Black-Scholes Explained: Inputs, a Worked Calculation, Put-Call Parity, and Its Limits

The Black-Scholes model is a formula that calculates a theoretical price for a European-style option from five inputs: the stock price, the strike price, the time to expiry, the risk-free interest rate, and the stock’s volatility. Published in 1973, it is still the reference point for how options are quoted and hedged, even though nearly everyone who uses it knows its assumptions are not literally true.

In short: Black-Scholes says an option’s fair value is the cost of a portfolio of stock and cash that would exactly replicate the option’s payoff, if you could trade continuously and the stock’s volatility were constant. Four of its inputs are observable; the fifth, volatility, is not, so in practice traders run the model in reverse to get implied volatility from market prices. For a hypothetical $50 stock, a three-month $50 call at 30% volatility and a 4% rate is worth about $3.23, and the matching put about $2.73. The gap between them is fixed by put-call parity. The model’s main weaknesses are constant volatility, no sudden jumps, and its European-exercise design.

A short history

Fischer Black and Myron Scholes published “The Pricing of Options and Corporate Liabilities” in the Journal of Political Economy in 1973. Robert Merton, working in parallel, published “Theory of Rational Option Pricing” in the Bell Journal of Economics and Management Science the same year and extended the framework. The timing was notable: the Chicago Board Options Exchange opened for listed stock option trading in April 1973. In 1997 Scholes and Merton received the Nobel Memorial Prize in Economic Sciences for the work; Black had died in 1995 and the prize is not awarded posthumously.

The core idea: replication

Black and Scholes showed that, under their assumptions, you can combine shares of the stock and borrowing or lending at the risk-free rate so that the portfolio’s value moves exactly like the option over the next instant. Adjust the mix continuously and the portfolio replicates the option all the way to expiry. If two things always have the same payoff, they must cost the same, or someone could profit risk-free by buying the cheaper one and selling the dearer one. So the option’s price equals the cost of the replicating portfolio.

A remarkable consequence: the stock’s expected return does not appear in the formula. Two investors who disagree about where the stock is heading, but agree on its volatility, should agree on the option’s value. The number of shares in the replicating portfolio is the option’s delta, which is why delta is also called the hedge ratio (see delta explained).

The five inputs

Input	Symbol	Observable?	Effect on call price if it rises	Effect on put price if it rises
Stock price	S	yes	up	down
Strike price	K	yes, fixed by the contract	down	up
Time to expiry (years)	T	yes	usually up	usually up
Risk-free rate	r	yes, roughly	up	down
Volatility	sigma	no	up	up

Dividends are a sixth input in the common extension of the model. A known dividend yield is handled by reducing the stock price by the present value of expected dividends, which lowers call values and raises put values.

The formula

For a European call (C) and put (P) on a stock that pays no dividends:

C = S x N(d1) - K x e^(-rT) x N(d2)
P = K x e^(-rT) x N(-d2) - S x N(-d1)

d1 = [ ln(S / K) + (r + sigma^2 / 2) x T ] / (sigma x sqrt(T))
d2 = d1 - sigma x sqrt(T)

N(x) is the cumulative standard normal distribution: the probability that a standard normal variable is below x. It turns d1 and d2 into numbers between 0 and 1.
e^(-rT) discounts the strike back to today: the strike is paid at expiry, so its present value is a bit less.
N(d1) is the call’s delta.
N(d2) is the model’s risk-neutral probability that the call expires in the money.

Read the call formula as: (shares held, valued at today’s price) minus (the present value of the strike, weighted by the chance you’ll pay it).

A worked calculation

Hypothetical Maple Widgets Inc. trades at $50. We price a call and a put with:

Strike K = $50
Time to expiry T = 0.25 years (three months)
Risk-free rate r = 4% (0.04)
Volatility sigma = 30% (0.30)
No dividends

Step 1: compute d1 and d2

ln(S / K) = ln(50 / 50) = 0
(r + sigma^2 / 2) x T = (0.04 + 0.09 / 2) x 0.25 = 0.085 x 0.25 = 0.02125
sigma x sqrt(T) = 0.30 x 0.5 = 0.15
d1 = (0 + 0.02125) / 0.15 = 0.1417
d2 = 0.1417 minus 0.15 = minus 0.0083

Step 2: look up the normal probabilities

N(d1) = N(0.1417) = 0.5563
N(d2) = N(minus 0.0083) = 0.4967

Step 3: discount the strike

K x e^(-rT) = 50 x e^(minus 0.01) = $49.5025

Step 4: assemble the call price

S x N(d1) = 50 x 0.5563 = $27.816
K x e^(-rT) x N(d2) = 49.5025 x 0.4967 = $24.587
C = 27.816 minus 24.587 = $3.23

Per 100-share contract, that is about $323.

Step 5: the put

N(minus d2) = 0.5033 and N(minus d1) = 0.4437
K x e^(-rT) x N(minus d2) = 49.5025 x 0.5033 = $24.916
S x N(minus d1) = 50 x 0.4437 = $22.184
P = 24.916 minus 22.184 = $2.73, or about $273 per contract

A second example: out of the money

Same inputs, but a $55 strike:

ln(50 / 55) = minus 0.0953
d1 = (minus 0.0953 + 0.02125) / 0.15 = minus 0.4937; d2 = minus 0.6437
N(d1) = 0.3107; N(d2) = 0.2599
S x N(d1) = $15.537; 55 x e^(minus 0.01) x N(d2) = 54.4527 x 0.2599 = $14.151
C = $1.39 ($139 per contract)

The model says this call has roughly a 26% risk-neutral chance of finishing in the money, and its delta is 0.31.

Put-call parity

Put-call parity is a relationship that must hold for European options with the same strike and expiry, whatever pricing model you use:

C - P = S - K x e^(-rT)

In words: a call minus a put on the same strike behaves exactly like owning the stock and owing the strike at expiry. At expiry, if the stock is above K, the call pays S minus K and the put is worthless; if below, the put costs you K minus S. Either way the combined position is worth S minus K.

Checking the numbers

Left side: C minus P = $3.2297 minus $2.7322 = $0.4975
Right side: S minus K x e^(-rT) = $50 minus $49.5025 = $0.4975

They match. For the $55 strike: the call is $1.3865 and parity gives a put of 1.3865 minus 50 + 54.4527 = $5.84, which is exactly what the put formula produces.

Why parity matters

If a call traded noticeably above its parity value relative to the put, a trader could sell the call, buy the put and the stock, borrow the present value of the strike, and lock in the difference. Market makers trade these relationships constantly, which keeps listed calls and puts tightly linked. Parity is why a call and put on the same strike show nearly the same implied volatility, and it is the arithmetic behind strategies like collars (see collars explained).

For American options, which can be exercised early, parity becomes a range rather than an exact equation, and dividends shift it further.

Greeks from the same formula

Differentiating Black-Scholes with respect to each input gives the Greeks, introduced in option Greeks explained. For the three-month $50 call above:

Greek	Value	Meaning per contract
Delta	0.556	behaves like about 56 shares
Gamma	0.053	delta rises about 0.05 per $1
Theta	minus $0.019 per day	loses about $1.90 per day
Vega	$0.099 per vol point	gains about $9.90 per point of IV
Rho	$0.062 per rate point	gains about $6.20 per point of rates

Running the model backwards: implied volatility

Because four inputs are known and the market supplies the option’s price, you can solve for the one unknown: volatility. If the three-month $50 Maple Widgets call trades at $3.50 instead of $3.23, the volatility that makes the formula produce $3.50 is about 32.7%. That is its implied volatility, covered in implied volatility explained. This is how the model is actually used: less as a price oracle and more as a translation layer that lets traders compare options across strikes, expiries, and stocks in a common unit.

The assumptions, and what they get wrong

Assumption	Reality
Volatility is constant over the option’s life and the same for every strike	Implied volatility differs by strike (the skew or smile) and by expiry, and changes daily
Stock prices follow a lognormal random walk with no jumps	Prices gap on earnings, news, and overnight; extreme moves happen more often than the normal distribution predicts
You can trade continuously, with no costs	Hedges are adjusted in discrete steps, and bid-ask spreads and commissions are real
There is a single, constant risk-free rate	Rates change and borrowing costs differ by participant
The option is European (exercise only at expiry)	Most US and Canadian equity options are American-style; see American vs European options
No dividends (in the original model)	Many stocks pay dividends, which affect both price and early exercise

The volatility smile

If Black-Scholes were exactly right, every option on the same stock and expiry would show the same implied volatility. They don’t. Since the October 1987 crash in particular, index options have consistently priced out-of-the-money puts at higher implied volatilities than at-the-money options. Traders keep using the formula but feed a different volatility into it for each strike. The smile is the market’s correction for the model’s thin-tailed assumption.

American options

For a call on a stock that pays no dividends, early exercise is never optimal (you would throw away time value), so the American call has the same value as the European one. American puts are different: deep in-the-money puts can be worth exercising early to collect the strike sooner. Pricing them needs a different method, such as a binomial tree. Using a 2,000-step binomial model with the same inputs, the American $50 put is worth about $2.77 versus $2.73 for the European put, and the $55 put about $5.95 versus $5.84. The early exercise right has value, and it grows the deeper in the money the put is.

Jumps and fat tails

The model’s normal distribution makes a one-day 20% fall in a broad index look nearly impossible. On October 19, 1987, the S&P 500 fell more than 20% in a single day. Option sellers who relied on the model’s tails were badly hurt. Extensions such as jump-diffusion and stochastic-volatility models exist, but none is a perfect description of markets.

What can go wrong when using it

Garbage in, garbage out. The output is only as good as the volatility input, and that is exactly the input nobody knows.
False precision. A model price to the cent can make an estimate feel like a fact.
Ignoring the spread. A model value of $3.23 means little if the market is $3.00 bid, $3.50 offered.
Wrong model for the contract. Using European formulas on American-style options with dividends or deep in-the-money puts understates their value.

Takeaway

Black-Scholes prices an option as the cost of replicating it with stock and cash, given five inputs. The arithmetic is short: compute d1 and d2, look up two normal probabilities, and combine them with the stock price and the discounted strike. Put-call parity ties calls and puts together independently of any model. The model’s assumptions break in well-known ways, and the market patches them by quoting implied volatility that varies by strike and expiry. It is best understood as a common language for options prices, not a forecast of what an option should be worth.