Systems and methods for valuing a derivative involving a multiplicative index
View Patent ↗Methods for calculating a value of a derivative based on a multiplicative index, where the multiplicative index is a multiple of an underlying index. The methods may comprise the steps of providing a first value of the underlying index and estimating a ratio of the multiplicative index over the underlying index as a first function of the underlying index over a first range of values of the underlying index. The method may also comprise the steps of calculating at least one parameter of the first function by comparing the first function to historical values of the multiplicative index and the underlying index. In addition, the method may comprise the steps of calculating a first value for the ratio given the first value of the underlying index, and calculating the value of the derivative for the first value of the ratio.
1. A computer system for calculating a value of a derivative, the system comprising:
at least one electronic processor and operatively associated memory, wherein the memory comprises instructions that when executed by the processor cause the processor to:
receive data indicating historical values of a first index, wherein the first index indicates an interest rate for tax-exempt municipal bonds;
receive data indicating historical values of a second index, wherein the second index indicates an interest rate for non-tax-exempt instruments;
receive data indicating a value of the second index;
estimate a ratio of the first index over the second index over at least a range of potential values of the second index as an index function, wherein:
the
index
function
=
(
a
T
F
T
2
+
b
T
F
T
+
c
T
)
Θ
(
F
T
)
+
(
c
T
-
b
T
2
4
a
T
)
(
1
-
Θ
(
F
T
)
)
the
index
function
=
1
,
if
F
T
≤
-
b
T
2
a
T
;
the
index
function
=
0
,
if
F
T
>
-
-
b
T
2
a
T
;
wherein F T is the second index, a T , b T and c T are parameters of the index function, and wherein estimating the ratio comprises calculating a T , b T and c T by comparing the index function to the data indicating historical values of the first index and the data indicating historical values of the second index;
calculate a value for the index function based on the calculated values for a T , b T and c T and the value of the second index; and
calculate the value of the derivative considering the calculated value of the index function and the value of the second index, wherein the value of the derivative is a function of the first index.
2. The system of claim 1 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to modify a characteristic of the derivative based on the value of the derivative for the first value of the index function.
3. The system of claim 1 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to hedge the derivative with a second derivative based on the second index.
4. The system of claim 1 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to estimate the index function as a constant over a second range of values of the second index.
5. The system of claim 1 , wherein calculating a T , b T and c T comprises calculating a T , b T and c T according to a linear regression method.
6. The system of claim 1 , wherein the first index is the BMA Municipal Swap Index and the second index is the 7 day LIBOR rate.
7. The system of claim 1 , wherein the first index is a four week rolling average of the BMA Municipal Swap Index and the second index is the 30 day LIBOR rate.
8. A computer system for calculating a value of a derivative, the system comprising:
at least one electronic processor and operatively associated memory, wherein the memory comprises instructions that when executed by the processor cause the processor to:
receive data indicating historical values of a first index, wherein the first index indicates an interest rate for tax-exempt municipal bonds;
receive data indicating historical values of a second index, wherein the second index indicates an interest rate for non-tax-exempt instruments;
receive data indicating a value of the second index;
estimate a ratio of the first index over the second index over at least a range of potential values of the second index as an index function, wherein:
the
index
function
=
(
a
T
F
T
2
=
b
T
F
T
+
c
T
)
Θ
(
F
T
)
+
(
c
T
-
b
T
2
4
a
T
)
(
1
-
Θ
(
F
T
)
)
the
index
function
=
1
,
if
F
T
<
_
-
b
T
2
a
T
;
the
index
function
=
0
,
if
F
T
>
-
b
T
2
a
T
;
wherein F T is the second index, a T , b T and c T are parameters of the index function, and wherein estimating the ratio comprises calculating a T , b T and c T by comparing the index function to the data indicating historical values of the first index and the data indicating historical values of the second index;
receive a first value function of the derivative, wherein the value function is in terms of the first index;
derive a second value function of the derivative in terms of the second index by substituting the index function into the first value function; and
calculate the value of the derivative by computing the second value function for the first value of the second index.
9. The system of claim 8 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to modify a characteristic of the derivative based on the value of the derivative.
10. The system of claim 8 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to hedge the derivative with a second derivative based on the second index.
11. The system of claim 8 , wherein calculating a T , b T and c T comprises calculating a T , b T and c T according to a linear regression method.
12. The system of claim 8 , wherein the first index is the BMA Municipal Swap Index and the second index is the 7 day LIBOR rate.
13. The system of claim 8 , wherein the memory further comprises instructions that when executed by the at least one processor cause the at least one processor to estimate the index function as a constant over a range of the second index.
14. A computer readable storage medium including instructions that, when executed by at least one processor, cause the at least one processor to:
receive data indicating historical values of a first index, wherein the first index indicates an interest rate for tax-exempt municipal bonds;
receive data indicating historical values of a second index, wherein the second index indicates an interest rate for non-tax-exempt instruments;
receive data indicating a value of the second index;
estimate a ratio of the first index over the second index over at least a range of potential values of the second index as an index function, wherein:
the
index
function
=
(
a
T
F
T
2
=
b
T
F
T
+
c
T
)
Θ
(
F
T
)
+
(
c
T
-
b
T
2
4
a
T
)
(
1
-
Θ
(
F
T
)
)
the
index
function
=
1
,
if
F
T
<
_
-
b
T
2
a
T
;
the
index
function
=
0
,
if
F
T
>
-
b
T
2
a
T
;
wherein F T is the second index, a T , b T and c T are parameters of the index function, and wherein estimating the ratio comprises calculating a T , b T and c T by comparing the index function to the data indicating historical values of the first index and the data indicating historical values of the second index;
calculate a value for the index function based on the calculated parameter calculated values for a T , b T and c T and the value of the second index; and
calculate the value of the derivative considering the calculated value of the index function and the value of the second index, wherein the value of the derivative is a function of the first index, and wherein the computer readable storage medium is a tangible computer readable medium.
15. A computer readable storage medium including instructions that when executed cause a processor to:
receive data indicating historical values of a first index, wherein the first index indicates an interest rate for tax-exempt municipal bonds;
receive data indicating historical values of a second index, wherein the second index indicates an interest rate for non-tax-exempt instruments;
receive data indicating a value of the second index;
estimate a ratio of the first index over the second index over at least a range of potential values of the second index as an index function, wherein:
the
index
function
=
(
a
T
F
T
2
=
b
T
F
T
+
c
T
)
Θ
(
F
T
)
+
(
c
T
-
b
T
2
4
a
T
)
(
1
-
Θ
(
F
T
)
)
the
index
function
=
1
,
if
F
T
<
_
-
b
T
2
a
T
;
the
index
function
=
0
,
if
F
T
>
-
b
T
2
a
T
;
wherein F T is the second index, a T , b T and c T are parameters of the index function, and wherein estimating the ratio comprises calculating a T , b T and c T by comparing the index function to the data indicating historical values of the first index and the data indicating historical values of the second index;
receive a first value function of the derivative, wherein the value function is in terms of the first index;
derive a second value function of the derivative in terms of the second index by substituting the index function into the first value function; and
calculate the value of the derivative by computing the second value function for the first value of the second index, and wherein the computer readable storage medium is a tangible computer readable medium.