# swaption model for forward swap rate

I have another question about interest rates. In this case it is about swaption and how to come up with a pricing formula. For the rest of my question I use the notation from Brigo. The payoff of a payer swaption discountad from the maturity $T_\alpha$ to the current time $t$ is given by

$$D(t,T_\alpha)N\left(\sum^\beta_{i=\alpha +1 }P(T_\alpha,T_i)\tau_i(F(T_\alpha;T_{i-1},T_i)-K)\right)^+$$

where

• $D(t,T_i)$ the discount factor at $t$ of time $T_i$
• N some notional
• $\tau_i$, general daycount convention for between $T_{i-1}$ and $T_i$
• $F(T_\alpha;T_{i-1},T_i)$ forward rate at $T_\alpha$ between $T_{i-1}$ and $T_i$
• strike rate $K$
• $P(T_\alpha,T_i)$ zero coupon bond at $T_\alpha$ with maturity $T_i$.

denoting with $S:=S_{\alpha,\beta}(0)$ the forward swap rate, i.e. that $K$ which makes the contract fair $(=0)$ we can come up with models for $S$. Assuming a log normal model we derive a Black like formula.

However, I'm interested in the case where $dS=\sigma dW$, i.e. $S$ is normally distributed (Bachelier model). How does a pricing forumla for a swaption look like? I just can find Black formula on the web. Many thanks for the reference / answer.

One can write for the payoff of an swaption $$\sum_i\tau_i P_{i+1}(S_{\alpha,\beta}(T_\alpha)-K)^+$$ and therefore the pricing equation follows Joshi's explainations. To derive the above equation use that the swap rate is given by $$S_{\alpha,\beta} = \sum_i \frac{\tau_iP_{i+1}}{\sum_i\tau_iP_{i+1}}F^i,$$ where $F^i$ are the corresponding forward rates.
well just take the Bachelier formula with $r=d=0$ $S_0 = S_{\alpha,\beta}$ and then multiply by the annuity.
The annuity will be $$\sum \limits_i \tau_i P_{i+1}.$$ where $P_{i+1}$ is the df for $t_{i+1}.$