Applied Mathematics, 2010, 1, 37-43
doi:10.4236/am.2010.11006 Published Online May 2010 (http://www.SciRP.org/journal/am)
Implied Bond and Derivative Prices Based on Non-Linear
Stochastic Interest Rate Models
Ghulam Sorwar1, Sharif Mozumder2
1Nottingham University Business School, Jubilee Campus, Nottingham, UK
2Department of Mathematics, University of Dhaka, Dhaka, Bangladesh
Received March 8, 2010; revised April 2, 2010; accepted April 30, 2010
Abstract
In this paper we expand the Box Method of Sorwar et al. (2007) to value both default free bonds and interest
rate contingent claims based on one factor non-linear interest rate models. Further we propose a one-factor
non-linear interest rate model that incorporates features suggested by recent research. An example shows the
extended Box Method works well in practice .
Keywords: Stochastic, Interest Rates, Derivatives, Box Method
1. Introduction
Stochastic differential equations are the foundations on
which modern option pricing methodology is based.
However, non-linear stochastic differential equations for
interest rate models have been proposed that captures the
non-linear dynamics of the spot interest rates. There are
two aspects to the modeling of interest rate term structure
models and interest rate contingent claims. The first
concerns the econometric aspects (see for example, [1])
and the second the numerical implementation of the re-
sulting models. With regard to the numerical aspects of
interest rate modeling, there exist three different ap-
proaches. The first is the lattice approach introduced by
Cox-Ross-Rubinstein (1979) [2]. However, as Ba-
rone -Adesi, Dinenis and Sorwar (1997) [3] have demon-
strated the lattice approach does not always lead to mea-
ningful bond and hence contingent claim prices. The
second approach is the Monte-Carlo simulation approach
introduced by Boyle (1977) is mainly used to value path
dependent European type contingent claims. To date no
single accepted Monte-Carlo simulation scheme has been
put forward for the valuation of American type contin-
gent claims. The third approach is the partial differential
equation (PDE) approach. With this approach, the par-
tial first and second order derivatives are discretized to
produce a system of equations which are then solved
iteratively to obtain the bond and contingent claim prices.
However, Sorwar et al. (2007) have shown that the usual
finite difference approach used to discretize the PDE
does not always lead to bond and contingent claim prices
that correspond with analytical prices where these prices
are available.
Sorwar et al. (2007) intro d uc ed the Box Method from
engineering to improve on the standard finite difference
approach. Sorwar et al. (2007) focused on the CKLS
(1992) model. Sorwar et al. (2007) did not attempt to
value bonds and contingent claims based on non-linear
interest rate models. Ait-Sahalia (1996) [4] non-and
Conley et al. (1997) [5] propose parametric linear
one-factor which allows non-linear parameterisation. Our
main objective in this paper is to expand the Box Method
of Sorwar et al. (1997) to price bonds and contingent
claims based on both linear and non-linear interest rate
models.
The outline of the paper is as follows: Section 2 the
general non-linear parametric model and the resulting
partial differential equation for default free bonds and
contingent claims is outlined. We then derive the Ex-
panded Box Method (EBM) for the valuation of default
free bonds and contingent claims. Using US estimates we
compute implied bond and contingent claims prices in
Section 3. Section 4 contains a summary and conclusion.
2. Expanded Box Method (EBM)
In this section we discuss the valuation of the bond and
contingent claim prices based on the extended
Ait-Sahalia (1996) [4] and Conley et al. (1997) [5]
framework. Following Sorwar et al. (2007) we let:
( )
*
t
Br ,t,T
: price of a discount bond at time t whi ch

G. SORWAR ET AL.
38
Table 1. lternative Parametric Specifications of the Spot Interest Rate Process
( )( )
2
t ttt
drrdtr dW µσ
= +
.
Drift function
( )
r µ
Diffusion function
( )
2
r σ
Refer en ce
01
r αα +
0
β
Vasicek (1977) [6]
01
r αα +
1
r β
Cox-Ingersoll-Ross(1985) [7]
Br o wn-Dybvig(1986) [8]
Gibbons-Ramaswamy(1993) [9]
01
r αα +
2
2
r β
Courtadon (1982) [10]
01
r αα +
3
2r
β
β
Chen et al. (1992)
23
01 2
rr
r
α
αα α ++ +
3
01 2
rr
β
ββ β ++
Ait-Sahalia (1996) [4]
3
5
4
01 2
rrr
α
α
α
αα α +++
3
01 2
rr
β
ββ β ++
matures at time
*
T
with the generated spot rate t
r .
( )
*
P t,T,T
: price of a contingent claim at time
t
which expires at time
T
based on a discount bond
which matures at time
*
T
subject to suitable boundary
conditions.
In a risk-neutral world, the drift rate is adjusted by the
market price of risk
r
λ
1 so that the short-term interest
process beco mes:
( )
( )
35
3
012 4
01 2
t
t
drrrr dt
rr dW
αα
β
α αλαα
ββ β
−
= +++++
++
(1)
The resulting partial differential equation is:
( )
3
35
2
01 22
012 4
1
2
0
U
rr r
UU
r rrrU
rt
β
αα
∂
ββ β∂
∂∂
α αλαα∂∂
−

++


+ ++++−+=

(2)
In equation (2)
( )
t
Ur ,t
may represent either
( )
*
t
Br ,t,T
or
( )
*
P t,T,T
subject to the appropriate
boundary conditions (see [10] for more details). Follow-
ing Sorwar et al. (2007) we transform the above pricing
equation such that either the bond or the contingent
claims evolves from the options expiration date or the
bonds maturity date to the present, i.e. we let
Tt τ = −
.
The above equation then becomes:
( )
35
3
2
012 4
2
01 2
2rr r
UU
r rrr
αα
β
α αλαα
∂∂
∂ ∂ββ β
−

++ ++
+−

++


33
01 201 2
22rU
U
rr rr
ββ
∂
∂τ ββ βββ β
=
++++ (3)
We now choose a general function
( )
R r,,
αβ
such
that:
( )
35
3
2
2
012 4
01 2
1
2
UU
R
Rr rr
rrr U
r
rr
αα
β
∂∂ ∂
∂∂ ∂
α αλαα∂
∂
ββ β
−

= +



++ ++

++


(4)
The above expression simplifies to yield:
( )
35
3
012 4
01 2
12rr r R
Rr rr
αα
β
α αλαα
∂
∂ββ β
−

++ ++
=
++


(5)
We now integrate from the general value
r
( )
11nn
r rr
−+
<< to the lower limit of integration
0r=
to obtain:
( )
( )
35
3
1
012 4
01 2
2
n
r
r
R r,,
rr r
exp dr
rr
αα
β
αβ
α αλαα
φββ β
−
−
=


++ ++

+

++




∫
where
( )
0ln R,, φ αβ =
. We further note that:
11UU
RQ
Rrr Qrr
∂∂ ∂∂
∂∂ ∂∂
 
=
 
 
where:
1Risk premium is treated differently by researchers. Vasicek (1977) [6]
takes
( )
r λλ =
, Chan et al. (1992) [1] take
( )
0r λ =
, ox et al. (1985)
,
we take
( )
rr λλ =
.

G. SORWAR ET AL.
39
( )
( )
35
3
012 4
001 2
2
r
Q r,,
rr r
exp dr
rr
αα
β
αβ
α αλαα
ββ β
−
=


++ ++



++




∫
So equation (3) becomes:
3
01 2
12Ur
QU
Qr rrr
β
∂∂
∂∂ββ β

−=

++

We now transform the interest rate as:
3
01 2
2U
rr
β
∂
∂τ ββ β ++
(6)
cr
cr
s+ =1
where c is a constant. (7)
This leads to the transformation of equation (6) as:
( )( )( )
( )()
( )
( )()
33
32
2
11
02 02
1 221
11
11 11
Us UU
s
Qs ssc scs
ss
ss
cs cscs cs
ββ
τ
ββ
ββ ββ
∂∂ ∂

Ψ− =

∂∂ ∂

−−
 
++ ++
 
−− −−
 
(8)
where:
( )()( )
( )( )
( )
( )( )( )
( )( )
35
3
2
1
02 4
2
01
02
1
11 1
2
1
11
s
scs Qs
s
ss
cs cscs
Q sexpdr
cs ss
cs cs
αα
β
αλ
αα α
β
ββ
−
Ψ=−



+


++ +

−− −




= 
−
 
++

 
−−




∫
Following the set-up of Sorwar et al. (2007) a grid of
size
MN×
is constructed for values of
( )
m
n
UU n r,m t= ∆∆
- the value of
U
at time increment
m
t
and interest rate increment
n
s
, for each method,
where:
0m
ttmt
=+∆
01m, ,....,M=
2
1
nn
s sa
+
∆ =∆+
1n ,....,N=
where a is an arbitrary constant.
Using the Euler backward difference for the time de-
rivative gives:
0
UU U
t
∂
∂τ
−
=∆
,
where
0
U
and
U
refers to bond or contingent claims
prices at time step m-1 and m respectively.
Integrating equation (8) from the point
1
1
2
2
nn
n
ss
s
−
−
+
=
to point
1
1
2
2
nn
n
ss
s
+
+
+
=
, we have:
( )() ( )()() ( )()
() ( )()
11 1
22 2
11 1
22 2
1
2
1
2
32
2
0
2
1
22
11
1
2
1
nn n
nn n
n
n
ss s
ss s
s
s
Us
tsdstQsfs UdsQsfs Uds
ss c scs
Qsfs U ds
cs
++ +
−− −
+
−
∂∂

−∆Ψ+ ∆+

∂∂
 −−
=−
∫∫ ∫
∫
(9)
Discretizing each of the above integrals, and rearrang-
ing gives us the following matrix equation:
1
11
mm mm
nnnnnn nn
UU UU αχ ηβ
−
−+
= ++
(10)
where:
( )
1
2
1
2
n
n
s
s
U
ts ds
ss
+
−
∂∂

−∆ Ψ+

∂∂

∫
() ( )( )
1
2
1
2
3
2
2
1
n
n
s
s
s
tQsfs Uds
cs
+
−
∆+
−
∫
2Where a and
0
s∆
are arbitrary constants. A derivation of this expres-
sion can be found in Settari and Aziz (1972) [11].

G. SORWAR ET AL.
40
() ( )( )
1
2
1
2
2
1
2
1
n
n
s
s
Qsfs Uds
cs
+
−−
∫
() ( )( )
1
2
1
2
0
2
1
2
1
n
n
s
s
Qsfs U ds
cs
+
−
=−
∫ (9)
Discretizing each of the above integrals, and rearrang-
ing gives us the following matrix equation:
1
11
mm mm
nnnnnn nn
UU UU αχ ηβ
−
−+
= ++
(10)
where:
()
( )
()
( )
()
( )
()
( )
( )
( )
()
( )
( )
()
1
2
1
2
11
22
11
22
11
22
1
1
1
01
11
03
2
12
2
22
1
1
n
n
n
nn n
n
n
nn n
nn
n
nnnn nn
nn
nn
n
n
nn
n
I
s
t
s sQs
s
t
ss Qs
ss
tt
tI I
ssQsss Qs
sf s
I ss
cs
fs
I ss
cs
α
χ
β
η
−
−
+
+
−+
−+
+−
+−
=
Ψ
−∆
=−
Ψ
−∆
=−
ΨΨ
∆∆
=++∆+
−−
= −
−
= −
−
The matrix equation linking all bond prices or contingent
claim prices between two successive time steps m-1 and
m is:
1
01
1
01
1
11 01
22 201
33 3
333
22 2
11 1
00 00
00 0
0 00
0
0
0
0 00
m
m
m
NN
m
m
NNN
NN N
m
NN NN
U
U
U
U
U
U
α
α
α
ηβ α
χηβ α
χηβ
χχβ
χηβ
χη α
−
−
−
−−−
−−−
−− −





=



































 
 
 

Sorwar et al. [12] used the following SOR iteration
process to determine bond and contingent claims prices:
( )
11
11
1
mmmm
nnnn nnn
n
zU UU α χβ
η
−−
−+
= −−
(11)
In particular they evaluated bond using the following
expression:
( )
1
1
mmm
nn n
Uz U ωω
−
= +−
(12)
Contingent claims were calculated using:
( )
1
1
m mm
n nn
UmaxZ ,zU ωω
−

= +−

(13)
where Z is the intrinsic value of the contingent claim
and for n=1,......,N-1, and
( ]
12, ω ∈
3.
3. Analysis of Results
In this section we apply the EBM using recent estimates
of the non-linear model of Ait-Sahalia (1996) [4] on
7-day Eurodollar deposit spot rate over 1973-1995 to
demonstrate the method. Ait -Sahalia (1996, Table 4) [4]
obtained the following estimates:
3
01
21
23
4 64310
4 333101 143102
.,
., .,,
αα
αα
−
−−
=−× =
×=−× =
4
45
1 304101., αα
−
=×=
.
4
01
33
23
1 10810
1 883109 681102 073
.,
.,. ,.
ββ
ββ
−
−−
=×=
−×= ×=.
Table 2 reports the bond prices for maturities ranging
from 6 months to 30 years and across interest rates of 2%
to 16%. Table III reports both the value of call and put
options across a wide range of interest rates. We consider
both short and long dated call and put options. The short
dated call and put options are based on a 5-year bond
with an expiry date of 1 year and is during the last year
before the bond matures. Similarly long dated options are
based on 10-year bond with an expiry date of 5 years
during the last 5 years of the bond. Finally both call and
put option prices are calculated across a wide range of
exercise prices. The exercise prices are chosen so as to
highlight variation of prices for both in-the-money and
out-of-the-money options. We assume
λ , the market
price of risk is zero.
Turning to Table 2, we find that at lower interest rate
bond prices decay slowly as the term to maturity in-
creases. For example, at 2% interest rate a 1 year matur-
ity bond is valued at 98.1119, whilst a 30 year bond is
valued at 74.8290. At high interest rates, the bond price
decay is more rapid for example at 16% interest rate, a 1
year maturity bond is valued at 85.2915, whist a 30 year
maturity bond is valued at 1.1770. Turning to Table 3,
we observe the following features. Short expiry call op
3
ω is determined by numerical experimentation. For all our calcula-
tions we took
1 85. ω =

G. SORWAR ET AL.
41
Table 2. All options written on zero coupon bonds with a face value of $100.00.
Interest Rate
Ma t ur ity
of Bond
2%
4%
6%
8%
10%
12%
14%
16%
0.5
99.028 6
98.037 0
96.985 5
96.088 5
95.131 5
94.184 4
93.250 6
92.340 3
1
98.111 9
96.143 4
94.080 5
92.340 6
90.505 0
88.705 9
86.956 6
85.291 5
5
92.240 0
83.303 5
74.341 3
67.468 5
60.862 3
54.901 0
49.732 4
45.621 2
10
87.043 1
71.953 5
56.701 7
46.171 7
37.275 0
30.119 3
24.783 4
21.303 8
15
83.108 9
64.153 8
44.665 1
32.180 0
22.949 1
16.526 7
12.393 3
10.131 7
20
79.922 8
58.647 3
36.472 3
22.964 4
14.317 8
9.0809
6.2237
4.8889
25
77.215 6
54.633 8
30.873 1
16.883 2
9.0110
5.0032
3.1400
2.3870
30 74.8290 51.602 1 27.0075 12. 8582 5.7491 2.7 679 1.5 921 1.1770
Table 3. All options written on zero coupon bonds with a face value of $100.00.
r(%)
Exercise-
Price
5 year ma-
turity
1 year ex-
piry
5 year ma-
turity
1 year ex-
piry
Exercise-
Price
10 year
maturity
5 year ex-
piry
10 year
maturity
5 year ex-
piry
(83.3035)
call
put
(71.9535)
call
put
4
70
16.003 1
0.0000
60
21.971 3
0.0007
75
11.195 9
0.0000
65
17.806 2
0.0493
80
6.3895
0.0050
70
13.641 8
0.6489
85
1.9369
1.6966
75
9.5270
3.1894
90
0.1421
6.6966
80
5.7979
8.0466
(67.4685)
(46.1717)
8
55
16.681 1
0.0000
35
22.557 8
0.0000
60
12.064 1
0.0000
40
19.184 3
0.0000
65
7.4471
0.0000
45
15.810 9
0.0058
70
2.8302
2.5315
50
12.437 5
3.8283
75
0.0203
7.5315
55
9.0641
8.8283
12
(54.9010)
(30.1193)
45
14.934 1
0.0000
20
19.139 5
0.0000
50 10.4996 0 .0000 25 16.3942 0 .0000
55
6.0652
0.1561
30
13.649 2
0.0183
60
1.6310
5.1561
35
10.904 2
4.8804
65
0.0000
10.156 1
40
8.1591
9.8804
16
(45.6212)
(21.3038)
35 15.7692 0 .0000 10 16.7416 0 .0000
40
11.504 6
0.0000
15
14.460 6
0.0000
45
7.2400
0.0005
20
12.179 5
0.0001
50
2.9755
4.3788
25
9.8985
3.6962
55
0.0129
9.3782
30
7.6174
8.6962
tions decay faster than longer expiry call options; for
example at r = 4%; the price of a call option decreases
from 16.0031 to 11.1959 when the exercise price in
creases from 70 to 75. For a similar 5 year call option the
price decreases from 21.9713 to 17.8062, when the exer-
cise price increases from 60 to 65. Furthermore, the call
option prices decrease at a slower rate at high interests.
This feature becomes more pronounced for longer expiry
call options. With regard to put options we find, the
prices are very close to zero, when the options are at-the-
money or out-of-the -money. Finally, we find that the
value of in-the-money put options is dominated by the
intrinsic-va lue .
4. Conclusions
The introduction of non-linear stochastic interest rate
models has led to the possibility of valuing interest con-
tingent claims that reflects the characteristics of the yield
curve more accurately. In this paper we have expanded
the Box Method to value both bond and American type
interest rate contingent claims based on single factor
non-linear interest rate models. We have found that the

G. SORWAR ET AL.
42
Expanded Box Method works well with the example
considered.
5. References
[1] K. C. Chan, G. A. Karolyi, F. A. Longstaff and A. B.
Sanders, “An Empirical Comparison of Alternative Mod-
els of the Short-Term Interest Rate,” Journal of Finance,
Vol. 47, No. 3, 1992, pp. 1209-1227.
[2] J. C. Cox and S. A. Ross, “Option Pricing: A Simplified
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G. SORWAR ET AL.
43
Appendix
( )
()()
1
2
11
22
11
1
22 2
n
nn n
s
nn
ss s
U UU
sds ss
sss s
+
+− −
+−
∂∂∂ ∂

Ψ ≈Ψ−Ψ

∂∂∂ ∂

∫
Furt he r :
1
2
1
2
1
1
1
1
n
n
mm
nn
s
nn
mm
nn
snn
UU
U
s ss
UU
U
s ss
+
−
+
+
−
−
−
∂≈
∂−
− ∂≈
∂−
Substitution of the above approximation yields:
( )
()
()()()
1
21
2
1
2
11 1
22 2
1
1
1
11 1
n
n
s
nm
n
nn s
nn n
mm
nn
nn nnnn
s
U
s dsU
ss ss
ss s
UU
ss ssss
+
−
+
+
+
+− −
−
+− −
Ψ
∂∂

Ψ≈ −

∂∂ −


ΨΨΨ

++

−− −


∫
() ( )()
1
2
1
2
3
2
2
1
n
n
s
s
s
tfs QsUds
cs
+
−
∆≈
−
∫
( )
() ( )
1
2
1
2
3
2
2
1
n
n
s
m
nn
s
s
tQsUfs ds
cs
+
−
∆−
∫
We further take:
() ( )
() ( )
()
1
2
11
22
1
2
33
22
11
n
n
s
n
nnn
sn
s
sfsdsfs ss
cs cs
+
−
+−
≈−
−−
∫
Similar approximation yields:
() ( )()
( )() ( )
()
() ( )()
( )() ( )
()
1
2
1
2
11
22
1
2
1
2
11
22
2
2
0
2
1
2
1
2
1
1
2
1
1
2
1
1
2
1
n
n
n
n
s
s
m
nnnnn
n
s
s
m
nnn nn
n
fs QsUds
cs
QsUfsss
cs
fs QsUds
cs
QsUf sss
cs
+
−
+
−
+−
−
+−
≈
−
−
−
≈
−
−
−
∫
∫