Smile Dynamics and the ATM Volatility Effect of Spot-Volatility Approximation

Summary

The document asks about an approximation discussed in a paper on delta hedging with a volatility smile: treating the sensitivity of implied volatility to the underlying price as equal to its sensitivity to strike. Under a downward-sloping smile, the cited passage describes a parallel downward shift as spot rises, which changes fixed-strike volatilities and moves at-the-money volatility by twice the slope amount.

The question tests this interpretation with a small set of strike volatilities around an at-the-money level. It asks whether the implied volatility assigned to a fixed strike becomes misleading when spot changes and the smile is shifted, and why an existing skew should imply further shifts. The document raises the distinction between a smile’s cross-sectional shape and its dynamics as spot moves, but offers no answer or evidence resolving the confusion. It does not specify a volatility model or provide a derivation, so it is best read as a conceptual question about smile assumptions in hedging.

Key ideas

• A volatility smile describes implied volatility across strikes at a given time.
• The discussed approximation equates spot sensitivity of implied volatility with strike sensitivity.
• Under a downward-sloping smile, that assumption implies a parallel shift as spot changes.
• The question challenges how this shift affects fixed-strike and at-the-money volatility.
• No derivation or resolution is provided, and the result depends on the assumed smile dynamics.

Tags

# Why approximating dSigma/dS with dSigma/dK changes the ATM volatility at twice the rate?


# Why approximating dSigma/dS with dSigma/dK changes the ATM volatility at twice the rate?












I'm referring to the paper "Delta Hedging With a Smile", Sami Vahamaa (2004). It mentions:

> By approximating ∂σ/∂S with ∂σ/∂K, it is assumed that as S changes by one unit, there is a parallel shift of ∂σ/∂K units in the volatility smile. If the current smile is downward sloping, the approximation assumes that the volatility smile is shifted downwards as the price of the underlying stock increases, and thus, all fixed-strike volatilities decrease by the amount defined by the slope of the current smile and at-the-money volatility decreases twice as much.

Let's say I have 5 strikes with it's 5 implied volatilities, with ATM strike being 100:

My understanding is that while S = 100, the implied vol of 25.5 in K=99 means that the average volatility of all the paths until maturity that end with S=99 will be 25.5. Now, if S goes down to 99 according to the paper the whole curve would shift upwards by 0.5 volatility points leaving us with:

Now the average volatility of all the paths ending with S=99 is 26, half a vol point higher. Doesn't it mean that the implied volatility of K=99 when S=100 was trivial/wrong/useless?

I guess what I am trying to understand is: if the skew is already there to account for how changes in S affects the ATM volatility, why are we shifting the whole curve upwards by an amount equal to the slope when S decreases? That just seems to me as having a skew that provides no or wrong information.

Thanks

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