Pricing Swaptions with a Normally Distributed Forward Swap Rate
Summary
The document explains how to price a payer swaption when the forward swap rate follows a normal, or Bachelier, model rather than a lognormal Black model. It expresses the swaption payoff as the positive part of the difference between the swap rate and strike, multiplied by the swap annuity, which is the sum of discounted accrual periods.
Under the normal-rate assumption, the ordinary Bachelier call value is applied to the forward swap rate and strike, then multiplied by the annuity. An alternative answer derives the same payoff representation by expressing the swap rate as a weighted average of forward rates. The note gives a pricing recipe but does not state a volatility convention or a worked numerical example; the formula’s inputs and discounting conventions must match the swap’s schedule and market setup.
Key ideas
- A payer swaption payoff can be represented as the positive part of the forward swap rate minus strike, times the annuity.
- With normally distributed swap rates, apply the Bachelier call formula to the forward rate and strike.
- The swap annuity is the sum of discounted accrual periods over the underlying swap schedule.
- The forward swap rate can be written as a weighted average of the component forward rates.
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Full text
# swaption model for forward swap rate
# 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.
## Answer by Phun (score 3, accepted)
https://quant.stackexchange.com/a/16563
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 by Mark Joshi (score 3)
https://quant.stackexchange.com/a/16203
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 $t_{i+1}.$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.