Method and system for simulating risk factors in parametric models using risk neutral historical bootstrapping

• US 7,228,290 B2
• Filed: 06/29/2001
• Issued: 06/05/2007
• Est. Priority Date: 06/29/2001
• Status: Active Grant

• Current Assignee: Goldman Sachs & Co. LLC (Goldman Sachs Group Incorporated)
• Original Assignee: Goldman Sachs & Company (Goldman Sachs Group Incorporated)
• Inventors: Browne, Sid, Maghakian, Arthur
• Primary Examiner(s): Colbert, Ella
• Assistant Examiner(s): Apple, Kirsten
• Application Number:
  US09/896,660
• Publication Number:
  US 20030014356A1
• Time in Patent Office: 2,167 Days
• Field of Search: 705/36, 705/36.R, 705/35
• US Class Current: 705/36.00R
• 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

Abstract

An improved method for simulating noise-varying risk factor values in a parametric simulation comprises analyzing historical data to determine the actual value of the risk factors and other attributes in the model and using this data to generate historical residual values which reproduces the historical price when used in the model with corresponding historical attribute values. The set of historical residual values is standardized and can be bootstrapped to increase the number of members in the set or vary the sets properties. Values of the historical residuals are then selected, e.g., at random, and used in place of the random noise components to produce simulated risk factor values which are used in the parametric model to simulate the evolution of the instrument price.

Claims

1. A method for simulating the value of an attribute of a financial instrument comprising the steps of:

providing a parametric model for the attribute having at least one noise-varying parameter α

_n with a corresponding noise component ε

_n;

determining values for the at least one parameter and the attribute at various time indices i using historical data;

deriving a set of historical residual values ε

_n,i for each noise component ε

_n, the historical residual value ε

_n,i at index i, when applied to the model with the determined parameter values at index i, at least substantially reproducing the determined attribute value at index i;

standardizing each set of historical residual values ε

_n; and

using values selected from the set of standardized historical residual values ε

_n as the noise component during a simulation of the attribute value via the model.

2. The method of claim 1, wherein the step of using comprises the steps of:

generating, for each noise-varying parameter α

_n in the model, a set of simulated parameter values using values selected from the corresponding set of standardized historical residual values ε

_n;

storing the simulated values for each parameter in a respective value matrix; and

applying values from the matrices to the model to produce simulated attribute values.

3. The method of claim 1, wherein the attribute is the price of a financial instrument.

4. The method of claim 3, wherein the step of using values is repeated for a plurality of scenarios for each instrument in a portfolio to produce a corresponding set of simulated instrument price scenarios for the respective instruments.

5. The method of claim 4, further comprising the steps of:

aggregating the simulated instrument prices to produce a set of simulated portfolio value scenarios; and

analyzing the simulated portfolio value scenarios to determine a value at risk for the portfolio.

6. The method of claim 1, further comprising the step of applying a multi-day bootstrapping procedure to the set of historical residual values to increase the quantity of historical residual values available in the using step.

7. The method of claim 6, wherein a two-day bootstrapping procedure is used.

8. The method of claim 1, further comprising the step of symmetrizing the set of historical residual values prior to the execution of the using step.

18. The system of claim 1, wherein the processor is further configured to symmetrize the set of historical residual values.

9. A method for simulating the price of a financial instrument, comprising the steps of:

providing a parametric pricing model having at least one parameter α

_n with a corresponding noise component ε

_n;

determining values for the parameters at various time indices i using historical data;

deriving a set of historical residual values ε

_n,i for each noise component ε

_n, the historical residual value ε

_n,i at index i, when applied to the model with the determined parameter values at index i, at least substantially reproducing the attribute price at index i;

standardizing each set of historical residual values ε

_n;

applying a multi-day bootstrapping procedure to each set of historical residual values to increase the quantity of historical residual values in each set; and

using values selected from the set of historical residual values as the noise component for the corresponding parameter during a simulation of the instrument price.

10. The method of claim 9, wherein the risk factor comprises volatility.

11. A system for simulating an attribute of a financial instrument, comprising:

a computer system having a processor and at least one data store, the data store having a parametric model for the attribute stored therein, the parametric model having at least one noise-varying parameter α

_n with a corresponding noise component ε

_n;

the data store further having a computer program stored therein to configure the processor to;

determine values for the at least one parameter and the attribute at various time indices i using historical data;

derive a set of historical residual values ε

_n,i for each noise component ε

_n, the historical residual value ε

_n,i at index i, when applied to the model with the determined parameter values at index i, at least substantially reproducing the determined attribute value at index i;

standardize each set of historical residual values ε

_n; and

use values selected from the set of standardized historical residual values ε

_n as the noise component during a simulation of the attribute value via the model.

12. The system of claim 11, wherein the processor is configured to use values by:

generating, for each noise-varying parameter α

_n in the model, a set of simulated parameter values using values selected from the corresponding set of standardized historical residual values ε

_n;

storing the simulated values for each parameter in a respective value matrix;

applying values from the matrices to the model to produce simulated attribute values.

13. The system of claim 11, wherein the attribute is the price of a financial instrument.

14. The system of claim 13, wherein the processor is configured to repeatedly use values selected from the set of standardized historical residual values ε

_n, for a plurality of scenarios for each instrument in a portfolio to produce a corresponding set of simulated instrument price scenarios for the respective instruments.

15. The system of claim 14, wherein the processor is further configured to:

aggregate the simulated instrument prices to produce a set of simulated portfolio value scenarios; and

analyze the simulated portfolio value scenarios to determine a value at risk for the portfolio.

16. The system of claim 11, wherein the processor is further configured to apply a multi-day bootstrapping procedure to the set of historical residual values to increase the quantity of historical residual values available for use.

17. The system of claim 16, wherein a two-day bootstrapping procedure is used.

19. A system for simulating the price of a financial instrument comprising:

a computer system having a processor and at least one data store, the date store having a parametric model for the attribute stored therein, the parametric model having at least one noise-varying parameter α

_n with a corresponding noise component ε

_n;

the data store further having a computer program stored therein to configure the processor to;

determine values for the parameters at various time indices i using historical data;

derive a set of historical residual values ε

_n,i for each noise component ε

_n, the historical residual value ε

_n,i at index i, when applied to the model with the determined parameter values at index i, at least substantially reproducing the attribute price at index i;

standardize each set of historical residual values ε

_n;

apply a multi-day bootstrap procedure to each set of historical residual values to increase the quantity of historical residual values in each set; and

use values selected from the set of historical values as the noise component for the corresponding parameter during a simulation of the instrument price.

20. The system of claim 19, wherein the parameter comprises volatility.