How Local Volatility Differs from Implied Volatility
Summary
The document introduces the distinction between implied and local volatility through a learner’s interpretation of option prices and diffusion models. Implied volatility maps strike and maturity to the volatility that reproduces a market option price under Black–Scholes. Local volatility instead describes the instantaneous volatility used in a price process, conditional on the underlying level and time.
The question presents a formula that derives local variance from derivatives of call prices with respect to maturity and strike, then asks why strike appears when the simulated asset process itself has no strike input. This points to the key conceptual bridge: market option prices across strikes encode information about the distribution of future underlying prices, which can be used to infer a state-dependent volatility function. The document offers no derivation or resolution and does not discuss assumptions, data quality, or practical calibration limits; its value is framing the distinction and the source of the learner’s confusion.
Key ideas
- Implied volatility expresses an option’s market price through a pricing model using strike and maturity.
- Local volatility specifies instantaneous diffusion volatility by underlying price level and time.
- The local variance formula uses strike derivatives of call prices to infer state-dependent volatility from the option surface.
- The document poses this conceptual question but does not provide a derivation or answer.
Tags
Full text
# Local volatility Formula and How To use it
# Local volatility Formula and How To use it
I'm new to Volatility Modelling, so the content of this question may be completely wrong and th question naive. I'm reading "The volatility surface" by Gatheral. I'm trying to get a sense of the first chapters about Local Volatility models.
For the moment I'm trying to get the intuition since the math proved to be a bit tough. My general intuition failed when I saw this formula to compute the Local Volatility surface from market prices.
$$ \sigma ^{2}\left ( K,T,S_{0}\right ) = \frac{ \frac{\partial C}{\partial T}}{ \frac{1}{2}K^{2}\frac{\partial^2 C}{\partial K^2}} $$
While at the beginning of the model, we were looking for $\sigma \left ( S_{t},t;S_{0}\right ) $ that had to be used in
$$\frac{\mathrm{d}S }{S} = \mu_{t} dt + \sigma(S_{t},t;S_{0})dZ$$
So now I'm a bit puzzled. Here is my general intuition about Local Volatility and Implied Volatility (surfaces). I hope that this helps in understanding the flaws in my reasoning.
So, ImpliedVolatiltiy Surface is a mapping from $(K,T)$ to ImpVol such that ImpVol reflects the current options market price when used in a model $(BS)$. Therefore it is nothing else than a way to express option prices in terms of the corresponding volatility implied by their market price in the BS model. Till here, everything is quite clear.
On the other hand, LocalVolatiltySurface is a map from $(S,T)$ to Vol. Where Vol is used in a numerical method (Monte Carlo) to simulate at each step the corresponding level of volatility that has to used to simulate the next step (in this sense, when plotted the local volatility surface should be plotted on the space $(S,T,Vol)$ and not $(K,T,Vol)$.
Or in theoretical terms, it is the instantaneous volatility of the price diffusion process for each level of $S$ and $T$. In other terms $\sigma(S_{t},t;S_{0})$. Moreover, such mapping is consistent with the current market expectation of volatility extrapolated from the Implied Volatiltiy Surface.
I don't see then how $ \sigma ^{2}\left ( K,T,S_{0}\right ) $ can be dependent from $K$, since $dS$ should not depend on the strike price of the option.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.