### European Swaption Price Formula

As a reminder, a European swaption gives the buyer of the swaption the right to enter, at the swaption maturity, into a swap, payer or receiver depending on the swaption type, at a fixed rate of K (the strike rate). Now if the swap rate at maturity is \( S_T \), and the difference is in the swapion buyer's favour, then the buyer will recieve the difference between\(S_T\) and K times the notional times the accrual factor at every fixed leg cash flow date (coupon date) of the underlying swap.

An easy way to remember this is to note that one can exercise the swaption and enter into a swap exchanging fixed rate K for a variable/floating. One can then take an opposite position in the same swap at the then market rate \( S_{T} \), reverse exchanging fixed rate \( S_T \) for the variable/floating. The floating legs payaments should be the same (indexed to the same underlying), leaving one with entitlement to the difference between the fixed cash flows. Surely, the option buyer will exercise the option only when the difference is in the buyer's favour: buyer of a Payer Swaption will exercise if K is lower than \( S_{T} \), and buyer of Receiver Swaption will exercise when K is higher than \( S_{T} \).

To simplify the presentation, we derive the formula for the price of a Payer Swaption, which can then be easily generalised to cater for for both a Receiver and a Payer Swapations by inserting the \( \phi \) symbol we have been using. Now the payoff of a Payer swaption at maturity of the swaption can be written as:

$$ Payoff_T = \sum_{i=1}^{fM}{{(S_T - K)}^{+} Notional (t_i-t_{i-1}) DF_{T, t_i} } $$ $$ \quad = \left( Notional \sum_{i=1}^{fM}{(t_i-t_{i-1}) DF_{T, t_i}} \right) {(S_T - K)}^{+} $$

Where M represents the maturity of the underlying swap, f the frequency of the swap's fixed leg, fM the number of cash flows of the underlying swap (frequency per year times maturity in years), \(t_i\) time of cash flow in years, \( DF_{T,t_i}\) the discount factors for time \(t_i\) as at time T, \( (t_i-t_{i-1})\) the accrual factor, K the strike rate, and \(S_T\) the swap rate at time T.

We assume Notional is equal to 1 to simplify the presentation. We also let \( \sum_{i=1}^{fM}{(t_i-t_{i-1}) DF_{T, t_i}} = A (T,f,M) \), the annuity factor as it is simply the present value at time T of an insturment paying 1 unit at the coupon date of a swap with fM cash flows according to the daycount conventions of the swap. The payoff at maturity can thus be written as:

$$ Payoff_T = A (T,f,M) {(S_T - K)}^{+} $$

The price of the swaption at any time t before the Maturity T is just the present value of its expectation, where the expectation is taken under the measure associated with the annuity numeraire:

$$ Payer \; Swaption_t = A (t,f,M) E^{A} \left[ \frac{1}{A (T,f,M)} Swaption_T \right]$$ $$ \quad = A (t,f,M) E^{A} \left[ \frac{1}{A (T,f,M)} A (T,f,M) {(S_T - K)}^{+} \right]$$ $$ \quad = A (t,f,M) E^{A} \left[ {(S_T - K)}^{+} \right]$$

As the forward swap rate is martingale under the measure associated with the annuity numeraire, its dynamic can be written as lognormal (as per Black's model), and the price of the swaption can be written in terms of Black's formula as follows:

$$ Payer \; Swaption_t = A (t,f,M) E^{A} \left[ {(S_T - K)}^{+} \right]$$ $$ \quad = A (t,f,M) e^{r(T-t)} e^{-r(T-t)} E^{A} \left[ {(S_T - K)}^{+} \right]$$ $$ \quad = A (t,f,M) e^{r(T-t)} Black \left(t, S_t,K, \sigma_{S_t},T \right) $$ $$ \quad = A (T,f,M) e^{-r(T-t)} \left[ \phi S N{\left( \phi d_{1} \right)}-\phi K N{\left( \phi d_{2} \right)} \right] $$

Where we used \( A (T,f,M) = e^{r(T-t)} A (t,f,M)\), which is just the relationship between the value of the annuity at time t and T when one one assumes a constant dicount rate r between t and T.

The above formula is easy to compute; however, the computation of A factor will require daycount fraction and discount rate for each coupon date, so we simplify it by assuming that the accrual factor is constant and there is a single discount rate. Naturally, we set the accrual factor equal to \( \frac{1}{f} \) and the discount rate equal to the forward swap rate. Hence:

$$ A (T,f,M) = \sum_{i=1}^{fM}{(t_i-t_{i-1}) DF_{T, t_i}} $$ $$ \quad \quad = \sum_{i=1}^{fM}{\frac{1}{f} \frac{1}{{(1+S/f)}^i}} $$ $$ \quad \quad = \frac{1}{f} \frac{1}{(1+S/f)} \frac{1-\frac{1}{{(1+S/f)}^{fM}}}{1-\frac{1}{{(1+S/f)}}} $$ $$ \quad \quad = \frac{1}{f} \frac{1}{(1+S/f)} \frac{1-\frac{1}{{(1+S/f)}^{fM}}}{\frac{S/f}{{(1+S/f)}}} $$ $$ \quad \quad = \frac{1}{f} \frac{1}{S/f} {\left( 1-\frac{1}{{(1+S/f)}^{fM}} \right)} $$ $$ \quad \quad = \frac{1}{S} {\left( 1-\frac{1}{{(1+S/f)}^{fM}} \right)} $$

The simplified formula for the price of a European Swaption can thus be written as:

$$ Swaption_t = A (T,f,M) Black \left(t, S_t,K, \sigma_{S_t},T \right) $$ $$ \quad = \frac{1}{S} {\left( 1-\frac{1}{{(1+S/f)}^{fM}} \right)} Black \left(t, S_t,K, \sigma_{S_t},T \right) $$ $$ \quad = A \; Black \left(t, S_t,K, \sigma_{S_t},T \right) $$