Deriving the Short-Maturity Link Between Implied and Local Volatility

Summary

This note asks how to solve a differential equation relating implied volatility across log-moneyness to local volatility in the short-maturity limit. It states the proposed relationship and a claimed solution: implied volatility at a given log-moneyness is the reciprocal of an integral of reciprocal local volatility over a scaled range of moneyness. The setup defines log-moneyness using strike, spot, and a constant interest rate.

The document provides no derivation, numerical example, or validation; it is a question requesting step-by-step reasoning. Its learning value is the stated connection between the two volatility surfaces and the differential equation posed for it. The result is presented specifically as maturity approaches zero, so it should not be read as a general identity for options with arbitrary maturities. The equation and notation also require careful interpretation when solving, including boundary conditions and the meaning of the volatility function along the integration range.

Key ideas

• The note concerns the short-maturity relationship between implied volatility and local volatility.
• It gives a differential equation in log-moneyness as the starting point for a derivation.
• The proposed solution expresses implied volatility through an integral of reciprocal local volatility.
• The stated result is limited to the maturity approaching zero and is not derived in the document.

Tags

# Implied Volatility is the harmonic average of Local Volatility


# 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?

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