I'm attempting to calculate local volatility given a set of option prices using $$ \sigma(T,K)=\sqrt{2\frac{\frac{\partial C}{\partial T}+rK\frac{\partial C}{\partial K}}{K^2\frac{\partial^2C}{\partial K^2}}}.$$
Let's say I'm given the following call strikes and maturities and prices:
Strike 1 Month 2 Month
10 0.50 0.75
11 0.35 0.50
12 0.25 0.35
Let's say we try to calcalate the 1 month 11 strike local volatility with a risk-free rate of $r=0.01$.
We can estimate theta, $\frac{\partial C}{\partial T}$, as 0.35/30(days) = 0.01.
Next, we have $\frac{\partial C}{\partial K}$ as the difference between the 10 and 11 strike: $\frac{0.50-0.35}{1} = 0.15$.
Next, we calculate $\frac{\partial^2 C}{\partial K^2}$ as the difference between the 12-11 call and the 11-10 call which calculates the rate of change of the call price by strike effectively: (0.50-0.35)-(0.35-0.25) = 0.05.
Then, we plug in as follows for the numerator: $2\cdot(0.01+0.01\cdot11\cdot0.15) = 0.053$. Then, for the denominator we have: $11^2\cdot0.05= 6.05$.
Then, if we divide and take the square root: we get $0.0935$, so a volatility of $9.35\%$.
Am I on the right track here? Most of the times you look up local volatility a lot of it is above my math ability, but I want to understand if I'm at least on the right track?