Black-Scholes Formula for a Call Option

Q: Why does the formula use the number e?

The number $e$, about 2.718, appears when interest compounds continuously. $e^{-rT}$ is the continuous discount factor that brings the strike back from the expiry date to today. A calculator gives it for any rate and time.

In one sentence: The Black-Scholes formula works out what the right to buy a stock later is worth today, using the stock price, the agreed price, the interest rate, the time left and the stock’s ups and downs.

$C = S_0 N(d_1) - K e^{-r T} N(d_2)$

Read it out loud: “C equals S 0 times N of d 1, minus K times e to the power minus r T, times N of d 2. C is the call option price, S 0 is the stock price today, K is the strike price, r is the risk-free interest rate, T is the time to expiry in years, and N is the standard normal cumulative probability. The numbers d 1 and d 2 are found first, from the formulas for d 1 and d 2.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Call option and strike price: A call option is a contract that gives its owner the right, but not the duty, to buy a stock at a fixed price, called the strike price K, on a set date. The owner pays a price for this right, and the formula calculates what that right is worth. A European option can be used only on the set date.
Natural logarithm, $\ln$: The natural logarithm, written $\ln$ , answers the question: what power of $e$ gives this number? So $\ln(1) = 0$ , because $e^{0} = 1$ . A ratio above 1 has a positive logarithm, and a ratio below 1 has a negative one.
Euler’s number $e$ and powers of $e$: The number $e$ is about 2.718. It is a fixed constant, like $\pi$ , and it appears wherever things grow or shrink continuously. $e^{x}$ is read ‘e to the power x’, and a calculator finds it. A negative power means one divided by the positive power, so $e^{-0.05} = 1 \div e^{0.05} \approx 0.9512$ .
Square root: The square root of a number is the number that, multiplied by itself, gives the original number. So $\sqrt{0.04} = 0.2$ , because $0.2 \times 0.2 = 0.04$ . The square root of 1 is 1, so $\sqrt{T} = 1$ when T is one year.
Volatility, $\sigma$: Volatility, written with the Greek letter sigma, $\sigma$ , is the yearly size of a stock’s ups and downs, a standard deviation of its returns. It is written as a decimal per year, so 0.2 means about 20% a year. Higher volatility means a wider range of possible future prices.
The standard normal distribution and $N(x)$: The normal distribution is the bell-shaped spread of many natural and financial quantities, with most values near the middle. The standard normal has its middle at 0 and its spread set to 1. $N(x)$ is its cumulative probability: the chance that a standard normal value is below x. $N(0) = 0.5$ , because half of the bell lies below 0.

What each symbol means

Symbol	Meaning
$C$	The value of the call option today, in dollars per share. The formula gives this number. It is the model’s value, not a quoted market price.
$S_0$	The stock price today, in dollars per share. The 0 is a label meaning ‘at the start’, the price now.
$K$	The strike price, the fixed price written into the contract, in dollars per share. The owner may buy the stock at K on the set date.
$r$	The risk-free interest rate per year, as a decimal. It is the return on a very safe asset. A value of 0.05 means 5% a year.
$T$	The time to expiry in years, the time left until the option’s set date. T = 1 means one year.
$e$	Euler’s number, about 2.718. In $e^{-rT}$ it discounts a payment due in T years back to today, at a continuous rate r.
$\sigma$	The volatility of the stock, the yearly size of its ups and downs, as a decimal. A value of 0.2 means 20% a year. It is used in the formulas for $d_1$ and $d_2$ .
$N(\cdot)$	The standard normal cumulative probability, a function that turns a number into a chance between 0 and 1. $N(x)$ is the chance that a standard bell-curve value is less than x.
$d_1$	An intermediate number, with no unit, that compares the stock price with the strike, adjusted for interest and drift, in units of the typical swing over the time. It is calculated as $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2) T}{\sigma \sqrt{T}}$ .
$d_2$	A second intermediate number, with no unit, equal to $d_1$ minus the volatility over the time: $d_2 = d_1 - \sigma \sqrt{T}$ .

The formula piece by piece

$S_0 N(d_1)$: The stock part. The stock price today, multiplied by $N(d_1)$ , which reflects how closely the option moves with the stock.
$K e^{-r T} N(d_2)$: The strike part. The strike price, brought back to today by the factor $e^{-rT}$ , then weighted by $N(d_2)$ , a probability-style number.
$C = S_0 N(d_1) - K e^{-r T} N(d_2)$: The call option’s value equals the stock part minus the strike part, both in today’s money. The inputs d_1 and d_2 are calculated first from the stock price, strike, rate, time and volatility.

What the formula is doing, and why

A European call option gives its owner the right, but not the duty, to buy a stock at the strike price $K$ on the expiry date. Its value depends on five things: the stock price $S_0$ , the strike $K$ , the interest rate $r$ , the time left $T$ and the volatility $\sigma$ . The Black-Scholes formula combines these five numbers into one value $C$ . Fischer Black and Myron Scholes published the model in 1973.

What the two big terms mean. The first term, $S_0 N(d_1)$ , is the stock part. It is the value of the shares the option lets you buy, scaled by $N(d_1)$ , a number that reflects how closely the option follows the stock. The second term, $K e^{-rT} N(d_2)$ , is the strike part. It is the money you would pay for the shares, brought back to today, and weighted by $N(d_2)$ , a probability-style number.

Why the minus sign? The option is worth the stock you would receive, less the strike you would pay, with both measured in today’s money. The stock part is added, and the strike part is taken away. Taking the strike away is what makes the call cheaper than the stock itself.

Why $e^{-rT}$ ? The strike is paid at expiry, one year away in the example, so it must be brought back to today. With positive interest, a payment due later is worth less now. The factor $e^{-rT}$ does that for interest that compounds continuously, and a higher rate makes it smaller.

Why volatility raises the price. Larger volatility means the stock can end up much higher or much lower. A call gains from the higher outcomes and does not lose more on the lower ones, because the owner can walk away. So a more volatile stock gives a more valuable call. The formula’s use of $N$ turns the distance between the stock and the strike, measured in units of volatility, into the probabilities used in the two terms.

What the formula assumes. The formula applies to European options, which are used only at expiry. It also assumes no dividends, a constant volatility, a constant interest rate and no trading costs. When these assumptions do not hold, real option prices can differ from the model’s value.

An everyday picture

Think of paying a small fee today for a reservation: the right, but not the duty, to buy a concert ticket at a set price next month. If the concert becomes popular, that right to buy at the set price becomes worth more. The Black-Scholes formula is a careful way to value that right. It weighs how far the price might move, how long you wait, and what the money you would pay later is worth today.

Worked example: Value of a 1-year call option with a stock at 100 dollars

A European call option on a made-up stock has a strike price of 100 dollars, and the stock costs 100 dollars today. The risk-free rate is 5% a year, the volatility is 20% a year, and the option runs for 1 year. In the formula’s symbols, $S_0 = 100$ , $K = 100$ , $r = 0.05$ , $\sigma = 0.2$ and $T = 1$ . These are round, illustrative numbers.

Compare the stock price with the strike. Divide the stock price by the strike to get $S_0 / K = 100 / 100 = 1$ . The natural logarithm of 1 is 0, so $\ln(S_0/K) = \ln(1) = 0$ . When the stock and the strike are equal, this first part of $d_1$ is zero.
Work out the drift part. This part is $(r + \sigma^2/2) T$ , which adds the interest rate to half the squared volatility, then multiplies by the time. Here $\sigma^2 = 0.2 \times 0.2 = 0.04$ , half of that is 0.02, and $(0.05 + 0.02) \times 1 = 0.07$ .
Find the size of the swing over the year. The denominator is $\sigma \sqrt{T} = 0.2 \times \sqrt{1} = 0.2$ , which is the typical size of the stock’s move over one year. Dividing the drift part by it gives $d_1 = \frac{0 + 0.07}{0.2} = 0.35$ .
Find d_2. Subtract the same swing from $d_1$ : $d_2 = d_1 - \sigma \sqrt{T} = 0.35 - 0.2 = 0.15$ . The second number is always a little lower than the first, by exactly one swing.
Look up N of each number. $N(0.35)$ is the chance that a standard bell-curve value is below 0.35, which is about 0.6368. $N(0.15)$ is the chance it is below 0.15, which is about 0.5596. A statistics table or a calculator gives these values.
Bring the strike back to today. The strike of 100 dollars is paid in one year, so its value today is $K e^{-rT} = 100 \times e^{-0.05} = 100 \times 0.9512 = 95.12$ dollars. The factor $e^{-0.05}$ is the discount for 5% interest over one year.
Find the stock part. Multiply the stock price by $N(d_1)$ : $S_0 N(d_1) = 100 \times 0.6368 = 63.68$ dollars. This is the stock part of the price, scaled to reflect how closely the option follows the stock.
Find the strike part. Multiply the discounted strike by $N(d_2)$ : $K e^{-rT} N(d_2) = 95.12 \times 0.5596 = 53.23$ dollars. This is what it costs, in today’s money, to pay the strike, weighted by the model’s probability.
Subtract to get the call price. The call value is the stock part minus the strike part: $C = 63.68 - 53.23 = 10.45$ dollars.

Answer: Under these made-up inputs, the call option is worth about 10.45 dollars per share. The model says the right to buy the 100 dollar stock in one year, at a strike of 100 dollars, is worth about 10.45 dollars today. This is a model value, not a price quoted by a market.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Mixing up the stock price $S_0$ and the strike $K$ . The strike is the fixed price written into the contract, and $S_0$ is the market price of the stock today.
Typing a percent as a whole number. A 5% interest rate is 0.05 in the formula, and plugging in 5 changes the answer completely.
Forgetting the square root in the volatility term. The swing over the time is $\sigma \sqrt{T}$ , not $\sigma T$ . With T = 1 the two look the same, so this error hides itself in the example, which makes it easy to repeat with other times.
Using the formula for American options or for stocks that pay dividends. The formula assumes the option is used only at expiry and that the stock pays no dividend.
Reading the call price as a forecast that the stock will rise. The formula does not use the stock’s expected growth, and it prices the option under a set of assumptions.

Where you will meet it

Pricing listed stock options on exchanges, where it is a common reference point for traders.
Working out implied volatility, the volatility that makes the formula match a market price.
Valuing employee stock options for accounting, using a model of this type.
Teaching the basic ideas of option pricing and hedging in university finance courses.

Questions people ask

Does the formula predict whether the stock will go up?

No. The formula prices the option under its assumptions. The expected growth of the stock does not appear in the final formula, which surprises many beginners. The model prices the option relative to the stock, not by forecasting the stock.

What exactly is N(d_1)?

N is the standard normal cumulative probability. $N(x)$ is the chance that a standard bell-curve value falls below x. Half of the bell lies below 0, so $N(0) = 0.5$ , and $N(0.35) \approx 0.6368$ .

Why does the formula use the number e?

The number $e$ , about 2.718, appears when interest compounds continuously. $e^{-rT}$ is the continuous discount factor that brings the strike back from the expiry date to today. A calculator gives it for any rate and time.

Put-Call Parity: Put-call parity says that a call and a put on the same stock, with the same strike and expiry, are priced so that their gap equals the stock price minus the discounted strike.
Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.
Continuous Compounding: Continuous compounding says money grows as if interest were added every instant, and the number e, about 2.718, sets the size of that growth.