swaption model for forward swap rate

Question

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.

Answer 1

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.

You can find this derivation in any book on option pricing, since it is model free. Look for example in Brigo D. and Mercurio F. Interest Rate Models - Theory and Practice.

Answer 2

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 $t_{i+1}.$