Local volatility computations

Market data suggests that the fair price of an option does not depend on a constant volatility ${\sigma }^2$, but rather on a volatility that varies with strike $K$ and maturity $T$. In the previous section, we fitted this dependence by a smooth implied volatility surface, obtained by inverting the Black—Scholes pricing formula for observed market prices. Based on this surface, it is possible to recover the local volatility via Dupire’s formula:

$${\sigma }_{\mathrm{loc}}^2(K,T,S_0)\frac{\partial C/\partial T}{\frac{1}{2}{K}^2,{\partial }^{2}C/\partial K^2}.$$

[@gatheralVolatilitySurfacePractitioners2006], Eq. 1.6

Defining the implied total variance

$$w(S_0,K,T):={\sigma }_{BS}^2(S_0,K,T),T$$

and the log moneyness

$$y:=\log \left(\frac{K}{F_T}\right),$$

where $F_T$ denotes the forward price at maturity $T$, Dupire’s formula can be rewritten in terms of the total variance surface as

$${\sigma }_{\mathrm{loc}}^2(K,T,S_0)=\frac{\partial w/\partial T}{1-\frac{y}{w}\frac{\partial w}{\partial y}+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right){\left(\frac{\partial w}{\partial y}\right)}^2+\frac{1}{2}\frac{{\partial }^{2}w}{\partial y^2}}.$$

[@gatheralVolatilitySurfacePractitioners2006], Eq. 1.10

Computational issues

The practical difficulty of this formula is its severe numerical instability, since it involves first- and second-order derivatives of the fitted variance surface. We illustrate this by a simple numerical experiment.

Let

$$w(y,T)=(\alpha +\beta y+\gamma y^2)T.$$

Then the corresponding derivatives are

$$\begin{array}{rl}{\partial }_{T}w& =\alpha +\beta y+\gamma y^2,\\ {\partial }_{y}w& =(\beta +2\gamma y),T,\\ {\partial }_{yy}w& =2\gamma T.\end{array}$$

Substituting these expressions into Dupire’s formula yields the analytic local variance

$${\sigma }_{\mathrm{loc}}^2(K,T,S_0)\frac{\alpha +\beta y+\gamma y^2}{1-\frac{y}{w}(\beta +2\gamma y)T+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right)(\beta +2\gamma y{)}^2{T}^2+\gamma T}.$$

To mimic the effect of noisy market calibration, we perturb the total variance on a discrete grid $y_0,\dots ,y_N$. Specifically, we define a noisy approximation

$$\tilde{w}(y_i)=w(y_i,T)(1+{\epsilon }_i),{\epsilon }_i\sim \mathcal{N}(0,{\sigma }_{\text{noise}}),i=0,\dots ,N,$$

where the ${\epsilon }_i$ are independent.

Assuming a uniform grid with spacing $h=y_{i+1}-y_i$, the derivatives of $\tilde{w}$ are approximated by central finite differences:

$${\partial }_y\tilde{w}(y_i)\approx \frac{\tilde{w}(y_{i+1})-\tilde{w}(y_{i-1})}{2h},$$

and

$${\partial }_{yy}\tilde{w}(y_i)\approx \frac{\tilde{w}(y_{i+1})-2\tilde{w}(y_i)+\tilde{w}(y_{i-1})}{h^2}.$$

Using these finite-difference approximations in Dupire’s formula produces a noisy estimate of the local variance. Empirically, the resulting error scales approximately linearly with the noise level ${\sigma }_{\text{noise}}$, as shown in Figure 1. This demonstrates that the map from the implied variance surface to the local volatility surface is highly ill-conditioned: even small perturbations in $w$ can lead to large errors in the recovered local volatility.

Figure 1: Result of the numerical experiment exhibiting instability.

### Remedies for the numerical issues Several remedies are commonly used in practice. First, one avoids differentiating raw implied volatility quotes directly and instead fits a smooth, arbitrage-free parameterization of the implied variance surface, such as SSVI or a suitably regularized spline. Second, derivatives should be computed analytically from the fitted parameterization whenever possible, rather than by finite differences on noisy grid data. Third, regularization or smoothing may be applied to suppress high-frequency noise, especially in the second derivative $\partial_{yy} w$, which is the main source of instability. Finally, it is often necessary to impose positivity constraints or denominator floors in the Dupire formula to avoid spurious explosions of the local variance.

References

J. Gatheral, The Volatility Surface: A Practitioner’s Guide. Wiley Finance Series. Hoboken, NJ: John Wiley & Sons, 2006. doi: 10.1002/9781119202073.