Estimating Local Volatility from Option Surfaces
Summary
The document asks how to estimate the strike and maturity derivatives needed to apply the Dupire equation to a dense, arbitrage-consistent grid of option prices. It raises central finite differences as a possible approach, with small bumps in maturity and strike, but does not establish that this is a reliable implementation method.
The accepted response recommends fitting a smooth surface, such as one built with smoothing splines, and calculating derivatives from that fitted surface. It says the process is generally more stable when Dupire is applied to implied volatility rather than option prices, and gives an implied-volatility form of the equation involving maturity and strike derivatives, the stock drift, and the Black–Scholes d1 term. The explanation offers no numerical comparison, parameter guidance, or validation procedure. Surface choice, smoothing, boundary behavior, and derivative stability therefore remain practical issues for implementation.
Key ideas
- Fit a smooth surface to option observations before calculating Dupire derivatives.
- Smoothing splines are offered as one possible surface-fitting approach.
- The response says applying Dupire to implied volatility can be more stable than applying it to option prices.
- The implied-volatility formulation uses derivatives with respect to both maturity and strike.
- The document does not specify smoothing parameters or a validation method.
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Full text
# Local Volatility implementation
# Local Volatility implementation
The Dupire equation is well-known and mentioned in thousands of articles. Although I could not find a lot of documentation about a consistent and proper way of implementing the formula (The difficulty is mainly estimating correctly the derivatives).
The question I want to raised is especially how to estimate properly the derivatives used in this formula.
For instance I have derived a grid of option prices for different strikes and expiries. The grid is supposed to be very dense with more than 200 prices for a given expiry. Lets say the grid is consistent and does not admit any arbitrages. My understanding is estimating derivatives with respect to T and K by using central finite difference scheme, with a bump of epsilon of the forward/expiry:
```
1. Estimate the derivative of option with respect to T by a bump of T
2. Estimate the second order derivative with respect to K
3. Apply the Dupire formula.
```
Are there any methods of derivative calculation to develop a full and consistent local volatility pricer?
## Answer by Antoine Conze (score 2, accepted)
https://quant.stackexchange.com/a/38971
The usual way is to fit a surface (e.g. smoothing splines) to the grid and to compute derivatives off the surface. Note however that the entire process tends to be more stable when applying the Dupire formula directly to the implied vol surface rather than to the option price surface. The Dupire formula when applied to the implied vol surface $\Sigma(K,T)$ is $$ \sigma_{\text{loc}}(K,T)^2= \frac{\Sigma^2 + 2T \Sigma\left(\frac{\partial \Sigma}{\partial T} + \mu_T K \frac{\partial \Sigma}{\partial K} \right)}{\left(1+d_1 K \frac{\partial \Sigma}{\partial K} \sqrt{T} \right)^2 + T \Sigma K^2 \left(\frac{\partial^2 \Sigma}{\partial K^2} - d_1 \sqrt{T}(\frac{\partial \Sigma}{\partial K})^2 \right)} $$ where $\mu_T$ is the stock drift term (possibly time dependent when fitted to the forwards) and $d_1$ is as in the BS formula.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.