Using a Replicating Portfolio to Understand Dupire’s Formula

Summary

The document asks whether Dupire’s formula can be understood through the same replicating-portfolio framework used to explain Black–Scholes. The answer distinguishes the purpose of the two frameworks: Dupire’s formula is presented as a calibration relationship that derives a local volatility surface consistent with market option prices, rather than as a standalone option-pricing formula.

A tree analogy explains how the replication idea carries over. In a basic Black–Scholes tree, the volatility assumption is constant across possible future nodes. In a Dupire or Derman–Kani setup, each node uses the volatility implied by the market for that point in the tree. The answer says the underlying hedging strategy remains similar, with an option position hedged using the underlying. It offers only a conceptual explanation, not a derivation or detailed portfolio construction, so it does not establish the formula algebraically or discuss practical calibration and hedging limitations.

Key ideas

• Dupire’s formula is used to calibrate local volatility to market option prices.
• A replicating hedge can still use an option position offset by exposure to the underlying.
• The tree analogy assigns market-consistent volatility across future nodes.
• The document gives a conceptual comparison rather than a mathematical derivation.

Tags

# Dupire's Formula by a Replicating Portfolio


# Dupire's Formula by a Replicating Portfolio












I understand that the BS equation can be explained by a replicating portfolio, e.g., short an option and long $\Delta$ shares of the underlier [Bergomi's Stochastic Volatility Modeling]. I also understand how to derive the Dupire's formula using Fokker-Planck equation or via a probabilistic approach [Derman and Kani (1998)].

My question is: Is there a replicating portfolio method to arrive at the Dupire's formula?

## Answer by KT8 (score 3, accepted)

https://quant.stackexchange.com/a/68735

As @user121416 mentioned, Dupire's formula is meant to be a calibration equation rather than a pricing formula. However, note that when using it, you're still able to use a similar replicating portfolio method as in BS.

The difference is the following, and lets think of a tree for simplicity, whereas for determining your delta in BS the same constant volatility is assumed at every node in the tree (i.e. the possible outcomes or future possible scenarios), Dupire (and then Derman and Kani) are telling you that every node of the tree shouldn't use that same volatility, but the volatility assumed by the market.

However, in both approaches the underlying hedging strategy is the same.

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

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.