Interpreting Local Volatility from the Implied Volatility Surface
Summary
The document offers an intuitive account of how patterns in out-of-the-money put implied volatility may relate to local volatility across stock levels and maturities. One response describes a forward line as the path of forward levels across expiries and links a strike's option premium to paths that reach regions where the option has substantial gamma, weighted by the chance of those paths. A second response uses a path-based analogy: if a more distant strike has higher implied volatility than a nearer strike, the additional volatility is attributed to the later segment of the path toward that farther level.
These explanations aim to make the shape of the local volatility surface easier to picture, but they are informal rather than a derivation. The document supplies no equations, market data, or empirical test, and the response itself notes that fuller source context would help clarify the textbook passage. Implied volatility summarizes an option price, while local volatility is a state- and time-dependent model input; the path analogy should therefore be treated as intuition, not a direct rule for recovering local volatility from a single strike or path.
Key ideas
- Implied volatility varies by strike and maturity, while local volatility depends on the underlying level and time.
- A forward line describes how forward levels change across maturities.
- Option prices reflect both paths through relevant stock-price regions and the probability of those paths.
- The path-segment analogy offers intuition for how volatility differences across strikes may arise.
- The explanations are qualitative and do not provide a rigorous calibration method.
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# Local volatility implied by implied vol surface # Local volatility implied by implied vol surface In his book volatility and correlation, Rebonato tries to explain intuitively the shape of local volatility surface (depending on stock level and time) from the implied volatility surface in the OTM put side. See below. However his explanation isn’t clear to me (the last paragraph especially), can someone shed more light? Thanks, ## Answer by Ezy (score 1) https://quant.stackexchange.com/a/42249 In equity options a no-arbitrage argument shows that implied volatility has to be continuous along the forward line The "forward line" is the path of forward levels as a function of the maturity T conditional on a certain value of the spot at T=0 or conditional on the terminal value $F_T$ at maturity. The forward line that terminates at $F_T = K$ is the path that gives most contribution to the option's premium with expiry T and strike K in a local volatility setup (this includes both the fact that the spot has to wander in the region where the option has most gamma to contribute to its premium and the probability of the spot to do this). Now to give more details on the text snipped you quoted it would be helpful if you could give a full citation including which book you are referring to and which section so we can get a bit more context ;) ! ## Answer by Attack68 (score 0) https://quant.stackexchange.com/a/41775 He is basically saying that the main path from A to B prices the option at strike B with respect to A, and if it has an implied vol of $x$, then that main path also has a vol of $x$. If you have an option at strike C (which is further away than B from A) then its main path from A to C has to also have a vol reflective of the implied vol, i.e a vol of $y > x$ greater than the vol of B. The key is that the main path A -> B -> C is the same in both cases for the part A -> B, so if C has a higher vol it must be the part of the path B -> C that takes up the slack and has a higher volatility to statistically account for the difference. And this volatility is local to C since it is beyond B.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.