Bond option modeling
View Patent ↗Systems and methods for determining a present value of an option on a security having a fixed cash flow leg based upon a Martingale. The Martingale may be based upon a ratio of the present value of the option and a numeraire. The numeraire may be a coupon annuity which may be based on coupons of the security post expiry of the option, accrual periods of the coupon, and spread-adjusted discount factors for coupon dates of the option. The spread-adjusted discount factor may be based on an instantaneous forward rate and a time-varying spread. The present value of the option may be determined based upon a spread, a notional value of the security, and an expectation of a maximum value of (1) a difference between an artificial strike coupon and a forward swap rate and (2) zero. This spread may equal a difference between the forward swap rate and a strike coupon or the strike coupon divided by the forward swap rate.
1. A computer-implemented method for determining a present value of an option on a security having a fixed cash flow leg, the method comprising:
computing, with a computer system that comprises a processor and memory, a numeraire for the option;
computing, with the computer system, a strike coupon rate for the option based on a strike price of the option, a notional of the security, coupon dates of the security, and discount factors for the security; and
computing, with the computer system, the present value of the option based on a product of factors, the factors comprising:
the numeraire;
the notional of the security; and
an expected value of the greater of a plurality of values, wherein the plurality of values comprises:
a first value that is a constant; and
a second value that is computed based on at least a coupon rate of the security and the strike coupon rate for the option, wherein the strike coupon rate is a Martingale.
2. A computer system comprising:
a processor; and
a memory in communication with the processor, wherein the memory stores instructions which when executed by the processor causes the processor to determine a present value of an option on a security having a fixed cash flow leg, by:
computing a numeraire for the option;
computing a strike coupon rate for the option based on a strike price of the option, a notional of the security, coupon dates for the security, and discount factors for the security; and
computing the present value of the option based on a product of factors, the factors comprising:
the numeraire;
the notional of the security; and
an expected value of the greater of a plurality of values, wherein the plurality of values comprises:
a first value that is a constant; and
a second value that is computed based on at least a coupon rate of the security and the strike coupon rate for the option, wherein the strike coupon rate is a Martingale.
3. A computer system comprising:
a processor; and
a database in communication with the processor, wherein the database stores financial data for the securities, and wherein the processor is programmed to determine a present value of an option on a security having a fixed cash flow leg, by:
computing a numeraire for the option;
computing a strike coupon rate for the option based on a strike price of the option, a notional of the security, coupon dates for the security, and discount factors for the security; and
computing the present value of the option based on a product of factors, the factors comprising:
the numeraire;
the notional of the security; and
an expected value of the greater of a plurality of values, wherein the plurality of values comprises:
a first value that is a constant; and
a second value that is computed based on at least a coupon rate of the security and the strike coupon rate for the option, wherein the strike coupon rate is a Martingale.
4. The method of claim 1 , wherein a forward swap rate for a swap market is also used to compute the second value, wherein the forward swap rate is related to the strike coupon rate.
5. The method of claim 4 , wherein the second value is computed based on the assumption that a difference between the forward swap rate and the strike coupon rate for the option is constant throughout the life of the option.
6. The method of claim 5 , wherein the second value is computed as (C+f tn −C k )−f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where to is a swap rate tenor.
7. The method of claim 4 , wherein the second value is computed based on the assumption that a ratio of the strike coupon rate for the option to the forward swap rate is constant throughout the life of the option.
8. The method of claim 7 , wherein the second value is computed as C·(f tn /C k ) t=t 0 −f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where tn is a swap rate tenor.
9. The method of claim 4 , wherein first value is zero.
10. The method of claim 1 , wherein the security comprises a bond.
11. The method of claim 10 , wherein the bond comprises a government bond.
12. The method of claim 10 , wherein the bond comprises a municipal bond.
13. The method of claim 10 , wherein the bond comprises a corporate bond.
14. The method of claim 1 , wherein the strike coupon rate for the option is computed as
C
K
=
K
/
L
-
Λ
(
t
exp
,
t
N
)
∑
i
=
1
N
Δ
i
Λ
(
t
exp
,
t
i
)
,
where C k is the strike coupon rate, K is a strike price of the option, L is the notional of the security, Δ i is an accrual period for an ith coupon of the security, and Λ represents the discount factors.
15. The computer system of claim 2 , wherein a forward swap rate for a swap market is also used to compute the second value, wherein the forward swap rate is related to the strike coupon rate.
16. The computer system of claim 15 , wherein the second value is computed based on the assumption that a difference between the forward swap rate and the strike coupon rate for the option is constant throughout the life of the option.
17. The computer system of claim 16 , wherein the second value is computed as (C+f tn −C k )−f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where tn is a swap rate tenor.
18. The computer system of claim 2 , wherein the second value is computed based on the assumption that a ratio of the strike coupon rate for the option to the forward swap rate is constant throughout the life of the option.
19. The computer system of claim 18 , wherein the second value is computed as C·(f tn /C k ) t=t 0 −f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where tn is a swap rate tenor.
20. The computer system of claim 2 , wherein first value is zero.
21. The computer system of claim 2 , wherein the security comprises a bond.
22. The computer system of claim 21 , wherein the bond comprises a government bond.
23. The computer system of claim 21 , wherein the bond comprises a municipal bond.
24. The computer system of claim 21 , wherein the bond comprises a corporate bond.
25. The computer system of claim 2 , wherein the strike coupon rate for the option is computed as
C
K
=
K
/
L
-
Λ
(
t
exp
,
t
N
)
∑
i
=
1
N
Δ
i
Λ
(
t
exp
,
t
i
)
,
where C k is the strike coupon rate, K is a strike price of the option, L is the notional of the security, Δ i is an accrual period for an ith coupon of the security, and Λ represents the discount factors.
26. The computer system of claim 3 , wherein a forward swap rate for a swap market is also used to compute the second value, wherein the forward swap rate is related to the strike coupon rate.
27. The computer system of claim 26 , wherein the second value is computed based on the assumption that a difference between the forward swap rate and the strike coupon rate for the option is constant throughout the life of the option.
28. The computer system of claim 27 , wherein the second value is computed as (C+f tn −C k )−f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where tn is a swap rate tenor.
29. The computer system of claim 3 , wherein the second value is computed based on the assumption that a ratio of the strike coupon rate for the option to the forward swap rate is constant throughout the life of the option.
30. The computer system of claim 29 , wherein the second value is computed as C·(f tn /C k ) t=t 0 −f tn , where C is the coupon rate of the security, C k is the strike coupon rate, and f tn is the forward swap rate, where tn is a swap rate tenor.
31. The computer system of claim 3 , wherein first value is zero.
32. The computer system of claim 3 , wherein the security comprises a bond.
33. The computer system of claim 32 , wherein the bond comprises a government bond.
34. The computer system of claim 32 , wherein the bond comprises a municipal bond.
35. The computer system of claim 32 , wherein the bond comprises a corporate bond.
36. The computer system of claim 3 , wherein the strike coupon rate for the option is computed as
C
K
=
K
/
L
-
Λ
(
t
exp
,
t
N
)
∑
i
=
1
N
Δ
i
Λ
(
t
exp
,
t
i
)
,
where C k is the strike coupon rate, K is a strike price of the option, L is the notional of the security, Δ i is an accrual period for an ith coupon of the security, and Λ represents the discount factors.