A new trend: uncertain parameter models

Finally, we will say a few words on a new trend that has been recently introduced in
academic research that could help in assessing model risk for a given position: the
uncertain volatility models. Work in the area was pioneered by Avellaneda, Levy, and
Paras (1995) for stocks and by Lhabitant, Martini, and Reghai (1998) for interest rate
contingent claims.
The new idea here is to build pricing and hedging models that work regardless of
the true world’s underlying model. As an example, should we consider the volatility
as constant, deterministic, or stochastic? This is difficult to answer, since the
volatility itself is a non-observable quantity! Rather than using a specific and probably
misspecified model, the uncertain volatility models take a very pragmatic view: if you
are able to bound the volatility between two arbitrary values, they will provide you
with valid prices and hedging parameters, whatever the underlying model. Rather
than specifying the behaviour of the volatility, you just specify a confidence interval
for its value. The volatility may evolve freely between these two bounds15. The
resulting pricing and hedging models are therefore much more robust than the
traditional ones, as they do not assume any particular stochastic process or require
going through an estimation procedure.

Conclusions
The application of mathematics to important problems related to financial derivatives
and risk management has expanded rapidly over the past few years. The increasing
complexity of the financial products coupled with the vast quantity of available
financial data explains why both practitioners and academics have found that the
language of mathematics is extremely powerful in modeling the returns and risks in
the application of risk management techniques. The result from this global trend is a
profusion of highly technical mathematical models, rather confusing for the financial
community. Which model to adopt? The past decade is full of examples where undue
reliance on inadequate models led to losses. The collapse of Long Term Capital
Management, which relied heavily on mathematical models for its investment strategy,
has raised some important concerns among model users. The mathematics in a
model may be precise, but they are useless if the model itself is inadequate or wrong.
Over-reliance on the security of mathematical models is simply an invitation for
traders to build up large and directional positions!
The most dangerous risks are where we do not even think about them. And they
tend to be there when we least expect them. Model risk is one of those. Furthermore,
it now appears to be present everywhere, and that we have to live with it. Virtually
all market participants are exposed to model risk and must learn to deal with it.
Regulators are also aware of it. Proposition 6 of the Basel Committee Proposal (1997)
is directly in line: ‘It is essential that banks have interest rate risk measurement
systems that capture all material sources of interest rate risk and that assess the
effect of interest rates changes in ways which are consistent with the scope of their
activities. The assumptions underlying the system should be clearly understood by
risk managers and bank management.’ Similar recommendations apply to any other
market.
However, uncertainty about the appropriate model should not necessarily reduce
the willingness of financial institutions to take risk. It should simply add one star in
the galaxy of risks because this risk is only an additional one, which can be priced,
hedged, and managed. 442