The Black and Bachelier Models

Traders who handle volatilities and prices of interest rate options, valuation and risk staff, and the system staff who load volatility data all have to tell whether a figure comes from the Black model or the Bachelier model. How do the two models differ in what they assume about the forward's moves?

How each model assumes the forward moves

The Black model and the Bachelier model (normal model) are both valuation models for options whose underlying is a value for a future date that is set in today's market. Examples of the underlying are the price of a futures contract and a forward rate, the rate for a future period¹. This page calls forward prices, futures prices, and forward rates together the forward, and writes it as $F$. The two models differ in how they assume the forward moves.

The Black model assumes that the rate of change of the forward follows a normal distribution. The forward then follows a lognormal distribution and never takes a value at or below zero¹. The forward is as likely to double as to halve, so upward moves are larger than downward ones. The Bachelier model assumes that the change in the forward itself follows a normal distribution. The forward moves up and down in the same way, and it can take values at or below zero². Louis Bachelier presented this model in 1900, and practitioners also call it the normal model³.

The two assumptions can be written as follows, using the change $dF$ in the forward over a short period $dt$¹. $dW$ is a movement that follows a normal distribution with mean 0 and variance $dt$ (the increment of a Wiener process, or Brownian motion)⁴.

$$dF = \sigma_B F \, dW$$ $$dF = \sigma_N \, dW$$

The first is the Black model and the second is the Bachelier model. $\sigma_B$ is the Black model volatility: the standard deviation of the rate of change of the forward, expressed as an annual figure and shown in %. $\sigma_N$ is the normal volatility. It is the standard deviation of the change in the forward, expressed as an annual figure and shown in bp for interest rates¹. In the Black model, the size of the change is $\sigma_B F$, which is proportional to the level of the forward. With a Black model volatility of 30% and a normal volatility of 30bp, the standard deviation of the change (annual) at each level of the forward is as follows.

In the Black model, the lower the rate, the smaller the moves, and the higher the rate, the larger the moves. In the Bachelier model, the rate moves by the same amount whatever its level.

The difference also shows in the shape of the distribution of the forward at expiry. With a forward of 1.00%, an expiry of 2 years, a Black model volatility of 30% and a normal volatility of 30bp, the two distributions overlaid look like Figure 1.

Figure description

The figure shows the distribution of the forward at expiry (2 years from now), with the rate in % on the horizontal axis. The Bachelier model is a symmetric normal distribution centered on 1.00%, with a standard deviation of 0.30% × √2 ≈ 0.42%. Its tail extends below 0%, and the area (probability) of that part is about 0.9%. The Black model is a lognormal distribution with the same mean of 1.00% and a standard deviation of the logarithm of 0.30 × √2 ≈ 0.42. It takes no value at or below 0%. Its peak is at about 0.76%, to the left of 1.00%, and the right tail is longer. You can see that in the Black model, upward moves are larger.

Pricing formulas

Both models find the expected payoff at expiry under the model's distribution and multiply it by the discount factor to the payment date to get today's price⁵. In the formulas, $F$ is the forward and $K$ is the strike price (the strike rate for interest rate options). Here, $T$ is the number of years to expiry, and $P$ is the discount factor to the payment date. Also, $\Phi$ is the cumulative distribution function of the standard normal distribution, and $\varphi$ is its probability density function.

In the Black model, the price $C$ of a call option (call), which pays the amount by which the forward is above the strike at expiry, is given by the following formulas⁶.

$$C = P \left[ F \Phi(d_1) - K \Phi(d_2) \right]$$ $$d_1 = \frac{\ln(F/K) + \sigma_B^2 T/2}{\sigma_B \sqrt{T}}$$ $$d_2 = d_1 - \sigma_B \sqrt{T}$$

In the Bachelier model, the price of a call is given by the following formulas⁶.

$$C = P \left[ (F - K) \Phi(d) + \sigma_N \sqrt{T} \varphi(d) \right]$$ $$d = \frac{F - K}{\sigma_N \sqrt{T}}$$

The price of a put option (put), which pays the amount by which the forward is below the strike, follows from the relationship that the call price minus the put price equals $P(F - K)$ (put-call parity). Written with the discount factor $P$, it is $P[K\Phi(-d_2) - F\Phi(-d_1)]$ in the Black model and $P[(K - F)\Phi(-d) + \sigma_N\sqrt{T}\varphi(d)]$ in the Bachelier model.

The Black model formula is the call formula in The Black-Scholes Model rewritten in terms of the forward price $F$ of the underlying⁵. Today's price of the underlying, interest rates, and dividends enter the price only through the forward price $F$ and the discount factor $P$. This means you can get the price from the forward and the discount factor seen in the market, without separate assumptions about rates or dividends.

What goes into $F$ and $P$ depends on the product. For JGB futures options, $F$ is the price of JGB futures, and $P$ is the discount factor to expiry, $e^{-rT}$⁷. $r$ is the interest rate to expiry. For a caplet, you multiply by the notional amount and the length of the period, and use the discount factor to the interest payment date⁸. For a swaption, instead of $P$, you multiply by the sum of the discount factors for each payment date of the underlying swap (the annuity)⁶.

Worked example: an option on a rate set in 2 years

Consider a hypothetical example in which a call has a notional amount of 10 billion yen. It pays, in 3 years, the amount by which a 1-year rate set in 2 years is above the strike rate. This trade is equivalent to a single caplet⁸. The forward rate is 1.00%, the discount factor to year 3 is 0.97, and the expiry is 2 years. The Black model volatility is 30%, and the normal volatility is 30bp (1.00% × 30%). The amount received is 10 billion yen × 1 year × (the rate set − the strike rate), and the price is found by putting 10 billion yen × 0.97 into $P$ in the formula.

The Black model gives the following⁹.

The Bachelier model gives the following⁹.

With a strike rate of 1.00%, the same as the forward rate, the two prices differ by about 120,000 yen, about 0.8% of the Black model price. When the strike rate is raised to 1.50%, the Black model price of 4,502,066 yen is close to twice the Bachelier model price of 2,412,440 yen. Because moves in the Black model grow as the rate rises, it puts a higher probability than the Bachelier model on the rate reaching a level above the forward rate. Under either model, the buyer receives the difference if the rate set in 2 years is above the strike rate. Otherwise the buyer receives nothing and loses the price paid. The seller's profit and loss is the reverse.

Volatility units and how to convert between them

In practice, the two volatilities are linked by the following approximation¹⁰.

$$\sigma_N \approx F \sigma_B$$

If the forward is 1.00% and the Black model volatility is 30%, the normal volatility is about 30bp. This formula is an approximation. Even when the strike equals the forward (at the money), the gap grows with a longer expiry and a higher volatility. At the money, $F\sigma_B(1 - \sigma_B^2 T/24)$ is a better approximation¹¹. Under the conditions of the worked example, this gives 30bp × (1 − 0.09 × 2 ÷ 24) = 29.78bp, which matches the normal volatility of 29.78bp that makes the two model prices equal¹².

Deep diveDeriving the at-the-money approximation

When $F = K$, the Black model has $d_1 = \sigma_B\sqrt{T}/2$ and $d_2 = -d_1$, and the call price is $PF[2\Phi(\sigma_B\sqrt{T}/2) - 1]$. The Bachelier model has $d = 0$, and the price is $P\sigma_N\sqrt{T}\varphi(0) = P\sigma_N\sqrt{T}/\sqrt{2\pi}$.

Expanding $\Phi(x)$ around $x = 0$ as $\Phi(x) \approx 1/2 + (x - x^3/6)/\sqrt{2\pi}$ and putting in $x = \sigma_B\sqrt{T}/2$, the Black model price is close to the following value.

$$\frac{PF\sigma_B\sqrt{T}}{\sqrt{2\pi}} \left( 1 - \frac{\sigma_B^2 T}{24} \right)$$

Setting this equal to the Bachelier model price gives $\sigma_N \approx F\sigma_B(1 - \sigma_B^2 T/24)$. If $\sigma_B^2 T$ is small, the term in parentheses is close to 1, and $\sigma_N \approx F\sigma_B$.

The gap widens further as the strike moves away from the forward. Under the conditions of the worked example (forward 1.00%, 2 years, Black model volatility 30%), the normal volatility that gives the same price as the Black model at each strike rate is as follows¹².

If the Black model volatility is 30% at every strike rate, the same prices expressed as normal volatility rise with the strike rate. So if you hold the volatility constant across strike rates in one model, the volatility in the other model varies with the strike rate.

The level of the forward also enters the conversion. Even if the Black model volatility stays the same, normal volatility falls when rates fall. The two volatilities broadly move alike, but there are many periods when they do not move together¹³. As the forward approaches 0, the Black model volatility that corresponds to a given normal volatility grows, roughly as $\sigma_N / F$.

You can see the difference between the two models in the simulator below. The initial values are a payer swaption (the right to become the party that pays the fixed rate) with an expiry of 2 years on a swap with a term of 3 years (2-year × 3-year). The curve is the simulator's default curve (market rates from 0.5% overnight to 1.5% at 10 years) with every rate lowered by 0.5 percentage points, and the forward swap rate is 0.84%. At a strike rate of 0.85%, close to at the money, the Black model volatility of 30% and the normal volatility of 25bp (0.84% × 30%) are set to match. The prices are almost the same: 39,841,337 yen under the Black model and 39,631,286 yen under the Bachelier model¹⁴. Raising the strike rate to 1.25% widens the gap, to 11,652,027 yen under the Black model and 6,239,954 yen under the Bachelier model.

Next, keep the volatilities unchanged, lower the curve by 1 percentage point, and set the strike rate to 0.35%, close to the forward swap rate (0.33%). The Black model then gives 15,041,083 yen and the Bachelier model 40,003,497 yen. A Black model volatility of 30% corresponds to only about 10bp of normal volatility, so the Black model price falls sharply. If you lower the curve by 1.5 percentage points so that the forward swap rate is at or below zero, the Black model price is shown as not available.

Negative rates and shifted Black

The Black model formula contains $\ln(F/K)$, so if the forward or the strike is at or below zero, the logarithm cannot be taken and the price cannot be calculated⁶. Under negative rates, information vendors stopped providing Black model volatilities for swaptions, and the Bachelier model came into wide use¹⁵.

The other approach is to add a fixed amount $s$ (the shift) to both the forward and the strike to make them positive, and then use the Black model. This model replaces the movement of the forward with the following formula, and is called the shifted lognormal model (shifted Black)¹⁶.

$$dF = \sigma_B (F + s) \, dW$$

In this model, $F + s$ follows a lognormal distribution, so the price is found by replacing $F$ with $F + s$ and $K$ with $K + s$ in the Black model formula. The forward can only take values above $-s$, and, as in the Black model, the distribution has a longer tail on the upside. For example, take a forward of −0.10%, a strike rate of 0.00%, and a shift of 1%, with the same 10 billion yen, discount factor of 0.97, 2-year expiry, and 30% volatility as in the worked example. Putting 0.90% and 1.00% into the Black model formula gives a price of 11,098,131 yen¹⁷. The simulator above cannot produce this price.

Deep diveDeriving the shifted Black pricing formula

Let $X = F + s$. Because $s$ is constant, $dX = dF = \sigma_B X\,dW$, so $X$ moves in the same way as the forward in the Black model. The payoff at expiry is $\max(F - K, 0) = \max(X - (K + s), 0)$, the same as a call with forward $X$ and strike $K + s$. So the price is the value of the Black model formula with $F + s$ and $K + s$ put in.

The problem with shifted Black is how large a shift to use. The shift must be large enough to make the forward and the strike positive, but how deeply negative rates go changes over time. As of 2021, the shift depends largely on market convention, and the author's impression is that round values such as 1% or 2% are used for yen rates¹⁶. For the same price, a different shift gives a different volatility.

The Bachelier model, on the other hand, treats rates as equally likely to rise or fall even when they are negative. If you consider this unrealistic, shifted Black is an alternative. Another reason shifted Black is used is that it is easy to support in systems that have long used the Black model¹⁶.

Which model each market uses

The conventions below are as of the dates of the sources: 2022 for JGB futures options and 2021 for swaptions.

For JGB futures options, the market convention is for participants to use the Black model (the Black 76 model) as the standard model for calculating implied volatility. The reasons include that it is close to the Black-Scholes model that practitioners are familiar with, and that JGB futures options have short expiries¹⁸. The Black-Scholes framework is applied to options on stock index, stock, and currency futures, and to some commodity options. It has also been used to quote the prices of caplets and swaptions as implied volatilities, even though their underlyings can hardly be said to follow geometric Brownian motion¹⁹. As shown in "Pricing formulas," the Black model is this framework written in terms of forward or futures prices⁵¹.

For swaptions, before negative rates, implied volatility was almost always calculated with the Black model. Swaptions after the period of negative rates are quoted with both the Bachelier model and shifted Black¹⁵. Partly because data on shifted Black volatilities is hard to obtain, the author of the source gives the impression that academics and practitioners mostly use the Bachelier model when they analyze implied volatility²⁰.

When option prices are expressed as implied volatilities, the values differ by model, and if firms use different models, the values are hard to compare¹⁸.

Assumptions and limitations

Both the Black model and the Bachelier model assume that volatility is constant regardless of the strike. In actual markets, there is a volatility smile, in which implied volatility rises as the strike moves away from at the money²². A single volatility cannot explain the prices of options with different strikes all at once. The shape of volatility across strikes, and the models that handle it, are covered in Volatility Smiles and Surfaces.

Neither model is used because it is the correct one. In part, they are used as conversion formulas that turn prices into volatilities for easier comparison, and whether the resulting volatility is plausible depends on whether the model's assumptions fit reality²³. The Black model treats moves as small when rates are low, and cannot handle rates at or below zero. The Bachelier model assumes that rates move by the same amount whatever their level, and puts probability on rates however low they go.

Notes

1. 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; hereafter Hattori (2021)), Sections 2.1 and 2.2 (pages 56–58), formulas (1) and (2). The source explains the models with swaptions, and the unit bp is based on Figure 2 on page 59. ↩ ↩² ↩³ ↩⁴ ↩⁵

2. Hattori (2021), Section 2.1 (page 57; the probabilities of doubling and halving) and Section 2.2 (page 58; rates are equally likely to rise or fall). ↩

3. Hattori (2021), page 58 and note 5 on page 58. ↩

4. Martin Haugh, "Generating Random Variables and Stochastic Processes" (Columbia University, 2017), Definition 1 in Section 7 (page 17). $dW$ is the increment of standard Brownian motion over the period $dt$. ↩

5. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), equation (31) in Section 7 (pages 16–17). $P$ on this page corresponds to the source's $e^{-r(T-t)}$, with $t = 0$. ↩ ↩² ↩³

6. Hattori (2021), Section 3.2 (pages 60–61), formulas (4) and (5) and note 12. The source's formulas are for a payer swaption, and its annuity $A$ corresponds to $P$ and its cumulative distribution function $N$ to $\Phi$ on this page. ↩ ↩² ↩³ ↩⁴

7. Hattori (2021), page 61, the paragraph after formula (5). It says that JGB futures options correspond to the case $A = e^{-rT}$. ↩

8. Martin Haugh, "Market Models" (Term Structure Models: IEOR E4710, 2010), page 2, formula (3) under "Black's Formula for Caplets." The source's formula is per unit of notional and multiplies by $\delta$ and the discount factor to the payment date. ↩ ↩²

9. Step 7 (step 5 for the Bachelier model) and the prices are calculated without rounding intermediate values, using the same calculation code as the rate-options simulator (src/lib/simulators/rate-options.ts). The values of $\Phi$ and $\varphi$ are shown rounded to 6 decimal places, so recalculating from the values in the table shifts the last digit of step 7 (step 5) slightly. Prices are rounded to the nearest yen. ↩ ↩²

10. Hattori (2021), Section 3.1, formula (3) (pages 59–60). The document says practitioners gauge the difference between the two volatilities with this approximation, and cautions that it is only an approximation. ↩

11. Hattori (2021), note 10 on page 60. The document presents it as a better approximation introduced by Corb (2012). The document's $\sigma_{LN}$ corresponds to $\sigma_B$ on this page. ↩

12. The normal volatility at which the Bachelier model formula gives the same price as the Black model, found by bisection using the same calculation code as the rate-options simulator (src/lib/simulators/rate-options.ts) and rounded to 2 decimal places. ↩ ↩²

13. Hattori (2021), Section 3.1 (page 59). ↩

14. The amounts are the results of the rate-options simulator, rounded to the nearest yen. The forward swap rate is 0.8377%, and 25bp is 0.8377% × 30% ≈ 25.1bp, rounded. ↩

15. Hattori (2021), Section 2.1 (page 57). The use of both models is in Section 2.4 (page 59). ↩ ↩²

16. Hattori (2021), Section 2.4 (pages 58–59). The document's "shift" corresponds to $s$ on this page. ↩ ↩² ↩³

17. Calculated by putting 0.90% and 1.00% into the Black model function of the same calculation code as the rate-options simulator (src/lib/simulators/rate-options.ts), and rounded to the nearest yen. ↩

18. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, 2022, "BOX 1 ブラック76 モデル（ブラック・モデル）" (Box 1: The Black 76 model (Black model)), page 7 (page 8 of the PDF). ↩ ↩²

19. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), Section 8, "Extensions of Black-Scholes" (page 17). ↩

20. Hattori (2021), Section 2.4 (page 59) and note 7. ↩ ↩²

21. Hattori (2021), Section 1 (page 56). ↩

22. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, 2022, page 15 (page 16 of the PDF), the explanation of Figure 7. ↩

23. Hattori (2021), Section 2.3 (page 58). ↩