Simulation Methods

Abstract

In the calculation of the value at risk by means of Monte Carlo simulations, all of the risk factors influencing a portfolio are simulated over the liquidation period δt as stochastic processes satisfying, for example, Equation 3.13 or even more general processes of the form 3.15. The value at risk of the risk factors themselves are taken into complete consideration using Equation 21.15 sometimes neglecting the drift in the simulation if the liquidation period is short:

$$\begin{gathered} Va{R_{long}}\left( c \right) \approx NS\left( t \right)\left[ {1 - \exp \left( { + Q_{1 - c}^{N\left( {0,1} \right)}\sigma \sqrt {\delta t} } \right)} \right] \hfill \\ Va{R_{short}}\left( c \right) \approx - NS\left( t \right)\left[ {1 - \exp \left( {t - Q_{1 - c}^{N\left( {0,1} \right)}\sigma \sqrt {\delta t} } \right)} \right] \hfill \\ \end{gathered} $$

As explained in Section 21.1, the value at risk of a long position in an underlying is only then equal to that of a short position if the drift is neglected and the linear approximation has been used. Since the linear approximation is usually not assumed in the Monte Carlo method, the VaRs of a long position will not equal that of a short position on the same underlying.