Theory 04 Variance and accuracy
Numerical refinements
The estimators of Conditioning: Vibrato and Vibrato Saltando are exact in theory; the following refinements make the inner expectation accurate in practice. In all of them the inner law is Gaussian, $X_T | X_{\tau} \sim \mathcal{N}(\mu, s^2)$, and the inner expectation is computed on the standardized variable $Z = (X_T - \mu)/s$.
Payoff Centering
For every $\theta$ the density integrates to one:
\[\int_{\mathbb{R}} f(x,\theta) dx = 1\]
hence all of its derivatives vanish and, for $n \geq 1$:
\[\mathbb{E}(B_n(\{\partial^{i}_{\theta} \log{f(X_T,\theta)}\}_{i = 1 : n})) = 0\]
Therefore any constant $c$ not depending on $\theta$ can be subtracted from the payoff without changing the sensitivities:
\[\partial^{n}_{\theta} V_0 = \mathbb{E}((\Phi(X_T) - c) B_n(\{\partial^{i}_{\theta} \log{f(X_T,\theta)}\}_{i = 1 : n}))\]
We take $c = \Phi(\mu)$, the payoff at the centre of the inner density, and add it back to the price. The price is unchanged, while the score terms are now multiplied by $\Phi(\mu + s Z) - \Phi(\mu) \approx \Phi'(\mu) s Z$ instead of $\Phi(\mu + s Z)$. In vibrato $s = \sigma \sqrt{T - \tau}$ is small and the score terms grow like $s^{-n}$, so without centering the variance of the delta estimator grows like $s^{-2}$; with centering it stays bounded. Together with antithetic pairs $\pm Z$, the second order term becomes $\frac{1}{2}(\Phi(\mu + s Z) + \Phi(\mu - s Z)) - \Phi(\mu) \approx \frac{1}{2}\Phi''(\mu) s^2 Z^2$, which compensates the $s^{-2}$ of the gamma score. In practice this reduces the noise of the gamma estimated with inner Monte Carlo by a factor 10 to 80.
Independent Inner Samples
The inner expectation is estimated with $M$ samples $Z_1, \dots, Z_M$ for each of the $N$ outer paths. If the same samples were used for every outer path, the inner error would be the same for all paths and would not average out: the estimator would converge, as $N \to \infty$, to a biased value with error of order $M^{-1/2}$ (deltas 8-15% off with $M = 200$). Drawing new inner samples for every outer path makes the inner errors independent, and the total error decreases with both $N$ and $M$. With quasi Monte Carlo inner samples, the same Sobol points are randomly shifted modulo 1 for every outer path (Cranley-Patterson rotation (Cranley and Patterson, 1976)), which keeps each inner estimate unbiased.
Quadrature
Alternatively, the inner expectation can be computed as a deterministic integral on a uniform grid in $Z$. The grid must reach far enough in the tails: for a call the payoff grows exponentially in the upper tail, and truncating at $\pm 5$ standard deviations loses enough of the second order score terms to bias the volga by 3% to 45% on Variance Gamma. The default grid spans $\pm 7$ standard deviations.