ATM Call Delta in a Local Volatility Model
Summary
The document examines a claim about the delta of an at-the-money European call in a local volatility model. It gives relationships between at-the-money implied and local volatility, describes implied volatility as depending on strike, and asks for a proof that the local-volatility delta can be expressed as the Black–Scholes delta plus a vega adjustment for how implied volatility changes with spot.
The answer explains that a vanilla option’s implied volatility can depend on both strike, maturity, and the current spot when the local volatility surface is held fixed. Applying the chain rule to the option value separates the direct spot sensitivity, evaluated at fixed implied volatility, from the sensitivity through the implied volatility; the latter is Black–Scholes vega multiplied by the spot derivative of implied volatility. This gives the structure behind the proposed correction. The response does not establish that the strike derivative in the original claim is generally interchangeable with the required spot derivative, nor does it work through the stated at-the-money identities, so that distinction needs care.
Key ideas
- In a local volatility model, a vanilla option’s implied volatility can vary with current spot as well as strike and maturity.
- The total spot sensitivity includes both a direct price effect and an effect transmitted through implied volatility.
- The chain rule expresses the second contribution as Black–Scholes vega times the spot sensitivity of implied volatility.
- The response describes a spot derivative, so it does not by itself prove a formula using an implied-volatility strike derivative.
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Full text
# Proof for ATM delta with Local col
# Proof for ATM delta with Local col
I am looking at a time-homogeneous local volatility model where
- ATM implied volatility equals ATM local volatility: $\sigma_{imp}(S_0)=\sigma_{local}(S_0)$
- ATM IV Skew = half of LV slope
- In general $\sigma_{imp}(K) = \sqrt{\frac{\sigma_{local}^2(S_0)+\sigma_{local}^2(K)}{2}} $
LV is a function of spot and and is Calibrated to IV which is a function of strike (we are working with European call option in this case).
Claim: Delta for an ATM European Call Option in the LV model is given by: $$ \Delta_{LV} = \Delta_{BS} + \text{vega}_{BS}*\frac{\partial \sigma_{imp}(K)}{\partial K}\bigg\rvert_{K=S_0} $$ where $\Delta_{BS}$ and $\text{vega}_{BS}$ is the Black Scholes vega and Delta.
What is the proof for this claim?
Bassically I don't really know exactly how volatilites look in each of the delta term and that is why I can't construct a proof. So by explaining the model and the last equation thoroughly I will probably be able to reach the proof myself.
## Answer by Quantuple (score 2)
https://quant.stackexchange.com/a/39287
In a local volatility model, which is inhomogeneous in space, you'll end up with having that implied volatility of a vanilla option $(K,T)$ is a function of the spot price $S$, i.e. $$ \Sigma = \sigma(T,K,S) $$ As such when you compute the (total) derivative of the option price with respect to the spot price you'll have: \begin{align} \frac{d V}{d S} &= \left.\frac{\partial V}{\partial S}\right\vert_{\Sigma} + \left.\frac{\partial V}{\partial \Sigma}\right\vert_{S} \frac{\partial \sigma(T,K,S)}{\partial S} \\ &= \Delta_{BS}(S,\Sigma) + \nu_{BS}(S,\Sigma) \frac{\partial \sigma(T,K,S)}{\partial S} (S) \end{align} The last term should be how the implied volatility moves as the spot moves but the local volatility function is kept unchanged.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.