# Is $N(d_1)$ a good approximation that a swap enters in the money?

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?

• 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? Commented May 26, 2023 at 11:17