# Reduced volatility in local stochastic volatility model

in Local Stochastic Volatility models I always read or hear "first the stochastic volatility model is calibrated to reduced vols and then the local volatility model corrects it" also I head that by reducing the volatility surface means flattening it.

My question, why do you reduced or flatten the volatility surface in order to calibrate the stochastic volatility ? and what is the intuition behind the parameter or function used to flatten the volatility surface?

I also seen the same thing is done for Mixed Local Volatility models (MLV). Any specific inputs regarding the MLV in this case?

why do you reduce or flatten the volatility surface in order to calibrate the stochastic volatility ?

This is a crude way of creating a partial volsurface, which is lower than the actual market vol surface. In the LSV process*, this is then merged with a partial vol surface created from the local vol surface. The two vol surfaces, when merged together, should correctly price european options BUT the dynamics of the new vol surface will be between stochastic (sticky delta) and local (stick strike). A quicker way is to actually reduce the stochastic parameters that generate the skew and Kurtosis in the stochastic model, by using a 'mixing fraction' [0.0 - 1.0].

Any specific inputs regarding the MLV in this case?

As for MLV, this is described in several Mathfinance presentations. The approach is simpler, but different. You start with a reduced local vol surface and introduce a fixed Stochastic multiplier on the local vol process. The multiplier e.g 0.10 is randomly positive or negative, so the local vol process is multiplied by either 1.1 (1.00*1.10) or 0.909 (1.00/1.10).

There is a brief discussion of an MLV like process here (section 8):

Austing, Peter, Finite Difference Schemes with Exact Recovery of Vanilla Option Prices (August 2, 2019). Risk, November 2020,

Available at SSRN: [https://ssrn.com/abstract=3530561] or [http://dx.doi.org/10.2139/ssrn.3530561]

• Note that this is a very heuristic description of what the LSV process is. A good overview, again from Peter Austing, is in:

Austing, P. (2014). Local Stochastic Volatility. In: Smile Pricing Explained. Financial Engineering Explained. Palgrave Macmillan, London. [https://doi.org/10.1057/9781137335722_9]