14 Pages Posted: 6 Apr 2014 Last revised: 17 Jun 2016
Date Written: April 10, 2016
This paper describes a fast analytic formula to obtain the basis point volatility 'b.p. vol' for a given option price under the Bachelier normal model with near machine accuracy. The b.p. vol is simply the implied volatility of an option under the Bachelier (or normal) model. In the Black-Scholes world, there is no simple formula to obtain the Black implied volatility out of the option price for a given strike and expiry. One has to rely on a rough approximation (Li, 2008), which can eventually be used initial guess for a solver (Li and Lee, 2011). A robust algorithm with near machine precision is given in (Jackel, 2013). In the Bachelier world, things are much simpler. The problem can be reduced to a single variable, allowing an analytic representation of the implied volatility (Choi et al., 2009). We will follow the same approach as Choi et al. (2009), but relying directly on the call option price instead of the straddle price, and using a more precise numerical representation. It handles near-the-money as well as very far out-of-the-money options and low volatilities.
Keywords: implied volatility, Bachelier, b.p. vol
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