System and methods for modeling a multiplicative index

• US 20090222385A1
• Filed: 04/10/2008
• Published: 09/03/2009
• Est. Priority Date: 02/28/2008
• Status: Active Grant

• Current Assignee: Morgan Stanley Services Group Inc. (Morgan Stanley)
• Original Assignee: Morgan Stanley
• Inventors: Chen, Mike
• Granted Patent: US 8,255,302 B2
• Time in Patent Office: Days
• US Class Current: 705/36.T
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis
• CPC Class Codes: G06Q 40/10 Tax strategies

Abstract

Computer-implemented methods for valuing a derivative based on the BMA rate: the methods may comprise generating a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic noise function, and a seasonality process. The methods may also comprise solving the model for at least one value of the LIBOR index, and estimating a value of the derivative given the solution of the model. The value of the derivative may then be stored.

Claims

1. A computer-implemented method for valuing a derivative based on the BMA rate, the method comprising:

generating a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic noise function, and a seasonality process;

solving the model for at least one value of the LIBOR index;

estimating a value of the derivative given the solution of the model; and

storing the value of the derivative.

2. The method of claim 1, wherein the derivative is a short-term derivative.

3. The method of claim 1, wherein the model is of the form:

BMA/LIBOR=ƒ

(LIBOR(t))+N+s(t)

wherein s(t) is the seasonality process; and

wherein N is the noise function.

4. The method of claim 3, wherein s(t) is a polynomial of the form:

$s<br><br>\left(t\right)=\underset{i=0}{\overset{P}{\sum <br><br>}}<br><br>c_i<br><br>x^i<br><br>\left(t\right)$

wherein P is the order of the polynomial;

wherein c is a constant; and

wherein xⁱ(t) represents a fraction of a year between the time t and the beginning of a year including t.

5. The method of claim 4, wherein P is greater than or equal to ten.

6. The method of claim 5, wherein P is equal to 10, and wherein c₀=−

0.0260;

c₁=0.5455;

C₂=−

32.5873;

C₃=296.7654;

C₄=−

951.9348;

C₅=1081.07;

C₆=258.3131;

C₇=−

1061.1486;

C₈=−

311.0940;

c₉=1262.5281; and

c₁₀=−

542.4610.

7. The method of claim 4, wherein s(0)=s(1).

8. The method of claim 1, wherein the noise function is expressed as a function of independent and identically distributed (IID) random variables.

9. The method of claim 8, wherein the noise function is expressed as a Normal density.

10. The method of claim 9, wherein the Normal density is a best fit density with a set of residuals between observed values of the BNA/LIBOR ratio and the BNA/LIBOR ratio generated by the model without considering the noise function.

11. The method of claim 1, wherein the noise function is expressed utilizing an auto-correlated model of a set of residuals between observed values of the BNA/LIBOR ratio and the BNA/LIBOR ratio generated by the model without considering the noise function.

12. The method of claim 11, wherein the auto-correlated model comprises a model selected from the group consisting of an Autoregressive Moving Average (ARMA) model;

a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model; and

a combination of an ARMA model and a GARCH model.

13. The method of claim 1, wherein the model of the BMA/LIBOR ratio is also a function of a stochastic tax-regime process.

14. The method of claim 13, wherein the tax-regime process comprises a jump-process model.

15. The method of claim 13, wherein the tax-regime process comprises a Markov-Chain with quarterly transitions, wherein the Markov-Chain comprises a first tax-exempt state and a second exemption-repealed state.

16. The method of claim 15, further comprising determining at least one transition probability of the Markov-Chain considering a value of a derivative based on at least one factor selected from the group consisting of the BMA rate and the BMA/LIBOR ratio.

17. The method of claim 16, wherein the derivative pays when a difference between the BMA rate and the LIBOR rate is less than the top marginal tax rate.

18. A system for valuing a derivative based on the BMA rate, the system comprising at least one processor programmed to:

generate a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic noise function, and a seasonality process;

solve the model for at least one value of the LIBOR index;

estimate a value of the derivative given the solution of the model; and

store the value of the derivative.

19. A computer readable medium having instructions thereon that when executed by at least one processor cause the at least one processor to:

generate a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic noise function, and a seasonality process;

solve the model for at least one value of the LIBOR index;

estimate a value of the derivative given the solution of the model; and

store the value of the derivative.

20. A computer-implemented method for valuing a derivative based on the BMA rate, the method comprising:

generating a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic tax-regime process, and a seasonality process;

solving the model for at least one value of the LIBOR index;

estimating a value of the derivative given the solution of the model; and

storing the value of the derivative.

21. The method of claim 20, wherein the derivative is a long-term derivative.

22. The method of claim 20, wherein the model is of the form:

BMA/LIBOR=ƒ

_r(LIBOR(t))+S_r(t)

wherein s(t) is the seasonality process; and

wherein r is the tax-regime process.

23. The method of claim 20, wherein the tax-regime process comprises a jump-process model.

24. The method of claim 20, wherein the tax-regime process comprises a Markov-Chain with quarterly transitions, wherein the Markov-Chain comprises a first tax-exempt state and a second exemption-repealed state.

25. The method of claim 20, further comprising determining at least one transition probability of the Markov-Chain considering a value of a derivative based on at least one factor selected from the group consisting of the BMA rate and the BMA/LIBOR ratio.

26. The method of claim 20, wherein the model is also a function of a noise function.

27. The method of claim 26, wherein the noise function is expressed as a function of independent and identically distributed (IID) random variables.

28. The method of claim 27, wherein the noise function is expressed utilizing an auto-correlated model of a set of residuals between observed values of the BNA/LIBOR ratio and the BNA/LIBOR ratio generated by the model without considering the noise function.

29. A system for valuing a derivative based on the BMA rate, the system comprising at least one processor programmed to:

generate a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic tax-regime process, and a seasonality process;

solve the model for at least one value of the LIBOR index;

estimate a value of the derivative given the solution of the model; and

store the value of the derivative.

30. A computer readable medium having instructions thereon that when executed by at least one processor cause the at least one processor to:

generate a model of the BMA/LIBOR ratio as a function of the LIBOR index, a stochastic tax-regime process, and a seasonality process;

solve the model for at least one value of the LIBOR index;

estimate a value of the derivative given the solution of the model; and

store the value of the derivative.