The SABR Model

When a swaption volatility cube is built, the model interpolates the smile between the points whose prices the market shows.

The forward and volatility, moving with correlation

The SABR model assumes that the forward $F_t$ and its volatility $\alpha_t$ are driven by two separate Brownian motions. The correlation between them is $\rho$¹.

$$dF_t = \alpha_t F_t^{\beta} \, dW^{(1)}_t, \qquad d\alpha_t = \nu \, \alpha_t \, dW^{(2)}_t, \qquad dW^{(1)}_t \, dW^{(2)}_t = \rho \, dt$$

There are four parameters: $\alpha$, $\beta$, $\rho$, and $\nu$. The initial values are $F_0 = f$ (today's forward) and $\alpha_0 = \alpha$, so the parameter $\alpha$ is the initial value of the volatility.

The equations are written under the measure in which the forward has no drift (the forward measure in the original paper, and the level measure for swaptions)². The name is an acronym of Stochastic Alpha Beta Rho³.

$\nu$ measures how much the volatility $\alpha_t$ itself varies, and is called the volatility of volatility (vol of vol). It is the same kind of quantity as $\xi$ in the Heston model. The equation for $\alpha_t$ has no drift term, so there is no mean reversion back to a level. The original paper describes the SABR model as the simplest of the stochastic volatility models that are homogeneous in the forward and the volatility⁴.

The SABR model fits the volatility smile of options with a single maturity. It suits markets in which each underlying has one maturity, such as swaptions, caplets, and floorlets⁵. You set the parameters for each maturity, and the model fills in the volatilities at strikes that the market does not show. The shape of the smile and the no-arbitrage conditions are covered in Volatility Smiles and Surfaces.

The original paper says that when the forward moves in the SABR model, the implied volatility curve moves in the same direction, and that this agrees with market experience⁶.

The four parameters set the shape of the smile

The four parameters each set a different feature of the smile. $\alpha$ mainly sets the height of the curve. $\rho$ mainly sets the slope. $\nu$ mainly sets how much smile the curve shows⁷. $\beta$ also sets the slope. It sets how the volatility at the money (ATM) moves when the forward moves.

The effects of $\beta$ and $\rho$ in the table follow the explanations in the original paper and in the Ministry of Finance paper⁸⁹. The original paper calls the path that the ATM volatility traces as the forward moves the backbone. The backbone is determined almost entirely by $\beta$. It is nearly flat at $\beta = 1$ and slopes steeply downward at $\beta = 0$⁸.

$\beta$ has two special cases. At $\beta = 1$, $dF_t = \alpha_t F_t \, dW^{(1)}_t$, which has the same form as the Black model. At $\beta = 0$, $dF_t = \alpha_t \, dW^{(1)}_t$, which has the same form as the normal model (the Bachelier model). As $\beta$ moves from 1 toward 0, the behavior shifts from that of the Black model toward that of the normal model¹⁰. Setting $\nu = 0$ freezes the volatility and gives the CEV model, $dF_t = \alpha F_t^{\beta} \, dW^{(1)}_t$⁹.

The Hagan approximation and its use

The derivation is in an appendix of the original paper; this page covers only how to use the formula. The price of a European option with maturity $T$ under the SABR model can be approximated in a form that uses Black's formula. The implied volatility $\sigma_B(K, f)$ to put into Black's formula is given in closed form by the following expression (the Hagan approximation)¹¹.

$$\sigma_B(K, f) = \frac{\alpha}{(fK)^{\frac{1-\beta}{2}} \left\{ 1 + \frac{(1-\beta)^2}{24} \ln^2 \frac{f}{K} + \frac{(1-\beta)^4}{1920} \ln^4 \frac{f}{K} \right\}} \cdot \frac{z}{x(z)} \cdot \left\{ 1 + \left[ \frac{(1-\beta)^2}{24} \frac{\alpha^2}{(fK)^{1-\beta}} + \frac{\rho \beta \nu \alpha}{4 (fK)^{\frac{1-\beta}{2}}} + \frac{2 - 3\rho^2}{24} \nu^2 \right] T \right\}$$ $$z = \frac{\nu}{\alpha} (fK)^{\frac{1-\beta}{2}} \ln \frac{f}{K}, \qquad x(z) = \ln \left\{ \frac{\sqrt{1 - 2\rho z + z^2} + z - \rho}{1 - \rho} \right\}$$

$K$ is the strike. The higher-order "$+\cdots$" terms in the original paper's formula are also dropped here. The symbols are the same as in the original paper (except that the maturity is written $T$). At the money ($K = f$), $z/x(z) = 1$, and the formula simplifies to the following¹².

$$\sigma_{\text{ATM}} = \frac{\alpha}{f^{1-\beta}} \left\{ 1 + \left[ \frac{(1-\beta)^2}{24} \frac{\alpha^2}{f^{2-2\beta}} + \frac{\rho \beta \nu \alpha}{4 f^{1-\beta}} + \frac{2 - 3\rho^2}{24} \nu^2 \right] T \right\}$$

The advantage of this formula is that it is cheap to compute. The parameters, the forward, the strike, and the maturity determine $\sigma_B$, and you then pass that value to Black's formula to get the price¹³. The author of the 2021 Ministry of Finance paper gives two reasons the SABR model is used to interpolate swaption smiles. The first is that the approximation formula makes the computation fast and the relationship to the Black model is clear. The second is that the added parameter improves the fit to actual values¹⁴. If you drop the correction term in $T$ at the end of the formula, the relative error exceeds 3% in extreme cases, so you cannot omit it¹⁵.

The steps for interpolating the smile and getting a price are as follows. How to fit is covered in Model Calibration.

1. Collect the market volatilities for each strike.
2. Choose $\beta$, and fit $\alpha$, $\rho$, and $\nu$ to the market values.
3. Use the Hagan approximation to find $\sigma_B$ for strikes that are not in the market.
4. Put $\sigma_B$ into Black's formula to get the price.

Deep diveApproximation near the forward, and the origin of the slope and the curvature

When the strike is not far from the forward, the Hagan approximation reduces to the following form. The original paper says this form should not be used for pricing; it is meant for seeing the qualitative shape of the smile¹⁶.

$$\sigma_B(K, f) \approx \frac{\alpha}{f^{1-\beta}} \left\{ 1 - \frac{1}{2}(1 - \beta - \rho\lambda) \ln \frac{K}{f} + \frac{1}{12} \left[ (1-\beta)^2 + (2 - 3\rho^2)\lambda^2 \right] \ln^2 \frac{K}{f} \right\}, \qquad \lambda = \frac{\nu f^{1-\beta}}{\alpha}$$

$\lambda$ measures how large the variation in the volatility, $\nu$, is relative to the local volatility at the current forward, $\alpha / f^{1-\beta}$. The first term is the ATM volatility, the second is the slope, and the third is the curvature¹⁶. The slope term splits into two parts.

The slope from $\beta$ ($-\frac{1}{2}(1-\beta)\ln\frac{K}{f}$) always slopes downward (a negative skew) when $\beta < 1$, because the local volatility $\alpha / F^{1-\beta}$ is lower when the forward is higher. The slope from $\rho$ ($\frac{1}{2}\rho\lambda\ln\frac{K}{f}$) slopes downward when the volatility and the underlying price are negatively correlated. Large moves in the forward tend to occur when volatility is high, and at strikes far from ATM, the high-volatility environment has a larger effect. This is the origin of the curvature term $\frac{1}{12}(2-3\rho^2)\lambda^2\ln^2\frac{K}{f}$, which is called the volga effect¹⁶.

Calculation example

Consider a hypothetical example in which an option has forward $f = 2.00\%$ and maturity $T = 1$ year, with parameters $\alpha = 0.03$, $\beta = 0.5$, $\rho = -0.3$, and $\nu = 0.4$. First, for a strike $K = 2.50\%$, the intermediate values of the approximation formula are shown step by step. The values are recomputed from the formula.

Here $\alpha = 0.03$ does not mean a volatility level of 3%. With $\beta = 0.5$, a guide to the level is $\alpha / f^{1-\beta} = 0.03 / 0.02^{0.5} \approx 21.2\%$, which is close to the ATM value of 21.40% (below)¹⁶.

The implied volatility at a strike of 2.50% is 19.53%. The value from the ATM formula ($K = 2.00\%$) is 21.40%.

The next table shows the implied volatility (in %) at five strikes when one parameter is moved at a time. The first row is the example above. In the other rows, only the changed parameter differs.

Raising $\alpha$ lifts the whole curve. Setting $\rho$ to 0 removes the negative skew and leaves a curved smile whose lowest point is at the 2.5% strike. Increasing $\nu$ raises both ends. Lowering $\beta$ steepens the negative skew. The difference between the 1.0% and 3.0% strikes is 7.4 percentage points at $\beta = 1$, 13.0 percentage points at $\beta = 0.5$ (the example above), and 18.8 percentage points at $\beta = 0$. In the $\beta = 1$ and $\beta = 0$ rows, $\alpha$ is reset so that the ATM values are close.

Next, consider how the price changes when you use the smile. The discount factor is 0.980199 for 1 year at a zero rate of 2% (continuously compounded). The price of a call on the forward is computed with Black's formula, per unit of notional.

If you use the ATM volatility for all strikes, the price is about 0.000125 lower at 1.5% and about 0.000090 higher at 2.5%. The price from the ATM volatility at a strike of 2.5% is about 1.33 times the SABR price. Without the downward-sloping skew, the prices come out too low at low strikes and too high at high strikes.

Choosing β and its use in interpolation

$\beta$ and $\rho$ both produce a negative skew. From a single smile on a single day, it is hard to tell the two apart, and either $\beta = 0$ or $\beta = 1$ fits the market smile equally well. The original paper says that choosing $\beta$ from the smile amounts to fitting the noise in the market¹⁷. There are two ways to choose $\beta$. One is to estimate it from past observations of the backbone (a plot of the logarithm of the ATM volatility against the logarithm of the forward). The other is to pick in advance from $\beta = 1$ (close to the Black model), $\beta = 0$ (the normal model), and $\beta = 0.5$¹⁸.

According to a 2021 Ministry of Finance paper, market convention tends to set $\beta$ to 0.5. The swaption volatility cube tools of financial data vendors set $\beta = 0.5$, impose a constraint that matches the ATM value exactly, and fit the rest to the market values (as of 2021)¹⁹.

After $\beta$ is chosen, the fitting of the parameters can be organized by using the ATM volatility $\sigma_{\text{ATM}}$ in place of $\alpha$. You obtain $\alpha$ by solving the ATM formula for $\alpha$ numerically²⁰. The inputs are the volatilities of swaptions with the same expiry and tenor and different strikes, and the parameters are fitted to minimize the sum over strikes of the squared difference between the market values and the approximation formula's values²¹.

The volatilities of swaptions form a three-dimensional object (the volatility cube) with three axes: strike, expiry, and tenor of the underlying swap²². Prices are visible for only some of the points in the market, so the smile is built by interpolation. The SABR model is widely used in the industry to interpolate swaption smiles, and it is also the default setting in vendor tools (as of 2021)²³. As an example of an actual fit, the original paper shows US dollar swaption values for April 28, 2000. $\nu$ was larger for shorter expiries and became smaller as expiry lengthened. $\rho$ was close to 0 for short expiries and became a large negative value as expiry lengthened. In both cases, the differences across the tenor of the underlying swap were small²⁴.

Handling negative interest rates

The SABR model cannot be used when the forward is negative, because $F^{\beta}$ is not necessarily a real number when $F$ is negative. The exception is $\beta = 0$, which has the same form as the normal model and so works with a negative forward²⁵. The original paper from 2002 says that some take the view that $\beta = 0$ is essential in markets where the forward is negative or near zero, such as yen interest rates¹⁸.

To use the model at negative rates without setting $\beta$ to 0, you can add a shift to the underlying. This shifted SABR model is widely used in practice (as of 2021). The equations are as follows²⁶.

$$dF_t = \alpha_t (F_t + s)^{\beta} \, dW^{(1)}_t, \qquad d\alpha_t = \nu \, \alpha_t \, dW^{(2)}_t, \qquad dW^{(1)}_t \, dW^{(2)}_t = \rho \, dt$$

$s$ is the size of the shift. Consider a hypothetical example in which the forward is $-0.20\%$ and the shift is 2.00%. Then $F_t + s = 1.80\%$, which is positive. In principle, any shift that makes the forward positive will do, but the depth of negative rates changes over time. In practice the size is set by market convention, and the default setting of one vendor's tool is 200bp (as of 2021)²⁷. Some data does not show the size of the shift²⁸. SABR models that have been rebuilt so that no arbitrage arises even at negative rates have also been developed²⁹.

Assumptions and limits

The Hagan approximation is an approximation, not an exact solution of the SABR model. The original paper says that the "$+\cdots$" terms of the approximation are smaller still, but it does not implement the higher-order terms¹⁵.

In low-rate or high-volatility environments, the density implied by the approximation can turn negative, which allows arbitrage. Later research discusses this as a problem near the origin, that is, on the low-strike side³⁰. The accuracy of the approximation drops for long maturities, when $\beta$ is near 0, and when $\nu$ is large³¹.

The SABR model is suited to a market with a single maturity, and fitting across several maturities needs the dynamic SABR model⁵. Quotes at thinly traded strikes can be stale, so take care when fitting the smile³².

When a model's assumptions fail, the model price and the market price diverge. The SABR model assumes that the volatility has no mean reversion and that the correlation between the forward and the volatility is constant, and it expresses the slope and curvature of the smile with three parameters ($\alpha$, $\rho$, and $\nu$, once $\beta$ is chosen). At maturities and strikes where the market smile departs from this shape, the approximation's values differ from the market's. When the model is used to interpolate a volatility cube, the interpolated values also rest on the SABR shape. In the worked example, setting $\rho$ to 0 moves the implied volatility at the 3.0% strike from 19.1% to 21.4%.

In practice, models that combine local volatility and stochastic volatility are also often used³³.

Notes

1. Hagan, Kumar, Lesniewski, Woodward, "Managing Smile Risk" (Wilmott magazine, 2002), equations (2.15a)–(2.15c) (printed page 88; page 5 of the PDF). The original paper writes $\hat F$ and $\hat\alpha$; the text here writes $F_t$ and $\alpha_t$. ↩

2. The same paper, section 2 (printed pages 85–86). In equations (2.1c) and (2.2c), the forward price and the forward swap rate are martingales under the respective measures. ↩

3. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), section 4.2 "SABR モデルは確率的ボラティリティ・モデル" (the SABR model is a stochastic volatility model) (printed pages 62–63). ↩

4. Hagan et al., section 2.3 (printed page 88). It describes the model as the simplest stochastic volatility model that is homogeneous in the forward and the volatility. ↩

5. Hagan et al., section 2.3 (printed page 88). It says the dynamic SABR model is treated in an appendix. ↩ ↩²

6. Hagan et al., section 3.2 (printed page 93). It writes "the implied volatility curve shifts in the same direction and by the same amount as the price f". ↩

7. Hagan et al., section 3.1 (printed page 91). $\alpha$ mainly sets the height of the curve, $\rho$ the slope, and $\nu$ how much smile the curve shows. ↩

8. Hagan et al., section 3 (printed page 90). The backbone is almost entirely determined by $\beta$; it slopes steeply downward at $\beta = 0$ and is nearly flat at $\beta = 1$. The slope arises from both $\beta$ and $\rho$ (printed pages 90–91). ↩ ↩²

9. Hattori (2021), Box 2 (printed page 66). $\alpha$ affects the level, $\beta$ the slope, $\nu$ the curvature, and $\rho$ the slope. When $\nu = 0$, the model becomes the CEV model. ↩ ↩²

10. Hattori (2021), section 4.2 (printed page 63). $\beta = 0$ is the normal model and $\beta = 1$ is the Black model. ↩

11. Hagan et al., equations (2.16)–(2.17) (printed page 89). The original paper writes the maturity as $t_{\text{ex}}$. Hattori (2021), Box 1 (printed page 65), gives the same formula and calls it the Hagan approximation. ↩

12. Hagan et al., equation (2.18) (printed page 89). At the money, $z/x(z) \to 1$. ↩

13. Hagan et al., end of section 2.3 (printed page 90). "once this formula is programmed, we just need to send the options to a Black pricer". ↩

14. Hattori (2021), section 4.1 (printed page 62). ↩

15. Hagan et al., end of section 2.3 (printed page 90). It says that omitting the last line gives a relative error above 3% in extreme cases. ↩ ↩²

16. Hagan et al., equations (3.1a)–(3.1b) (printed pages 90–91). It writes "should not be used to price real deals". The explanation of the origin of the slope and the curvature is in the same section 3 (printed pages 90–91). ↩ ↩² ↩³ ↩⁴

17. Hagan et al., section 3.1 (printed page 91). Figure 3.3 is an example in which the smile fits about equally well at both $\beta = 0$ and $\beta = 1$. ↩

18. Hagan et al., section 3.1 (printed page 91). Foreign exchange option desks and the like are said to favor $\beta = 1$; markets such as yen interest rates, where the forward is negative or near zero, to favor $\beta = 0$; and US interest rate desks to favor $\beta = 1/2$. ↩ ↩²

19. Hattori (2021), section 4.3 (printed page 63) and Box 2 (printed page 66). Box 2 says that market convention tends to set $\beta$ to 0.5. ↩

20. Hagan et al., section 3.1 (printed pages 91–92). It says to solve the ATM formula for $\alpha$ (inverting equation (2.18) on the spot when needed). ↩

21. Hattori (2021), section 4.3 (printed page 63). Equation (6) (the sum of squared differences over strikes). ↩

22. Hattori (2021), section 4.4 (printed pages 64–65). ↩

23. Hattori (2021), section 4.1 (printed page 61). ↩

24. Hagan et al., the text that describes Tables 3.1 and 3.2 in section 3.1 (printed page 92). The data are "m into n" swaptions on April 28, 2000. ↩

25. Hattori (2021), section 4.5 and note 23 (printed page 65). ↩

26. Hattori (2021), section 4.5 (printed page 65). ↩

27. Hattori (2021), section 2.4 (printed pages 58–59) and section 4.5 (printed page 65). ↩

28. Hattori (2021), note 7 (printed page 59). ↩

29. Hattori (2021), note 16 (printed page 62). It cites Hagan et al. (2014), "Arbitrage-Free SABR", among others. ↩

30. Gulisashvili, Horvath, Jacquier, "Mass at zero and small-strike implied volatility expansion in the SABR model", beginning of section 1 (page 1). ↩

31. The same paper, section 1 (page 1). "typically loses accuracy for long-dated derivatives, when the CEV exponent (the parameter β) is close to zero, or when the volatility of volatility ν is large". ↩

32. Hagan et al., section 3.1 (printed page 91). It says quotes at strikes far from the forward may be stale. ↩

33. Hattori (2021), note 17 (printed page 62). ↩