Fast Monte Carlo Greeks for financial products with discontinuous pay-offs (Q2847241)

Whilst Monte Carlo simulation has long been a standard technique for the evaluation of high-dimensional integrals, there are still many subtleties to be explored.NEWLINENEWLINEIn the present paper two new schemes for discretizing processes driven by Brownian motions are introduced. They allow to accelerate the computation of the sensitivities of discontinuous integrals using pathwise methods even when the underlying densities post discretization are singular. These two methods allow Greeks for financial product with trigger features to be computed in the LIBOR market model with similar speed compared to that obtained using the adjoint method for continuous pay-offs.NEWLINENEWLINEIn the introduction of the paper a brief review of the pathwise methods for the computation of the sensitivities of the integrals with respect to the parameters is given.NEWLINENEWLINEIn Section 2 the numerical schemes and the notations about the problems of the paper are outlined. The main stochastic differential equation which is solved in the paper is presented and the dependence on the parameters is discussed. An Euler discretization scheme for solving this equation is presented.NEWLINENEWLINEIn Section 3 a class of numerical schemes, which are quasi mean-shifted proxy schemes, is introduced. In Definition 3.1 the notion of quasi mean-shifted proxy simulation scheme is developed. The computation of Monte Carlo Greeks using quasi mean-shifted proxy schemes is developed. A review of the quasi mean-shifted proxy simulation schemes for the purposes of Greeks computations, like full proxy scheme, the PP scheme, the localized PP scheme and the MPP simulation scheme, is presented. In Subsections 3.2 and 3.3 the computing techniques of the PP simulation scheme and the MPP simulation scheme are developed. The problem for minimizing the variance of the Monte Carlo weights is discussed.NEWLINENEWLINEIn Section 4 it is shown that under the quasi mean-shifted proxy schemes the expectation and the differentiation when computing price sensitivities can be interchanged. Four assumptions are introduced. In Theorem 4.1 the expectation of the partial derivative \(\displaystyle {\partial \alpha \over \partial \theta}\) is shown.NEWLINENEWLINEIn Section 5 the pathwise derivatives for the PP and the MPP simulation schemes are derived. In Proposition 5.1 under PP simulation scheme the pathwise derivative with respect to \(\theta_0\) for the financial contract \(D\) is given. In Proposition 5.2 this is done under the minimal PP simulation scheme.NEWLINENEWLINEIn Section 6 a naive algorithm to evaluate the pathwise derivative is presented. Under this approach, the partial derivatives which take part in Propositions 5.1 and 5.2 are calculated using matrix recursions. In Subsections 6.1 and 6.2 a naive approach is applied to evaluate PP and MPP pathwise derivatives.NEWLINENEWLINEIn Section 7 the adjoint method is used to evaluate the PP and MPP pathwise derivatives.NEWLINENEWLINEIn Section 8 it is shown that under the LIBOR market model, all deltas and vegas can be evaluated with a computational order proportional to the number of rates times the number of factors at each step of the simulation using the pathwise PP method and the pathwise MPP method with linear proxy constraint functions.NEWLINENEWLINEIn Section 9 the relationships between both the pathwise PP method and the pathwise MPP method with the standard pathwise method and the LRM method are discussed. It is shown that the pathwise PP method and the pathwise MPP method are hybrids of the standard pathwise method and LRM.NEWLINENEWLINEIn Section 10 the computing of the deltas and vegas is considered for target redemption notes with LIBOR floater and target redemption notes with CMS floater and the LMM is used as the benchmark model.NEWLINENEWLINEIn Section 11 the standard errors of deltas and vegas which are calculated using the bump and revalue method, the pathwise PP and MPP methods are compared.NEWLINENEWLINEThe paper finishes with a conclusion, where a general review on the PP and the MPP methods for rapid computation of Greeks is given.