One of the keys to the successful implementation of a VaR system is a precise
financial-engineering design. There is a temptation to specify the design by providing
simple examples and leave the details of the treatment of system details to the
implementers. The result of this will often be inconsistencies and confusing behavior.
To avoid this, it is necessary to provide an almost ‘axiomatic’ specification that
provides an exact rule to handle the various contingencies that can come up.
This will typically require an iterative approach, amending the specification as the
implementers uncover situations that are not clearly specified. Thus, while the
descriptions that follow may appear at first unnecessarily formal, experience suggests
that a high level of precision in the specification pays off in the long run.
The fundamental problem of VaR, based on information known at the anchor time,
t0, is to estimate the probability distribution of the value of one’s financial position
at the target date, T. In principle, one could do this in a straightforward way by
coming up with a comprehensive probabilistic model of the world. In practice, it is
necessary to make heroic assumptions and simplifications, the various ways in
which these assumptions and simplifications are made lead to the various VaR
methodologies.
There is a key abstraction that is fundamental to almost all of the various VaR
methodologies that have been proposed. This is that one restricts oneself to estimating
the profit and loss of a given trading strategy that is due to changes in the value
of a relatively small set of underlying variables termed risk factors. This assumption
can be formalized by taking the risk factors to be the elements of a m-dimensional
vector mt that describes the ‘instantaneous state’ of the market as it evolves over
time. One then assumes that the value of one’s trading strategy at the target date
expressed in base currency is given by a function vt0,T :Rmî[t0, T ] R of the trajectory
of mt for t é [t0, T ]. We term a valuation function of this form a future valuation
function since it gives the value of the portfolio in the future as a function of the
evolution of the risk factors between the anchor and target date.3 Note that vt0,T is
defined so that any dependence on market variables prior to the anchor date t0, e.g.
due to resets, is assumed to be known and embedded in the valuation function vt0,T.
The same is true for market variables that are not captured in mt . For example, the
value of a trading strategy may depend on credit spreads or implied volatilities that
are not included in the set of risk factors. These may be embedded in the valuation
function vt0,T, but are then treated as known, i.e. deterministic, quantities.
To compute VaR, one postulates the existence of a probability distribution kT on
the evolution of mt in the interval [t0, T ]. Defining M[t0,T ]•{mt : t é [t0, T ]}, the quantity
vt0,T(M[t0,T ] ), where M[t0,T ] is distributed according to kT, is then a real-valued random
variable. VaR for the time horizon Tñt0 at the 1ña confidence level is defined to be
the a percentile of this random variable. More generally, one would like to characterize
the entire distribution of vt0,T(M[t0,T ] ).
The problem of computing VaR thus comes down to computing the probability
distribution of the random variable vt0,T(M[t0,T ] ). To establish a VaR methodology, we
need to
1 Define the market-state vector mt .
2 Establish a probabilistic model for M[t0,T ] .
3 Determine the parameters of the model for M[t0,T ] based on statistical data.
4 Establish computational procedures for obtaining the distribution of vt0,T(M[t0,T ] ).
In fact, most existing procedures for computing VaR have in common a stronger set
of simplifying assumptions. Instead of explicitly treating a trading strategy whose
positions may evolve between the anchor and target date, they simply consider the
existing position as of the anchor date. We denote the valuation function as of the
anchor date as a function of the risk factors by vt0 :Rm R. We term a valuation
function of this form a spot valuation function. Changes in the risk factors, which
we term perturbations, are then modeled as being statistically stationary. The VaR
procedure then amounts to computing the probability distribution of vt0 (mt0ò*m),
where *m is a stochastic perturbation.
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