Swaptions: Pricing and Trading
A swaption is an option on entering an interest-rate swap, with payoff determined by a forward swap rate and its annuity. It is the central liquid option for trading rate volatility, but “a 10Y swaption” is incomplete: expiry, underlying swap tenor, strike, settlement, collateral, and volatility convention all determine the contract.
Contract and payoff
A payer swaption gives the right to pay fixed and receive floating; it benefits when swap rates rise. At expiry T, its cash-settled approximation is
payoff_payer = N * A(T) * max(S(T) - K, 0)
payoff_receiver = N * A(T) * max(K - S(T), 0)
S(T) is the forward swap rate, K the strike, and A(T) = sum alpha_i P(T,Ti) the fixed-leg annuity. Physical settlement creates the actual swap; cash settlement needs a specified annuity convention. The latter distinction changes the payoff when the curve moves sharply and cannot be ignored for large or long-dated trades.
| Quote component | Example | Why it matters |
|---|---|---|
| expiry x tenor | 2Y x 10Y | option ends in 2 years on a 10-year swap |
| strike | ATM, 25bp OTM | defines moneyness and delta |
| type | payer/receiver | rate-up versus rate-down exposure |
| settlement | physical/cash | determines payoff annuity |
| vol convention | normal/Black | determines pricing transformation |
Black and normal prices
Under Black's lognormal swap-rate model, the payer price is
V = N * A * [S * N(d1) - K * N(d2)]
d1 = [ln(S/K) + 0.5*sigma^2*T] / (sigma*sqrt(T))
d2 = d1 - sigma*sqrt(T)
For a normal (Bachelier) quote:
V = N * A * [(S-K)*N(d) + sigma_N*sqrt(T)*n(d)]
d = (S-K)/(sigma_N*sqrt(T))
Black requires positive S and K; normal volatility is defined in rate units and remains usable around zero. Never convert an implied volatility with a constant multiplier. Convert through price under the same forward, annuity, expiry, and settlement convention.
from math import log, sqrt
from scipy.stats import norm
def black_payer(S, K, vol, T, annuity, notional=1.0):
st = vol * sqrt(T)
d1 = (log(S / K) + 0.5 * vol * vol * T) / st
return notional * annuity * (S * norm.cdf(d1) - K * norm.cdf(d1 - st))
def bachelier_payer(S, K, vol_n, T, annuity, notional=1.0):
st = vol_n * sqrt(T)
d = (S - K) / st
return notional * annuity * ((S-K)*norm.cdf(d) + st*norm.pdf(d))
Smile, surface, and calibration
The market quotes a grid by expiry, tenor, and strike or delta, often as ATM, risk reversals, and butterflies. Pricing a nonstandard strike requires interpolation that avoids static arbitrage: total variance should generally increase in expiry, and call prices must be convex in strike. SABR is a compact way to interpolate a smile; its caveats and parameters are covered in the SABR model.
The forward swap rate and annuity come from discount and projection curves. Rebuild them from the same market snapshot as volatility. A stale curve paired with a current vol surface can make an apparent vol P&L that is actually forward P&L. The curve construction and schedule conventions are detailed in yield-curve bootstrapping.
Trading exposures
An ATM payer swaption is long rate convexity and long volatility. A receiver can hedge downside yields, but is not simply the opposite trade after smile and skew. For a portfolio, decompose daily P&L as
dV approximately delta*dS + 0.5*gamma*dS^2
+ vega*dvol + theta*dt + vanna*dS*dvol
Delta is commonly hedged with the underlying forward-starting swap or a swap future. Vega is hedged across the expiry-tenor surface, not with a single arbitrary option. Bermudan exposure cannot be fully hedged with Europeans because exercise optionality introduces path-dependent gamma and vega.
| Trade | Primary view | Residual risk |
|---|---|---|
| long payer | rates rise or realized vol exceeds implied | theta decay, payer skew |
| long receiver | rates fall / convexity hedge | receiver skew, carry |
| payer spread | capped upside-rate view | strike interpolation, gap risk |
| calendar | relative volatility maturity | correlation of surface nodes |
Controls that matter
Report delta as PV01 of the underlying swap and distinguish a parallel curve shock from a quote bump followed by recalibration. Report vega in price per one basis point of normal vol or per one vol point of Black vol, explicitly. Check finite-difference Greeks against automatic differentiation or a second bump size. Large disagreements at deep strikes usually signal a convention, smile, or numerical issue rather than free alpha.
Stress joint curve and surface shifts. A payer's delta changes when rates rise; vol and skew also tend to move. Isolated delta and vega explain is therefore incomplete in selloffs. Relate the option’s local Greek behavior to options Greeks, but use full repricing for risk limits.
Key takeaways
- A swaption payoff is an option on the forward swap rate times the swap annuity.
- Expiry, tenor, settlement, and volatility convention are contract-defining data.
- Price normal and Black quotes through their respective formulas, not a naive vol conversion.
- Hedge delta with rate instruments and vega across the relevant surface nodes.
- Full repricing under joint curve-volatility stresses is more informative than isolated Greeks.