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In Stochastic Volatility Modeling (Lorenzo Bergomi) the Dupire's formula is:

$\sigma (t,S)^2$ $=$ $2$${dC\over dT}$ $+$ $qC$ $+(r-q)K$${dC \over dK}$ $x$ ${1 \over K^2 {d^2C \over dK^2}}$

with $K=S$ and $T=t$

Then he says this equation expresses that the local volatility for the spot S and time t is reflected in the differences of option prices with strikes straddling S and maturities straddling T.

I don't understand why he uses the term straddling

Axel Haddar
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  • I assume that your last $t$ should be a $T$? – LocalVolatility Feb 22 '17 at 15:04
  • Exactly sorry I am going to edit this. – Axel Haddar Feb 22 '17 at 15:05
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    The "straddling" simply refers to the local volatility being linked to the first two derivatives w.r.t. $K$ and the first derivative w.r.t. $T$. In practice you don't observe a call price surface for a continuum of strikes and maturities but would need to infer the the derivatives from the discrete set of neighboring strikes and maturities. – LocalVolatility Feb 22 '17 at 15:08
  • @LocalVolatility does that mean it is hard to get the static smile property with local volatility model ? – Axel Haddar Feb 22 '17 at 15:16
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    It means that for example $\frac{dC}{dT}$ would be replaced by $\frac{C_2-C_1}{T_2-T_1}$ and so on, since only discrete options can be observed – nbbo2 Feb 22 '17 at 15:22
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    Ha! Ha! Actually I "straddle" between brilliant and idiot https://www.merriam-webster.com/dictionary/straddle – nbbo2 Feb 22 '17 at 15:51

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