System and method for pricing default insurance

• US 20030115125A1
• Filed: 10/17/2002
• Published: 06/19/2003
• Est. Priority Date: 12/14/2001
• Status: Active Grant

• Current Assignee: Morgan Stanley Services Group Inc. (Morgan Stanley)
• Original Assignee: Morgan Stanley
• Inventors: Zhang, Zhifeng, Cotton, Peter, Pang, Kin, Lee, Shinghoi
• Granted Patent: US 7,774,255 B2
• Time in Patent Office: Days
• US Class Current: 705/36
• CPC Class Codes: G06Q 40/00 Finance; Insurance; Tax strategies; Processing of corporate or income taxes
• CPC Class Codes: G06Q 40/03 Credit; Loans; Processing thereof
• CPC Class Codes: G06Q 40/04 Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis
• CPC Class Codes: G06Q 40/08 Insurance

Abstract

A system for modeling a basket of securities containing a plurality of securities is provided and includes a default/recovery model database for storing default/recovery data regarding the plurality of securities. Also included is a default simulation engine for calculating a default time for at least one of the plurality of securities based on the default/recovery data. Finally, a cash flow engine is included for generating cash flows for the basket of securities based on the default times.

Claims

1. A method for calculating a default time for at least one of a plurality of securities contained in a basket of securities, the method comprising the steps of:

identifying a plurality of hazard rates, each of said plurality of hazard rates corresponding to one of said plurality of securities;

formulating a plurality of compensators, each of said plurality of compensators based on one of said plurality of hazard rates corresponding to each of said plurality of securities;

selecting a plurality of barriers, each of said plurality of barriers corresponding to one of said plurality of securities;

determining a proposed default time for said at least one of said plurality of securities;

calculating a value of said compensator and said barrier corresponding to said at least one of said plurality of securities based on said proposed default time; and

determining that said proposed default time is said default time for said at least one of said plurality of securities.

2. The method of claim 1, further comprising the step of:

repeating the steps of selecting a plurality of barriers, determining a proposed default time for said at least one of said plurality of securities, calculating a value of said compensator and said barrier and determining that said proposed default time is said default time a plurality of times.

3. The method of claim 1, wherein the step of determining a proposed default time includes the steps of:

calculating values of said plurality of hazard rates and said plurality of barriers calculated at time zero;

forming a plurality of ratios wherein each of said plurality of ratios equals one of said plurality of barriers calculated at time zero divided by a corresponding one of said plurality of hazard rates calculated at time zero; and

selecting a smallest one of said plurality of ratios as said proposed default time.

4. The method of claim 3, wherein the step of determining that said proposed default time is said default time includes the step of:

(e) calculating said compensator of said at least one of said plurality of securities and said barriers of said at least one of said plurality of securities using said proposed default time; and

(f) determining that said proposed default time is said default time if said compensator of said at least one of said plurality of securities calculated using said proposed default time is within a selected percentage of said barriers of at least one of said plurality of securities calculated using said proposed default time.

5. The method of claim 4, wherein if said compensator of at least one of said plurality of securities calculated using said proposed default time is not within a selected percentage of said barriers of at least one of said plurality of securities calculated using said proposed default time, the method includes the steps of:

(g) calculating an adjustment to said proposed default time based on said compensator, said barrier and said hazard rate of said at least one of said plurality of securities.

6. The method of claim 5, further comprising the step of:

repeating steps (e)-(g) until it is determined that said compensator of said at least one of said plurality of securities calculated using said proposed default time is within a selected percentage of said barriers of at least one of said plurality of securities calculated using said proposed default time.

7. The method of claim 5, further comprising the step of:

repeating steps (e)-(g) until it is determined that said compensator of said at least one of said plurality of securities calculated using said proposed default time exceeds said barriers of at least one of said plurality of securities calculated using said proposed default time.

8. The method of claim 5, wherein said step of calculating includes the step of:

calculating said adjustment according to;

(B−

H)/h

wherein B is said barrier of said at least one of said plurality of securities evaluated at said proposed default time, H is said compensator of said at least one of said plurality of securities evaluated at said proposed default time and h is said hazard rate of said at least one of said plurality of securities evaluated at said proposed default time.

9. The method of claim 1, wherein said basket of securities has a time horizon and wherein the step of determining a default time includes the step of:

setting said proposed default time to said time horizon.

10. The method of claim 1, wherein said basket of securities has a time horizon and wherein the step of determining a default time includes the step of:

setting said proposed default time as a fraction of said time horizon.

11. The method of claim 10, wherein the step of determining that said proposed default time is said default time includes the steps of:

(h) calculating said compensator of at least one of said plurality of securities and said barriers of at least one of said plurality of securities using said proposed default time; and

(i) determining that said proposed default time is said default time if said compensator of said at least one of said plurality of securities calculated using said proposed default time is in a relative relationship with said barriers of at least one of said plurality of securities calculated using said proposed default time.

12. The method of claim 11, wherein said relative relationship exists when said compensator of said at least one of said plurality of securities calculated using said proposed default time is within a selected percentage of said barriers of at least one of said plurality of securities calculated using said proposed default time.

13. The method of claim 11, wherein said relative relationship exists when said compensator of said at least one of said plurality of securities calculated using said proposed default time exceeds said barriers of at least one of said plurality of securities calculated using said proposed default time.

14. The method of claim 11, wherein if said compensator of at least one of said plurality of securities calculated using said proposed default time is not within a selected percentage of said barriers of said at least one of said plurality of securities calculated using said proposed default time, the method includes the steps of:

(j) incrementing said proposed default time by an additional fraction of said time horizon; and

repeating steps (h)-(j) until it is determined that said compensator of at least one of said plurality of securities calculated using said proposed default time is within a selected percentage of said barriers of at least one of said plurality of securities calculated using said proposed default time.

15. The method of claim 1, wherein each of said plurality of barriers is a random variable.

16. The method of claim 15, wherein said barriers are dependent.

17. The method of claim 16, wherein said dependence between said barriers is specified by a Copula function.

18. The method of claim 17, wherein said Copula function is gaussian.

19. The method of claim 1, wherein the step of selecting a plurality of barriers includes the steps of:

determining normally distributed random numbers X_k; and

setting the barriers B_k according to the formula B_k=−

log(1−

invnormcdf(X_k))

20. The method of claim 15, wherein the step of selecting a plurality of barriers includes the step of:

randomly selecting each of said plurality of barriers.

21. The method of claim 20, wherein the step of selecting a plurality of barriers includes the step of:

selecting each of said plurality of barriers so that said plurality of barriers have an exponential distribution.

22. The method of claim 1, wherein each of said plurality of hazard rates is a random variable.

23. The method of claim 1, wherein each of said plurality of hazard rates is a stochastic process.

24. The method of claim 1, wherein the step of identifying a plurality of hazard rates includes the step of:

determining each one of said plurality of hazard rates for said corresponding one of said plurality of securities based on default swap prices for said corresponding one of said plurality of securities.

25. The method of claim 1, wherein the step of identifying a plurality of hazard rates includes the step of:

determining each one of said plurality of hazard rates for said corresponding one of said plurality of securities based on a bond rating for said corresponding one of said plurality of securities.

26. The method of claim 1, wherein the step of identifying a plurality of hazard rates includes the step of:

calibrating a specification of dependence between hazard rates based on correlation information.

27. The method of claim 1, wherein the step of identifying a plurality of hazard rates includes the step of:

calibrating the specification of dependence between barriers based on correlation information.

28. The method of claim 26, wherein said correlation information is based on historical time series of default swap.

29. The method of claim 26, wherein said correlation information is based on historical time series of bond spreads.

30. The method of claim 26, wherein said correlation information is based on historical time series of bond ratings.

31. The method of claim 26, wherein said correlation information is based on market prices of basket default swaps.

32. The method of claim 26, wherein said correlation information is based on option market pricing.

33. The method of claim 1, wherein the step of formulating a plurality of compensators includes the step of:

integrating each of said one of said plurality of hazard rates associated with the one of said plurality of compensators.

34. The method of claim 5, wherein the step of calculating includes the step of:

calculating an adjustment according to;

E(e<sup>−

H(T*)</sup>)=e<sup>−

B</sup>

where T* is the estimated time, H is said at least one of said plurality of compensators and B is said corresponding one of said plurality of barriers.

35. The method of claim 5, wherein the step of calculating an adjustment to said proposed default time includes the steps of:

calculating said adjustment T_adj according to;

T_adj=(B−

H(T*))/h

where T* is the previous proposed default time, H(T*) is said at least one of said plurality of compensators at the previously estimated time, B is said corresponding one of said plurality of barriers, and h is an estimate of the average or indicative value of said corresponding one of said hazard rates; and

forming a new proposed default time T_new according to;

T_new=T*+T_adj.

36. The method of claim 5, wherein the step of calculating an adjustment to said proposed default time includes the step of:

calculating said adjustment so that a portion of said plurality of compensators evaluated at the adjusted proposed default time are less than a corresponding portion of said plurality of barriers.

37. The method of claim 1, wherein the step of deriving a proposed default time for said at least one of said plurality of securities based on said estimated time includes the step of:

multiplying said estimated time by a multiplier.

38. The method of claim 1, wherein the step of calculating a value of said compensator and said barrier corresponding to said at least one of said plurality of securities based on said proposed default time includes the step of:

calculating the value of said compensator based on a combination of said compensator and said one of said plurality of hazard rates associated with said compensator.

39. The method of claim 38, wherein said calculation involves interpolation.

40. The method of claim 38, wherein said calculation involves extrapolation.

41. The method of claim 1, further comprising the steps of:

(k) calculating a calculated price for insuring against a default of said at least one of said plurality of securities at said default time;

(l) deriving from a market a derived price for insuring against said default of said at least one of said plurality of securities;

(m) adjusting a specification of dependence between barriers corresponding to said at least one of said plurality of securities based on said difference between said calculated price and said derived price.

42. The method of claim 41, further comprising the step of:

repeating steps (k)-(m) a plurality of times.

43. The method of claim 42, wherein the plurality of times is greater than 5.

44. The method of claim 41, further comprising the step of:

repeating steps (k)-(m) until the difference between said calculated price and said derived price is less than a specified tolerance.

45. The method of claim 1, further comprising the step of:

calculating an insurance premium associated with a default of said at least one of a plurality of securities based on said default time.

46. The method of claim 1, further comprising the steps of:

calculating a default time for each of said securities in said basket of securities; and

calculating cash flows associated with said basket of securities based on said default times.

47. The method of claim 1, further comprising the steps of:

calculating a default time for each of said securities in said basket of securities; and

performing a valuation of said basket of securities based on said default times.

48. The method of claim 1, further comprising the steps of:

calculating a default time for each of said securities in said basket of securities; and

performing a risk assessment on said basket of securities based on said default times.

49. A system for modeling a basket of securities containing a plurality of securities, comprising:

a default/recovery model database where said database stores default/recovery data regarding said plurality of securities;

a default simulation engine where said default simulation engine calculates a default time for at least one of said plurality of securities based on said default/recovery data; and

a cash flow engine where said cash flow engine generates cash flows for said basket of securities based on said default times.

50. The system of claim 49, wherein said default simulation engine calculates the default time for said at least one of a plurality of securities according to a method comprising the steps of:

identifying a plurality of hazard rates, one of said plurality of hazard rates corresponding to each of said plurality of securities;

formulating a plurality of compensators, each of said plurality of compensators based on said one of said plurality of hazard rates corresponding to each of said plurality of securities;

(a) selecting a plurality of barriers, one of said plurality of barriers corresponding to each of said plurality of securities;

(b) determining a proposed default time for said at least one of said plurality of securities;

(c) calculating a value of said compensator and said barrier corresponding to said at least one of said plurality of securities based on said proposed default time; and

(d) determining that said proposed default time is said default time for said at least one of said plurality of securities.

51. The system of claim 50, wherein said default simulation engine repeats steps (a)-(d) a plurality of times.

52. The system of claim 49, further comprising a valuation/pricing module for performing a valuation of said basket of securities based on said default time.

53. The system of claim 49, further comprising a risk assessment module for performing a risk assessment on said basket of securities based on said default times.

54. The system of claim 53, wherein said risk assessment module calculates an insurance premium associated with a default of said at least one of a plurality of securities based on said default time