Local volatility surface corresponding to the implied volatility surface

Question

In Derman/Kani/Zou paper about local vol they rebuilt a local vol surface from an implied vol surface. Each implied volatility depicted in the surface of the "implied Vol" is the Black-Scholes implied volatility. Bascially the volatility you have to enter into the Black-Scholes formula to have its theoretical option value match the option’s market price.

Derman Paper

Now, in the local vol model, they extract the market’s consensus for future local volatilities σ(S,t), as a function of future index level S and time t, from the spectrum of available options prices as quoted by their implied Black-Scholes volatilities. The model fits a consistent implied tree to these quoted option prices, and then allows the calculation of the fair values and exposures of all (standard and exotic) options, consistent with all the initial liquid options prices.

Question: If I compare the graphs in the paper of the implied vol surface and the local vol surface why is it so different? The local vol should be consistent with the liquid option prices. i.e. Term 1.0, 550 level: implied surface 13.5% vol, local vol surface 18% vol. If there is a liquid strike in the market they should have the same vol, am i right?

Many thanks

Answer 1

You should not expect the local vol to be equal to the implied vol except in the trivial case where both are constant (Black-Scholes model). I haven't read the Derman articles but it is quite clear using Dupire's formula (see Gatheral's book for example).

Local volatility can be computed in terms of call prices using Dupire's formula $$ \sigma^2(T,K) = \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{ \frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$ To get the relationship with implied volatility, it is better to think in terms of the log-moneyness forward $y = \ln(K/F_0^T)$ rather than strike. Writing $w(T,y) = T\Sigma^2(T,y)$ for the total implied variance, the Black-Scholes formula reads $$ C(T,K) = C_{BS}(T,K,\Sigma(T,\ln(K/F)),r,q) = S_0 \left( N(-\frac{y}{\sqrt{w}} + \frac{1}{2}\sqrt{w}) - e^y N(-\frac{y}{\sqrt{w}} - \frac{1}{2}\sqrt{w}) \right) $$ Plugging it into the Dupire formula, one gets
$$ \sigma_{\mathrm{Dup}}(T,K)^2 = \frac{ \frac{\partial w}{\partial T} }{1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{4}\left( - \frac{1}{4} + \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial y^2} } $$

This general formula can be simplified in limit cases:

No skew: in this case, $\sigma_{\mathrm{Dup}}(T)^2 = \frac{\partial w}{\partial T} = \Sigma(T)^2 + 2T\Sigma\frac{\partial \Sigma}{\partial T}$. Local vol is already different from implied vol unless they are both constant (Black-Scholes model).

Short maturities: When $T\to 0$, and the derivatives of the implied vol stay bounded, one can check that
$$ \Sigma(0,y) = \frac{1}{\int_0^1 \frac{dt}{\sigma(0,ty)}} $$ so that
$$ \frac{\partial \Sigma}{\partial y} (0,0) = \frac{1}{2}\frac{\partial \sigma_{Dup}}{\partial y} (0,0) $$ in other words, at the money, for very short maturities, the implied volatility skew is half the local volatility skew.

Also note that if you start with a twice differentiable implied vol surface, the local vol will only be continuous.

Answer 2

Loosely speaking:

Local volatility is the instantaneous volatility after time T if the spot is S at that time.

Implied volatility is the expected integrated volatility from today up to time T if the spot ends up at S at that time.

Answer 3

Adding as answer as I don't have enough reputation to comment -- there is a typo in AFK's local vol formula, it should be: $$\sigma_{\mathrm{Dup}}(T,K)^2 = \frac{ \frac{\partial w}{\partial T} }{1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{4}\left( - \frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial y^2} }$$ see Gatheral's "The Volatility Surface", p.13, eq. (1.10) (i.e., the sign of $\frac{1}{w}$ in the denominator is wrong).

Answer 4

One interesting fact that explains a lot at least to me is that implied vol has to do with only one option price. (One data point) Whereas local volatility is not determined from one unique call price in particular, but rather by many data points. To make things simple consider 4 neighbor data points in the surface. From this four is derived one local vol. The idea here is that it has to do with how we get from one call price to another so the local vol doesn't belong to a particular datapoint.