# 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 $t_{i+1}.$