FRANÇOIS-SERGE LHABITAN T 203between the drift of the underlying asset and the risk-free rate. Depend-
ing on the sign of these differences, at maturity, the hedging strategy
may yield a profit or a loss. Therefore, the trader may end-up with
a replicating portfolio that is far from what he should have in order
to fulfil his liabilities. For some exotic options, delta hedging can
actually even increase the risk of the option writer, as evidenced by
Gallus (1996).The third term results again from a difference between the true delta
parameter and the delta given by the model. In addition, it depends
on a stochastic term (dW(t)), making the hedging strategy result both
stochastic and path-dependent. Last, but not least, it also depends on the
true level of volatility.
Clearly, in the presence of model risk, even though we assume frictionless
markets, the delta hedging strategy of our trader is no longer replicating or
self-financing. Even worse, it becomes path-dependent.
How can we one deal with model risk in practice when delta hedging a
position? The answer is not straightforward. Rebalancing the hedge more
frequently does not help, because there is still a difference between the true
hedging parameters and those given by the model. A possible solution con-
sists of looking for a super-hedging strategy, for example, a strategy such
that the hedging result is guaranteed with a given probability whatever
the true model^6. Unfortunately, super-hedging strategies become rapidly
expensive as the probability of being hedged increases. Another solution
consists of specifying a loss function to be minimized by the hedging strat-
egy. In this case, perfect hedging is transformed into minimum residual
risk hedging. But as a consequence, pricing is not uniquely determined
and risk neutrality cannot be used – different agents may have different
loss functions, and therefore, reach different prices for the option being
considered.
10.7 Eleven rules for managing model risk
Managing and controlling model risk is a difficult and complicated pro-
cess, which should generally be performed almost on a case-by-case basis
and cover at least three distinct areas: (a) the choice, testing and safe-
keeping of the mathematics and computer code that form the model;
(b) the choice of inputs and calibration of models to market data; and
(c) the management issues associated with these activities. The success
of model risk management and control will often depend crucially on
personal judgement and experience. Therefore, the following set of rules
should not be considered as a series of recipes, but rather as a minimum
checklist.