Modeling option price dynamics

• US 7,065,475 B1
• Filed: 10/31/2000
• Issued: 06/20/2006
• Est. Priority Date: 10/31/2000
• Status: Expired due to Term

• Current Assignee: Goldman Sachs & Co. LLC (Goldman Sachs Group Incorporated)
• Original Assignee: Goldman Sachs & Company (Goldman Sachs Group Incorporated)
• Inventors: Brundobler, Stefan
• Primary Examiner(s): Paladini, Albert W.
• Application Number:
  US09/702,423
• Time in Patent Office: 2,058 Days
• Field of Search: 703/2, 705/36, 705/37, 705/400, 705/35, 705/10
• 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 40/04 Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange

Abstract

This invention presents a method for pricing an option. The steps for this method include configuring a general option pricing model with parameters to conform the model to a market behavior of an underlying asset. A price for an option is then calculated using the model. The configured model can be calibrated to implied volatility data describing the current state of the market. The underlying asset can include commodity prices, interest rates, and currency exchange rates. More than one general option pricing models can be used to price the option. Additionally, correlations between the general option pricing models can be included in the calculation. The configuring of the parameters can be done through the formula:

$\frac{\mathrm{dF}\left(t,T\right)}{F\left(t,T\right)}=\sum _a\left[\sum _i{y}_{\mathrm{ia}}{B}_i\left(t,T\right)g_i\left(T\right)\right]{\sigma }_a\left(t\right){\mathrm{dz}}_a\left(t\right),$
wherein F(t, T) represents the value of the underlying asset and dF(t, T) represents a change in the value of the underlying asset; a represents randomness factors; i represents an amount of mean reversion factors used in the model; t represents the current time; T represents the forward time; y_ia represents the move shape coefficient; B_i(t, T) represents the mean reversion factor; g_i(T) represents the volatility adjustment factor; σ_a(t) represents the instantaneous factor volatility; and dz_a(t) represents the random increment.

Claims

1. A computer-implemented method for pricing an option comprising:

(a) calculating in a computer the formula;

$\frac{\mathrm{dF}<br><br>\left(t,<br><br>T\right)}{F<br><br>\left(t,T\right)}=\underset{a}{\sum <br><br>}<br><br>\left[\underset{i}{\sum <br><br>}<br><br>y_{\mathrm{ia}}<br><br>B_i<br><br>\left(t,T\right)<br><br>g_i<br><br>\left(T\right)\right]<br><br>{\mathrm{\sigma <br><br>}}_a<br><br>\left(t\right)<br><br>{\mathrm{dz}}_a<br><br>\left(t\right),$

wherein F(t, T) represents the value of the underlying asset and dF(t, T) represents a change in the value of the underlying asset;

i represents an amount of mean reversion factors used in the model;

t represents the current time;

T represents the forward time;

y_ia represents the move shape coefficient;

B_i(t, T) represents the mean reversion factor;

g_i(T) represents the volatility adjustment factor;

σ

_a(t) represents the instantaneous factor volatility; and

dz_a(t) represents the random increment;

a represents the index enumerating the random increments dz_a(t); and

(b) calculating in a computer a price of an option based on the calculated results of the formula.

2. The method of claim 1, wherein the mean reversion factor further comprises the formula:

$B_i<br><br>\left(t,T\right)=\exp <br><br>\left[-{\int <br><br>}_t^T<br><br>{\mathrm{\beta <br><br>}}_i<br><br>\left(u\right)<br><br>d<br><br>u\right],$

wherein β

_i(u) represents a time dependency of a mean reversion rate.

3. The method of claim 1, wherein the instantaneous factor volatility further comprises the formula:

σ

_a(t)=σ

_0a(t)+ξ

(t)dσ

_a(t),

wherein σ

_0a(t) represents a base value for the instantaneous factor volatility;

ξ

(t) represents a calibration coefficient; and

dσ

_a(t) represents a calibration gradient determining how much each volatility is affected by the calibration coefficient.

4. A computer-implemented apparatus for pricing an option comprising:

(a) means for calculating the formula;

$\frac{\mathrm{dF}<br><br>\left(t,<br><br>T\right)}{F<br><br>\left(t,T\right)}=\underset{a}{\sum <br><br>}<br><br>\left[\underset{i}{\sum <br><br>}<br><br>y_{\mathrm{ia}}<br><br>B_i<br><br>\left(t,T\right)<br><br>g_i<br><br>\left(T\right)\right]<br><br>{\mathrm{\sigma <br><br>}}_a<br><br>\left(t\right)<br><br>{\mathrm{dz}}_a<br><br>\left(t\right),$

wherein F(t, T) represents the value of the underlying asset and dF(t, T) represents a change in the value of the underlying asset;

i represents an amount of mean reversion factors used in the model;

t represents the current time;

T represents the forward time;

y_ia represents the move shape coefficient;

B_i(t, T) represents the mean reversion factor;

g_i(T) represents the volatility adjustment factor;

σ

_a(t) represents the instantaneous factor volatility; and

dz_a(t) represents the random increment;

a represents the index enumerating the random increments dz_a(t); and

(b) means for calculating a price of an option based on the calculated results of the formula.

5. The apparatus of claim 4, wherein the mean reversion factor further comprises the formula:

$B_i<br><br>\left(t,T\right)=\exp <br><br>\left[-{\int <br><br>}_t^T<br><br>{\mathrm{\beta <br><br>}}_i<br><br>\left(u\right)<br><br>d<br><br>u\right],$

wherein β

_i(u) represents a time dependency of a mean reversion rate.

6. The apparatus of claim 4, wherein the instantaneous factor volatility further comprises the formula:

σ

_a(t)=σ

_0a(t)+ξ

(t)dσ

_a(t),

wherein σ

_0a(t) represents a base value for the instantaneous factor volatility;

ξ

(t) represents a calibration coefficient; and

dσ

_a(t) represents a calibration gradient determining how much each volatility is affected by the calibration coefficient.

7. An article of manufacture for pricing an option, the article of manufacture comprising a computer-readable medium holding computer-executable instructions for performing the steps of:

(a) calculating in a computer the formula;

$\frac{\mathrm{dF}<br><br>\left(t,<br><br>T\right)}{F<br><br>\left(t,T\right)}=\underset{a}{\sum <br><br>}<br><br>\left[\underset{i}{\sum <br><br>}<br><br>y_{\mathrm{ia}}<br><br>B_i<br><br>\left(t,T\right)<br><br>g_i<br><br>\left(T\right)\right]<br><br>{\mathrm{\sigma <br><br>}}_a<br><br>\left(t\right)<br><br>{\mathrm{dz}}_a<br><br>\left(t\right),$

wherein F(t, T) represents the value of the underlying asset and dF(t, T) represents a change in the value of the underlying asset;

i represents an amount of mean reversion factors used in the model;

t represents the current time;

T represents the forward time;

y_ia represents the move shape coefficient;

B_i(t, T) represents the mean reversion factor;

g_i(T) represents the volatility adjustment factor;

σ

_a(t) represents the instantaneous factor volatility; and

dz_a(t) represents the random increment;

a represents the index enumerating the random increments dz_a(t); and

(b) calculating in a computer a price of an option based on the calculated results of the formula.

8. The article of manufacture of claim 7, wherein the mean reversion factor further comprises the formula:

$B_i<br><br>\left(t,T\right)=\exp <br><br>\left[-{\int <br><br>}_t^T<br><br>{\mathrm{\beta <br><br>}}_i<br><br>\left(u\right)<br><br>d<br><br>u\right],$

wherein β

_i(u) represents a time dependency of a mean reversion rate.

9. The article of manufacture of claim 7, wherein the instantaneous factor volatility further comprises the formula:

σ

_a(t)=σ

_0a(t)+ξ

(t)dσ

_a(t),

wherein σ

_0a(t) represents a base value for the instantaneous factor volatility;

ξ

(t) represents a calibration coefficient; and

dσ

_a(t) represents a calibration gradient determining how much each volatility is affected by the calibration coefficient.