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?

  • $\begingroup$ The method described in here is much better than creating some smooth price/vol surface with splines/parametric models. In my opinion. $\endgroup$ – will Mar 25 '18 at 13:26
  • $\begingroup$ interesting paper. Perhaps a bit overkill if you already have a dense grid of option prices, presumably coming from some parametric model. Smoothing splines are nice to remove small non regularities that might come from slightly truncated prices - such as options prices being quoted with only a limited number of decimals. $\endgroup$ – Antoine Conze Mar 25 '18 at 14:26

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.

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  • $\begingroup$ Thank you for your answer. Do you have any reference papers to the fitting methods that you would recommend? $\endgroup$ – ALFRAM Mar 24 '18 at 19:37
  • 2
    $\begingroup$ nkahale.free.fr/papers/Interpolation.pdf is useful for understanding arbitrage free interpolation of $\Sigma(K,T)$ w.r.t. $T$, including when there are discrete dividends. Smoothing splines can be found in most numerical libraries for interpolating $\Sigma(K,T)$ w.r.t. $K$ Alternatives are fitting parameterization such as Gatheral's SVI, or SABR, etc. $\endgroup$ – Antoine Conze Mar 25 '18 at 8:04
  • $\begingroup$ The huge problem with this is that fitting splines requires arbitrary restraints on the derivatives - which are exactly what you're using in dupire's formula. $\endgroup$ – will Mar 25 '18 at 13:23

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