Computing ATM implied volatility analytically

I recently had to compute ATM implied bpvol (or normal volatility) as well as Black volatility from an ATM option premium. Immediately, I looked for a library function which, more often than not, are written for the general case and make use of a solving routine. To my later irritation, I had completely forgotten that both bpvol and Black vol can be determined analytically (and trivially) from the ATM premium. So this note, is as much for my own interest as it is for anyone else’s interest.

Suppose the dynamics of a forward rate, F, follow the assumption of the normal model; that is,

$$dF=\nu dW$$

Then the forward value of a European option call option with strike \(K\), expiring at time \(T\) is,

$$C(K) = (F-K) N(d) + n(d) \nu\sqrt{T}$$
where, \(d=\frac{F-K}{\nu \sqrt{T}}\) and \(n(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\). In the case of ATM, \(F=K\), and the formula collapses to \(C(F)=\nu\sqrt{\frac{T}{2\pi}}\), which is trivially invertible, so that,

$$\nu=C(F)\sqrt{\frac{2\pi}{T}}$$

Similarly, if one assumes the dynamics of a forward rate follow a lognormal process; that is,
$$dF=\sigma F dW$$

Then the one arrives at an implied Black vol of

$$\sigma = \frac{2}{\sqrt{T}} N^{-1} \left(\frac{1}{2}\left[\frac{C(F)}{F}+1\right]\right)$$
where, \(N^{-1}(x)\) is the inverse of the cumulative normal distribution function. A good survey of accurate implementations is found in William Shaw’s article, Refinement of the Normal Quantile.