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Implied Volatility Skew Bounds as Correlation Approaches One

Article Quant Q&A · Author: user34971
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Summary

The document asks whether model-independent bounds or accurate approximations exist for implied-volatility skew in stochastic-volatility models as the correlation between asset and volatility shocks approaches positive or negative one. It seeks results that do not depend on the particular drift and diffusion functions governing volatility, ideally expressed using observable Black–Merton–Scholes quantities.

The author proposes a relationship between the difference of two specially defined implied volatilities and an expectation involving average volatility and its stochastic shock. If that volatility difference were known in the limiting-correlation case, the author suggests it could help infer the expectation and then correlation. However, the document supplies no answer, bound, empirical evidence, or validation of this derivation. The proposed relationship is presented as approximate, and its usefulness depends on resolving the original question about the limiting skew.

Key ideas

  • The question concerns implied-volatility skew in stochastic-volatility models near extreme asset-volatility correlation.
  • The desired result would avoid dependence on the model’s specific volatility drift and diffusion functions.
  • The author proposes relating a difference between specially defined implied volatilities to an expectation involving average volatility and its shock.
  • The proposed relationship is approximate and is not supported by a result or empirical validation in the document.

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OptionsVolatilityDerivatives pricingStatistics
Full text
# Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$


# Robust bounds or approximations on implied volatility skew when $\lvert \rho \rvert \rightarrow 1$












Are there any robust / non-parametric results for pure stochastic volatility models, in terms of bounds or preferably accurate approximation, for the implied volatility skew $\partial IV(k) / \partial k$ when the correlation between the stochastic instantaneous volatility and the asset approaches $\pm 1$?

What I mean with "robust" is that if the SV process is $$ dS_t = \sigma_t S_t dZ_t $$ $$ d\sigma_t = a (t,\sigma_t) dt + b(t,\sigma_t) dW_t $$ $$ dW_t dZ_t = \rho dt $$ whatever $a$ and $b$ are, the bound/approximation on the skew should not depend on the particular form of $a$ and $b$, hence can the bound/approximation be expressed in terms of Black-Merton-Scholes quantities only (as these BMS quantities are directly observable in the market)?

EDIT:

I can derive that, if $\Sigma_{d_2}$ denotes the implied volatility where both the BMS vanna and volga of an option is zero, and $\Sigma_{d_1}$ is the implied volatility where only the BMS volga of a vanilla option is zero, then $$ \rho E_t \left[ \bar{\sigma} \int_t^T \sigma_u dW_u \right] \approx \Sigma_{d_1} - \Sigma_{d_2} $$ with $$ \bar{\sigma} = \sqrt{ \frac{1}{T-t} \int_t^T \sigma_u^2 du} $$ So if I know the value of $\Sigma_{d_1} - \Sigma_{d_2}$ as $\lvert \rho \rvert \rightarrow 1$, then I can back out $ E_t \left[ \bar{\sigma} \int_t^T \sigma_u dW_u \right]$. And given the observable value of $\Sigma_{d_1} - \Sigma_{d_2}$ I can then subsequently back out $\rho$.

Hence my question. Giving it a bounty.

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