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

Question

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 $\delta\to 0$?

Answer 1

To answer the last question first , both these models have a particular forward swap rate as their underlying. These models do not attempt to relate the forward swap rate to any other forward rates or spot rates. In addition, each swaption is considered a separate instrument with its own forward rate and volatility, and no attempt is made within these models to relate say the volatility of a 1y5y with that of a 1y10y.

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 $S(t)$.

Lastly the interpretation of the Bachelier vol of 136 relative to a forward rate of say 3.65% is that the standard deviation of the forward rate in one year is 1.36%. In the lognormal model with a vol of 40%, the standard deviation (to a first order approximation) is 40% of 3.65%.