Calculating local volatility from option prices?

Question

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?

Answer 1

One issue I see: $$2\cdot0.01+0.01\cdot11\cdot0.15 = 0.0365$$ must be replaced by

$$2\cdot \left(0.01+0.01\cdot11\cdot0.15\right) = 0.053$$

Edit: (Detailing my comments a bit) Dupire's equation, as you wrote it, is correct (assumes dividends are null):

$$ \frac{\partial C}{\partial T} = \frac{1}{2}\sigma^2 K^2\frac{\partial^2 C}{\partial K^2} -r K \frac{\partial C}{\partial K}, $$

where $\sigma = \sigma(S_t, t)$, that is, dependent on underlying and time, with underlying following the local volatility dynamics (aka generalized Black-Scholes dynamics):

$$ dS_t = rS_t dt +\sigma S_t dW.$$

A proof can be found here.

You can think of it as a 'dual' companion of Black-Scholes equation (usually uses $t$, not $T$, time to expiry, as variable):

$$ -\frac{\partial C}{\partial t} = \frac{1}{2}\sigma^2 S^2\frac{\partial^2 C}{\partial S^2} +r S \frac{\partial C}{\partial S} - rC.$$

Note that, if you assume $r=0$, we have:

$$ -\frac{\partial C}{\partial t} = \frac{1}{2}\sigma^2 S^2\frac{\partial^2 C}{\partial S^2}.$$

Edit 2: You are computing an instantaneous quantity from raw data using rough finite difference derivative approximations. Usually, one fills in the continuous space of calls parameterized by strike and time to expiry, $C(K,T)$, using smooth interpolations (more precisely, this 'filling' is first done in the BS-implied volatilty space), then gets first and second derivatives and the needed local volatility $\sigma(K,T)$.