How Negative Spot–Volatility Correlation Produces Implied Volatility Skew
Summary
The explanation links negative correlation between an asset’s spot price and its instantaneous volatility to a downward implied-volatility skew. It contrasts the result with the standard Black–Scholes assumption of constant volatility, under which out-of-the-money puts and calls have the same implied volatility in the idealized model.
With strongly negative spot–volatility correlation, paths ending far below the initial price tend to coincide with rising volatility, which increases the effective volatility reflected in put prices. Paths ending far above the initial price tend to coincide with falling volatility, lowering the effective volatility reflected in call prices. This produces higher implied volatility for low-strike puts than for high-strike calls. The note gives an intuitive path-based explanation, not a derivation or empirical test; the strength and shape of skew depend on model assumptions beyond this illustration.
Key ideas
- Constant volatility in standard Black–Scholes does not produce implied-volatility skew.
- Negative spot–volatility correlation links falling spot paths with rising instantaneous volatility.
- Low-strike puts are influenced by bearish paths with higher volatility in this setup.
- High-strike calls are influenced by bullish paths with lower volatility in this setup.
- The resulting pattern is a negative implied-volatility skew.
Tags
Full text
# Vol skew and spot-vol correlation # Vol skew and spot-vol correlation Why does the market assign a vol skew in the presence of a spot vs vol correlation. Both heuristic/fundamental answers as well as mathematical explanations welcomed ## Answer by Quantuple (score 2, accepted) https://quant.stackexchange.com/a/30304 Suppose you were to price 2 instruments: a strongly OTM put and a strongly OTM Call. In the standard BS settings, instantaneous volatility is assumed to be constant. Consequently, the implied volatility of these 2 instruments will be the same, resulting in an absence of implied volatility skew. Now, assume a negative spot/instantaneous volatility correlation. Make it very negative ($\rho \approx -1$) to make things clearer. - When you price the OTM put, it is the strongly bearish paths that will matter to you (think Monte Carlo if you like, only the paths that finish below the strike $ K << S_0 $ will contribute to the option price), that is, the paths where the spot is expected to strongly decrease. Because of the negative spot/vol correlation, along the latter "bearish" paths, instantaneous volatility is expected to increase, thus making "effective" volatility (i.e. the implied volatility) higher compared to the pure BS case. - When you price the OTM call, it is the strongly bullish paths that will matter to you (think Monte Carlo if you like, only the paths that finish above the strike $ K >> S_0 $ will contribute to the option price), that is, the paths where the spot is expected to strongly increase. Because of the negative spot/vol correlation, along the latter "bullish" paths, instantaneous volatility is expected to decrease, thus making "effective" volatility (i.e. the implied volatility) lower compared to the pure BS case. This illustrates how a negative spot/(instantaneous) volatility correlation leads to a negative implied volatility skew.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.