The Black-Scholes Model

In option pricing work, prices are sometimes quoted in terms of implied volatility (the volatility backed out of market prices). The Black-Scholes model is used as the framework for this way of quoting¹.

What the model assumes and what it calculates

The Black-Scholes model gives the theoretical price of a European option (an option that can be exercised only on the expiry date)² with a single formula based on five inputs. The inputs are the price of the underlying asset $S$, the strike price $K$, the time to expiry $T$, the risk-free rate $r$, and the volatility $\sigma$³. The risk-free rate is the rate for investing and borrowing with no credit risk or similar risks⁴. Of the five inputs, only the volatility cannot be observed directly in the market⁵.

Consider a hypothetical example in which Company D, which pays no dividends, has a call option (call) on its shares. With an underlying price (share price) of 100, a strike price of 100, a time to expiry of 1 year, an interest rate of 1%, and a volatility of 20%, the call has a theoretical price of 8.43. This is the theoretical value of the option price (premium) that the buyer pays the seller⁶.

The formula holds under the following assumptions⁷.

• The underlying price follows geometric Brownian motion. Returns over short periods follow a normal distribution, and the price at expiry follows a lognormal distribution.
• The volatility and the risk-free rate stay constant until expiry.
• The underlying can be traded continuously and sold short, with no transaction costs.
• There is no opportunity to make a certain profit through arbitrage.
• The underlying pays no dividends.

Geometric Brownian motion is a stochastic process in which the logarithm of the price follows a Brownian motion⁸. Its properties and its relation to the lognormal distribution are covered in Random Walks and Brownian Motion.

The Black-Scholes formula was presented by Fischer Black and Myron Scholes in a 1973 paper. The paper is titled "The Pricing of Options and Corporate Liabilities" and appeared in the Journal of Political Economy⁹. In the same year, Robert Merton derived the formula by another method and extended it. For this reason, the model is also called the Black-Scholes-Merton model⁹¹⁰.

Formulas for the call and put prices

The price of a call $C$ is the underlying price times $N(d_1)$, minus the present value of the strike price times $N(d_2)$¹¹. The interest rate $r$ is continuously compounded, $e^{-rT}$ is the discount factor for the time to expiry $T$, and $Ke^{-rT}$ is the present value of the strike price.

$$C = S\,N(d_1) - Ke^{-rT} N(d_2)$$

$d_1$ and $d_2$ are given by the following formulas. $\ln$ is the natural logarithm, and $\sigma\sqrt{T}$ is the standard deviation of the log return up to expiry.

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$

$N(\cdot)$ is the cumulative distribution function of the standard normal distribution, and $N(d)$ is the probability that a value drawn from a normal distribution with mean 0 and standard deviation 1 is below $d$¹². The price of a put option (put) $P$ uses the same $d_1$ and $d_2$ in the following formula¹³.

$$P = Ke^{-rT} N(-d_2) - S\,N(-d_1)$$

$N(d_1)$ is the delta of the call (the ratio of the change in the call price to a small change in the underlying price). It is also the amount of the underlying to hold to produce the same changes in value as the call¹⁴. Splitting the formula into its two terms, you can read the call as a combination of two positions. One is "buying $N(d_1)$ units of the underlying," and the other is "borrowing $Ke^{-rT}N(d_2)$." The call and put prices are linked by put-call parity, and either formula can be derived from the other¹⁵.

Worked example: an option on Company D's shares

Take the Company D example. The underlying price is 100, the strike price is 100, the time to expiry is 1 year, the continuously compounded interest rate is 1%, the volatility is 20%, and there are no dividends. Following the formulas step by step, the call is worth 8.43 and the put 7.44¹⁶.

The put uses $N(-d_1) = 1 - N(d_1) = 0.44038231$ and $N(-d_2) = 1 - N(d_2) = 0.51993881$. The present value of the strike price, 99.004983, times 0.51993881 is 51.4765. The underlying price of 100 times 0.44038231 is 44.0382. Subtracting the second from the first gives a put price of 7.4383.

The difference between the call and the put, 0.9950 (8.4333 − 7.4383), equals the underlying price of 100 minus the present value of the strike price, 99.0050. For an underlying that pays no dividends, this relation holds regardless of the formula.

How the price changes with the inputs

The price of a call rises when the underlying price, the volatility, or the interest rate rises or the time to expiry is longer, and falls when the strike price rises¹⁷. In this example, the put falls when the underlying price or the interest rate rises, and rises when the volatility rises. The table shows the prices when one input at a time is changed from the Company D example¹⁸.

As the time-to-expiry rows of the table show, the put is also worth more with a longer time to expiry in this example, but the price of a European put does not always rise with the time to expiry¹⁹.

Plotting the call and put prices of the same example against the underlying price gives Figure 1.

Figure description

The figure plots the call and put prices of the Company D example (strike price 100, time to expiry 1 year, interest rate 1%, volatility 20%) for underlying prices from 70 to 130. The call slopes upward and the put slopes downward. At the two ends, the call is worth 0.28 at 70 and 31.90 at 130, and the put 29.28 at 70 and 0.90 at 130. The two curves cross where the underlying price equals the present value of the strike price, 99.00, and above that point the call is worth more than the put. The vertical line marks the strike price of 100, and the crossing point lies a little to its left. The figure shows that when the underlying price moves, the call and put prices move in opposite directions.

When the underlying price is 100, the intrinsic value of the call (the payoff if exercised now)²⁰ is 0, and the entire price of 8.43 is time value. If the underlying price is 110, the intrinsic value is 10 and the time value is 4.94 (14.94 − 10). If it is 90, the intrinsic value is 0 and the time value falls to 3.86. The ratios of the change in the price to small changes in the inputs are called the Greeks, and they include delta and vega.

Dividend yields and currency options

When the underlying pays dividends at a constant dividend yield $q$ (continuously compounded), multiply the underlying price by $e^{-qT}$ and replace the interest rate in $d_1$ with $r - q$²¹.

$$C = Se^{-qT} N(d_1) - Ke^{-rT} N(d_2)$$ $$d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}$$

$d_2$ is $d_1 - \sigma\sqrt{T}$, as without dividends, and the put is $P = Ke^{-rT}N(-d_2) - Se^{-qT}N(-d_1)$¹³. With a dividend yield of 2% in the Company D example, $d_1 = 0.05$ and $d_2 = -0.15$. The call falls from 8.43 to 7.36, and the put rises from 7.44 to 8.35²². This is because the dividends paid on the underlying lower its forward price.

Rewriting this form with the forward price of the underlying, $F = Se^{(r-q)T}$, gives a call price of $e^{-rT}\{F N(d_1) - K N(d_2)\}$. The current price of the underlying affects the option price only through the forward price. The formula written in terms of the forward price is called the Black model formula²³. The Black-Scholes model can also be applied to currency options¹. The formula then has the foreign interest rate in place of the dividend yield.

Deriving the formula from replication

The Black-Scholes formula follows from the idea of combining the underlying with risk-free investing and borrowing, and continuously rebalancing the mix to produce the same changes in value as the option (replicating it)²⁵. If the amount of the underlying in the combination matches the slope of the call price with respect to the underlying price (delta), the gains and losses from moves in the underlying offset each other over a short time. The slope changes as the underlying price and time move, so the combination has to be rebalanced continuously²⁶. The idea of replication is covered in No-Arbitrage and Replication.

The formula derived this way does not contain the expected return of the underlying (how fast investors expect it to rise in value)²⁷. Setting the expected return equal to the risk-free rate gives the same formula, so this way of pricing is called risk-neutral valuation. The Black-Scholes formula can also be obtained as the limit of the binomial model as the step length shrinks to zero²⁸.

Deep diveThe Black-Scholes partial differential equation

Suppose the underlying price follows the geometric Brownian motion $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$, where $\mu$ is the expected return and $W_t$ is a standard Brownian motion. Use Ito's lemma to express how the call price $C(S, t)$ moves. Next, write how a combination of $\partial C/\partial S$ units of the underlying and cash (growing at the risk-free rate) moves, with no money added or withdrawn along the way. Setting the random terms of the two equal, and the non-random terms equal, gives the following partial differential equation²⁵.

$$\frac{\partial C}{\partial t} + rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} - rC = 0$$

Solving this equation with the condition at expiry $C(S, T) = \max(S - K, 0)$ gives the call formula in the section "Formulas for the call and put prices." Because $\mu$ does not appear in the equation, the expected return does not enter the price.

Where the assumptions differ from the market

The assumptions of the Black-Scholes model differ from real markets in several ways. As a result, a formula with a single volatility cannot explain all the market prices of options with different strike prices and expiries at once.

The first is how prices move. Real share prices and exchange rates sometimes jump sharply, and the distribution of returns has fatter tails than a normal distribution²⁹. A paper by the Institute for Monetary and Economic Studies of the Bank of Japan shows the kurtosis of daily returns for the yen-dollar exchange rate (August 1998 to July 1999) and the Nikkei Stock Average (October 2000 to September 2001). The kurtosis was 12.01 and 4.93, respectively, both above the 3 of a normal distribution³⁰.

The second is that volatility is not constant. If the model were correct, the volatility backed out of market prices (implied volatility) would be the same regardless of strike price or expiry. In practice, it takes different values by strike price and expiry, and its shape also changes over time³¹. The relation to the strike price is called the volatility smile.

The third is the rebalancing needed for replication. In practice, the combination cannot be rebalanced continuously and is rebalanced at intervals, so replication leaves an error³². The 1973 formula was also derived under ideal conditions that ignore transaction costs, taxes, and the possibility that the counterparty defaults³³.

The fourth is the exercise style. The formula prices European options, and early exercise of American options, which can be exercised before expiry, can be handled with a binomial tree²⁴. According to Columbia University lecture notes from 2013, options on individual stocks are generally American options, and put-call parity does not apply to American options³⁴.

To capture these differences, models have been developed such as jump-diffusion models, which add price jumps, and models in which volatility moves randomly¹. Even so, this framework is still used to quote prices in terms of implied volatility, including for products the model does not fit well¹.

Notes

1. Martin Haugh, "Black-Scholes and the Volatility Surface," Section 8 (page 17). ↩ ↩² ↩³ ↩⁴

2. Martin Haugh, "Forwards, Swaps, Futures and Options" (Columbia University, 2016), Definition 2 in Section 4 (page 11). ↩

3. Steven Skiena, "Lecture 12: The Black-Scholes Model" (Stony Brook University), "The Black-Scholes-Merton Model" (slide 2). The source lists six inputs, adding the dividend yield. ↩

4. Martin Haugh, "Forwards, Swaps, Futures and Options," Section 4.2 (page 12), and 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング" (Pricing financial instruments that reflect credit risk), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (March 1999), "（２）割引レート算定のためのリスクプレミアムの推定" (Estimating the risk premium for calculating the discount rate), page 71 (in Japanese). ↩

5. Steven Skiena, "Lecture 12: The Black-Scholes Model," "Impact of Volatility" (slide 12). ↩

6. Osaka Exchange, "理論" (Theory), Kitahama Investment School (in Japanese), "ブラック-ショールズ・モデル" (Black-Scholes model). The source describes the model as a representative model used to calculate premiums. ↩

7. Skiena, "Lecture 12," "Assumptions of the Model" (slide 14) and "Risk Neutral Valuation" (slide 15). The assumption of a stock that pays no dividends is from Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), Section 1 (page 1). ↩

8. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (1) in Section 1 (page 1) and the formula for the price at expiry $S_T$ in Section 2 (page 3). ↩

9. The Royal Swedish Academy of Sciences, "The Prize in Economic Sciences 1997 - Press release" (October 14, 1997), the main text and "Further reading." The papers are Black and Scholes (1973, Journal of Political Economy, vol. 81, pages 637–654) and Merton (1973, vol. 4, pages 141–183). ↩ ↩²

10. 久田祥史 (Yoshifumi Hisata), "ジャンプ拡散過程を用いたオプション価格付けモデルについて" (On option pricing models using jump-diffusion processes), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2003), "１．はじめに" (1. Introduction), page 52 (in Japanese). ↩

11. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (9) in Section 1 (page 2). The source writes $N(\cdot)$ as $\Phi(\cdot)$ and the time to expiry as $T - t$. This page sets $t = 0$ and writes the source's $\log$ as $\ln$. ↩

12. Steven Skiena, "Lecture 12: The Black-Scholes Model," "Formulae Details" (slide 4). ↩

13. Steven Skiena, "Lecture 12: The Black-Scholes Model," "The Pricing Formula" (slide 3) and "Computing the Call and Put Price" (slide 7). The source writes the dividend yield as $y$. The formula without dividends is the form with $y = 0$. ↩ ↩²

14. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (6) in Section 1 (page 2) and the formula for delta in Section 4 (page 8). The amount borrowed is calculated by subtracting $S\,N(d_1)$ from the call formula. ↩

15. Martin Haugh, "Black-Scholes and the Volatility Surface," Section 1 (page 2) and equation (21) in Section 4 (page 7). ↩

16. Values calculated with the functions in src/lib/simulators/black-scholes.ts. In the table, $N(\cdot)$ and the discount factor are rounded to eight decimal places, the present value of the strike price to six, and the other amounts to four. Recalculating from the values in the table also gives 8.4333 for the call and 7.4383 for the put. ↩

17. The Royal Swedish Academy of Sciences, "The Prize in Economic Sciences 1997 - Press release" (October 14, 1997), "The Black-Scholes formula." ↩

18. Values calculated with the functions in src/lib/simulators/black-scholes.ts and rounded to two decimal places. ↩

19. Steven Skiena, "Lecture 12: The Black-Scholes Model," "Impact of Expiration Date" (slide 11). ↩

20. Martin Haugh, "Forwards, Swaps, Futures and Options," Section 4 (page 11). ↩

21. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (13) under "Dividends" in Section 2 (page 3). ↩

22. $d_1 = (0 + (0.01 - 0.02 + 0.02) \times 1) \div 0.2 = 0.05$. The prices are calculated with the functions in src/lib/simulators/black-scholes.ts and rounded to two decimal places. ↩

23. Martin Haugh, "Black-Scholes and the Volatility Surface," equation (31) in Section 7 (pages 16–17). ↩

24. Martin Haugh, "Black-Scholes and the Volatility Surface," "Dividends and Option Pricing" in Section 7 (page 17). ↩ ↩² ↩³

25. Martin Haugh, "Black-Scholes and the Volatility Surface," equations (1) to (9) in Section 1 (pages 1–2). ↩ ↩²

26. The Royal Swedish Academy of Sciences, "The Prize in Economic Sciences 1997 - Press release," "The method." ↩

27. Martin Haugh, "Black-Scholes and the Volatility Surface," after equation (8) in Section 1 (page 2). ↩

28. Steven Skiena, "Lecture 12: The Black-Scholes Model," "Where Does the Black-Scholes Formula Come From?" (slide 16). ↩

29. 久田祥史 (Yoshifumi Hisata), "ジャンプ拡散過程を用いたオプション価格付けモデルについて," "２．ブラック＝ショールズ･モデルが捉え切れない市場の特徴" (2. Features of the market that the Black-Scholes model does not fully capture), pages 53–54 (in Japanese), and Skiena, "Lecture 12," "Limitations of the Black-Scholes Model" (slide 21). ↩

30. 久田祥史 (Yoshifumi Hisata), "ジャンプ拡散過程を用いたオプション価格付けモデルについて," "図表 2" (Figure 2), page 54 (in Japanese). ↩

31. Martin Haugh, "Black-Scholes and the Volatility Surface," Section 3 (page 4). ↩

32. Martin Haugh, "Black-Scholes and the Volatility Surface," Section 5 (page 12). ↩

33. 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング," "（１）プライシング理論の枠組み" (Framework of pricing theory) in "３．" (3.), page 77 (in Japanese). ↩

34. Martin Haugh, "Black-Scholes and the Volatility Surface," Section 3 (page 5). ↩

35. Martin Haugh, "Black-Scholes and the Volatility Surface," "Arbitrage Constraints on the Volatility Surface" in Section 3 (page 5). ↩