Black-Scholes-Merton Intro

Q: The BSM closed-form formula directly prices

European options only. The closed-form BSM equation prices European options, which can only be exercised at expiry. American options allow early exercise and generally require numerical methods such as binomial trees or finite-difference solvers. For a non-dividend American call, early exercise is never optimal, so the BSM call price applies to that case as well (Hull, 2022, §15.5).

Q: Which input does the BSM call price NOT depend on?

Stock's expected return µ. Risk-neutral pricing replaces the real-world drift µ with the risk-free rate r. The hedging argument cancels the stock's expected return out of the formula. This is the single most counterintuitive feature of BSM and one of the reasons the 1997 Nobel committee cited the work.

The Black-Scholes-Merton (BSM) model gives a closed-form price for European options on a non-dividend paying stock. The call formula is $C = S_0 N(d_1) - K e^{-rT} N(d_2)$ . The model rests on five assumptions, continuous trading with no transaction costs, lognormal stock prices, no dividends in the basic form, constant volatility $\sigma$ , and a constant risk-free rate $r$ with continuous compounding. Black and Scholes (1973) and Merton (1973) showed that, under these assumptions, a self-financing portfolio of stock and bond can replicate the option, so its price is unique and free of any risk premium.

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Black-Scholes-Merton, one formula that priced the option market

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Try it yourself

Black-Scholes explorer

Move the inputs and watch the call and put reprice. The smooth curve is the Black-Scholes value; the dashed kink is intrinsic value max(S−K, 0). The gap between them is time value (which can go slightly negative for a European put when rates are high).

Curve Black-ScholesDashed IntrinsicTime value A$6.63

Call valueA$6.63

Put valueA$4.65

d₁0.2121

d₂0.0707

N(d₁)0.584

N(d₂)0.528

Put-call parity: C − P = A$1.98 = S − Ke^−rT = A$1.98

Spot S100

Strike K100

Volatility σ20%

Maturity T0.50 yr

Rate r4.0%

Why it matters

BSM is what the binomial tree becomes when you take the number of steps to infinity. Each tiny step is a coin flip on the stock, and as the steps shrink to zero, the lognormal limit kicks in and the discrete tree converges to the BSM formula (Cox, Ross and Rubinstein, 1979). The key conceptual move is that the option price does not depend on the stock's expected return $\mu$ . Risk-neutral pricing replaces $\mu$ with the risk-free rate $r$ , because the writer can hedge away the directional risk by holding $N(d_1)$ shares against each short call.

Before you read on — recall

The BSM closed-form formula directly prices

Formulas

European call

C = S_0 N(d_1) - K e^{-rT} N(d_2)

S_0

is spot,

K

is strike,

r

is the risk-free rate,

T

is time to expiry in years, and

N(\cdot)

is the standard normal CDF.

European put

P = K e^{-rT} N(-d_2) - S_0 N(-d_1)

Follows from the call by put-call parity,

C - P = S_0 - K e^{-rT}

Worked examples

Scenario

A non-dividend stock trades at A$50 with $\sigma = 30\%$ , $r = 5\%$ , $K =$ A$50, $T = 1$ year. Price the European call using BSM.

Solution

Compute $d_1 = [\ln(1) + (0.05 + 0.045)(1)] / 0.30 = 0.3167$ and $d_2 = 0.3167 - 0.30 = 0.0167$ . From a z-table, $N(d_1) = 0.6242$ and $N(d_2) = 0.5067$ . Discount factor $e^{-rT} = e^{-0.05} = 0.9512$ . So $C = 50 \times 0.6242 - 50 \times 0.9512 \times 0.5067 \approx$ A$7.12.

Scenario

BSM transformed option markets. Before 1973 traders priced options by feel. After Fischer Black, Myron Scholes and Robert Merton published the formula, the Chicago Board Options Exchange (also founded in April 1973) had a consistent benchmark.

Solution

Within two years, Texas Instruments was selling a handheld calculator pre-loaded with the BSM formula. Scholes and Merton shared the 1997 Nobel Memorial Prize in Economic Sciences for the work. Black had died in 1995.

Common mistakes

✗BSM gives the true price. It gives the no-arbitrage price under its assumptions. Real markets have stochastic volatility, jumps, transaction costs, and discrete trading. The 1987 crash made the constant-volatility assumption visibly wrong, and traders have priced options off a volatility smile ever since.
✗BSM only works because of complex math. The derivation rests on one financial idea, a dynamic hedge of $\Delta$ shares against the option removes all market risk. The PDE and the normal CDF are just the machinery for turning that hedge into a number.

Revision bullets

•Closed-form price for European options
•Assumes constant $\sigma$ , constant $r$ , no dividends
•Call $C = S_0 N(d_1) - K e^{-rT} N(d_2)$
•Continuous limit of the binomial tree
•Price is independent of $\mu$ , the stock's expected return
•Black, Scholes, Merton published the result in 1973

Quick check

The BSM closed-form formula directly prices

Which input does the BSM call price NOT depend on?

Connected topics

Cont. Pricing Am. vs Eu.Time Value Delta Δ The Greeks Normal Dist.d1 & d2 BSM + Divs Impl. Vol Exam Hotspots

More in Black-Scholes-Merton

Z-table BSM Call BSM Put

In learning paths

Pricing Models Path Exam Revision Path

Sources

Black & Scholes (1973), JPE
Black, Fischer, and Myron Scholes. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81, no. 3 (1973): 637-654.
Original paper deriving the closed-form European call price under continuous trading and lognormal stock dynamics.
Merton (1973), Bell J. Econ.
Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4, no. 1 (Spring 1973): 141-183.
Companion paper extending Black-Scholes under weaker assumptions and introducing the dividend-yield generalisation used throughout modern option pricing.
Hull (2022), §15
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Standard undergraduate derivation of BSM, including the PDE, risk-neutral valuation, and worked numerical examples.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross, and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7, no. 3 (1979): 229-263.
Shows that the binomial tree converges to the BSM formula as the number of steps grows, the bridge that links the discrete and continuous approaches.

How to cite this page

Dr. Phil's Quant Lab. (2026). Black-Scholes-Merton Intro. Derivatives Atlas. https://phucnguyenvan.com/concept/bsm-intro

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