Finance / SABR (Hagan et al. 2002) stochastic-volatility implied-vol smile

SABR Model Calculator — Swaption Volatility Smile

Compute the SABR implied-volatility smile and swaption prices from alpha, beta, rho and nu using the Hagan et al. (2002) asymptotic formula, with the ATM limit.

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ZORVLABS · SABR SWAPTION

Stochastic Alpha Beta Rho Swaption Matrix

SABR forward-measure dynamics with Hagan's asymptotic Black implied-vol formula, backbone, smile/skew, and 2D / 3D views.

Numerical Integrity

asymptotic SABR

Hagan et al. Eq. (2.17) / (2.18) style implementation.

ATM vol

0.103971

σATM from Hagan eq. 2.18

Call price

0.002767

Black price × annuity

Put price

0.002767

Parity-consistent

z / x(z)

0.000000

Smile distortion factor

Backbone

0.099846

σATM at T→0

Skew slope

-2.19801243

Finite-difference dσ/dK

Vol-of-vol λ

107.858371

ν / (α F^{1−β})

Delta / Vega

0.518531 / 0.152378

Black sensitivity

View mode

3D surface

Forward F0

Annuity / PV01

ATM strike

Maturity T (years)

Alpha

Beta

Rho

Nu (vol-of-vol)

Smile min strike factor

Smile max strike factor

Surface strike min factor

Surface strike max factor

Surface maturity max

Surface strike resolution

Surface maturity resolution

ATM / Backbone

0.103971 / 0.099846

ATM vs T→0 backbone

Smile curvature

436.85674548

Second derivative proxy

z factor

0.000000

ν/α · (FK)^(1−β)/2 · ln(F/K)

x(z)

1.000000

Hagan transform

Core SABR SDE: dF_t = α_t F_t^β dW_1, dα_t = ν α_t dW_2, dW_1 dW_2 = ρ dt.

Black vol asymptotic: σ_B = α / (F K)^((1−β)/2) · (z/x(z)) · [1 + O(T)].

2D volatility smile

Three forward levels; values are Black implied vols from the SABR asymptotic formula

2D backbone and term structure

ATM volatility backbone across maturities

2D swaption price curve

Black price × annuity at selected maturity

2D ATM smiles at shifted maturities

Smile evolution over time, same forward

3D SABR surface

Implied volatility surface

Source equations

Hagan, Kumar, Lesniewski & Woodward (2002)

Dynamics: dF_t = α_t F_t^β dW_1, dα_t = ν α_t dW_2, dW_1 dW_2 = ρ dt.

Lognormal implied vol: σ_B = α/(FK)^((1−β)/2) · (z/x(z)) · [1 + ((1−β)^2 α^2)/(24(FK)^(1−β)) + (ρβνα)/(4(FK)^((1−β)/2)) + (2−3ρ²)ν²/24]T.

Transform: z = (ν/α)(FK)^((1−β)/2) ln(F/K), x(z)=ln((√(1−2ρz+z²)+z−ρ)/(1−ρ)).

ATM limit: σ_ATM = α/F^(1−β) · [1 + ((1−β)^2 α²)/(24 F^(2−2β)) + (ρβνα)/(4F^(1−β)) + (2−3ρ²)ν²/24]T.

finance methodology

A clear path from inputs to results.

Build a swaption volatility smile from the four SABR parameters that desks actually quote. SABR (stochastic alpha-beta-rho) gives a closed-form asymptotic for the Black implied volatility at every strike, so you can take a forward swap rate and a calibration and read off the lognormal volatility and swaption price across the strike grid. Enter the forward, the annuity (PV01), maturity, and alpha, beta, rho, and nu, and the matrix returns the at-the-money volatility, the full smile, and prices.

Step-by-step methodWorked exampleAssumptions explained

How to use

Set the forward swap rate and the maturity (option expiry) in years for the swaption you are quoting.
Enter the annuity, or PV01, of the underlying swap; this is the present value of a basis point that scales the Black price into a cash premium.
Set the strike (or strike grid center); at the money the strike equals the forward.
Enter the SABR parameters: alpha (the overall volatility level), beta (the backbone exponent, 0 for normal and 1 for lognormal dynamics, 0.5 here), rho (the forward/volatility correlation that tilts the smile), and nu (the volatility-of-volatility that curves it).
Read the at-the-money implied volatility from Hagan eq. 2.18 and the full smile from eq. 2.17 across the strike range.
Read the swaption price, computed with the Black formula using the SABR volatility at each strike and discounted by the annuity.

The formula

\sigma_B(K,F) = \frac{\alpha}{(FK)^{(1-\beta)/2}\left[1 + \frac{(1-\beta)^2}{24}\ln^2\!\frac{F}{K} + \frac{(1-\beta)^4}{1920}\ln^4\!\frac{F}{K}\right]}\cdot\frac{z}{x(z)}\cdot\left\{1 + \left[\frac{(1-\beta)^2}{24}\frac{\alpha^2}{(FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4(FK)^{(1-\beta)/2}} + \frac{2-3\rho^2}{24}\nu^2\right]T\right\}, \quad z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln\frac{F}{K}, \quad x(z) = \ln\frac{\sqrt{1-2\rho z + z^2} + z - \rho}{1-\rho}

Reference: Hagan, Kumar, Lesniewski & Woodward (2002), "Managing Smile Risk", Wilmott Magazine, eqs. (2.17a-2.17c) for the implied-volatility smile and (2.18) for the at-the-money limit.

Worked example

Five-year at-the-money payer swaption

Inputs

Forward swap rate 3.25% (0.0325), annuity/PV01 0.92, strike 3.25% (at the money), maturity 5 years; SABR parameters alpha = 0.018, beta = 0.5, rho = -0.25, nu = 0.35.

Calculation path

1
At the money, ln(F/K) = 0 so z/x(z) -> 1 and the smile formula reduces to the eq. 2.18 limit sigma_ATM = alpha/F^{1-beta} * {1 + [ ((1-beta)^2/24)(alpha^2/F^{2-2beta}) + (rho*beta*nu*alpha)/(4*F^{1-beta}) + ((2-3*rho^2)/24)*nu^2 ] * T}. The base alpha/F^{0.5} = 0.018/sqrt(0.0325) = 0.099846. The bracket is (0.25/24)(0.000324/0.0325) + (-0.25*0.5*0.35*0.018)/(4*0.180278) + (1.8125/24)(0.1225) = 0.0001039 - 0.0010921 + 0.0092519 = 0.0082637, times T = 5 gives a 4.132% uplift, so sigma_ATM = 0.099846 * 1.041319 = 0.103971. Pricing the payer with Black at that volatility, annuityPV01*(F*N(d1) - K*N(d2)) with F = K, gives 0.002767.

Result

At-the-money lognormal (Black) implied volatility 10.397%; at-the-money payer swaption price 0.002767 per unit notional, or 27.67 basis points. The eq. 2.17 smile value at the forward matches the eq. 2.18 limit to the printed digits, as it must.

Interpreting the result

The SABR output is a Black implied volatility, not a price: you feed sigma_B(K,F) into the Black swaption formula and discount by the annuity to get the premium. Alpha sets the overall height of the smile, beta fixes the backbone (how the at-the-money volatility moves as the forward moves), rho tilts the smile so that one wing is higher than the other, and nu controls the curvature or how fast volatility rises away from the money. Because the formula is an asymptotic expansion, it is accurate near the money and for moderate maturities but can misbehave at very low strikes, which is where practitioners switch to the normal or shifted-SABR variants.

Assumptions & limitations

The implied volatility is the Hagan et al. (2002) asymptotic expansion, accurate near the money and for moderate maturities; it loses accuracy at extreme strikes and long expiries and can produce arbitrage in the far wings.
Prices use the lognormal Black model with the SABR volatility, so the forward is assumed positive; near-zero or negative rates need the normal or shifted-lognormal version, which this lognormal form does not apply.
The annuity (PV01) is supplied as a single number; the full discount-curve and swap-schedule construction behind it is not modeled here.
The four SABR parameters are taken as given, not calibrated to a set of market quotes within the tool.
Beta is treated as an input rather than being inferred, since beta and rho are partly interchangeable in fitting the smile and beta is usually fixed by convention.

Frequently asked questions

What do the four SABR parameters mean?

Alpha is the current volatility level and sets the height of the smile. Beta is the backbone exponent that fixes how the at-the-money volatility moves when the forward moves: beta = 0 is normal (absolute) dynamics, beta = 1 is lognormal, and 0.5 is a common middle choice for rates. Rho is the correlation between the forward and its volatility and tilts the smile, while nu is the volatility-of-volatility that controls its curvature.

What is the annuity or PV01 for?

A swaption pays off on a swap, so its value scales with the present value of that swap's fixed-leg basis point, called the annuity or PV01. The Black formula gives a price per unit annuity, and multiplying by the annuity (0.92 in the example) converts it into an actual premium. A larger annuity, from a longer or higher-notional swap, means a proportionally larger swaption premium.

When does the SABR approximation break down?

It is an asymptotic expansion, so it is sharpest near the money and for shorter maturities. At very low or negative strikes the lognormal form can imply negative probabilities (butterfly arbitrage) in the deep wings, and long expiries degrade accuracy. In low-rate environments desks use the normal SABR or a shifted-lognormal version to keep the forward away from zero.

Why does SABR return a volatility instead of a price?

SABR is a smile model: its job is to give the Black implied volatility at every strike in closed form. You then plug that volatility into the standard Black swaption formula and multiply by the annuity to get the cash premium. Separating the two lets desks quote and risk-manage in volatility terms, which is how the swaption market actually trades.

Why does the at-the-money formula look simpler?

At the money the strike equals the forward, so ln(F/K) is zero, the z/x(z) ratio tends to one, and the log-moneyness correction terms vanish. What remains is eq. 2.18, alpha/F^{1-beta} times a single maturity-dependent correction. That is why the worked example evaluates the compact ATM form and still matches the general eq. 2.17 smile value at the forward.

Does rho or beta control the skew?

Both affect the slope of the smile, which is why they are partly interchangeable in calibration. Beta sets the backbone and contributes a structural skew through the (FK) power terms, while rho adds a direct correlation-driven tilt. In practice beta is fixed by convention (here 0.5) and rho is calibrated to match the observed skew, with nu then fitting the curvature.

Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.