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In one sense, it’s just an accounting convention, so it doesn't matter. In another sense, the implied volatility can be interpreted as the minimum realised volatility which implies that your option price was ≤ fair value (realized via dynamic hedging/gamma scalping, see Gamma Pnl vs Vega Pnl from BS.)

So does it matter, or does it not? Which is the correct hurdle realized volatilty?

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  • $\begingroup$ With the appropriate input vols, both models can match the market price as you say. However they will give radically differently deltas for you to hedge with. As a trader, you need to choose the model which best reflects the market dynamics you expect to see, to minimise hedging error. So it isn't just an accounting convention but one of risk management in my view. $\endgroup$ Apr 28 at 4:51
  • $\begingroup$ Replace 'deltas' with 'Greeks' to be most general. Wish there was an edit button! $\endgroup$ Apr 28 at 5:54
  • $\begingroup$ linkedin.com/feed/update/urn:li:activity:6926568087676743681 $\endgroup$ May 5 at 14:07

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Neither of them is correct or incorrect, these are just two different numerical inputs that one should plug-in into two different formulas to get the market price of an option given all other information.

The Black-76 formula (i.e. Black-Scholes in terms of forward price rather than a spot price) for pricing a call option is

$$C_{BS}(K) = F_0 N(d_1) - K N(d_2)$$ where $d_{1,2} = \frac{\log(F_0/K)}{\sigma_{BS}\sqrt{T}}\pm\frac{\sigma_{BS}\sqrt{T}}{2}$.

The Bachelier formula for a call is $$C_N(K) = (F_0-K) N(d_N) + \sigma_N\sqrt{T}n(d_N)$$ where $d_N = \frac{F_0-K}{\sigma_N\sqrt{T}}$.

It's clear that giving the same $K$, $F_0$, $T$ you have to use two different implied volatilites $\sigma_{BS}$ and $\sigma_N$ in order to match the market price of an option with two different formulas.

The more qualitative explanation is that volatility has different meanings in each model. The Black-Scholes model assumes a lognormal distribution of the underlying. The Bachelier model assumes a normal distribution. The Black-Scholes volatility $\sigma_{BS}$ measures the relative change in $F_t$, while the Bachelier volatility $\sigma_N$ measures the absolute change in $F_t$. In other words, the probability of the forward rate going from $1\%$ to $2\%$ is the same as the probability of it going from $2\%$ to $4\%$ in Black-Scholes and from $2\%$ to $3\%$ in Bachelier.

Thus these are just two different pricing conventions and one can more or less freely go back-and-forth between them. More details on conversion between Black-Scholes and Bachelier volatilities can be found in such articles as A Black-Scholes user's guide to the Bachelier model or Volatility conversion calculators if needed. Hope it helps.

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  • $\begingroup$ Hi Hasek, thank you for the prompt and detailed reply - I'm afraid this doesn't really answer what I was asking so I've edited the question to make it clearer. Thanks again $\endgroup$ Apr 27 at 16:15
  • $\begingroup$ Just like there are two ways to compute implied vol $\sigma$ (Bachelier and BS) there are two ways to compute realized vol (Bachelier and BS). Then you can compare like with like and there is no confusion possible. $\endgroup$
    – nbbo2
    Apr 28 at 12:11

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