# Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model

I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in Section 2.1 of the same paper or some kind of version of Heston. I am interested in this question because I want to do a Monte Carlo based calibration and my model (which has multiple factors) is too slow for this. A reasonable approximation with a simpler (one-factor) model would be much faster and would be acceptable in this context.

In more detail, I am mostly interested in a one-factor approximation of two-factor stochastic volatility models, such that most of the volatility dynamics behaviours modelled by the two-factor model are accurately reprices by the one-factor one. A one factor would have a single Brownian motion driving the stochastic volatility, whereas and $$n$$-factor model would have $$n$$ Brownian motions driving the stochastic volatiity.

Have such questions been studied in the literature? Have such questions been studied in some generality or only for some types of models?

I could not find any reference until now, but I am sure there must be such approaches out there.

• Approximating a 2-factor model by a 1-factor model is along the same lines as approximating a parallel shift and a twist by only a parallel shift. It defeats the purpose. Besides, you do not have to approximate, there are enough 1 factor SV models out there which you can modify/tweak to your liking as long as they remain arbitrage free.
– user34971
Sep 29, 2022 at 13:48
• In addition to Frido's point, am not sure it makes sense to use a reduced-form for calibration of your bigger model; the parameters masked out in the calibration phase won't be defined and therefore you cannot price anything with the full model, only the reduced-form one. Which is basically Frido's point - just use a simpler model for this purpose. Sep 30, 2022 at 5:33
• @JamesSpencer-Lavan I agree with you. Do you know where can I read about two-factor models and what they can achieve better than one-factor models? I was looking for some more readable reference. Oct 3, 2022 at 18:30
• @fwd_T not being facetious but Bergomi's paper summary provides an insight to what two-factor SVM brings over one-factor: "In a previous article we highlighted how traditional stochastic volatility and Jump/Lévy models impose structural constraints on how the short forward skew, the spot/vol correlation, and the term structure of the vol-of-vol are related. Here we propose a model that enables them to be controlled separately and also prices options on realized variance consistently." Oct 3, 2022 at 18:57
• @JamesSpencer-Lavan ok, I will read this paper to see what I get from it. Oct 11, 2022 at 13:48