equilibrium models are described here and from
those only the Vasicek model used will be covered in greater detail.
1.2.2 Equilibrium Model
Equilibrium models usually start with assumptions about economic variables and derive a
process for the short rate r . They explore what the process for r implies about bond prices and
option prices. The name equilibrium comes from the fact that these models aim to achieve a
balance between the supply of bonds and other interest rate derivatives and the demand of
investors. In Vasicek, the coefficient of the Wiener-Process is simply the volatility σ and thus
independent of the short rate r. These models should be arbitrage free otherwise the economy
would not be at equilibrium. The diffusion process proposed by Vasicek is a mean reverting
process. Vasicek’s goal is to choose simplified tractable version of stochastic differential
equation which fits the actual stochastic behavior of the short rate. The disadvantage of the
equilibrium models is that they do not automatically fit today’s term structure of interest rates
contrarily to the arbitrage free model where the bond prices are equivalent to the market bond
prices (i.e. the model yield curve perfectly fits to the market one). Ho-Lee, Hull White Extended
Vasicek, Hull-White Extended CIR and Black-Derman-Toy are some classical no-arbitrage
models.

7. 6 | P a g e
Figure1: Example of an arbitrage free model (Ho-Lee) fitting the yield Curve STRIPS 2002
2. Vasicek Interest Rate Model
The Vasicek model assumes that the current short interest rate is known and the instantaneous
risk-free rate must satisfy the following stochastic differential equation
d = ( - )dt + σdWt
(1)
Where α , γ and r0 are constants and dWt represents an increment to the standard Brownian
motion
( - ) is the drift factor that represents the expected change in the interest rate at time t
is speed of reversion: It gives the adjustment of speed and has to be positive in order to
maintain stability around for the long-term value.
is long term level of the mean : The long run equilibrium value towards which the interest rate
goes back.

8. 7 | P a g e
σ ,constant, is the volatility of the interest rate.
The Vasicek model uses a mean-reverting stochastic process to model the evolution of the short-
term interest rate. Mean reversion is one of the key innovations of the model .If the interest rate
is bigger than the long run mean
( > ), then the coefficient > 0 makes the drift become negative so that the rate will be pulled
down in the direction of and likewise when it drifts below the long term rate it is pushed up.
This feature of interest rates can also be justified with economic arguments: High interest rates
tend to cause the economy to slow down and borrowers require fewer funds. This causes the
rates to decline to the equilibrium long-term mean. In the opposite situation when the rates are
low, funds are of high demand on the part of the borrowers so rates tend to increase again
towards the long-term mean. (Zeytun & Gupta, 2007, p. 2)
r (t) is a continuous function of time and follows a Markovian process, which means that the
system has no memory. Since this is a continuous Markovian process it´s called a diffusion
process. The model assumes the market to be efficient. Solving the Ornstein- Uhlenbeck
Stochastic Differential Equation includes taking the derivative of which yields:
d( ) = dt + d
(2)
Rearrange the order of equation (2) gives
= ( ) − dt
(3)
Multiply both sides of equation (1) with is:
= ( − )dt + σd
(4)
By using equation (3) and substitute it into equation (4), yields:
( ) = dt + σd
(5)
It an integral is taken from time t=0 to t gives:

9. 8 | P a g e
= + ∫ ds + ∫ σd
(6)
In term of , equation (6) will yield to:
= + ∫ ds + ∫ σd
= + (1 − ) +∫ ( )
σd
(7)
The solution of the stochastic differential equation (1) between s and t, if 0≤s<t is:
= ( )
+ 1 − ( )
+σ ∫ σd
(8)
Inspecting the equation above (8) we observe that the expression for is made up of two
deterministic terms representing the drift ( ( )
+ 1 − ( )
and a stochastic
integral representing the diffusion of our process, from which we can reduce the first and second
conditional moments.
We note that:
E [(∫ ( )
σd ) ] = ∫ ( ( )
σ) ds
= (1- )
Thus, the conditional mean and variance of given are:
From the drift terms: [ ] = + ( - ) ( )
[ ] = (1- ( )
)
(9)
If time increases the mean tends to the long-term value and the variance remains bounded,
implying mean reversion. The long-term distribution of the Ornstein-Uhlenbeck process is
stationary and Gaussian with mean and variance σ /2 .One unfortunate consequence of a
normally distributed interest rate is that it is possible for the interest rate to become negative with

10. 9 | P a g e
positive probability. This isn´t a big problem with real interest rates since they often can be
negative but nominal ones are unlikely to be negative.
3. Parameter Estimation
Since Vasicek first introduced his model of short term risk free interest rate the discussion of the
parameters estimation continues. In this section we will discuss the well-known techniques for
parameter estimation
3.1 Least Square Regressions
3.1.1 Data
Let the time step Δt = 0.25, the mean reversion rate = 3.0, long term mean =1.0, and the
volatility =0.50. The table and figure below show a simulated scenario for the Ornstein-
Uhlenbeck process we will use this data to explain the model calibration steps.
I T
0 0.00 3.0000
1 0.25 1.7600 -1.0268
2 0.50 1.2693 -0.4985
3 0.75 1.1960 0.03825
4 1.00 0.9468 -0.8102
5 1.25 0.9532 -0.1206
6 1.50 0.6252 -1.9604
7 1.75 0.8604 0.2079
8 2.00 1.0984 0.9134
9 2.25 1.4310 2.1375
10 2.50 1.3019 0.5461
11 2.75 1.4005 1.4335
12 3.00 1.2686 0.4414
13 3.25 0.7147 -2.2912
14 3.50 0.9237 0.3249
15 3.75 0.7297 -1.3019
16 4.00 0.7105 -0.8995
17 4.25 0.8683 0.0281

11. 10 | P a g e
18 4.50 0.7406 -1.0959
19 4.75 0.7314 -0.8118
20 5.00 0.6232 -1.3890
Table 1 Data used for the model calibration in the example above
3.1.2 Parameter Estimation using Least Square Regression Technique
One possible way to calibrate the estimations of , , σ, and r(t) is to employ the least
squares approach such that the generated term structure can best fit data over historical time. In
this section we will discuss the least squares regression method for calibrating the model
parameters of the Vasicek Model.
The relation between two consecutive observations and is linear with independent
identical random values such that:
= a + b +
(10)
Where a= b= (1- ) = σ
(11)
We note here is the observation of the interest rate r at time t (r )
and the observation of the interest rate r at time t+1 (r )
Express these equations in terms of the parameters , and σ , we have
= - = σ =
( )
(12)
In order to simplify further calculations, a least square regression can easily be done by using the
quantities bellows:
= ∑ = ∑

12. 11 | P a g e
= ∑ = ∑ = ∑
(13)
From which we get the following estimated parameters a, b, of the least square fit
= = =
( )
( )
The figure below shows a simulated scenario for the Ornstein-Uhlenbeck process
Figure2: Relationship between consecutive observations
In this plot, rnext and r are respectively and in the equation (10), and we observe the
Least Square fitting of the line of data.
Applying the regression to the data in table 1, we will get the estimated parameters below:
Parameters Value
22.5302
20.1534
30.83392
25.19744
22.2223
A 0.4574064

13. 12 | P a g e
B 0.4923971
Sd 0.2072782
3.128732
0.9074879
Σ 0.5830761
Table 2 Estimated Parameters using Least Square Regression Technique
3.2 Maximum Likelihood Estimator
The maximum likelihood estimation is a powerful method. It handles bond prices and
therefore it is possible to estimate parameters of the interest rate process. The generalized
method of moments can be used for Vasicek and CIR models.
In order to estimate the parameters , and of the model we use a maximum likelihood
method as suggested in James & Webber (2004, p. 506-507). The maximum likelihood method
finds the parameter values so that the actual outcome has the maximum probability. An
advantage of the maximum likelihood method over for example generalized methods of
moments (GMM) is that it provides an exact maximum likelihood estimator. On the downside it
assumes that the variance of the stochastic variable is constant between the discrete observations.
Let’s define the likelihood function. Then we have to find the parameter values that maximize
the value of this likelihood function. These values are then the maximum likelihood estimated.
The conditional density functions for given is:
( / ; , σ) = √
exp[
( ∆ ∆ )
]
Where =
( ∆ )
(14)

14. 13 | P a g e
The density log-likelihood of a set of observation ( , , ………., ) can be derived from
density function
ln L( , σ ) = ∑ ln ( , , σ )
= - ln (2ᴨ ) –nln(σ) - ∑ ( − ∆
− (1 − ∆ ))
The log-likelihood function needs to be maximized by taking partial derivatives of equation ….
With respect to , σ which yield three equations all equal to zero.
Լ ( , )
 0
= ∑ [ − ∆
− (1 − ∆ )]
=
∑ [ ∆ ]
( ∆ )
=
∆
( ∆ )
Լ ( , )
 0
= -
∆ ∆
∑ [( − )( − ) − ∆
(( − ) ]
= - ∆
ln [
∑ (( )( )
∑ ( )
] = - ∆
ln [ ]
Լ ( , )
 0
= - ∑ [( − − ∆
(( − )] ]
σ = ∑ [( − − ∆
(( − )] ]
= [ − 2 ∆
+ ( ∆
) − 2 (1 − ∆ ) − ∆
+
(1 − ∆ ) ]
( ∆ )

15. 14 | P a g e
(The highlight part correspond to sigma2 in the code: See Appendix 8.2)
Table 3 Estimated Parameters using the Maximum Likelihood Estimation technique
4. Zero Coupon Bond Pricing
4.1 Data: Treasury Strips on January 8, 2002 (Pietro Veronesi, Valuation, Risk and Risk
Management P 543)
In this Section we are going to use the STRIPS prices on January 8, 2002 with maturity ranging
from May 15, 2002 to February 15, 2011 collected from the Wall Street Journal. We use this date
as the yield Curve was particularly steep, which in turns highlights some issues that arise when
using the Vasicek Model. The third column in the Table 3.3.1 below reports time to maturity T,
and the last column depict the continuously compounded yields.
PriceofStrips maturity yield
99.4375 0.3479 0.016212
98.9375 0.6000 0.017803
Parameters estimated Values
0.9074879
3.128732
Σ 0.5531545

16. 15 | P a g e
97.6250 1.1041 0.021770
95.7813 1.6000 0.026940
94.7813 1.8521 0.028940
---------- -------- -----------
--------- -------- -----------
64.4688 8.6000 0.051045
62.9063 9.1041 0.050914
Table 4 STRIPS on January 8, 2002 (just some data from the 28 observations)
4.2 Data: Treasury Strips on September 25, 2008
*The Table 5 below reports the treasury Strips on September 25, 2008. Let the short term rate be
= 2%. You can download data from the Federal Reserve Web site
(http://federalreserve.gov/releases/h15/data.htm)
time to
maturity ZCB Yield
0.139 99.918 0.6
0.389 99.498 1.31
0.689 98.999 1.59
0.889 98.493 1.72
1.139 98.002 1.78
1.389 97.507 1.83
1.639 96.899 1.93
1.889 96.314 2
2.139 95.742 2.05
2.389 94.85 2.23
2.639 94.324 2.23
……… ……….. ………
……… ……….. ………
7.889 73.613 3.92
8.139 72.106 4.06
Table 5 STRIPS on September, 2008 (just some data from the 33observations)

17. 16 | P a g e
Figure 3 Yield Curve and Vasicek Model from STRIPS (September 25,2008)
The graph shoes the yield curve and its fitted values according to the Vasicek model, we obtain
as estimated parameters: alpha = 0.2271832, gamma = 4.545345, sigma = 0.290494
4.3 Data: Euro Interbank Offered rate (Euribor) from January 18, 1999 to November 18, 2008
(Data from Bank of France, www.quandl.com/data/BOF)
Here, we are using the 3 months Euribor Interest rate data collected from January 18, 1999 to
November 18, 2008
0 2 4 6 8
0.51.01.52.02.53.03.54.0
Fitted Zero Coupon Yield Curves from STRIPS
Time to Maturity
Yield(%)
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
*
-
Vasicek
STRIPS

18. 17 | P a g e
Figure 4: Euribor 3 Months period daily rates (Jan 1999- March2008)
4.4 Data: Euro Interbank Offered rate (Euribor) from January 2005 to March 2006
(Data from Bank of France, www.quandl.com/data/BOF)
In this example, we calibrate the Vasicek Model to the current yield curve . The short rate now
r(0) is the one month rate r(0,1month) . We now find the estimated parameters alpha, gamma,
sigma so that the Vasicek model best fit the current term structure.
Parameters estimated Values
0.024
0.49
σ 0.06
Table 6 Estimated Parameters (EURIBOR Jan 2005-March 2006) using the least square
Regression

19. 18 | P a g e
Figure5: Fitted Vasicek Model to the Market yield curve (Euribor 01/2005 to 03/2006)
From the Figure 5, we see the Ornstein- Uhlenbeck model better fitted the Maket yield than the
previous cases, although it is obvious the model does fit the actual yield curve optimally.
Let’s denote the time to maturity ɽ = T-t, the term structure of interest rates is a function of the
current interest rate and ɽ. The yields can then be calculated easily for all maturities using the
price of the corresponding zero-coupon bonds
(ɽ) = -
( ( ,ɽ))
ɽ
= -
(ɽ)
ɽ
+
(ɽ)
ɽ
(19)
We start with the parameter estimates , , σ . We must also specify the interest rate prevailing
at time t=0 and the maturity of zero-coupon bond. In this case, T=10 years. The program then
calculates the values of A (t, T), B (t, T) and the zero coupon bond price Z (t, T)
The time step needs to be quite small in order to achieve accuracy with the Euler-discretization

20. 19 | P a g e
Figure 6 Fitted Zero Coupon Yield Curves from STRIPS (January 8, 2002)
As we can see, the Vasicek Model is not able to match the term structure of interest rates on
January 8, 2002, except a single point. That could be explain by the fact that the Vasicek Model
is a simple model, with only three parameters and one driving factor ( ). Also, this could be
created when the term structure is very steep (case where the Federal Reserve want to lower the
target Funds rate to boost the U.S economy)

21. 20 | P a g e
Figure 7 Fitted Zero Coupon Yield Curves from data in table 1
Figures 2 and 3 plot the zero coupon yields for both the shorts rates observed from the Market
( ) and those implied by the Vasicek Model (****).
We observed different patterns (shapes) of the term structure. In both cases (fig2 and fig 3) . The
different shapes represented here are the increasing shape (normal) in figure 2 and the inverted
yield curve in figure3. Vasicek Model is sensitive to its parameters in the sense; a small change
in parameter values may result in a large change in bond price.
Between the three parameters, , , σ with which to fit bond prices of all maturities, which
there are in principle an infinite number, one doesn’t always get a good fit. This difficulty can be
circumvented by letting the coefficients depend deterministically on t: this will allow us to fit at
least any initial yield curve. Since the Strips prices cannot be recon ciliated with the Vasicek
model, according to the model, there is an arbitrage opportunity.

22. 21 | P a g e
5. Monte Carlo Simulation for Vasicek Model Parameters
Monte Carlo simulation is a set of computational algorithms that rely on repeated random
sampling numerous times in order to achieve asymptotic results that are analytically hard.
Through Monte Carlo simulation, this process can be done too many times. In this way, Monte
Carlo simulation provides much more extensive information about what may happen and how
likely it is to happen. In the simulation process, the most important step is to estimate the
parameters of our model , ,σ accurately. Parameter estimation methods regarding the
Vasicek Model are described in the previous sections. In order to estimate distribution of the
parameters that fits Vasicek Model, Monte Carlo Simulation is used and in each run of the
Monte Carlo.
Figure 8 Sample paths generated from STRIPS: January 8, 2002
Here you can see that, this process starts at 0.02, but is pulled towards 0.05, which is the value
of . Also, you may observe, as noticed before, that the major problem with this process is that
the interest rate can drop below 0%. In the real world, this shouldn’t happen. The Cox-Ross-
Ingersoll model, actually corrects for this, but the process is no longer normally distributed.

23. 22 | P a g e
6. Term Structure
The term structure of interest rates, also known as a yield curve, illustrates the relationship
between interest rates and different times of maturity. The yield curve can be basically shape, but
there are some cases where it should look different. There are four main patterns that can
characterize the term structure: normal, flat, inverted and humped. The Vasicek model can
generate all of them. The yields curves implied by Vasicek Model in Figure 2 and Figure 3
highlighted consecutively the normal yield and the inverted yield curve.
Under normal market conditions, (figure 1) the investors think the economy will continue to
grow and expect higher yields for instruments with longer maturities.
The inverted yield curve (Figure 2) is the opposite. In such market conditions, bonds with
maturities dates further into the future are expected to yield less than bonds with shorter
maturity.
If we note that Z(0,T) = where is the continuous time yield for a zero coupon bond
with maturity T. We can denote the term structure of interest rates as a function of the current
interest rates and time to maturity ɽ. This is given by:
(ɽ) = -
( ( ,ɽ))
ɽ
= = -
(ɽ)
ɽ
+
(ɽ)
ɽ
with ɽ = T-t
To demonstrate some of the possible curve shapes allowed by the Vasicek model, I produced the
following chart. Notice that the yield curve can be both increasing and decreasing with maturity.
Figure 9 Five Spot curves implied by the Vasicek Model

24. 23 | P a g e
r = 3.5%
r = 3%
r = 2.5%
Figure10 Vasicek Yield vs maturity for three initial spot rate : Parameters = -0.2352252,
= 0.02509691
σ = 0.003787045
We observe (fig 9) that the level, slope and curvature change over the time. The figure shows
that if the current short term rate is low, the vasicek model implies a rising term structure of
interest rate. In case we have a high current short term interest, the model implies a decreasing
term structure. The term structure moves relatively to the interest rate. As mentioned above,
these movements are correlated. The two factor vasicek model will take of the problem.
Depite the model’s simplicit, the mean-reverting feature of the spot rate process allows for varied
behavior of the term structure of yields to maturity as we can see on figure 10
7. The two-factor Vasicek Model
Let’s assume that the short term interest rate is given by
r , + r ,
(20)
We consider two different factors denoted by r , and r , and assume they move according to the
process

25. 24 | P a g e
Where the risk neutral factor dynamics have the general form
dr , = ( r − r , ))dt + σ d ,
(21)
dr , = ( r − r , ))dt + σ d ,
(22)
With r and r the risk neutral parameters, , and , denote independent standard
Brownian motion
From the risk neutral factor dynamics be given by the equations 21 and 22, and the interest rate
be linear in the factors as in equation 20, the zero coupon bond price is given by
(ɽ ) = (ɽ ) (ɽ)
where i=1, 2
(23)
Z(r , , r , ,t,T) = ( ; ) ( ; ) , ( ; ) ,
(24)
Where
(t; T) = [1 − ( )
];
(25)
A(t; T) = ( (t; T) - (T-t)) (r − ( )
) - ( ; )
We can write the process for the short term rate as follows
d = [ ∗
(r ∗
- ) + ∗
r ∗
+ ( ∗
− ∗
) r , ]dt + σ d + σ d
(27)
Contrarily to the one factor model, the risk neutral drift here depends on the second factor r ,
with
dr , = ( − r , ) dt + σ (r , , t) d ,
(28)
The yields can then be calculated easily for all maturities using the price of the
corresponding zero-coupon bonds
(ɽ) = -
( , , , , , )
ɽ
(ɽ) = -
(ɽ)
ɽ
+
(ɽ)
ɽ
r +
(ɽ)
ɽ
r ,
(29)

26. 25 | P a g e
In practice, the components and of the short rate are not observables. An observable
quantity is the short rate
r = r , + r , and we note r , in equation (29) means the long interest rate is not
perfectly correlated with the short rate. r , and r , only affect the interest rater when
looking for his average level which correspond to equation (20).
The expectation and the variance under the real probability measure are:
E [ , / ]= ( )
, + (1 − ( )
) , i= 1,2
Var[ , / ]= (1 − ( )
) i=1,2
The limiting distributions of both Ornstein-Uhlenbeck processes forming the two-factor Vasicek
model are known to be normal distributions with density
(x) =
( )
i= 1,2
Let’s denote the long term yield , and the short term yield, thus the coefficients , ,
, , are for a long term period as well. From the equation (29) of the zero coupon spot rate,
we derive to the dynamics of long and short term yields below using Ito Lemma
d = ( + + , ) dt + σ d , + σ d ,
(30)
d , = ( , + , + , , ) dt + σ , d , + σ , d ,
(31)
As we can see in the equations 30 and 31, the short term rate and long term rate are feeding
themselves .
The volatilities of the long term yield and the short term yield are related as we can see in the
equation below
σ , = σ
∗ɽ
ɽ
; σ , = σ
∗ɽ
ɽ
(32)
We should notice that these volatilities are computed from historical data.

27. 26 | P a g e
7.1 Case studied
Let today be January 8, 2002. Now we use the two factor model to fit the coupon bonds traded
on that day and estimate the parameters using the nonlinear least square method. As data, we use
the same strips prices as in the example above. We select the 5 year yield as the long term zero
coupon yield ɽ = 5 added to the current short term free rate and 1-month T-bill rate for the short
term. It’s not easy to observe the 5 year zero coupon yield daily, but we should estimate from
fitting the discount curve Z(0,T). As notice earlier, these volatilities are computed from historical
data. Let’s fix the volatility of the short rate (1-month rate) and the long term rate (5-year yield)
respectively by σ , =0.0221 and σ , = 0.0125 and since r , and r , are not
identifiable as together, we could set r , = 0. As with the one factor model, we can search the
parameters ∗
, ∗
, ∗
, ∗
, σ and σ such that the following quantity is minimized.
J ( ∗
, ∗
, ∗
, ∗
, σ σ ) = ∑ ( − )
8. Comparative study of the Vasicek and the CIR Model
Vasicek and CIR are two of the historically most important short rate models used in the pricing
of interest rate derivatives. In CIR model (sometimes called square root model) under the risk-
neutral measure Q, the instantaneous short rate follows the stochastic differential equation
d = ( - )dt + σ dWt , r(0) = 0
(33)
Where , ,σ are positive constants, and W(t) is a Brownian motion. When we impose the
condition 2 > σ2 then the interest rate is always positive, otherwise we can only guarantee
that it is non-negative. The drift factor ( - ) is same as in the Vasicek model. Therefore, the
short rate is mean-reverting with long run mean and speed of mean-reversion equal to . The
volatility of the Vasicek model is multiplied with the term ,and this eliminates the main
drawback of the Vasicek model, a positive probability of getting negative interest rates. As the
short-rate rises, the volatility of the short-rate also raises .When the interest rate approaches zero
then σ approaches zero, cancelling the effect of the randomness, so the interest rate remains
always positive. When the interest rate is high then the volatility is high.

28. 27 | P a g e
Vasicek
CIR
Hist. Data
Figure 11: Calibration of Vasicek and CIR models to historic data (Data from Example 3.1)
Here, we give a plot of the yields curve to compare Vasicek and CIR models. We could see that
it is better to use the CIR model because the short rates cannot be negative

29. 28 | P a g e
9. Conclusion
In this study,I worked on the Vasicek, Comparative study of the Vasicek and the CIR
Model and the Two Factor Vasicek equilibrium models . I used the R language for
implementation and sometimes Matlab for reconciliation and check the results. Note that for the
two factor models, I haven’t expanded the details and the implementation I used MLE and Least
square methods for estimating the model parameters. Moreover, parameter estimation of affine
term structure is essentially a time-series problem. Most papers have been focused on Kalman
filter which doesn’t require that the variables be observable. the estimation of Vasicek model
using either MLE or Least Squares gives accurate estimates of the mean alpha gamma and
volatility sigma, but poor estimates of the mean reversion alpha Short-rate simulation and bond
pricing are not so difficult, as for the CIR model the exact distribution of rt is known and
analytical formulas exist for bond pricing. As elaborated in section 8, the CIR model estimates
the yield curve relatively much better compared to the Vasicek Model, It is better to use this
model since the short rates values cannot be negatives. In the other hand, we have another model,
the Nelson Siegel Model which fits the yield curve perfectly. I goes however without saying that,
understand complex interest rates models like vasicek Model requires a lot of practice and work
specially multiple factors models.

30. 29 | P a g e
10. Appendix
R code
10.1 Estimation of parameters in Vasicek Model using Least Square Regression Technique
## The values of interest rates observed
r <- c(3, 1.76, 1.2693, 1.196, 0.9468,
0.9532, 0.6252, 0.8604, 1.0984,
1.431, 1.3019, 1.4005, 1.2686,
0.7147, 0.9237, 0.7297, 0.7105,
0.8683, 0.7406, 0.7314, 0.6232)
n <- 20 ## number of observations
delta <- 0.25 ## time step delta = ti+1 - ti
## Least Square Calibration
function [alpha,gamma,sigma] = Vas_LS(r,delta)
n = length(r)-1;
rx <- sum(r[1:length(r)-1])
ry <- sum(r[2:length(r)])
rxx <- crossprod(r[1:length(r)-1], r[1:length(r)-1])
rxy <- crossprod(r[1:length(r)-1], r[2:length(r)])
ryy <- crossprod(r[2:length(r)], r[2:length(r)])
a <- (n*rxy - rx * ry ) / ( n * rxx - rx^2 )
b <- (ry - a * rx ) / n
sd <- sqrt((n * ryy - ry^2 - a * (n * rxy - rx * ry)) / (n * (n-2)))
## Now, the parameters estimated using the Least Square Calibration (with R)
alpha <- -log(a)/delta ## denote the speed of reversion
gamma <- b/(1-a) ##denote long term level of the mean
sigma <- sd * sqrt( -2*log(a)/delta/(1-a^2)) ## denote the volatility
end
## I estimate the parameters using the Least Square Calibration (with Matlab)
to check
r = [3.0000 1.7600 1.2693 1.1960 0.9468 0.9532 0.6252 0.8604 1.0984 1.4310
1.3019 1.4005 1.2686 0.7147 0.9237 0.7297 0.7105 0.8683 0.7406 0.7314
0.6232];
delta = 0.25;
[alpha, gamma, sigma] = VAS_LS(S,delta)
##Outputs
alpha = 0.90748
sigma = 0.58307

31. 30 | P a g e
gamma = 3.12873
rnext <- a * r + b + sd ## denote the interest rate at time ti+1 . The
relationship between consecutive observations rnext and r is linear with a
iid normal random term sd(epsilon)
## Plot Relationship between consecutive observations (Fig 2)
plot(rnext,r, col="blue")
lines(rnext,r)
fit <- lm(r ~ rnext)
abline(fit)
10.2 Maximum Likelihood Calibration
## sample of historic short rate data from federal reserve website
r <- c(3, 1.76, 1.2693, 1.196, 0.9468,
0.9532, 0.6252, 0.8604, 1.0984,
1.431, 1.3019, 1.4005, 1.2686,
0.7147, 0.9237, 0.7297, 0.7105,
0.8683, 0.7406, 0.7314, 0.6232)
n <- 20
delta <- 0.25
## Below, we estimate the parameters using the Maximum Likelihood Calibration
(With R)
rx <- sum(r[1:length(r)-1])
ry <- sum(r[2:length(r)])
rxx <- crossprod(r[1:length(r)-1], r[1:length(r)-1])
rxy <- crossprod(r[1:length(r)-1], r[2:length(r)])
ryy <- crossprod(r[2:length(r)], r[2:length(r)])
## Below, we estimate the parameters using the Maximum Likelihood Calibration
(With Matlab)
r = [ 3.0000 1.7600 1.2693 1.1960 0.9468 0.9532 0.6252 0.8604 1.0984 1.4310
1.3019 1.4005 1.2686 0.7147 0.9237 0.7297 0.7105 0.8683 0.7406 0.7314 0.6232
];
delta = 0.25;
[alpha, gamma, lsigma] = VAS_ML(S,delta)
##Outputs
alpha = 0.90748
gamma = 0.55315
sigma = 3.12873
Now, the parameters estimated using the Least Square Calibration
gamma <- (ry * rxx - rx * rxy) / (n* (rxx - rxy) - (rx^2 - rx*ry) )

32. 31 | P a g e
alpha <- -log((rxy - gamma * rx - gamma * ry + n * gamma^2) / (rxx - 2 *
gamma * rx + n * gamma^2)) / delta
sigma2 <-(ryy -2*a * rxy + a^2 *rxx -2*gamma *(1-a)*(ry - a * rx) + n *
gamma^2*(1 - a)^2)/n
sigma <-sqrt(sigma2*2 * alpha/(1 - a^2)) ## refer to the highlight part in
yellow ,sect. 3.2
R Code Fig 3 (extend)
r <- c(3, 1.76, 1.2693, 1.196, 0.9468,
+ 0.9532, 0.6252, 0.8604, 1.0984,
+ 1.431, 1.3019, 1.4005, 1.2686,
+ 0.7147, 0.9237, 0.7297, 0.7105,
+ 0.8683, 0.7406, 0.7314, 0.6232)
n <- 20 ## number of observations
##define the function B(t,T)
T<-seq(0.25,5,0.25)
B<- (1/alpha)*(1-exp(-alpha*T))
delta<-0.25 ##time step
##define the function A(t,T)
A <-(gamma-(sigma^2)/(2*(alpha^2)))*(B-T)-((sigma^2)*((B^2)/(4*alpha)))
r0 <-3
Z <-exp(A-(B*r0))
rvas<-(-A/T) + (B*r0)/T
T <-seq(0.25,5,0.25)
## Plot Vasicek and CIR Model
dev.new()
plot(T,r,type="l", lty=1, xlab="Time to Maturity", ylab="Yield(%)",
main="Comparison between Vasicek and CIR Model ")
points(T,rvas, pch="*")
Outputs:
gamma
0.9074879
alpha
3.128732
sigma
0.5531545
8.2 Fitted Zero Coupon Yield Curves from Strips: January 8, 2002
R Code
##give the strips data for january 2,2002
strips<-read.csv("C:/Users/cedric/Desktop/Data/Strips.csv")
dim(strips)
price<-strips[,1,1:28]
mat<-strips[,2,1:28] # assigning maturity as a column vector from column 2
of Strips
yield<-strips[,3,1:28] # assigning yield as a column vector from column 3 of
Strips matrix

33. 32 | P a g e
yvas<-function(T,strips,rt) {
alpha= (strips$alpha)
gamma= (strips$gamma)
sigma= (strips$sigma))^2
B=1/alpha*(1-exp(-alpha*T))
A=(gamma-sigma2/(2*alpha^2))*(B-T)-sigma2/(4*alpha)*B^2
yield=-(A-B*rt)/T
if(T[1]==0) { yield[1]=rt }
return( yield )
}
}
##Treasury Strips observed (September 2008)# data got from Veronesi, we could
find it from www.federalreserve.gov/releases/h15/data.htm as well
strips2008<-read.csv("C:/Users/cedric/Desktop/Data/strips2008.csv") ## gives
the strips data for september 25,2008
dim (strips2008)
price<-strips2008[,1,1:33]
maturity<-strips2008[,2,1:33]
yield<-strips2008[,3,1:33]
n<-33
delta<-0.25 ##time step
r0<-2 ##current short rate
rx <- sum(yield[1:length(yield)-1])
ry <- sum(yield[2:length(yield)])
rxx <- crossprod(yield[1:length(yield)-1], yield[1:length(yield)-1])
rxy <- crossprod(yield[1:length(yield)-1], yield[2:length(yield)])
ryy <- crossprod(yield[2:length(yield)], yield[2:length(yield)])
a <- (n*rxy - rx * ry ) / ( n * rxx - rx^2 )
b <- (ry - a * rx ) / n
sd <- sqrt((n * ryy - ry^2 - a * (n * rxy - rx * ry)) / (n * (n-2)))
## Parameters estimation
alpha <- -log(a)/delta ## denote the speed of reversion
gamma <- b/(1-a) ##denote long term level of the mean
sigma <- sd * sqrt( -2*log(a)/delta/(1-a^2)) ## denote the volatility
##define the function B(t,T)
T<-seq(0.25,8.25,0.25)
B<- (1/alpha)*(1-exp(-alpha*T))
##define the function A(t,T)
A <-(gamma-(sigma^2)/(2*(alpha^2)))*(B-T)-((sigma^2)*((B^2)/(4*alpha)))
r0 <-2 ## Current short rate
Z <-exp(A-(B*r0))
rvas<-(-A/T) + (B*r0)/T
alpha
0.2019314
gamma
0.06356295
sigma
0.002218611

34. 33 | P a g e
## Monte Carlo Simulation for Vasicek Model Parameters
## define model parameters (from Strips, January 8, 2002, Veronesi Example)
r0 <- 0.02 # current overnight rate
gamma <- 0.0509
alpha <- 0.3262
sigma <- 0.0221
## simulate short rate paths
n <- 1000 # MC simulation trials
T <- 8 # total time
m <- 200 # subintervals
dt<- T/m # difference in time each subinterval
r <- matrix(0,m+1,n) # matrix to hold short rate paths
r[1,] <- r0
for(j in 1:n){
for(i in 2:(m+1)){
dr <- alpha*(gamma-r[i-1,j])*dt + sigma*sqrt(dt)*rnorm(1,0,1)
r[i,j] <- r[i-1,j] + dr
}
}
## plot of sample paths
t <- seq(0, T, dt)
rT.expected <- gamma + (r0-gamma)*exp(-alpha*t)
rT.stdev <- sqrt( gamma^2/(2*alpha)*(1-exp(-2*alpha*t)))
matplot(t, r[,1:10], type="l", lty=1, main="Plot of sample Paths", ylab="rt")
abline(h=gamma, col="red", lty=2)
lines(t, rT.expected, lty=2)
lines(t, rT.expected + 2*rT.stdev, lty=2)
lines(t, rT.expected - 2*rT.stdev, lty=2)
points(0,r0)
---------Calibration of Vasicek and CIR models to historic data (Data from
Example 3.1)
mat=c(0.00, 0.25,0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50, 2.75,
3.00)
t=seq(from=min(mat),to=max(mat),length=500)
yestVas<-
optim(f=l2norm,p=log(c(exp(0.174),yield[1],0.004)),y=cbind(mat,yield),
familie="yvasicek", rt=yield[1],control=list(maxit=400))
yestCIR<-optim(f=l2norm,p=log(c(exp(0.16),yield[1],0.04)),y=cbind(mat,yield),
familie="ycir", rt=yield[1],control=list(maxit=400))
vas.par=list(lambda=exp(yestVas$par[1]),theta=exp(yestVas$par[2]),sigma=exp(y
estVas$par[3]),r=yield[1])
cir.par=list(k=exp(yestCIR$par[1]),theta=exp(yestCIR$par[2]),sigma=exp(yestCI
R$par[3]),r=yield[1])
vas.par
cir.par
plot(mat,yield,type="b",col="black",lwd=2,ylim=range(yield),xlab="time to
maturity",ylab="yield %",yaxt="n",main="Calibration of Vasicek and CIR
Models",cex=2)

35. 34 | P a g e
lines(t,yvas(t,list(lambda=yestVas$par[1],theta=yestVas$par[2],sigma=yestVas$
par[3]),yield[1]),lty=1,col="blue",lwd=2)
lines(t,ycir(t,list(lambda=yestCIR$par[1],theta=yestCIR$par[2],sigma=yestCIR$
par[3]),yield[1]),lty=1,col="red",lwd=2)
axis(side=2,at=yticks,labels=paste(format(100*yticks,digits=3),"%",sep=""))
legend(0.5,3.2,legend=c("Vasicek","CIR","STRIPS"),pch=c("blue","red"),"o")
## Figure 10 shows the effect of changing the values of current short rates
on the vasicek yield
r0<-3.0 ## Set up the first current short rate
r1<-3.5 ## second short rate
r2<-2.5 ## third short rate
rvas<-(-A/T) + (B*r0)/T ## vasicek yield for ro
rvas1<-(-A/T) + (B*r1)/T ## vasicek yield for r1
rvas2<-(-A/T) + (B*r2)/T ## vasicek yield for r2
## Plot differents curves for short rates r0,r1,r2
plot(T,rvas,type="l", lty=1, xlab="Maturity", ylab="rvas", main="Vasicek
Yield vs maturity for three values of initial spot rate ",)
par(new=TRUE)
lines(T,rvas1,col='red')
par(new=TRUE)
lines(T,rvas2,col='blue')

36. 35 | P a g e
11. References
 Zeytun & Gupta. 2007. A comparative Study of the Vasicek and the CIR Model of the
Short Rate
 Vasicek 1977. An Equilibrum Characterization of the Term Structure
 On the Simulation and Estimation of the Mean – Reverting Ornstein – Uhlenbeck
Process. William Smith Feb 2010
 Fixed Income Securities. Valuation, Risk, and Risk Management by Pietro Veronesi
 Risk Properties and Parameter estimation on Mean Reversion and Garch Models by
Roelf Sypkens
 Short rates and bond prices in one-factor models by Muhammad Naveed Nazir
 A comparative Study of the Vasicek and the CIR Model of the Short Rate by Serkan
Zeytun,Ankit Gupta
 Lecture Notes in Empirical Finance: Yield Curve Models 2 (MLE and GMM) by Paul
Soderlind
 Affine Interest Rate Models-Theory and Practice. By Dr Josef Teichmann
 Interest Rate Modelling. John Willey and Son by James,Jessica & Webber,Nick 2004
 Mathematical analysis of term structure models by Beata Stehlikova
 Affine Interest Rate Models – Theory and Practice by Christa Cuchievo
 Term Structure forecasting by Emile van Elen
 Affine Term Structure Models: Theory and Implementation by David Bolder
 Options : recent advances in theory and practices vol 2 by Stewart Hedges
 An Introduction to the Mathematics of Financial Derivatives (2nd
Edition) by Salith N.
Neftci
 Pricing derivative Securities by EPPS