In this paper we present many investigations into the results of simulating the process of hedging a vanilla option at discrete times. We consider mainly a ‘maxi’ option (paying Max(A, B)), though calls, puts and ‘minis’ are also considered. We show the sensitivity of the variability of the hedging error to the actual investment strategy adopted, and to the many ways in which the simulated real world can diverge from the assumed option pricing model. We show how prudential reserves can be calculated, using conditional tail expectations, and how net premiums or fair values (which we present as the same) can be calculated, allowing for the necessary prudential reserves. We use two bond models, the very simple Black-Scholes one and a less unrealistic one. We also use the Wilkie model as an even more realistic real-world model, allowing for many complications in it to make it more realistic. We make observations on the important difference between real-world models and option pricing models, and emphasise the latter as the way of getting hedging quantities, and not just option prices.

The use of classical bonus–malus systems entails very high maluses and other problems which, during recent years, have been criticised by actuaries. To avoid these problems, new bonus–malus models have been developed. For instance, it is well known that the use of an exponential loss function reduces the differences between overcharges and undercharges, solving the problem of high maluses. In order to measure the sensitivity of the exponential bonus–malus system, and according to robust Bayesian analysis, we first model the structure function by specifying a subclass of the generalised moments class. We then examine the range of relativities for each prior. Finally, we illustrate our method with a numerical example based on real data.