Method and system for simulating implied volatility surfaces for use in option pricing simulations

• US 7,149,715 B2
• Filed: 06/29/2001
• Issued: 12/12/2006
• 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): Kazimi, Hani M.
• Assistant Examiner(s): Subramanian, Narayanswamy
• Application Number:
  US09/896,488
• Publication Number:
  US 20030014355A1
• Time in Patent Office: 1,992 Days
• Field of Search: 703/2, 703/22, 705 35- 39
• 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 40/02 Banking, e.g. interest calculation or account maintenance credit or loans G06Q40/03
• 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

A method and system for simulating changes in volatility for a price of a particular option on an underlying financial instrument is disclosed. A volatility surface model having at least one surface parameter is provided along with a set of volatilities for a plurality of options on the underlying financial instrument. The set of volatilities is analyzed to determine an initial value for each surface parameter which, when used in the surface model, defines a surface approximating the set of volatilities. The values of the surface parameters are then evolved using an appropriate evolution function. A volatility value for a particular option is extracted from the volatility surface defined by the evolved surface parameter values. The extracted volatility value can then be used in an option pricing model to provide a price of the particular option.

Claims

1. A method for simulating changes in volatility for a price of a particular option on an underlying financial instrument comprising the steps of:

(a) providing a surface model having at least one surface parameter;

(b) providing a set of volatilities for a plurality of options on the underlying financial instrument;

(c) analyzing the set of volatilities to determine a surface parameter value;

(d) determining a surface parameter value in accordance with a beta evolution function; and

(e) defining a surface approximating the set of volatilities using the surface model and the surface parameter value;

(f) extracting a volatility for the particular option from a volatility surface defined by surface model using the next value of the at least one surface parameter value;

wherein the extracted volatility is used in an option pricing model to provide a price of the particular option.

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

generating calibration data representing offsets between at least some of the volatilities in the set of volatilities and the surface defined by an initial value for the at least one surface parameter when applied to the surface model; and

adjusting the extracted volatility in accordance with the calibration data.

3. The method of claim 2, further comprising the step of repeating step (d) to produce a sequence of values for the at least one surface parameter.

4. The method of claim 2, wherein volatility is specified relative to Δ

and T values for an associated option, the surface model has surface parameters β

₀ . . . β

_n, n0, and the surface model is of the form;

σ

_i(Δ

, T)=F(β

_0,i, . . . , β

_{n, i}, Δ

, T)

where σ

is a measure of the volatility for an option with a given Δ

, which is a ratio between a change in option price and security price, and T, which is a term remaining for the option, and F is a function of Δ

, T and the surface parameters β

₀ . . . β

_n.

5. The method of claim 2, wherein volatility is specified relative to Δ

and T values for an associated option, the surface model has surface parame ters β

₀ . . . β

_n, n0, and the surface model is of the form;

Inσ

(Δ

, T)=β

₀+β

₁(Δ

−

X₁)⁺+β

₂(T−

X₂)⁺+β

₃(T−

X₃)⁺

where σ

is a measure of the volatility for an option with a given Δ

, which is a ratio between a change in option price and security price, and T, which is a term remaining for the option;

where (x)⁺ is a piecewise linear function having a value equal to x when x>

0 and is otherwise equal to zero; and

X₁, X₂ and X₃ are constant terms.

6. The method of claim 5, wherein X₁, X₂ and X₃ are substantially equal to 0.5, 4.0, and 24, respectively.

7. The method of claim 1, wherein the beta evolution function is a function of at least one beta evolution parameter, a respective surface parameter value, and a noise component.

8. The method of claim 7, wherein the beta evolution function is a linear mean-reversion process.

9. The method of claim 8, wherein the surface model has surface parameters β

₀ . . . β

_n, n0 and the beta evolution function for a respective surface parameter β

_m, 0 m n, is of the form;

Δ

β

_m,i=α

_m(θ

_m−

β

<sub>m,i−

1</sub>)+V_mε

_m,i

where Δ

β

m,i is a change in beta value, α

_m is a mean-reversion speed, θ

_m is a mean value for β

_m, and V_m is volatility of β

_m, and ε

_m,i is a noise term.

10. The method of claim 7, wherein the step of simulating a future value for a surface parameter comprises the steps of:

selecting a value of ε

from a set of predetermined noise values associated with the surface parameter; and

calculating the surface parameter value using the beta evolution function with the selected value of ε

.

11. The method of claim 10, wherein the set of predetermined noise values is generated by:

applying a sequence of values for the associated surface parameter to the beta evolution function and determining residual values of the noise component ε

which recreates the sequence of values; and

standardizing the determined residual values to a range corresponding to that of the noise component ε

in the beta evolution function.

12. The method of claim 11, further comprising the step of applying a multi-day bootstrapping procedure to each respective set of predetermined noise values to increase the quantity of predetermined noise values.

13. A system for simulating changes in volatility for a price of a particular option on an underlying financial instrument comprising:

a computer having a processor and at least one data store;

the data store containing therein at least;

a surface model having at least one surface parameter; and

a set of volatilities for a plurality of options on the underlying financial instrument;

the processor being configured via computer software to;

analyze the set of volatilities to determine a surface parameter value;

determine a surface parameter value in accordance with a beta evolution function;

define a surface approximating the set of volatilities using the surface model and the surface parameter value, and

extract a volatility for the particular from a volatility surface defined by surface model using at least one surface parameter value;

wherein the extracted volatility is used in an option pricing model to provide a price of the particular option.

14. The system of claim 13, wherein the processor is further configured to:

generate calibration data representing offset between at least some of the volatilities in the set of volatilities and the surface defined by an initial value for the at least one surface parameter when applied to the surface model; and

adjust the extracted volatility in accordance with the calibration data.

15. The system of claim 14, wherein the processor is further configured to repeatedly determine each surface parameter value to produce a sequence of values for the at least one surface parameter.

16. The system of claim 14, wherein volatility is specified relative to Δ

and T values for an associated option, the surface model has parameters β

₀ . . . β

_n, n 0, and the surface model is of the form;

σ

_i, (Δ

, T)=F(β

_0,i, . . . , β

_0,i, Δ

, T)

where σ

is a measure of the volatility for an option with a given Δ

, which is a ratio between a change in option price and security price, and T, which is a term remaining for the option, and F is a function of Δ

, T, and the surface parameters β

₀ . . . β

_n.

17. The system of claim 14, wherein volatility is specified relative to Δ

and T values for an associated option, the surface model has surface parameters β

₀. . . β

_n, n 0, and the surface model is of the form;

Inσ

(Δ

, T)=β

₀+β

₁(Δ

−

X₁)⁺+β

₂(T−

X₂)⁺+β

₃(T−

X₃)⁺

where σ

is a measure of the volatility for an option with a given Δ

, which is a ratio between a change in option price and security price, and T, which is a term remaining for the option;

where (x)⁺ is a piecewise linear function having a value equal to x when x>

0 and is otherwise equal to zero; and

X₁, X₂ and X₃ are constant terms.

18. The system of claim 17, wherein X₁, X₂ and X₃ are substantially equal to 0.5, 4.0, and 24, respectively.

19. The system of claim 13, wherein the beta evolution function is a function of at least one beta evolution parameter, a respective surface parameter value, and a noise component ε

.

20. The system of claim 19, wherein the beta evolution function is linear mean-reversion process.

21. The system of claim 20, wherein the surface model has surface parameters β

₀. . . β

_n n 0 and the beta evolution function for a respective surface parameter β

m, 0 m n, is of the form;

Δ

β

_m,i=α

_m(θ

_m−

β

<sub>m,i,−

1</sub>)+V_mε

_m,i

where Δ

β

_m,i is a change in beta value, α

_m is a mean-reversion speed, θ

_m is a mean value for β

_m, and V_m is a volatility of β

_m and ε

_m,i is a noise term.

22. The system of claim 19, wherein the processor is configured to simulate the future value for a surface parameter by at least:

selecting a value of s from a set of predetermined noise values associated with the surface parameter; and

calculating the surface parameter value using the beta evolution function with the selected value of ε

.

23. The system of claim 22, wherein the processor is configured to generate the of predetermined noise values by:

applying a sequence of values for the associated surface parameter to the beta evolution function and determining residual values of the noise component ε

which recreates the sequence of values; and

standardizing the determined residual values to a range corresponding to that of the noise component ε

in the beta evolution function.

24. The system of claim 23, wherein the processor is further configured to apply a multi-day bootstrapping procedure to each respective ser of predetermined noise values to increase the quantity of predetermined noise values.