Methods and Systems for Creating a Government Bond Volatility Index and Trading Derivative Products Based Thereon
A computer system for calculating a government bond volatility index comprising memory configured to store at least one program; and at least one processor communicatively coupled to the memory, in which the at least one program, when executed by the at least one processor, causes the at least one processor to receive data regarding options on government bond derivatives; calculate, using the data regarding options on government bond derivatives, the government bond volatility index; and transmit data regarding the government bond volatility index.
1 . A computer system for calculating a government bond volatility index comprising:
memory configured to store at least one program; and
at least one processor communicatively coupled to the memory, in which the at least one program, when executed by the at least one processor, causes the at least one processor to:
receive data regarding options on government bond derivatives;
calculate, using the data regarding options on government bond derivatives, the government bond volatility index; and
transmit data regarding the government bond volatility index.
2 . The computer system of claim 1 , wherein the data regarding options on government bond derivatives includes data regarding prices of options on government bond derivatives.
3 . The computer system of claim 2 , wherein the data regarding prices of options on government bond derivatives includes data regarding prices of European style options on government bond forwards.
4 . The computer system of claim 2 , wherein the data regarding prices of options on government bond derivatives includes data regarding prices of options that are not European style options on government bond forwards.
5 . The computer system of claim 4 , wherein, when the data regarding prices of options on government bond derivatives includes data regarding prices of options that are not European-style options on government bond forwards, converting the data regarding prices of options that are not European-style options on government forwards to data regarding prices of European style options on government bond forwards.
6 . The computer systems of claim 1 , wherein calculating the government bond volatility index includes valuing a basket of options on the government bond derivatives required for model-independent pricing of a variance swap contract on the government bond derivatives.
7 . The computer systems of claim 3 , 4 , 5 , or 6 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
*
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
*
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
8 . The computer systems of claim 3 , 4 , 5 , or 6 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ; and
GB-VI bp (t,T,T D ,T N ) is the value of the government bond volatility index at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
9 . The computer systems of claim 3 , 4 , 5 , or 6 ,
wherein, in the absence of accrued coupons at time T, the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Y
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
P
^
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
×
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
where
P
^
(
y
)
≡
∑
i
=
1
N
C
i
n
(
1
+
y
n
)
-
i
+
100
(
1
+
y
n
)
-
N
;
and, wherein, in the presence of accrued coupons at time T with the next coupon due at t j , the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Y
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
×
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
where
P
^
T
(
x
)
≡
∑
i
=
j
N
C
i
n
(
1
+
x
)
-
c
(
t
i
-
T
)
c
(
year
)
+
100
(
1
+
x
)
-
c
(
t
n
-
T
)
c
(
year
)
and
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
and
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
t j is the first coupon payment on or after T;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
N denotes the total number of coupon payments of a government bond;
C i denotes the amount of the i th coupon out of N coupons of a government bond;
n denotes the frequency of coupon payments per annum of a government bond;
y denotes the yield of a government bond;
x denotes the yield of a government bond;
{circumflex over (P)}(y) is a bond price corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} −1 (y) is the functional inverse of {circumflex over (P)}(y);
{circumflex over (P)} T (x) is a bond price at time T corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} T −1 (x) is the functional inverse of {circumflex over (P)} T (x);
dc(year) is the number of days in a year based on a day count convention used for the government bond;
dc(T−t) is the number of days between t and T based on a day count convention used for the government bond;
GB-VI y bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point yield volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ;
GB-VI bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index in terms of percentage price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
10 . The computer systems of claim 3 , 4 , 5 , or 6 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Yd
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
(
1
+
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
)
×
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
∑
i
=
j
N
C
i
n
(
1
+
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
)
-
c
(
t
i
-
T
)
c
(
year
)
(
c
(
t
i
-
T
)
c
(
year
)
)
+
100
(
1
+
P
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
)
-
c
(
t
N
-
T
)
c
(
year
)
(
c
(
t
N
-
T
)
c
(
year
)
)
where
P
^
T
(
x
)
≡
∑
i
=
j
N
C
i
n
(
1
+
x
)
-
c
(
t
i
-
T
)
c
(
year
)
+
100
(
1
+
x
)
-
c
(
t
N
-
T
)
c
(
year
)
and
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
*
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
*
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
and
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call i (K i ,T,T D , T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
N denotes the total number of coupon payments of a government bond;
C i denotes the amount of the i th coupon out of N coupons of a government bond;
n denotes the frequency of coupon payments per annum of a government bond;
x denotes the yield of a government bond;
{circumflex over (P)} T (x) is a bond price corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} T −1 (x) is the functional inverse of {circumflex over (P)} T (x);
dc(year) is the number of days in a year based on a day count convention used for the government bond;
dc(T−t) is the number of days between t and T based on a day count convention used for the government bond;
t j is the first coupon payment on or after T;
GB-VI Yd bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point yield volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ;
GB-VI bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index in terms of percentage price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
11 . The computer system of claim 1 , wherein the at least one processor is further caused to:
create a standardized exchange-traded derivative instrument based on the government bond volatility index; and
transmit data regarding the standardized exchange-traded derivative.
12 . The computer system of claim 11 , wherein transmitting data regarding the standardized exchange-traded derivative instrument includes transmitting data regarding one or more of a settlement price, a bid price, an offer price, or a trade price of the standardized exchange-traded derivative instrument.
13 . A non-transitory computer readable storage medium having computer-executable instructions recorded thereon that, when executed on a computer, configure the computer to perform a method to calculate a government bond volatility index, the method comprising:
receiving data regarding options on government bond derivatives;
calculating, using the data regarding options on government bond derivatives, the government bond volatility index; and
transmitting data regarding the government bond volatility index.
14 . The non-transitory computer readable storage medium of claim 13 , wherein the data regarding options on government bond derivatives includes data regarding prices of options on government bond derivatives.
15 . The non-transitory computer readable storage medium of claim 14 , wherein the data regarding prices of options on government bond derivatives includes data regarding prices of European style options on government bond forwards.
16 . The non-transitory computer readable storage medium of claim 14 , wherein the data regarding prices of options on government bond derivatives includes data regarding prices of options that are not European style options on government bond forwards.
17 . The non-transitory computer readable storage medium of claim 16 , wherein, when the data regarding prices of options on government bond derivatives includes data regarding prices of options that are not European-style options on government bond forwards, converting the data regarding prices of options that are not European-style options on government bond forwards to data regarding prices of European style options on government bond forwards.
18 . The non-transitory computer readable storage medium of claim 13 , wherein calculating the government bond volatility index includes valuing a basket of options on the government bond derivatives required for model-independent pricing of a variance swap contract on the government bond derivatives.
19 . The non-transitory computer readable storage medium of claim 15 , 16 , 17 or 18 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
*
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
*
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
20 . The non-transitory computer readable storage medium of claim 15 , 16 , 17 or 18 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ; and
GB-VI bp (t,T,T 1 ,T N ) is the value of the government bond volatility index at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
21 . The non-transitory computer readable storage medium of claim 15 , 16 , 17 or 18 ,
wherein, in the absence of accrued coupons at time T, the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Y
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
P
^
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
×
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
where
P
^
(
y
)
≡
∑
i
=
1
N
C
i
n
(
1
+
y
n
)
-
i
+
100
(
1
+
y
n
)
-
N
;
and, wherein, in the presence of accrued coupons at time T with the next coupon due at t j , the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Y
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
×
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
where
P
^
T
(
x
)
≡
∑
i
=
j
N
C
i
n
(
1
+
x
)
-
c
(
t
i
-
T
)
c
(
year
)
+
100
(
1
+
x
)
-
c
(
t
N
-
T
)
c
(
year
)
and
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
*
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
*
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
and
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
t j is the first coupon payment on or after T;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i T,T D ,T N ) is a price at time t of a put option, struck at expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
N denotes the total number of coupon payments of a government bond;
C i denotes the amount of the i th coupon out of N coupons of a government bond;
n denotes the frequency of coupon payments per annum of a government bond;
y denotes the yield of a government bond;
x denotes the yield of a government bond;
{circumflex over (P)}(y) is a bond price corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} −1 (y) is the functional inverse of {circumflex over (P)}(y);
{circumflex over (P)} T (x) is a bond price at time T corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} T −1 (x) is the functional inverse of {circumflex over (P)} T (x);
dc(year) is the number of days in a year based on a day count convention used for the government bond;
dc(T−t) is the number of days between t and T based on a day count convention used for the government bond;
GB-VI Y bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point yield volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ;
GB-VI bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index in terms of percentage price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
22 . The non-transitory computer readable storage medium of claim 15 , 16 , 17 or 18 , wherein the government bond volatility index is calculated at time t according to the equation:
GB
-
VI
Yd
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
(
1
+
P
^
T
-
1
[
I
t
bp
I
t
perc
]
)
×
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
∑
i
=
j
N
C
i
n
(
1
+
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
)
-
c
(
t
i
-
T
)
c
(
year
)
(
c
(
t
i
-
T
)
c
(
year
)
)
+
100
(
1
+
P
^
T
-
1
[
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
]
)
-
c
(
t
N
-
T
)
c
(
year
)
(
c
(
t
N
-
T
)
c
(
year
)
)
where
P
^
T
(
x
)
≡
∑
i
=
j
N
C
i
n
(
1
+
x
)
-
c
(
t
i
-
T
)
c
(
year
)
+
100
(
1
+
x
)
-
c
(
t
N
-
T
)
c
(
year
)
and
GB
-
VI
(
t
,
T
,
T
D
,
T
N
)
≡
100
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
*
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
+
∑
i
:
K
i
≥
K
*
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
K
i
2
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
K
*
)
2
and
GB
-
VI
bp
(
t
,
T
,
T
D
,
T
N
)
≡
100
2
×
2
P
t
(
T
)
(
T
-
t
)
[
∑
i
:
K
i
<
K
0
Put
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
+
∑
i
:
K
i
≥
K
0
Call
t
(
K
i
,
T
,
T
D
,
T
N
)
Δ
K
i
]
-
(
F
t
(
T
D
,
T
N
)
-
K
*
)
2
wherein:
t denotes a time at which the government bond volatility index is calculated;
T denotes a time of expiry of options on government bond derivatives;
T D denotes a time of maturity of government bond derivatives underlying the options where T D ≧T;
T N denotes a time of expiry of government bonds;
Z+1 denotes a total number of options used in the index calculation;
K 0 denotes the lowest strike of the Z+1 options;
K i denotes the i th highest strike of the Z+1 options;
K Z denotes the highest strike of the Z+1 options;
Δ K i =½( K i+1 −K i−1 ) for i≧ 1, and Δ K 0 =( K 1 −K 0 ),Δ K Z =( K Z −K Z−1 );
if the price is observable at time t, then F t (T D ,T N ) is a price at time t of a government bond derivative contract underlying the put and call options, expiring at T D with an underlying government bond maturing at T N ;
if the price is not observable at time t, then F t (T D ,T N ) is the strike at which the difference between the put and call prices is smallest;
if there exists an option struck at F t (T D ,T N ), then K * equals F t (T D ,T N );
if there does not exist an option struck at F t (T D ,T N ), then K * is the first available strike below F t (T D ,T N );
P t (T) is a price at time t of a zero-coupon non-defaultable bond maturing at T;
Put t (K i ,T,T D ,T N ) is a price at time t of a put option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
Call t (K i ,T,T D ,T N ) is a price at time t of a call option, struck at K i , expiring at T, and having an underlying government bond derivative expiring at T D with an underlying bond maturing at T N ;
N denotes the total number of coupon payments of a government bond;
C i denotes the amount of the i th coupon out of N coupons of a government bond;
n denotes the frequency of coupon payments per annum of a government bond;
x denotes the yield of a government bond;
{circumflex over (P)} T (x) is a bond price corresponding to a bond yield of a coupon-bearing government bond;
{circumflex over (P)} T −1 (x) is the functional inverse of {circumflex over (P)} T (x);
dc(year) is the number of days in a year based on a day count convention used for the government bond;
dc(T−t) is the number of days between t and T based on a day count convention used for the government bond;
t j is the first coupon payment on or after T;
GB-VI Yd bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point yield volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ;
GB-VI bp (t,T,T D ,T N ) is the value of the government bond volatility index in terms of basis point price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N ; and
GB-VI(t,T,T D ,T N ) is the value of the government bond volatility index in terms of percentage price volatility at time t calculated based on options expiring at T on government bond derivatives expiring at T D with an underlying bond maturing at T N .
23 . The non-transitory computer readable storage medium of claim 13 , wherein the at least one processor is further caused to:
create a standardized exchange-traded derivative instrument based on the government bond volatility index; and
transmit data regarding the standardized exchange-traded derivative.
24 . The non-transitory computer readable storage medium of claim 23 , wherein transmitting data regarding the standardized exchange-traded derivative instrument includes transmitting data regarding one or more of a settlement price, a bid price, an offer price, or a trade price of the standardized exchange-traded derivative instrument.