Choosing Local, Stochastic, and Stochastic Local Volatility Models
Summary
The answer explains stochastic local volatility (SLV) as a hybrid that combines a local volatility surface calibrated to vanilla options with a stochastic process for volatility. The local component helps fit vanilla prices, while the stochastic component adds another source of randomness to volatility dynamics. The response frames the hybrid as a way to address limitations that appear when either approach is used alone. It says local volatility can produce a leverage surface that flattens with maturity, making forward smiles less convex and potentially mispricing exotics sensitive to forward skew and smile, such as cliquets. It also notes a directional valuation concern: stochastic volatility may overvalue barrier and touch options, while local volatility may undervalue them. Volatility of volatility and spot-volatility correlation govern the blend and can be calibrated to exotic-option quotes. This is a brief conceptual answer, not a full derivation or empirical comparison; its claims about pricing behavior are not accompanied by data or model assumptions.
Key ideas
- SLV combines a local volatility surface with a separate stochastic volatility process.
- Local volatility can fit vanilla options while producing less convex forward smiles at longer maturities.
- The answer describes opposite barrier and touch valuation biases for stochastic and local volatility models.
- Volatility of volatility and spot-volatility correlation help control the SLV blend and its calibration.
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Full text
# When to use a Local Vol model vs Stochastic Vol Model? # When to use a Local Vol model vs Stochastic Vol Model? I'm new to volatility modeling, I'm struggling to understand when to use a Local Vol model and when to use Stochastic Vol Model, Also now we use a hybrid model combining the two models ? Can someone please provide some explanation ? ## Answer by AKdemy (score 8) https://quant.stackexchange.com/a/70964 First, what is the SLV? It combines LV (not really a model, just uses vanilla surface to get a grid) with SV (in a nutshell, BSM with a separate stochastic process for vol, hence multiple dynamic factors). Shortcomings SLV tries to address?: - BS does not price exotic option well. - `LV` calibrates nicely to vanilla but the calibrated leverage surface is typically observed to flatten with maturity which means the forward volatility smile will be less convex than on the initial pricing date and you will not be pricing deals properly which are primarily sensitive to forward volatility skew and smile (cliquets and co). - `SV` prices for barriers and touches tend to be overvalued by SV (undervalued by LV). In SLV, mostly ($\xi$) vol of vol and correlation ($\rho$) control the mixing of LV and SV. Hence, appropriate calibration of the mixing parameters will allow you to closely match market quotes for exotic options. LV and stochastic SV are simply degenerate cases where the mixing fraction is such that only one or the other is used (e.g. if $\xi = 1$, SLV becomes purely SV). These questions/answers should also help - Local Volatility vs. Stochastic Volatility - What are the advantages/disadvantages of these approaches to deal with volatility surface?.
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