How Dupire’s Formula Relates Local Volatility to Option Prices

Summary

The document quotes a form of Dupire’s formula and asks what it means to say that local volatility at a given spot and time is reflected in option prices with strikes and maturities “straddling” those values. The key idea behind that wording is that the formula uses changes in option prices around the point of interest: strike derivatives are evaluated near the spot level, while maturity changes are evaluated near the time in question. These neighboring observations help infer the local volatility there.

The source provides the formula but no answer to the question, derivation, worked example, or discussion of assumptions. In particular, it does not explain the required smoothness of option prices, the handling of rates and dividends, or practical issues in estimating derivatives from market data. The text is therefore a prompt about interpreting the local-volatility relationship, rather than a complete guide to implementing or validating Dupire’s method.

Key ideas

• Dupire’s formula links local volatility to changes in option prices across strike and maturity.
• “Straddling” refers to using option data around the spot and time being studied.
• The document states the formula but does not derive it or answer the conceptual question.
• It does not address data quality, derivative estimation, or the formula’s assumptions.

Tags

# Dupire's formula explanation


# Dupire's formula explanation












In Stochastic Volatility Modeling (Lorenzo Bergomi) the Dupire's formula is:

$\sigma (t,S)^2$ $=$ $2$${dC\over dT}$ $+$ $qC$ $+(r-q)K$${dC \over dK}$ $x$ ${1 \over K^2 {d^2C \over dK^2}}$

with $K=S$ and $T=t$

Then he says this equation expresses that the local volatility for the spot S and time t is reflected in the differences of option prices with strikes straddling S and maturities straddling T.

I don't understand why he uses the term straddling

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