Why the Heston Quadratic Exponential Scheme Uses Two Variance Forms
Summary
This exchange explains why Andersen’s quadratic exponential scheme uses different distributions to simulate the next Heston variance value. At higher variance levels, the method approximates the transition with a quadratic function of a Gaussian variable. At lower levels, it uses a distribution with a point mass at zero and an exponential tail. The change reflects how the variance transition density shifts from strongly decreasing at low levels toward a more bell-shaped form at higher levels.
The answer also identifies moment matching as a practical constraint: the equations for matching two moments may fail for one form in some regions, while there is an overlap where both forms are available. A switching rule selects between them. The exchange describes QE as a useful weak approximation for short time steps, while cautioning that it may perform poorly for Greeks. It offers qualitative reasoning rather than a derivation, implementation details, or numerical comparisons.
Key ideas
- The Heston QE scheme uses a quadratic Gaussian form when variance is sufficiently high.
- At low variance, it uses a distribution with a probability mass at zero and an exponential tail.
- The transition density changes shape as the current variance changes.
- Moment matching can make either approximation unavailable in some regions.
- The method is described as useful for weak simulation but potentially less suitable for Greeks.
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# Quadratic exponential method (by Andersen) in Heston model
# Quadratic exponential method (by Andersen) in Heston model
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 by Mark Joshi (score 4, accepted)
https://quant.stackexchange.com/a/15590
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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.