Implied Volatility is the harmonic average of Local Volatility

I am trying to demonstrate the famous result that states that when $$T \rightarrow 0$$, the Implied Volatility is the harmonic average of Local Volatility.

I am st the final stage, and I have the following equation:

$$I(0,k)=\sigma(0,k)\left(1-\frac{k}{I}\frac{\partial I}{\partial k}\right)^2$$

This differentiation equation can be solved and yields the result:

$$I(0,k)=\left(\int_{0}^{1}\frac{dy}{\sigma(0,ky)}\right)^{-1}$$

$$I(T,k)$$ is implied volatility, $$\sigma(T,k)$$ is Local Volatility, $$k=\ln(K/S)-rT$$, $$K$$ is strike, $$S$$ is spot, $$r$$ is constant interest rate.

My question is: can you please give a step by step solution for the aforementioned differential equation?

• The differential equation comes from the Dupire local vol formula expressed in terms of the implied vol, after you take the T->0 limit. The solution is easy once you have the right ODE, yours above are missing squares on sigma and I. See Bergomi’s book, second chapter, p48. Mar 3 at 9:52