# How to interpret the implied vol for swaptions in a bachelier and black model and how forward pricing relates to it

I have two very simple question about the implied volatility of a swaption and how it relates to actual rates level. Suppose we have two famous models, Bachelier and Black. Under either model, the swap rate $$S(T)$$ for a standard swaption with maturity $$T$$ and Brownian Motion $$W(t)$$ is given by for Bachelier $$S(T) = S(t) + \sigma_N(W(T)-W(t))$$ and thus $$S(T)$$ is normally distributed,

$$S(T)\sim\mathcal{N}(S(t),\sigma^2_N(T-t))$$

On the other hand for the Black model, $$S(T)$$ is lognormally distributed

$$\log(S(T))\sim\mathcal{N}(-\frac{1}{2}\sigma^2_B(T-t),\sigma^2_B(T-t))$$.

Interpretation of implied vol in both models

If a market plattform is showing a quoted implied vol (Bachelier model) of a 1y5y swaption (US SOFR ATM) as $$136.17$$ how does this number translate into actual swap rates in one year? That means what is the unit of this measure and how can I relate this to a current 5y SOFR rate $$(S(t))$$ of $$3.65$$?

Same question if I see a implied black volatility of $$40.86$$ for the same swaption structure?

Interpretation of Forward rate

From the models I see what the expected value of $$S(T)$$ both models are, namely $$S(t)$$ and $$S(t)exp(-\frac{1}{2}\sigma_B^2(T-t))$$. How does this relate to a very short forward rate, i.e. $$S(t,T,T+\delta)$$ where the latter is the forward swap rate at time $$t$$ spanning from $$T$$ to $$T+\delta T$$. I observe these forward swap rates in the market too, and should in theory the model implied expected value be equal these market forward swap rates if $$\delta\to 0$$?

Secondly there are a couple of errors in your statement. The mean of $$log(S(T))$$ is $$log((S(t)) - \sigma^2(T-t)/2$$. Also, in both models the expectation of $$S(T)$$ is $$S(t)$$.