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3

This was also discussed in Derman's book The Volatility Smile (see Chapter 16). Specifically, he approximated the local volatility by a linear function of the form \begin{align*} \sigma(S) = \sigma_0 -2 b(S-S_0), \end{align*} and then approximated the implied volatility $\Sigma(S, K)$ for an option with strike $K$ by the average of $\sigma(S)$ between S and ...

7

They are not the same, but they are related. Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are "long vol" and saying "I am a buyer of vol/gamma/vega" means that you are taking a position that benefits from a rise in volatility (either ...

1

Thanks to all for the input. After a bit of research, I replaced the Black-Scholes pricer with a binomial tree pricer that includes early exercise and the known dividend in September, using what's explained in van der Hoek (2006) I chose a drift rate such that ATM vols match and put-call vols now match pretty closely even toward the wings. I definitely ...

1

The procedure you have specified in your last paragraph is the only reasonable way to do it. Clearly the cap volatility is some sort of weighted average of the constituent caplet volatilities, but the weighting is complex , having strike dependence as well as maturity dependence.

0

You don't need an approximation, i.e., if you have the Black's vols, you can simply compute the corresponding price and then invert Bachelier model (normal model) to get implied normal volatility. In the case of the transition from Normal (Bachelier) to Lognormal (Black-Sholes) you need to be more careful if you have negative forwards.

1

You basically have it. $$Normal Vol= Black Vol * Forward Swap Rate$$. Normal vol is usually quoted as an annual vol , not converted to daily by dividing by sqrt(252). The forward swap rate is the fair market rate for the swap that underlies the swaption. So one might have 1yr 10yr normal vol =70bp, forward swap rate = 1.40% and Black vol = 50%. ...

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