System and method for estimating portfolio risk using an infinitely divisible distribution

A computer-implemented method for estimating portfolio risk using an infinitely divisible distribution, comprising steps of:

• storing a time series comprising a plurality of risk factors applicable over at least one time horizon, a portfolio comprising a plurality of financial assets, and one or more risk adjusted return points for the financial assets;
• associating the financial assets with the risk factors;
• estimating the parameters of one or more risk factors through an infinitely divisible tempered stable distribution model exhibiting leptokurtic behavior having the measure v(dx) defined in polar coordinate according to
• v(dx)=M(dr,du)=r−α−1q(r,u)dr ζ(du),
• where q(r,u) is a positive function on (0,∞)×Sd−1 such that limr→0+q(r,u)>0 and limr→∞q(r,u)=0 and where q(r,1)=e−λ+2r2/2, q(r,−1)=e−λ−2r1/2, and ζ(1)=C+, ζ(−1)=C−, wherein C+,C−,λ+,λ−>0;
• generating scenarios for the model;
• determining at least one of value at risk, average value at risk, option price, and portfolio optimization from the estimated parameters and the generated scenarios;
• and
• providing the output generated in the application module to a user,
• wherein the steps are performed by a processor.

A computer-implemented system for estimating portfolio risk using a multivariate model, comprising:

• a memory containing a database configured to store a time series comprising a plurality of risk factors applicable over at least one time horizon, a portfolio comprising a plurality of financial assets, and one or more risk adjusted return points for the financial assets;
• an input device to receive input from a user;
• a network operatively coupled and providing communication between the memory, the input device, an output device, and a processor;
• where the processor is programmed to execute program modules, the program modules comprising:
• an association module configured to associate the financial assets with the risk factors;
• a parameter estimation module configured to estimate parameters of one or more risk factors through an infinitely divisible tempered stable distribution model exhibiting leptokurtic behavior according to
• X=(X1, X2, . . . , XN)=μ+γ(T−1)+√{square root over (T)}Z,
• where μ=(μ1,μ2, . . . , μN) ∈ RN, γ=(γ1,γ2, . . . ,γN) ∈ RN,
• Z=(Z1,Z2, . . . ,ZN)˜N(0,Σ), Σ=[σk,l]k,l∈{1,2, . . . ,N}, and
• T=1-dim GRDTS subordinator with parameters (α,C,θ,p);
• a scenario generation module to generate scenarios for the model;
• an application module to determine output comprising at least one of value at risk, average value at risk, option price, and portfolio optimization from the estimated parameters and the generated scenarios;
• and
• an output device configured to provide the output generated in the application module to the user.

A computer-implemented method for estimating portfolio risk using a multivariate model, comprising steps of:

• storing a time series comprising a plurality of risk factors applicable over at least one time horizon, a portfolio comprising a plurality of financial assets, and one or more risk adjusted return points for the financial assets;
• associating the financial assets with the risk factors;
• estimating parameters of one or more risk factors through an infinitely divisible tempered stable distribution model exhibiting leptokurtic behavior according to
• X=(X1, X2, . . . , XN)=μ+γ(T−1)+√{square root over (T)}Z,
• where μ=(μ1, μ2, . . . , μN) ∈ RN, γ=(γ1, γ2, . . . , γN) ∈ RN,
• Z=(Z1, Z2, . . . , ZN)˜N(0,Σ), Σ=[σk,l]k,l∈{1,2, . . . ,N}, and
• T=1-dim GRDTS subordinator with parameters (α,C,θ,p);
• generating scenarios for the model;
• determining at least one of value at risk, average value at risk, option price, and portfolio optimization from the estimated parameters and the generated scenarios;
• and
• providing the output generated in the application module to a user,
• wherein the steps are performed by a processor.