Reduced Variance Sensitivity Estimators for Gaussian Systems

Suppose we have a system of independent Gaussian random variables Xi ∼ N (μ, σ2 i ), i ∈ 1, . . . , N in (Ω,F ,P). We can determine the derivative estimator of D(X; ·) w.r.t a parameter, σ, of a system L(X, ·) if on (Ω,F), D(X; ·), L(X; ·) is given by the expression ∂ ∂σEP[L(X;σ)] = EQ[D(X;σ)]. This thesis provides an analysis to Gaussian random variables w.r.t three types of unbiased derivative estimators: Infinitesimal Perturbation Analysis (IPA), Score Function/Likelihood Ratio method (SF/LR), and the Weak Derivative/MVD derivative estimator, with particular focus to the last estimator. In this thesis, we provide a derivation of the three estimation schemes for analyzing these estimators for Gaussian systems. Here, the author provides provides a proof for zero mean monomial and general exponential performance functions, t hat the WD estimator coupled with a simulation scheme yields the least variance. This is contrary to the general notion that IPA is the best estimator. We then provide two applications for derivative estimation. Firstly, a Stochastic Activity Network is analyzed. Using each derivative estimation scheme, the sensitivity of the first two moments of the completion times are analyzed w.r.t a common standard deviation of the activity arcs in the activity network before providing a financial application. In the Black-Scholes framework, the author derives a WD derivative estimator to determine vega, the sensitivity of the price of the option w.r.t the spot volatility of the model. We then compare the result of this estimator to the analytical expression of vega for European Call and Put contract, and Lookback call, before providing an estimate for vega for Asian options.