Robust VaR

Just how robust is VaR? In most financial applications we choose fairly simple
models and then abuse the input data outside the model to fit the market. We also
build a set of rules about when the model output is likely to be invalid. VaR is no
different. As an example, consider the Black–Scholes–Merton (BSM) option pricing
model: one way we abuse the model is by varying the volatility according to the
strike. We then add a rule not to sell very low delta options at the model value
because even with a steep volatility smile we just can’t get the model to charge
enough to make it worth our while to sell these options. A second BSM analogy is
the modeling of stochastic volatility by averaging two BSM values, one calculated
using market volatility plus a perturbation, and one using market volatility minus a
perturbation, rather than building a more complicated model which allows volatility
to change from its initial value over time.
Given the uncertainties in the input parameters (with respect to position, liquidation
strategy, time horizon and market model) and the potential mis-specification of
the model itself, we can estimate the uncertainty in the VaR. This can either be done
formally, to be quoted on our risk reports whenever the VaR value is quoted, or
informally, to determine when we should flag the VaR value because it is extremely
sensitive to the input parameters or to the model itself.
Here is a simple analysis of errors for single risk factor VaR. Single Risk factor VaR
is given by Exposure*NumberOfStandardDeviations*StandardDeviation* Horizon.
If the exposure is off by 15% and the standard deviation is off by 10% then relative
error of VaR is 15ò10ó25%! Note that this error estimate excludes the problems of
the model itself. The size of the error estimate does not indicate that VaR is
meaningless – just that we should exercise some caution in interpreting the values
that our models produce.