The first and most crucial step consists of choosing a particular stochastic model for the behaviour of prices.
A commonly used model in Monte carlo simulation is the Geometric Brownian motion model which assumes movements in the market price are uncorrelated over time and that small movements in prices can be described by:
dS t = μ t S t dt + σ t S t dz
dz is a random variable distributed normally with mean zero and variance dt.
If the stochastic process chosen for the price is unrealistic, so will be the estimate of VAR.
For example, the geometric Brownian motion model adequately describes the behaviour of stock prices and exchange rates but not that of fixed income securities.
In Brownian motion models, price shocks are never reversed and prices move as a random walk.
This cannot be the price process for default free bond prices which must converge to their face value at expiration.
V A R Applications Passive Reporting risk Disclosure to shareholders Management reports Regulatory requirements Defensive Controlling risks Setting risk limits Active Allocating risk Performance valuation Capital allocation , Strategic business decisions.
EVT extends the central limit theorem which deals with the distribution of the average of identically and independently distributed variables from an unknown distribution to the distribution of their tails.
The EVT approach is useful for estimating tail probabilities of extreme events.
For very high confidence levels (>99%), the normal distribution generally underestimates potential losses.
A properly working model would still produce two to three exceptions a year.
But – the existence of clusters of exceptions indicated that something was seriously wrong.
Credit Suisse reported 11 exceptions at the 99% confidence level in the third quarter, Lehman brothers three at 95%, Goldman Sachs five at 95%, Morgan Stanley six at 95%, Bear Stearns 10 at 99% and UBS 16 at 99%.
With the benefit of hindsight, the type of VAR model that would actually have worked best in the second half of 2007 would most likely have been a model driven by a frequently updated short data history.
Or any frequently updated short data history that weights more recent observations more heavily than more distant observations.
In the wake of the recent credit crisis, there is a strong case for increasing the frequency of updating.
Monthly, quarterly or even weekly updating of the data series would improve the responsiveness of the model to a sudden change of conditions.