Short-Maturity Implied Volatility from a Local Volatility Function

Summary

The document asks how to apply a short-maturity result from local volatility theory: in the limiting case of time to maturity approaching zero, implied volatility is related to a harmonic average of local volatility along a path indexed by log-moneyness. It gives a particular local-volatility function with constant parameters and asks how to compute the implied volatility from it, including how to choose the integration limits.

The response points to the cited research paper but provides no derivation, integration bounds, or evaluated formula. As a result, the text introduces a useful asymptotic connection between local and implied volatility while leaving the practical calculation unresolved. The stated result applies in a short-maturity limit, so it should not be read as a general closed-form conversion for arbitrary maturities. Applying it requires careful attention to the path variable, coordinate conventions, and boundary behavior of the local-volatility function.

Key ideas

• In the short-maturity limit, implied volatility is related to a harmonic average of local volatility along log-moneyness.
• The document supplies a parametric local-volatility function and asks how to insert it into the integral.
• The cited response gives a source rather than solving the integration or specifying practical bounds.
• The limiting relation should not be assumed to provide implied volatility at arbitrary maturities.

Tags

# Closed formula for computing Implied Volatility from Local Volatility function


# Closed formula for computing Implied Volatility from Local Volatility function












The main result of this paper (Asymptotics and Calibration in Local Volatility Models, Berestycki, Busca, and Florent. Quantitative Finance, 2002) is equation (16) on page 63, that states that:

In the limit $\tau \rightarrow 0$, the implied volatility is the harmonic mean of the local volatility, namely

$$\lim_{\tau \rightarrow} \frac{1}{\phi(x,\tau)}=\int_{0}^{1}\frac{ds}{\sigma(sx,0)}$$

where $\phi(x,\tau)$ is the implied volatility, $x=\ln(S/K)+r\tau$, $\tau$ is time to maturity, $\sigma(K,T)$ is the Local Volatility, $K$ is strike, $S$ is spot.

Suppose that for $\tau \rightarrow 0$ the Local Volatility is given by the following equation.

$$f(x)=\sqrt{\sigma_0^{2}+2\rho\alpha\sigma_0 x+\alpha^{2}x^{2}}$$

where $\sigma_{0}, \rho,\alpha$ are constants.

How can I compute the Implied Volatility from this Local Volatility equation, based on the formula above provided in the aforementioned paper? I do not see how to apply the formula in practice. For example, what are the integration limits that I should use?

## Answer by Jerome Busca (score 2)

https://quant.stackexchange.com/a/81834

See https://onlinelibrary.wiley.com/doi/10.1002/cpa.20039

“Computing the implied volatility in stochastic volatility models” Berestycki-Busca-Florent

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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