Quadratic exponential method (by Andersen) in Heston model

Question

I am having trouble understanding the reasons that led Andersen to define his QE scheme to efficiently simulate Heston Stochastic volatility model (you may check the celebrated scheme here).

The gist of it is that for "sufficiently large" values of the process modelling the variance, the scheme adopts the form: $$ V(t+dt) = a(b + Z)^2 $$ where $a,b$ are certain constants and $Z$ is a standard Gaussian random variable. For low values of $V$ he thinks that it is better to use: $$ V(t+dt) = \Psi^{-1}(U_V;p,\beta) $$ where $U_V$ is drawn from a uniform distribution, and $$ \Psi^{-1}(u) = \Psi^{-1}(u;p,\beta) = \left\{ \begin{array}{rl} 0 &\mbox{ if $0\leq u \leq p$} \\ \beta^{-1}\ln\left(\frac{1-p}{1-u} \right) &\mbox{ if $p<u\leq1$} \end{array} \right. $$

Andersen deduces this scheme by saying:

The first step is based on an observation that a non- central chi-square with moderate or high non-centrality parameter can be well-represented by a power-function applied to a Gaussian variable.

But then he gives little detail about why this observation is useful to develop his scheme, and why he thinks that when the variance his low the scheme needs to change its form. If somebody knows the mechanics behind his reasoning, I would be very glad if you could provide more insight into the subject.

Answer 1

There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable for the exponential form. Fortunately, the domain where both are solvable is non-empty and so there is always at least one available. It is then simply a question of when to transit from one to other. QE is probably the best short-stepping weak approximation. It is not so good for Greeks, however. (see http://ssrn.com/abstract=1718102)