Let $dS/S = \sigma(S,t) dW$.

Let the local volatility be known, i.e, we know the formula $\sigma(S,t)$.

How do I derive $\Delta$ of a regular call in this model?

Let BS be the Black Scholes price, and let $\sigma_{imp}$ be the implied volatility induced by above local volatility. Then, does $\sigma_{imp}$ depend on $S$ as well?

  • 1
    $\begingroup$ No closed form formula for the price, and a fortiori the Greeks. You should resort to compute the delta numerically. Indeed the LV model embeds a spot/IV dynamics. But it is not clear what you are asking IMHO (what is your precise definition of $\sigma_{imp}$) $\endgroup$ – Quantuple Mar 31 '17 at 7:10
  • $\begingroup$ @quantuple - if you have constructed a parametric form for the call prices, and then convert it to a parametric form for local via via dupire, and then fit that, then you can have a closed form for the price. $\endgroup$ – will Apr 2 '17 at 11:48
  • $\begingroup$ Sure but that's a special case $\endgroup$ – Quantuple Apr 2 '17 at 11:55

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