Dupire’s Formula, Local Volatility, and Implied Volatility

Summary

The document asks how to interpret Dupire’s formula after building a surface of call prices across strikes and maturities. Its proposed intuition is that market option prices can be used to infer a local volatility function, then that function describes the dynamics consistent with the observed option surface under a local volatility model. It also asks how implied volatility fits into this picture.

The text is a question rather than a resolved explanation: it gives no derivation, empirical evidence, or answer about the role of implied volatility. Its framing also leaves open important qualifications, including assumptions about smooth option prices and the model dependence of the inferred local volatility. Treat it as a useful prompt about the relationship between option prices, local volatility, and implied volatility, not as a complete pricing method or a demonstrated test for whether an individual option is correctly priced.

Key ideas

• A surface of call prices across strikes and maturities is the input considered for Dupire’s formula.
• The question proposes using that surface to infer local volatility as a function of time and the underlying price.
• It asks whether the inferred local volatility specifies option price dynamics consistent with the observed market surface.
• The document raises, but does not answer, how implied volatility relates to local volatility.

Tags

# Motivation behind Dupire Formula


# Motivation behind Dupire Formula












I am currently in the process of studying the book "Stochastic Volatility Modeling" by Bergomi to get a more practical point of view on volatility. While the math makes perfect sense to me I struggle to understand the moral of the story behind Dupire Formula. The explanation I have given to myself is this

- A practitioner gathers data on market prices of different call options depending on maturity $T$ and strike prices $K$.

- He hence got a (differentiable) function $C(T,K)$ which express the market price of a call option as a function of $K$ and $T$.

- Via the Dupire formula he can compute the quantity $\tilde{\sigma}(t,S_{t})$.

- Now given an option with an arbitrage free price process $P(t,S_t)$ the practitioner can check if it is priced in the correct way because this statement will be true if and only if its dynamic coincide with the one given by the Black and Scholes formula with $\sigma = \tilde{\sigma}(t,S_{t}) $.

Is this intuition correct? If so how does implied volatility fits into the story?

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