Method and System for Stress Testing Simulations of the Behavior of Financial Instruments

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 under normal market conditions. The values of the surface parameters are then evolved using an appropriate evolution function. Prior to applying the surface parameters to the model, the parameter values can be adjusted to introduce changes in offset, skew, term, or other parameters of the volatility surface to allow for simulation of unusual market conditions. A volatility value for a particular option is extracted from the volatility surface defined by the evolved and stress-adjusted surface parameter values. The extracted volatility value can then be used in an option pricing model to provide a price of the particular option.

1 . A method for simulating the behavior of a financial instrument in response to unusual market conditions comprising the steps of:
 (a) providing a volatility surface model to be used during simulation of the instrument's behavior, the surface model defining a volatility surface using a plurality of surface parameters β 0  . . . β n , n≧0, each surface parameter being associated with at least one attribute of the modeled volatility surface;   (b) determining a value for surface parameters β 0,normal  . . . β n,normal  for a step of the simulation under normal market conditions;   (c) varying at least one of the surface parameters β x,normal  by a respective stress value β x,stress , 0≦x≦n; and   (d) extracting a volatility from a volatility surface defined by surface model using the determined normal surface parameter values as varied by the stress value;   wherein the extracted volatility can be used in a pricing model to provide a price of the particular instrument.   
     
     
         2 . The method of  claim 1 , wherein the financial instrument is an option on an underlying financial instrument and the volatility surface model represents implied volatility for the option relative to Δ and T values, the surface model having 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; and
 wherein the plurality of surface parameters comprise at least one surface parameter associated with an offset of the volatility surface relative to the Δ and T axes, at least one surface parameter associated with changes in the volatility surface with respect to the Δ of the option, and at least one surface parameter associated with changes in the volatility surface with respect to the term of the option. 
 
     
     
         3 . The method of  claim 2 , wherein the surface model is of the form:
   ln σ(Δ, T )=β 0 +β 1 (Δ− x   1 )+β 2 ( T−x   2 ) + +β 3 ( T−x   3 ) + 
   
       where x 1 , x 2 , and x 3  are constant terms. 
     
     
         4 . The method of  claim 3 , wherein x 1 , x 2 , and x 3  are substantially equal to 0.5, 4.0, and 24, respectively. 
     
     
         5 . The method of  claim 2 , further comprising the steps of
 providing a set of volatilities for a plurality of options on the underlying financial instrument;   analyzing the set of volatilities to determine an initial value β 0,initial,normal  . . . β n,initial,normal  for the surface parameters which, when used in the surface model, define a surface approximating the set of volatilities under normal market conditions; and   determining a next value for each surface parameter in accordance with a beta evolution function.   
     
     
         6 . The method of  claim 5 , 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 the initial values β 0,initial,normal  . . . β n,initial,normal  for the at least one surface parameter when the initial values are applied to the surface model; and   adjusting the extracted volatility in accordance with the calibration data.   
     
     
         7 . The method of  claim 5 , further comprising the step of repeating the step of determining a next value to produce a sequence of values for the at least one surface parameter for normal market conditions. 
     
     
         8 . The method of  claim 5 , wherein the beta evolution function for a respective surface parameter β m  is of the form:
   Δ m,i =α m (θ m −β m,i-1 )+ν m ε m,i  
 
 
       where α m  is a mean-reversion speed, θ m  is a mean value, and ν m  is a volatility of β m , and ε m,i  is a noise term. 
     
     
         9 . A system for simulating the behavior of a financial instrument in response to unusual market conditions comprising:
 a computer having a processor and at least one data store;   the data store containing therein at least:
 a volatility surface model to be used during simulation of the instrument's behavior, the surface model defining a volatility surface using a plurality of surface parameters β 0  . . . β n , n≧0, each surface parameter being associated with at least one attribute of the modeled volatility surface; 
   the processor being configured via computer software to:
 determine a value for surface parameters β 0,normal . . .  β n,normal  for a step of the simulation under normal market conditions; 
 vary at least one of the surface parameters β x,normal  by a respective stress value β x,stress , 0≦x≦n; and 
 extract a volatility from a volatility surface defined by surface model using the determined normal surface parameter values as varied by the stress value; 
 wherein the extracted volatility can be used in a pricing model to provide a price of the particular instrument. 
   
     
     
         10 . The system of  claim 9 , wherein the financial instrument is an option on an underlying financial instrument and the volatility surface model represents implied volatility for the option relative to Δ and T values, the surface model having 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; and
 wherein the plurality of surface parameters comprise at least one surface parameter associated with an offset of the volatility surface relative to the Δ and T axes, at least one surface parameter associated with changes in the volatility surface with respect to the Δ of the option and at least one surface parameter associated with changes in the volatility surface with respect to the term of the option. 
 
     
     
         11 . The system of  claim 9 , wherein the surface model is of the form:
   ln σ(Δ, T )=β 0 +β 1 (Δ− x   1 )+β 2 ( T−x   2 ) + +β 3 ( T−x   3 ) + 
   
       where x 1 , x 2 , and x 3  are constant terms. 
     
     
         12 . The system of  claim 11 , wherein x 1 , x 2 , and x 3  are substantially equal to 0.5, 4.0, and 24, respectively. 
     
     
         13 . The system of  claim 10 , wherein the data store further comprises data representing a set of volatilities for a plurality of options on the underlying financial instrument;
 the processor being further configured to:
 analyze the set of volatilities to determine an initial value β 0,initial,normal  . . . β n,initial,normal  for the surface parameters which, when used in the surface model, define a surface approximating the set of volatilities under normal market conditions; and 
   determine a next value for each surface parameter in accordance with a beta evolution function.   
     
     
         14 . The system of  claim 13 , wherein the processor is further configured to:
 generate calibration data representing offsets between at least some of the volatilities in the set of volatilities and the surface defined by the initial values β 0,initial,normal  . . . β n,initial,normal  for the at least one surface parameter when the initial values are applied to the surface model; and   adjust the extracted volatility in accordance with the calibration data.   
     
     
         15 . The system of  claim 13 , wherein the processor is further configured to repeatedly determine a next value to produce a sequence of values for the at least one surface parameter for normal market condition and store the sequence of values in the data store. 
     
     
         16 . The system of  claim 13 , wherein the beta evolution function for a respective surface parameter β m  is of the form:
   Δβ m,i =α m (θ m −β m,i-1 )+ν m ε m,i  
 
 
       where α m  is a mean-reversion speed, θ m  is a mean value, and ν m  is a volatility of β m , and ε m,i  is a noise term.