Black-Scholes Model: Find Overpriced and Underpriced Options

What the Black-Scholes model computes

The Black-Scholes model gives a theoretical price for a European-style option: one that can be exercised only at expiry.

Fischer Black and Myron Scholes published it in 1973 in the Journal of Political Economy.

Robert Merton published "Theory of Rational Option Pricing" the same year and extended the model.

That is why many textbooks call it Black-Scholes-Merton (BSM). The 1997 economics prize in memory of Alfred Nobel went to Merton and Scholes "for a new method to determine the value of derivatives", per the press release of 14 October 1997.

The release says that "in 1973, Black and Scholes published what has come to be known as the Black-Scholes formula".

It adds that Merton "also generalized the formula in many directions". The release notes that Fischer Black, who helped develop the method, died in 1995.

The core idea is replication. In the model's ideal market, you can copy an option's payoff with a changing mix of stock and borrowing. If the copy costs a known amount, the option must cost the same, or there would be a riskless profit.

That is why the stock's expected return does not appear in the formula. Only volatility matters, and that is the input you cannot see directly.

The formula in plain language

With a continuous dividend yield q, the call value C and put value P are:

C = S·e^−qT·N(d1) − K·e^−rT·N(d2)
P = K·e^−rT·N(−d2) − S·e^−qT·N(−d1)
d1 = [ln(S ÷ K) + (r − q + σ² ÷ 2)·T] ÷ (σ·√T)
d2 = d1 − σ·√T

N(x) is the standard normal cumulative distribution: the chance that a normal variable is below x. Spreadsheets call it NORM.S.DIST(x, TRUE).

Read the call formula as two pieces:

• S·e^−qT·N(d1) is what you expect to receive in stock, adjusted for dividends you miss by holding the option instead of the share.
• K·e^−rT·N(d2) is the present value of the strike you expect to pay. N(d2) is the probability the call ends in the money, under the model's "risk-neutral" world.

N(d2) is not the chance of making a profit. A call can finish in the money and still lose money if it ends below strike plus premium. It is also not a real-world probability, because it uses r, not the stock's expected return.

Set q = 0 for a stock without dividends. With q = 0 you get the 1973 Black-Scholes formula; the q term is Merton's extension.

Worked example: the textbook $4.76 call

John Hull's textbook Options, Futures, and Other Derivatives uses this example in its Black-Scholes-Merton chapter (Chapter 15 in the 9th edition). A university course page that follows that chapter reproduces it with the same results.

Step 1: d1 and d2

ln(42 ÷ 40) = 0.04879
(0.10 + 0.20² ÷ 2) × 0.5 = 0.06000
σ·√T = 0.20 × 0.70711 = 0.14142
d1 = (0.04879 + 0.06000) ÷ 0.14142 = 0.7693
d2 = 0.7693 − 0.1414 = 0.6278

Step 2: the normal probabilities

N(0.7693) = 0.7791 and N(0.6278) = 0.7349. For the put you also need N(−0.7693) = 0.2209 and N(−0.6278) = 0.2651. The discounted strike is 40 × e^−0.05 = 38.0492.

Step 3: the prices

Call = 42 × 0.7791 − 38.0492 × 0.7349 = 32.72 − 27.96 = $4.76
Put = 38.0492 × 0.2651 − 42 × 0.2209 = 10.09 − 9.28 = $0.81

Both match the textbook: call $4.76, put $0.81. The call has $2.00 of intrinsic value (42 − 40) and $2.76 of time value. For a US equity contract on 100 shares, $4.76 a share is $476 a contract.

You can check these numbers, and try your own, in the Black-Scholes calculator. It also accepts a dividend yield and solves for implied volatility.

The Python used for every figure on this page is short. A cut-down version:

from math import log, sqrt, exp, erf
N = lambda x: 0.5 * (1 + erf(x / sqrt(2)))
def bs(S, K, T, r, sig, q=0.0):
  d1 = (log(S/K) + (r - q + sig**2/2)*T) / (sig*sqrt(T))
  d2 = d1 - sig*sqrt(T)
  c = S*exp(-q*T)*N(d1) - K*exp(-r*T)*N(d2)
  p = K*exp(-r*T)*N(-d2) - S*exp(-q*T)*N(-d1)
  return c, p
print(bs(42, 40, 0.5, 0.10, 0.20))  # (4.7594, 0.8086)

What a dividend yield does

Dividends lower call values and raise put values, because the share price drops when a dividend is paid. Keep every other input of the example and add a dividend yield:

Leaving out a 2% yield overstates this call by $0.32. A call that looks "expensive" against a no-dividend value may be fairly priced once dividends are in.

Put-call parity check

For European options on the same stock, strike and expiry, calls and puts are tied together. With dividend yield q:

C − P = S·e^−qT − K·e^−rT
Example: 4.7594 − 0.8086 = 3.9508
42 − 38.0492 = 3.9508 ✓

Parity does not depend on volatility, so it is a quick sanity check.

If your call and put values break it, an input is wrong. If market quotes break it by more than the spread, check for dividends, borrow costs or American early exercise before assuming a mistake.

The Greeks in one table

The Greeks measure how the value changes when one input moves.

The Options Industry Council (OIC) defines vega as "the sensitivity of option value to changes in implied volatility" and theta as "the sensitivity of an option's premium to change in time".

Vega is the Greek that matters most for this guide. A one-point error in your volatility estimate moves this call's value by about $0.09 a share, or $9 a contract. For a portfolio view, use the options Greeks spreadsheet.

Assumptions and where they break

The model assumes an idealised market. Each assumption fails in a known way, and each failure changes how you should read a "mispricing".

Early exercise, in one paragraph

For a call on a stock that pays no dividend before expiry, early exercise is never optimal, so the American and European values match. That is Merton's 1973 result. Puts are different: deep in-the-money American puts can be worth exercising early.

Calls can also be worth exercising just before an ex-dividend date. For those cases, binomial trees or other American models are used instead of the plain formula.

The volatility smile

If markets fully believed Black-Scholes, every strike on one expiry would show the same implied volatility. They do not. CMC Markets' explainer and most textbooks describe the equity "skew" or "smirk", where lower strikes carry higher IV.

So the market uses Black-Scholes as a quoting language, not as a belief. Traders quote options in IV, and the shape of IV across strikes holds the information the model leaves out.

What "overpriced" really means

Every input except volatility is either observed or close to it. So if your Black-Scholes value differs from the market price, the gap almost always comes from volatility. Plugging in a different σ moves the value; that is all.

The useful question is turned around: given the market price, which volatility makes the formula match it?

That number is the implied volatility. OIC describes IV as "a measure of how much the marketplace expects the asset price to move based on the price of the option".

An option is "rich" if its IV is high compared with the volatility you expect over its life. It is "cheap" if its IV is low. Without your own forecast, there is no such thing as overpriced. There is only a price.

Historical (realized) vs implied volatility

OIC defines historical volatility as "a measure of actual asset price changes over a specific period". The usual estimate is the standard deviation of daily log returns, annualised:

Daily log return = ln(P_t ÷ P_t−1)
Realized σ (annual) = stdev(daily log returns) × √252
Example: 1.26% a day × √252 = 20.0% a year

Realized volatility looks back. Implied volatility looks forward. A gap between them is normal, not proof of mispricing.

For the S&P 500, Cboe reports that "from 1990 to 2018, the average implied volatility, as measured by the Cboe Volatility Index (VIX), was 19.3%", while "the average realized volatility of the S&P 500 Index was 15.1%".

That 4.2-point gap is called the volatility risk premium. Option sellers demand it for taking crash risk. So index IV above realized volatility is the usual state, and it does not by itself make options "overpriced".

IV percentile and IV rank

Two common context numbers compare today's IV with the same option series' own past:

• IV percentile: the share of days in the past year when IV was lower than today. 80 means IV was lower on 80% of days.
• IV rank: where today's IV sits between the year's low (0) and high (100).

Both tell you whether IV is high for this stock, not whether it is too high for what will happen next. A high reading before earnings is expected; the event itself is what is being priced.

For a market-wide view, the VIX is "designed to measure the market's expectation of 30-day forward looking volatility" of the S&P 500, per Cboe's VIX methodology (version 6.0, revised 26 Feb 2026). It is built from SPX option prices.

Step by step: is this option rich or cheap?

The steps below use the textbook option and a hypothetical quote of $4.90 bid, $5.30 ask. The quote is made up for illustration; it is not a market price.

Step 1: get the quote, not just the last trade

Write down the bid, the ask and the mid. A last trade can be hours old. Here the mid is $5.10 and the spread is $0.40, or 7.8% of the mid. Wide spreads are common in thin strikes and far expiries.

Step 2: solve IV at the bid, the mid and the ask

Enter each price in the Black-Scholes calculator with the right dividend yield and rate.

Step 3: build your own volatility estimate

Measure realized volatility over a few windows that bracket the option's life, for example 20, 60 and 250 trading days. Note any earnings date, dividend or scheduled event inside the option's life.

Your estimate is a judgement, not a measurement. If the 20-day number is 18% but earnings fall before expiry, recent quiet days understate what is ahead.

Step 4: compare with the right side of the spread

• Cheap to buy only if your forecast is above the IV at the ask (25.8%).
• Rich to sell only if your forecast is below the IV at the bid (21.6%).
• No trade if your forecast falls between 21.6% and 25.8%. The spread has eaten the gap.

Step 5: turn the gap into dollars

Say your forecast is 28%. The model value at 28% is $5.51, so buying at $5.30 is $0.21 a share "cheap", or $21 a contract. Vega gives the same rough answer: 2.2 points × $0.088 ≈ $0.19.

Now subtract commissions and fees for opening and closing. Compare the remainder with how wrong your forecast could be. A 2-point forecast error on this option is worth about $18 a contract, so a $21 edge is thin.

Step 6: check what you are actually betting on

A volatility edge pays off cleanly only if you hedge the stock exposure (delta) as the price moves. Without hedging, a cheap call still loses if the stock falls. Delta-hedging needs stock trades, which add their own costs.

Use the options profit calculator or the options profit spreadsheet to see the outcome at expiry over a range of prices. For the size of a typical move, see the expected move calculator.

Step 7: compare across strikes before you decide

Look at IV for nearby strikes and the next expiry.

If one strike sits well off the smooth smile shape of its neighbours, that is a more meaningful signal than a gap against a single model value. If the whole chain is high, the market is pricing an event.

Limits of the rich/cheap test

• Your forecast can be wrong. Realized volatility over the option's life is unknown until it is over. A history-based forecast misses news, earnings surprises and regime changes.
• The model is not the market. Skew, jumps and early exercise mean a single-σ Black-Scholes value is a reference, not a fair price.
• Costs are real. Spreads, commissions and hedging costs can exceed small gaps. The edge must survive the bid-ask, not the mid.
• Selling "rich" options carries tail risk. The volatility risk premium exists because sellers sometimes take large losses. An uncovered short call has no cap on the loss.
• Few trades prove nothing. A handful of good results cannot separate skill from luck, because single option outcomes are noisy.

The SEC's Investor.gov bulletin on options is direct about the buyer's side: "If a holder's option expires 'out-of-the-money' the entire premium will be lost."

Read the Characteristics and Risks of Standardized Options booklet before trading, as FINRA and the SEC advise.

If you write calls against shares you own, the covered call calculator shows break-even, maximum gain and the return if the shares are called away.

Checklist

To plan around a typical move into earnings, the expected move spreadsheet converts IV into a one-standard-deviation price range.

FAQ

What is the Black-Scholes model in simple terms?

It is a formula that gives a theoretical price for a European call or put. It uses the stock price, strike, time to expiry, interest rate, dividend yield and volatility. It assumes you could copy the option with stock and borrowing at no cost.

Is the Black-Scholes model still used?

Yes, mainly as a quoting convention. Markets quote options in implied volatility, which is defined through Black-Scholes. Traders then adjust for skew, jumps and early exercise with other models or adjustments.

How do you know if an option is overpriced?

Solve its implied volatility at the price you would trade, then compare it with your own forecast of volatility over the option's life. If IV at the bid is clearly above your forecast after costs, it looks rich. Your forecast can still be wrong.

What is the difference between implied and historical volatility?

Historical (realized) volatility measures how much the price actually moved in the past. Implied volatility is the volatility that makes the model match the current option price. One looks back; the other is the market's forward-looking price.

Why doesn't Black-Scholes work for American options?

The formula assumes exercise only at expiry. American options can be exercised earlier. For calls on non-dividend stocks this does not change the value, but American puts, and calls before a dividend, can be worth more. Binomial models handle that.

What does N(d2) mean?

It is the probability, in the model's risk-neutral world, that the option finishes in the money. It is not the real-world chance of a profit, because it ignores the premium and uses the risk-free rate instead of the stock's expected return.

Can Black-Scholes be used for crypto options?

The math works for any asset, and crypto option venues commonly quote implied volatility. Crypto prices jump often and trade around the clock, so the constant-volatility assumption fits worse. Treat the result as a rough reference.

Bottom line

Black-Scholes turns six inputs into a theoretical option value. In the textbook example, a $40 call on a $42 stock with six months left, 10% rates and 20% volatility is worth $4.76, and the put is worth $0.81. Put-call parity confirms both.

"Overpriced" and "underpriced" are claims about volatility. Solve IV at the bid and the ask, compare with a realistic forecast, and count costs. If your forecast sits inside the spread, there is no edge. Even with one, risk and model error remain.

Related free tools and templates:

• Black-Scholes calculator (prices, Greeks, implied volatility)
• Options Greeks spreadsheet
• Expected move calculator
• Options profit calculator