Dupire Formula question

Question

I want to calculate the local volatility from Dupire's formula:

$\sigma _{VL}^{2} (K,T,S_{0}) = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2} K^{2} \frac{\partial^2 C}{\partial K^2}}$

So I use the aproximation forms as :

$\frac{\partial C }{\partial T} \cong \frac{C(K, T + \Delta T) - C(K,T - \Delta T}{2 \Delta t}$

$\frac{\partial^2 C}{\partial K^2} \cong \frac{C(K- \Delta K, T) - 2C(K,T) + C(K + \Delta K,T )}{(\Delta K)^{2}}$

With this date to get the local volatility of the option SPX(288.5,April 15), should I do this?:

$\frac{\partial C }{\partial T} \cong \frac{2.02 - 1.51}{2 * 3}$

$\frac{\partial^2 C}{\partial K^2} \cong \frac{2.03 -2*1.65 + 1.4}{0.5^2}$

Answer 1

I first want to clarify one statement. You write "the local volatility of the option ...". A local volatility is, unlike an implied volatility, not a property of an option but instead a function of time and the underlying assets price. I.e. an option does not have an implied volatility but the local volatility function describes the dynamics of the underlying asset and is found such as to match the market prices of all European vanilla options jointly.

Regarding your actual problem. Typically you want to be able to evaluate the local volatility surface at arbitrary (time, spot)-points, independent of the market strikes and expiries. In a finite-difference scheme for example, you need to be able to evaluate the local volatility at all grid points. A common approach is thus to construct an a surface interpolator first from the existing European options prices and then evaluate the partial derivatives of this interpolator. While you could interpolate prices directly, it is more common to construct an interpolated implied volatility surface. A crucial property in order for local volatilities to exist is for this surface to be free of butterfly and calendar spread arbitrage.

Here are some references to get you started:

1. Fengler, Matthias R. (2009) "Arbitrage-Free Smoothing of the Implied Volatility Surface", Quantitative Finance, Vol. 9, No. 4, pp. 417-428

2. White, Richard (2013) "Local Volatility," Open Gamma Quantitative Research