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I'm looking for an easy method to approximate the probability of the forward swap rate that is implied by the swpation market. One possibility would be to fit a certain model, e.g. SABR, and extract the risk neutral density.

On the other hand I know from the equity case that $N(d_1)$, the delta, is used an approximation that the underlying ends in the money. Is there a similar approximation in the swaption case? I.e. could we use normal vol (bachelier model), and use $N(d_1)$, where $d_1 = \frac{F-K}{\sigma \sqrt{T}}$ for forward swap rate $F$, strike $K$, and implied normal vol $\sigma$ and time to maturity $T$. Or is there any other approximation used frequently?

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It is best to , as you say, extract the risk neutral density from a model that fits the market skew , such as sabr. Then you can compute the probability directly. The problem with N(d) is that you are assuming constant volatility , either constant normalized volatility in the Bachelier model or constant lognormal volatility in the BS model. In a market where there is significant skew , this will give you the wrong answer.

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  • $\begingroup$ Thanks for your answer. As a small follow up. Do you have a explanation or reference why the $N(d)$ takes a constant vol into account? I observe in the market different implied vols for different strikes, $\sigma_1,\dots,\sigma_N$. I could then use $N(d_{\sigma_1}),\dots,N(d_{\sigma_N})$, no? $\endgroup$
    – swissy
    Commented May 26, 2023 at 11:17
  • $\begingroup$ Whatever vol you choose , the underlying n(d) model is assuming the same vol for all strikes. Yes you can pick a different vol , but it will still give you the wrong answer. $\endgroup$
    – dm63
    Commented May 26, 2023 at 22:37

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