Local volatility

Given the price of the asset ${\displaystyle S_{t}}$ governed by the risk neutral SDE

${\displaystyle dS_{t}=(r-d)S_{t}dt+\sigma (t,S_{t})S_{t}dW_{t}}$

The transition probability ${\displaystyle p(t,S_{t})}$ conditional to ${\displaystyle S_{0}}$ satisfies the forward Kolmogorov equation (also known as Fokker–Planck equation)

${\displaystyle p_{t}=-[(r-d)s\,p]_{s}+{\frac {1}{2}}[(\sigma s)^{2}p]_{ss}}$

Because of the Martingale pricing theorem, the price of a call option with maturity ${\displaystyle T}$ and strike ${\displaystyle K}$ is

${\displaystyle {\begin{aligned}C&=e^{-rT}\mathbb {E} ^{Q}[(S_{T}-K)^{+}]\\&=e^{-rT}\int _{K}^{\infty }(s-K)\,p\,ds\\&=e^{-rT}\int _{K}^{\infty }s\,p\,ds-K\,e^{-rT}\int _{K}^{\infty }p\,ds\end{aligned}}}$

Differentiating the price of a call option with respect to ${\displaystyle K}$

${\displaystyle C_{K}=-e^{-rT}\int _{K}^{\infty }pds}$

and replacing in the formula for the price of a call option and rearranging terms

${\displaystyle e^{-rT}\int _{K}^{\infty }s\,p\,ds=C-K\,C_{K}}$

Differentiating the price of a call option with respect to ${\displaystyle K}$ twice

${\displaystyle C_{KK}=e^{-rT}p}$

Differentiating the price of a call option with respect to ${\displaystyle T}$ yields

${\displaystyle C_{T}=-r\,C+e^{-rT}\int _{K}^{\infty }(s-K)p_{T}ds}$

using the Forward Kolmogorov equation

${\displaystyle C_{T}=-r\,C-e^{-rT}\int _{K}^{\infty }(s-K)[(r-d)s\,p]_{s}\,ds+{\frac {1}{2}}e^{-rT}\int _{K}^{\infty }(s-K)[(\sigma s)^{2}\,p]_{ss}\,ds}$

integrating by parts the first integral once and the second integral twice

${\displaystyle C_{T}=-r\,C+(r-d)e^{-rT}\int _{K}^{\infty }s\,p\,ds+{\frac {1}{2}}e^{-rT}(\sigma K)^{2}\,p}$

using the formulas derived differentiating the price of a call option with respect to ${\displaystyle K}$

${\displaystyle {\begin{aligned}C_{T}&=-r\,C+(r-d)(C-K\,C_{K})+{\frac {1}{2}}\sigma ^{2}K^{2}C_{KK}\\&=-(r-d)K\,C_{K}-d\,C+{\frac {1}{2}}\sigma ^{2}K^{2}C_{KK}\end{aligned}}}$