Dupire Local Volatility Is Defined Across Strikes and Maturities

Summary

The document answers whether local volatility can be estimated using only the spot price at a particular time. It explains that Dupire’s approach derives local variance as a function of option maturity and strike from a surface of call prices, rather than estimating a single value for one selected spot and time. The formula uses derivatives of call prices with respect to maturity and strike, together with rates and dividends.

After constructing this function, a practitioner can evaluate it at a chosen maturity and strike, including values set equal to a current time and spot under a chosen convention. The answer also notes that the calculation can be expressed using the full implied-volatility surface. It is a concise conceptual explanation rather than a calibration guide: it gives no market data, numerical example, interpolation procedure, or treatment of the sensitivity of Dupire estimates to noisy option quotes and numerical differentiation.

Key ideas

• Dupire local volatility is a surface indexed by option maturity and strike.
• The calculation uses call-price derivatives with respect to maturity and strike.
• The resulting surface can be evaluated at a selected maturity and strike after it is computed.
• The implied-volatility surface can also be used to express the local-volatility function.
• The explanation does not cover numerical calibration or the effects of noisy option data.

Tags

# Local volatility parametrization using the spot


# Local volatility parametrization using the spot












Is it possible to estimate the local volatility using the spot price S at time t instead of the strike price K and the expiry date T ?

Any help would be appreciated.

## Answer by AFK (score 2, accepted)

https://quant.stackexchange.com/a/18504

One does not estimate the local volatility at a given $T$ and $K$. Instead, Dupire's formula actually gives $\sigma(T,K)$ for all $T$ and $K$. $$ \sigma^2(t_0,S_0;T,K)= \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{\frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$ where $C(t_0,S_0;T,K)$ are the call prices for maturity $T$ and strike $K$. You can also express this directly in terms of the whole implied volatility surface $\Sigma : (T,K) \mapsto \Sigma(T,K)$.

Once you computed the function, you can evaluate at $T = t$ and $K = S$ or any other value you want.

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.