European Options Sensitivities via Monte Carlo Techniques

This paper proposes a unified approach to Monte Carlo estimation of sensitivity of European option premiums with respect to some arbitrary parameter. The classical framework assumes that the underlying parameter is some intrinsic parameter of the model, e.g., interest rate, volatility or time to maturity, in which case sensitivities are also known as ”Greeks”. Intrinsic parameters only induce variability in the dynamic of the stock-price(s) under consideration. The present approach allows the parameter under discussion to induce variability in the payoff function and also in the exercise rule of the option. Our leading examples come from the family of the so-called digital options, i.e., financial options which pay off some (apriori) fixed amount of money provided that the stock-price(s) at maturity lie in some certain region, such as (finite intersections of) polyhedra and/or spheres, in the n-dimensional space, where n ≥ 1 denotes the number of underlying assets. For such options, the payoff and the exercise rule can be chosen independently. This approach essentially relies on differentiation of multiple integrals with parameter and appropriate formulas are established in a rather general setting. In our leading examples the underlying parameter dictates the exercise rule, i.e., integrals on moving domains have to be considered, and a direct appeal to surface integrals must be made. General Monte Carlo integration techniques for evaluating such integrals will be presented and illustrated by some examples. Finally, the connection between the main results of this paper and the concept of weak differentiation will be discussed.