Nan Chen, Columbia University, IEOR

We discuss Monte Carlo methods for two problems central to the pricing and hedging of derivative securities: (i) calculation of "Greeks" (price sensitivities) and (ii) valuation of American options. Both are based on joint work with Paul Glasserman. 
Malliavin Greeks without Malliavin calculus: Standard methods for estimating sensitivities differentiate paths or differentiate measures. A recent line of work derives estimators using Malliavin calculus. In implementing a discrete-time approximation of a continuous-time model, one may discretize first and then differentiate or vice-versa. In several important cases the first route produces the same estimators produced by Malliavin calculus, but using only traditional elementary techniques.

Additive and multiplicative duality: Pricing an American option entails solving an optimal stopping problem. Dual formulations replace maximization over stopping times with minimization over martingales. We compare duals based on additive and multiplicative decompositions of positive supermartingales. We establish an equivalence in the quality of the bounds achieved by the two methods, but show that the variance of the multiplicative method is typically much larger.