Forward Skew in Local- and Stochastic-Volatility Models

Summary

The document explores how implied forward skew relates to the shape of the market implied-volatility surface and to calibration choices in local-volatility (LV) and stochastic-volatility (SV) models. It presents a formula for deriving forward implied volatility from two maturities and asks whether flatter long-dated spot skew necessarily implies flattening in implied forward skew. It also questions whether LV models inherit that behavior from the market surface.

The remaining questions concern whether an SV model can fit the entire implied-volatility surface while preserving forward skew, or whether preserving that skew may require sacrificing fit at longer expiries. These are posed as conceptual alternatives rather than answered claims. The document provides no calibration results, model specifications, or evidence to resolve them, and the displayed volatility calculation alone does not establish how a calibrated model’s forward skew will behave. Readers should treat the text as a set of modeling questions, not as a demonstrated comparison of LV and SV performance.

Key ideas

• The document relates implied forward skew to the maturity structure of the implied-volatility surface.
• It asks whether flat long-dated spot skew leads to flattening in implied forward skew.
• It questions how local-volatility models reflect forward-skew behavior in the market surface.
• It asks whether stochastic-volatility models can preserve forward skew while fitting the full surface.
• No calibration evidence or model-specific answer is provided.

Tags

# 81239


# To preserve the forward skew in a stochastic volatility model, are we required to restrict the calibration to part of the Implied Vol surface?












I've understood in my learning journey that LV models have flattening of forward skew especially at longer expiries, and SV models are able to preserve this forward skew. I've the following questions in that context:

1] If we plot the implied forward skew from market implied vols, they display a flattening behavior, because the spot smile has flat skew for large expiries. Would this be a correct statement? $$ \sigma_{12,\text{IV}} = \sqrt{\frac{\sigma_{2,\text{IV}}^2 T_2 - \sigma_{1,\text{IV}}^2 T_1}{T_2 - T_1}} $$ 2] Is it correct to say that LV models have flattening of forward skew because of the nature of the market implied vol surface itself? For example, if the market did not have skew flattening of spot smile for large expiries, then there wouldn't be flattening of implied forward skew, and subsequently the forward skew from the LV model also wouldn't show flattening behavior. 3][a] For an SV model, how is it possible to calibrate to entire implied vol surface, and also preserve forward skew, at the same time? If this is indeed possible, does this mean that the forward vol from an SV model does not match the same from the implied vol surface? 3[b] Or is it that if an SV model preserves the forward skew, then it cannot calibrate to the entire implied vol surface. For example, it might calibrate to the short end of the surface, and preserve the forward skew, and subsequently the skew of spot smile for large expiries would be larger than that implied by the market, and hence the SV model doesn't calibrate to the long end of the surface in this case.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.