One option on the whole swap rate, discounted with the annuity: the forward swap rate, Black-76, and why it costs less than a cap at the same strike.
A swaption is priced with Black's model applied to the forward swap rate, then multiplied by the notional and by the annuity, the present value of 1 paid on each fixed date of the underlying swap. On an illustrative 1-year into 4-year payer swaption with a 50,000,000 notional, a 4.50 per cent strike, a forward swap rate of 4.2698 per cent and 20 per cent volatility, the premium is 428,065, or 85.6 basis points of notional paid upfront.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A payer swaption gives the right, on one date, to enter a swap paying a fixed rate. A borrower expecting to draw a four-year floating rate loan in a year's time buys it to cap the rate it will fix at: if the four-year swap rate in a year is above 4.50 per cent, it exercises and pays 4.50; if below, it lets the option lapse and swaps at the market. It is a single option on an average rate, which is what makes it different from a cap, priced caplet by caplet in how to price an interest rate cap with Black's model.
| Maturity, years | Zero rate | Discount factor |
|---|---|---|
| 1 | 3.80% | 0.963391 |
| 2 | 3.95% | 0.925446 |
| 3 | 4.05% | 0.887715 |
| 4 | 4.12% | 0.850870 |
| 5 | 4.18% | 0.814851 |
Notional 50,000,000. Expiry in one year, into a four-year swap with fixed payments at years 2, 3, 4 and 5. Strike 4.50 per cent. Black volatility of the forward swap rate 20 per cent. The discount factor is 1 ÷ (1 + zero)t.
Step 1, the annuity. The present value of 1 paid at each fixed date of the underlying swap:
A = DF2 + DF3 + DF4 + DF5 = 0.925446 + 0.887715 + 0.850870 + 0.814851 = 3.478883
Step 2, the forward swap rate. The fixed rate that makes the forward-starting swap worth zero today. The floating leg from year 1 to year 5 is worth DF1 − DF5, the same identity used in how the fixed rate on a swap is calculated:
F = (DF1 − DF5) ÷ A = 0.148540 ÷ 3.478883 = 4.2698%
Step 3, Black-76 on the rate.
d1 = [ln(F ÷ K) + σ2T ÷ 2] ÷ (σ√T) = (−0.052519 + 0.02) ÷ 0.20 = −0.1626
d2 = d1 − σ√T = −0.3626
N(d1) = 0.4354; N(d2) = 0.3585
Rate value = F × N(d1) − K × N(d2) = 1.859136% − 1.613043% = 0.2461%, or 24.61 bp a year
Step 4, to money. The rate value is earned on each of the four fixed dates, so it is discounted with the annuity, not with a single factor:
Premium = N × A × rate value = 50,000,000 × 3.478883 × 0.002461 = 428,065
Excel, with F, K, σ, T, N and A in B1:B6: =B5*B6*(B1*NORM.S.DIST((LN(B1/B2)+B3^2*B4/2)/(B3*SQRT(B4)),TRUE)-B2*NORM.S.DIST((LN(B1/B2)-B3^2*B4/2)/(B3*SQRT(B4)),TRUE))
The borrower pays 428,065 today, 85.6 bp of notional. Spread over the swap's fixed dates, the premium divided by the annuity, that is a running cost of 24.61 bp a year, the rate value from step 3; dividing 85.6 by four years instead would ignore discounting. The strike is 23 bp above the forward, so the option is out of the money; at a strike equal to the forward, 4.2698 per cent, it would cost 591,603. The quick approximation for an at-the-money swaption, 0.4 × σ × √T × F × A × N, gives 594,160, a useful check on any dealer quote.
Parity ties the payer to the receiver at the same strike. The receiver costs 828,547, and the difference, minus 400,482, is exactly 50,000,000 × 3.478883 × (4.2698% − 4.50%), the value of a forward swap at the strike.
| Strike | 15% vol | 20% vol | 25% vol | 30% vol |
|---|---|---|---|---|
| 4.00% | 704,596 | 837,655 | 974,198 | 1,112,260 |
| 4.50% | 283,364 | 428,065 | 575,080 | 722,989 |
| 5.00% | 90,625 | 196,282 | 320,498 | 455,008 |
Five points of volatility move the base premium by about 145,000 to 147,000, roughly a third of it. Volatility is the input to negotiate; the curve is observable, the volatility is the dealer's.
A cap at 4.50 per cent on the same four annual periods, fixing at years 1 to 4 and paying at years 2 to 5, at the same 20 per cent volatility, costs 746,037, built from caplets of 82,776, 167,381, 223,763 and 272,116. The swaption costs 57.4 per cent of that, 317,973 less. The saving is real but it buys something different. The cap pays in any year the rate is high, even if the others are low. The swaption is exercised once, on the average, and only into a swap fixed for the whole term: if the borrower is not certain to borrow, or wants to keep floating, the cheaper option is the wrong one.
Dealers increasingly quote swaption volatility in normal terms, in basis points. The 20 per cent here corresponds to roughly 87.7 bp of normal volatility, which in the Bachelier model gives 428,975. Check which convention a quote uses before putting it into a formula.
Premium = notional × annuity × [F N(d1) − K N(d2)], with F = (DFstart − DFend) ÷ annuity. On this curve: F = 4.2698 per cent, A = 3.478883, premium 428,065. Black-76 written out in cells, so a strike can be re-struck and the premium recomputed, is in the free workbook for this case. All figures are illustrative.
A payer swaption is the right to enter a swap paying fixed, so it gains when rates rise; a receiver swaption is the right to receive fixed and gains when they fall. They are linked by parity: payer minus receiver equals notional x annuity x (forward minus strike). In the worked case 428,065 minus 828,547 gives -400,482, which is 50,000,000 x 3.478883 x (4.2698 minus 4.50 per cent).
A cap is a strip of separate options, one on each period's rate, and each can pay on its own. A swaption is one option on the average rate across all periods, exercised once. At the same strike, volatility and dates, the 1y into 4y swaption costs 428,065 and the matching four-caplet cap 746,037: the swaption is 57.4 per cent of the price, but protects only if the borrower is willing to lock the whole remaining term on one date.
Dealers now quote most swaption volatility in normal (basis point) terms, which handles low and negative rates. The two give the same price when they are consistent: the 20 per cent lognormal volatility here corresponds to roughly 87.7 bp of normal volatility, and the Bachelier model on that figure gives 428,975, within 0.2 per cent of the Black price.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: corporate finance, valuation and markets → · All 1340 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.