# SABR Model

> The SABR model describes how forward prices and volatility move together and fits smiles with four parameters. Learn the model, each parameter and its uses in rates.

Source: https://learn.tradelabsai.com/options/sabr-model/  
Track: Options · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "SABR Model", https://learn.tradelabsai.com/options/sabr-model/

The SABR model, introduced by Patrick Hagan, Deep Kumar, Andrew Lesniewski and Diana Woodward in 2002, is the standard way to model volatility smiles in interest rate options. Its name stands for "stochastic alpha, beta, rho", three of its parameters. SABR describes how a forward rate or price and its volatility move together, and it comes with a simple approximate formula for implied volatility at each strike. That formula made it practical for traders to fit and interpolate smiles for swaptions, caps and floors, and it is widely used in FX and commodity markets too.

## The model

SABR models a forward F and its volatility α:

```
dF = α × F^β dW1
dα = ν × α dW2
corr(dW1, dW2) = ρ
```

The forward has no drift because it is a forward under its own pricing measure. Volatility follows a lognormal process with no mean reversion.

## The four parameters

| Parameter | Role | Effect on the smile |
|---|---|---|
| α (alpha) | Current volatility level | Shifts the whole smile up or down |
| β (beta) | Backbone: how volatility relates to the level of F | Sets how the smile moves when F moves; often fixed in advance |
| ρ (rho) | Correlation between forward and volatility | Controls skew (slope) |
| ν (nu) | Volatility of volatility | Controls curvature (smile shape) |

**β** deserves special attention:

- **β = 1:** lognormal behaviour, like Black 76.
- **β = 0:** normal behaviour, like Bachelier, allowing negative rates.
- **β = 0.5:** a common compromise in rates markets.

See [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/).

## Hagan's formula

The key practical result is an approximate closed form expression for Black (or normal) implied volatility at any strike, given the four parameters. Traders calibrate α, ρ and ν (with β fixed) to market volatilities for each expiry and tenor, then use the formula to read implied volatility at any strike.

**Example: Fitting a swaption smile**
For a 1 year option on a 10 year swap, the market quotes normal volatilities at several strikes around the forward swap rate of 3.5%. A trader fixes β = 0.5 and fits α, ρ and ν. The calibrated ρ of minus 0.3 produces higher volatility for low strikes (receiver swaptions), and ν of 0.4 produces curvature so that far strikes on both sides are higher than at the money. The fitted smile is then used to price swaptions and caps at strikes that are not quoted.

## Why SABR caught on

- **Realistic smile dynamics:** when the forward moves, the smile moves with it in a way that matches market behaviour better than local volatility, which improves hedging. This was the main motivation of Hagan's paper, "Managing Smile Risk". See [Local Volatility](https://learn.tradelabsai.com/options/local-volatility/).
- **Simple, fast formula** for implied volatility.
- **Few, intuitive parameters** that traders can relate to level, skew and curvature.
- **Flexibility** across rates, FX and commodities.

## Known problems and fixes

- **Negative densities at low strikes:** Hagan's approximation can imply negative probabilities for very low strikes, especially for long expiries, which creates arbitrage. Fixes include arbitrage free SABR methods solving the model numerically.
- **Negative rates:** the original model with β > 0 cannot handle negative forwards. Shifted SABR adds a constant to rates; normal SABR uses β = 0.
- **No mean reversion:** volatility can drift without bound over long horizons, which matters for long dated products.
- **One expiry at a time:** SABR is usually calibrated separately for each expiry, so it is a smile model rather than a full term structure model.

## SABR vs Heston

| | SABR | Heston |
|---|---|---|
| Main markets | Interest rates, FX | Equities |
| Mean reversion in volatility | No | Yes |
| Calibration | Per expiry | Across expiries |
| Pricing | Approximate implied volatility formula | Semi closed form prices |
| Negative rates | With shifted or normal versions | Not designed for them |

See [Stochastic Volatility and the Heston Model](https://learn.tradelabsai.com/options/heston-model/).

## Frequently asked questions

### What is the SABR model?

A stochastic volatility model for forward prices or rates, with four parameters that control the level, backbone, skew and curvature of the volatility smile.

### What does beta mean in SABR?

It sets how volatility relates to the level of the forward: β = 1 is lognormal, β = 0 is normal, and values in between blend the two.

### Why is SABR used in interest rate markets?

Because it fits swaption and cap smiles well with a fast formula and gives realistic smile movements for hedging, with shifted or normal versions handling negative rates.

Next, learn how traders fill gaps between quoted options in [Volatility Interpolation and Extrapolation](https://learn.tradelabsai.com/options/volatility-interpolation/).

## Sources

- Wikipedia, [SABR volatility model](https://en.wikipedia.org/wiki/SABR_volatility_model)

## Continue learning

- Next lesson: [Volatility Interpolation and Extrapolation](https://learn.tradelabsai.com/options/volatility-interpolation/)
- Previous lesson: [Stochastic Volatility and the Heston Model](https://learn.tradelabsai.com/options/heston-model/)
- Related: [Stochastic Volatility and the Heston Model](https://learn.tradelabsai.com/options/heston-model/): The Heston model treats volatility as a random, mean reverting process linked to price. Learn its five parameters, how it creates skew and how it is calibrated.
- Related: [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/): Black 76 prices options on futures and forwards; Bachelier assumes normal price changes and handles negative prices. Learn the formulas, uses and differences.
- Related: [Interest Rate Swaps](https://learn.tradelabsai.com/bonds-credit/interest-rate-swaps/): An interest rate swap exchanges fixed interest payments for floating ones on a notional amount. Learn how swaps work, SOFR, swap rates, valuation, uses and risks.
- Related: [Volatility Smile and Skew](https://learn.tradelabsai.com/volatility/volatility-smile-and-skew/): Implied volatility differs by strike, forming a smile or skew. Learn the shapes in equities, FX and commodities, why they exist and how to measure skew.
- Related: [Local Volatility](https://learn.tradelabsai.com/options/local-volatility/): The local volatility model, from Dupire and Derman Kani, makes volatility depend on price and time so it fits every listed option. Learn how it works and its limits.