Advances in Risk Management

GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 361

0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

100%
80%
60%
40%
20% 0% 20% 40% 60% 80% 100%

Figure 18.1Hedging error probability density function in the B–S case.

The “real” asset dynamics is given by equation (18.1). All errors are

expressed as a percentage cost of the option

Table 18.1Evaluation of the hedging errors

Model parameters

rabα x ρ

True model 0.02 0.508 0.151 0.4 0.8 −0.2

Trader’s model 0.02 0.534 0.75 0.201 0.65 0

Notes: The table shows the values of the parameters used to evaluate the hedging errors. “True”

model parameters have been selected to provide a realistic shape of the smile. Trader’s model

parameters are obtained by minimizing the difference between the smiles generated by the two

models. Calibration errors are within the typical implied volatility bid/ask spread of 1%.

This example clearly demonstrates that if the trader’s model is a bad
representation of reality, the hedging error can be very large.
As a second test, we now assume that the trader uses a stochastic volatility
model to hedge the contingent claim which is structurally similar to the
“real” one but he has a bad assessment of some parameters. In this case,
equation (18.5) holds. In particular he believes that correlation is zero; that
isρ′=0, while in realityρ =0. In our example, we fixedρ=−20% and to
ensure that all market information is captured we calibrate model (18.2)
to the smile generated by equation (18.1). In this procedure we determine
the unknown parameters by minimizing the squared difference of option
prices between the two models for different values of the strike, following a
standard practice. The resulting errors are within the typical bid–ask spread
and amount to a fraction of a percent in log-normal units (Table 18.1).