Why LSV Models Calibrate Stochastic Volatility to Reduced Surfaces
Summary
The document explains why local stochastic volatility (LSV) calibration may first fit a stochastic volatility model to a reduced version of the market volatility surface. This creates a lower partial surface that can be combined with one derived from local volatility. The combined surface is intended to preserve European option prices while producing dynamics between stochastic volatility’s sticky-delta behavior and local volatility’s sticky-strike behavior.
As a quicker alternative, the answer suggests reducing the stochastic model’s skew and kurtosis through a mixing fraction. For mixed local volatility (MLV), it describes starting from reduced local volatility and applying a fixed stochastic multiplier that randomly raises or lowers the local volatility. These are heuristic descriptions, not a full calibration recipe: the document gives no derivation or empirical comparison, and it points readers to further treatments for details.
Key ideas
- A reduced volatility surface is described as a lower partial surface used in combining stochastic and local volatility effects.
- The merged LSV surface is intended to price European options correctly while yielding intermediate volatility dynamics.
- A mixing fraction can reduce the skew and kurtosis generated by the stochastic volatility model.
- The MLV description applies a randomly positive or negative fixed multiplier to reduced local volatility.
- The answer characterizes its LSV explanation as heuristic and provides no detailed calibration procedure.
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# Reduced volatility in local stochastic volatility model # 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? ## Answer by JulianRCook (score 4) https://quant.stackexchange.com/a/71627 > 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]
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