Value at risk for linear and non-linear derivatives
Frei, Clemens U
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This paper examines the question whether standard parametric Value at Risk approaches, such as the delta-normal and the delta-gamma approach and their assumptions are appropriate for derivative positions such as forward and option contracts. The delta-normal and delta-gamma approaches are both methods based on a first-order or on a second-order Taylor series expansion. We will see that the delta-normal method is reliable for linear derivatives although it can lead to signif-icant approximation errors in the case of non-linearity. The delta-gamma method provides a better approximation because this method includes second order-effects. The main problem with delta-gamma methods is the estimation of the quantile of the profit and loss distribution. Empiric results by other authors suggest the use of a Cornish-Fisher expansion. The delta-gamma method when using a Cornish-Fisher expansion provides an approximation which is close to the results calculated by Monte Carlo Simulation methods but is computationally more efficient.