System and method for determining Value-at-Risk using FORM/SORM
First Claim
1. A computer-implemented method for determining Value-at-Risk (VAR), said method comprising the steps of:
- (a) determining a probability preserving transformation between a set of correlated price returns of one or more financial instruments and of standard Gaussian variates using a probability model;
(b) creating a set of loss threshold values at which a lower tail of a probability distribution of portfolio value change is to be evaluated;
(c) selecting a value from the set of loss threshold values;
(d) determining a limit-state surface on which the portfolio value change is equal to the selected loss threshold value of step (c) by expressing a limit-state-equation in terms of one or more standard Gaussian variates using the probability preserving transformation calculated in step (a);
(e) finding one or more design points on the limit-state surface closest to an origin of a standard Gaussian space;
(f) calculating a probability of portfolio value change not exceeding the selected loss threshold value using one or more methods from the group consisting of (First-Order Reliability Method, Second-Order Reliability Method, or importance sampling around the one or more design points);
(g) repeating Steps (c) through (f) for each selected loss threshold value of step (b), whereby a lower tail of the cumulative probability distribution of portfolio value change is created; and
(h) determining a Value-at-Risk as a desired quantile of the lower tail of the cumulative probability distribution of portfolio value change.
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Abstract
A system and method are presented for the determination of Value-at-Risk (VAR) and other tail-risk measures for a portfolio of derivative securities. The present invention determines the tail of the probability distribution of portfolio returns based on first- and second order structural reliability (FORM/SORM) methods. As used herein, the present inventive method is referred to as “Reliability VAR.” The inventive system and method of calculating VAR is not restricted to representation of positions in a portfolio as “delta-gamma” sensitivities to the underlying price returns. Additionally, the inventive system and method lends itself to the determination of VAR in the presence of underlying price returns with so-called “fat tails.” In particular, a probability preserving transformation using a Hermite-model based correlation-mapping technique, previously used only in structural reliability analysis, has been applied to transform the VAR-related probability-estimation problem with non-Gaussian risk factors to an equivalent probability estimation problem in the standard Gaussian space.
57 Citations
11 Claims
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1. A computer-implemented method for determining Value-at-Risk (VAR), said method comprising the steps of:
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(a) determining a probability preserving transformation between a set of correlated price returns of one or more financial instruments and of standard Gaussian variates using a probability model;
(b) creating a set of loss threshold values at which a lower tail of a probability distribution of portfolio value change is to be evaluated;
(c) selecting a value from the set of loss threshold values;
(d) determining a limit-state surface on which the portfolio value change is equal to the selected loss threshold value of step (c) by expressing a limit-state-equation in terms of one or more standard Gaussian variates using the probability preserving transformation calculated in step (a);
(e) finding one or more design points on the limit-state surface closest to an origin of a standard Gaussian space;
(f) calculating a probability of portfolio value change not exceeding the selected loss threshold value using one or more methods from the group consisting of (First-Order Reliability Method, Second-Order Reliability Method, or importance sampling around the one or more design points);
(g) repeating Steps (c) through (f) for each selected loss threshold value of step (b), whereby a lower tail of the cumulative probability distribution of portfolio value change is created; and
(h) determining a Value-at-Risk as a desired quantile of the lower tail of the cumulative probability distribution of portfolio value change. - View Dependent Claims (2, 3, 4, 5, 6)
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8. A system for determining Value-at-Risk (VAR), said system comprising of:
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a computer; and
a software program being executable by the computer, the software program for executing the steps;
(a) determining a probability preserving transformation between a set of correlated price returns of one or more financial instruments and of standard Gaussian variates using a probability model;
(b) creating a set of loss threshold values at which a lower tail of a probability distribution of portfolio value change is to be evaluated;
(c) selecting a value from the set of loss threshold values;
(d) determining a limit-state surface on which the portfolio value change is equal to the selected loss threshold value of step (c) by expressing a limit-state-equation in terms of one or more standard Gaussian variates using the probability preserving transformation calculated in step (a);
(e) finding one or more design points on the limit-state surface closest to an origin of a standard Gaussian space;
(f) calculating a probability of portfolio value change not exceeding the selected loss threshold value using one or more methods from the group consisting of (First-Order Reliability Method, Second-Order Reliability Method, or importance sampling around the one or more design points);
(g) repeating steps (c) through (f) for each selected loss threshold value of step (b), whereby a lower tail of the cumulative probability distribution of portfolio value change is created; and
(h) determining a Value-at-Risk as a desired quantile of the lower tail of the cumulative probability distribution of portfolio value change. - View Dependent Claims (9)
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10. A computer-usable medium having computer-readable program code embodied therein for causing a computer to perform the steps of:
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(a) determining a probability preserving transformation between a set of correlated price returns of one or more financial instruments and of standard Gaussian variates using a probability model;
(b) creating a set of loss threshold values at which a lower tail of a probability distribution of portfolio value change is to be evaluated;
(c) selecting a value from the set of loss threshold values;
(d) determining a limit-state surface on which the portfolio value change is equal to the selected loss threshold value of step (c) by expressing a limit-state-equation in terms of one or more standard Gaussian variates using the probability preserving transformation calculated in step (a);
(e) finding one or more design points on the limit-state surface closest to an origin of a standard Gaussian space;
(f) calculating a probability of portfolio value change not exceeding the selected loss threshold value using one or more methods from the group consisting of (First-Order Reliability Method, Second-Order Reliability Method, or importance sampling around the one or more design points);
(g) repeating Steps (c) through (f) for each selected loss threshold value of step (b), whereby a lower tail of the cumulative probability distribution of portfolio value change is created; and
(h) determining a Value-at-Risk as a desired quantile of the lower tail of the cumulative probability distribution of portfolio value change. - View Dependent Claims (11)
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Specification