# 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,

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 ;) !

• Could you provide a reference for the continuity statement? – LocalVolatility Nov 17 '18 at 13:33
• I don't know of a reference but it's a very basic consequence of the relationship between the terminal payoffs. Suppose for example we look through a dividend date. Call $T^-$ and $T^+$ the cum and ex dividend dates of stock S and consider 2 call options with strike $K^-$ and $K^+$. Then clearly $S(T^-)-K^-=S(T^+)-K^+$ iff $K^+=K^--d$. By martingale property this means the options values at time 0 are equal too so we get $C(S_0,K^-,T^-)=C(S_0,K^+,T^+)$ – Ezy Nov 17 '18 at 15:42

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.

• @ Attack68, not sure I agree with the part where you say the A->B part is the same in both « main paths ». I don’t see how we can infer this as these paths will generally cross B in different points in time. – ababoua Sep 17 '18 at 19:39