Analytic Implied Basis Point Volatility

Normal Distribution pdfBasis point volatility, or simply ‘BP Vol’, refers to the volatility parameter of the Normal, or Bachelier, model, $$dF=\sigma dW$$ I was interested to implement the implied volatility calculation using the analytic approximation of J. Choi, K Kim and M. Kwak (2009), “Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion”, Applied Math. Finance, 16(3), pp. 261-268. Section 4 summarises the analytic approximation, and it looks easy to implement in code, however there are a few minor points to note.


Let \(\nu = \frac{F_0-K}{C+P}\). In theory \(-1 < \nu < 1\), yet in practice for large \(d\) (\(\approx 7.6\)), \(\text{abs}(\nu)\) can numerically equal or even creep over 1. I have not been able to rearrange the calculations to guarantee \(-1 < \nu < 1\) at such extremes, so for safety we force the issue by defining $$\nu:=\max\left(-1+\epsilon, \min\left(\frac{F_0-K}{C+P}, 1-\epsilon\right)\right)$$ where, \(\epsilon\) is machine epsilon.

Now near the ATM strike, \(\nu \rightarrow 0\) and \(\eta = \nu/\tanh^{-1}(\nu) \rightarrow 1\), in which case define $$\eta := \cases{\nu/\tanh^{-1}(\nu),&if $\text{abs}(\nu)\ge \sqrt{\epsilon}$;\cr 1,&otherwise. \cr}$$

Finally, to sharpen the answer one can ‘polish the root’; that is, make one Newton-Raphson iteration using vega, which in most cases will improve the answer.

Update 7th April 2013
I had a look again at why \(\nu\) was able to creep over 1. In my test case I was generating the straddle prices from known set of parameters. Unfortunately I had not structured the straddle formula to avoid evaluation of \(N(d)\) near 1.0. When it is structured to concentrate evaluation of \(N(d)\) and \(n(d)\) near 0.0, then the numerical problem is avoided and no guarding is needed. Specifically, the straddle prices can be constructed using the following formula;
In the case of \(F\lt K (d \lt 0)\);
$$ 2((F-K)N(d)+\sigma\sqrt{T}n(d))-(F-K)$$
whilst in the case of \(F \gt K(d \gt 0)\)
$$ (F-K) + 2((K-F)N(-d)+\sigma\sqrt{T}n(-d))$$

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