Notes on Options, Hedging, Prudential Reserves and Fair Values

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.