Method and system for simulating implied volatility surfaces for basket option pricing

• US 20030074167A1
• Filed: 05/31/2002
• Published: 04/17/2003
• 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
• Granted Patent: US 7,440,916 B2
• Time in Patent Office: Days
• US Class Current: 703/2
• 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 40/04 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 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. The volatility of a basket options valued relative to the performance of multiple components can be simulated by determining the value of surface parameters for options on the component securities and then combining the component surface parameters to determine surface parameters for a volatility surface of the basket.

Claims

1. A method for simulating the volatility of basket options having a value based on the performance N underlying component instruments, N>

1, the method comprising the steps of;

(a) providing a volatility surface model defining a volatility surface using a plurality of surface parameters β

₀ . . . β

_n, n≧

0;

(b) providing values for surface parameters β

_0,k . . . β

_n,k 1≦

k≦

N that define, for each particular component instrument k, a respective volatility surface via the surface model for options on that component instrument;

(c) determining values for surface parameters β

_B,0 . . . β

_B,n defining a volatility surface for the basket option using the surface parameters β

_0,k . . . β

_n,k associated with each component instrument; and

(d) using a volatility value extracted from the volatility surface defined by surface parameters β

_B,0 . . . β

_B,n to simulate the performance of the basket option.

2. The method of claim 1, wherein the surface model has a form

σ

(Δ

,T)=F(β

₀, . . . β

_n, Δ

,T)

where (i) σ

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

T and (ii) F is a function of Δ

, T and the surface parameters β

₀ . . . β

_n.

3. The method of claim 2, wherein the surface model is of the form:

ln σ

(Δ

,T)=β

₀+β

₁(Δ

−

x₁)+β

₂(T−

x₂)⁺+β

₃(T−

x₃)⁺

where x₁, x₂, and x₃ are constant terms.

4. The method of claim 3, wherein x₁, x₂, and x₃ are substantially equal to 0.5, 4.0, and 24, respectively.

5. The method of claim 4, wherein:

6. The method of claim 5, wherein:

7. The method of claim 6, wherein a relationship between (i) the surface parameters β

_B,0 . . . β

_B,n defining the volatility surface for the basket option and (ii) the surface parameters β

_0,k . . . β

_n,k defining volatility surfaces for options on the components of the basket option, can be expressed as;

8. A system for simulating the volatility of basket options having a value based on the performance N underlying component instruments, N>

1 comprising;

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

the data store containing therein at least;

a volatility surface model defining a volatility surface using a plurality of surface parameters β

₀ . . . β

_n, N≧

0; and

values for surface parameters β

_0,k . . . β

_n,k 1≦

k≦

N, the values defining a respective volatility surface for options on each component k via the surface model;

the processor being configured via computer software to;

determine values for surface parameters β

_B,0 . . . β

_B,n defining a volatility surface for the basket option using the surface parameters β

_0,k . . . β

_n,k associated with each component instrument; and

use a volatility value extracted from the volatility surface defined by surface parameters β

_B,0 . . . β

_B,n simulate the performance of the basket option.

9. The system of claim 8, wherein the surface model has a form

σ

(Δ

,T)=F(β

₀, . . . , β

_n, Δ

,T)

where (i) σ

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

T and (ii) F is a function of Δ

, T and the surface parameters β

₀ . . . β

_n.

10. The system of claim 9, wherein the surface model is of the form:

ln σ

(Δ

,T)=β

₀+β

₁(Δ

−

x₁)+β

₂(T−

x₂)⁺+β

₃(T−

x₃)⁺

where x₁, x₂, and x₃ are constant terms.

11. The system of claim 10, wherein x₁, x₂, and x₃ are substantially equal to 0.5, 4.0, and 24, respectively.

12. The system of claim 11, wherein the processor is configured to determine the basket option surface parameters according to:

13. The system of claim 12, wherein:

14. The system of claim 13, wherein a relationship between (i) the surface parameters β

_B,0 . . . β

_B,n defining the volatility surface for the basket option and (ii) the surface parameters β

_0,k . . . β

_n,k defining volatility surfaces for options on the components of the basket option, can be expressed as;