Black-Scholes-Merton Option Pricing

Derivative Pricing · Continuous-time · Sell-side

Family 02 Pricing PDE $\mathbb{Q}$-measure

Black & Scholes (1973) and Merton (1973) solved the European option pricing problem by constructing a self-financing replicating portfolio. The result is a closed-form price for European calls and puts, a parabolic PDE that any derivative must satisfy, and five partial derivatives — the Greeks — that describe how the price moves with its inputs. The work won the 1997 Nobel Memorial Prize in Economic Sciences and remains the single most-taught result in mathematical finance.

Educational Purpose

This page is a mathematical demonstration of the Black-Scholes-Merton model. It is not investment advice, options-trading guidance, or a hedging recommendation. The assumptions below (constant volatility, continuous trading, no transaction costs) are systematically violated in real markets, which motivates the many extensions cited at the end of this page.

Why option pricing matters

Three anchor facts

$600 T+

notional value of OTC derivatives outstanding globally (2023) — a market whose pricing infrastructure rests squarely on the Black-Scholes-Merton framework and its descendants.

BIS Derivatives Statistics · bis.org

1997

Nobel Memorial Prize in Economic Sciences awarded to Robert C. Merton and Myron S. Scholes “for a new method to determine the value of derivatives”. Black had died in 1995.

Nobel Committee · nobelprize.org

Δ Γ Θ ν ρ

the five first- and second-order sensitivities — the Greeks — that every options desk monitors continuously. All have closed-form expressions under BSM.

Hull (2021) ch.19 · pearson.com

The model

Underlying dynamics, PDE, and closed-form price

OR family: Derivative pricing Solver class: Closed-form / PDE Measure: Risk-neutral $\mathbb{Q}$ Realism: ★★★ Exact (under assumptions)

Notation

Symbol	Meaning
$S_t$	Price of the non-dividend-paying underlying asset at time $t$
$K$	Strike price of the option
$T$	Maturity date of the option
$\tau = T - t$	Time to maturity
$r$	Continuously-compounded risk-free rate (constant)
$\sigma$	Volatility of the underlying (constant, annualised)
$W^{\mathbb{Q}}_t$	Standard Brownian motion under the risk-neutral measure $\mathbb{Q}$
$V(S,t)$	Price of a European contingent claim with payoff $V(S_T,T) = g(S_T)$
$C, P$	European call and put price respectively
$N(\cdot), \phi(\cdot)$	Standard-normal cumulative and density functions

Dynamics

Under the risk-neutral measure $\mathbb{Q}$, the underlying follows a geometric Brownian motion with drift equal to the risk-free rate:

Underlying SDE (under $\mathbb{Q}$) $$ dS_t = r \, S_t \, dt + \sigma \, S_t \, dW^{\mathbb{Q}}_t $$

Geometric Brownian motion with constant drift $r$ and constant volatility $\sigma$.

Black-Scholes PDE

Any self-financing replicating portfolio argument on an option $V(S,t)$ written on this underlying shows that $V$ must satisfy a parabolic partial differential equation (no arbitrage):

Black-Scholes PDE (Black & Scholes 1973, Merton 1973) $$ \frac{\partial V}{\partial t} \;+\; \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \;+\; r S \frac{\partial V}{\partial S} \;-\; r V \;=\; 0 $$

With terminal condition $V(S,T) = g(S)$ (the contract payoff).

European call & put — closed form

For a European call with payoff $C_T = (S_T - K)^+$ and a European put with payoff $P_T = (K - S_T)^+$, the PDE solves in closed form:

European call price $$ C(S, t) \;=\; S \, N(d_1) \;-\; K e^{-r\tau} \, N(d_2) $$
European put price $$ P(S, t) \;=\; K e^{-r\tau} \, N(-d_2) \;-\; S \, N(-d_1) $$
where $$ d_1 \;=\; \frac{\ln(S/K) + (r + \tfrac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 \;=\; d_1 - \sigma\sqrt{\tau} $$

Put-call parity: $C - P = S - K e^{-r\tau}$. Both prices satisfy the BSM PDE and the appropriate terminal condition.

Greeks

Five closed-form sensitivities. For a call:

$$ \Delta = \frac{\partial C}{\partial S} = N(d_1), \qquad \Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{\tau}} $$ $$ \mathrm{Vega} = \frac{\partial C}{\partial \sigma} = S\,\phi(d_1)\sqrt{\tau} $$ $$ \Theta = \frac{\partial C}{\partial t} = -\frac{S\,\phi(d_1)\sigma}{2\sqrt{\tau}} - rK e^{-r\tau} N(d_2) $$ $$ \rho = \frac{\partial C}{\partial r} = K \tau e^{-r\tau} N(d_2) $$

$\Gamma$ and Vega are the same sign for calls and puts; $\Delta_{put} = \Delta_{call} - 1$. Convention: $\Theta$ per year; many desks quote $\Theta/365$ per calendar day.

Assumptions & limitations

All violated in real markets — which is why the extensions exist

Black-Scholes model assumptions

Geometric Brownian motion. Log-returns are normally distributed with constant drift and volatility. Real-market returns exhibit heavy tails, skew, and volatility clustering.
Constant volatility $\sigma$. Empirically, implied volatility varies by strike (smile / skew) and by maturity (term structure). This is the single most-violated assumption; Heston (1993) and local-volatility models relax it.
Constant risk-free rate $r$. Extensions: Merton (1973) already allows deterministic $r(t)$; stochastic-rate models use HJM / Vasicek curves.
No dividends. Continuous dividend yield $q$ adds $-qS\,\partial_S V$ to the PDE. Discrete dividends require different treatment.
No transaction costs. Real hedging incurs bid-ask, commissions, and market impact; transaction-cost models (Leland 1985) adjust the effective volatility.
Continuous trading. The replicating portfolio rebalances continuously — impossible in practice. Discrete hedging errors scale with $\sqrt{\Delta t}$.
No arbitrage; frictionless markets; unlimited borrow/lend at $r$. Foundation of the no-arbitrage pricing argument.
European exercise only. American options (early exercise) have no closed form; they are priced via binomial trees, PDE with free boundary, or Monte Carlo with Longstaff-Schwartz regression.
Single underlying. Multi-asset / basket / rainbow options require multi-dimensional extensions (correlated GBMs).

Interactive solver

Set $(S, K, T, r, \sigma)$ and read the price and Greeks

Black-Scholes Calculator

★★★ Exact under BSM assumptions

Underlying $S$

100.00

Strike $K$

100.00

Time to maturity $T$ (years)

1.00

Risk-free rate $r$

3.00%

Volatility $\sigma$

25.00%

Option type

Price

—

Delta ($\Delta$)

—

Gamma ($\Gamma$)

—

Vega

—

Theta ($\Theta$)/yr

—

Rho ($\rho$)

—

Thick line: option value $V(S,t)$ at present. Dashed line: payoff $g(S_T)$ at maturity. Vertical marker: current spot.

Desk ↔ model mapping

On the desk	In the model
Spot price of the underlying	$S$ (or $S_0$)
Contract strike	$K$
Days to expiry / 365	$\tau$
Implied volatility	$\sigma$ (the only unobservable; back-solved from market price)
Risk-free curve at tenor $\tau$	$r$
Per-unit-of-underlying hedge ratio	Delta ($\Delta$)
Change in Delta per $\$$ change in underlying	Gamma ($\Gamma$)
Time decay (per year)	Theta ($\Theta$)
Sensitivity to volatility	Vega

Extensions & cousins

The Black-Scholes assumptions relaxed, one axis at a time

Stochastic volatility — Heston (1993): $d\sigma_t^2$ is its own mean-reverting SDE; produces volatility smiles.
Jump-diffusion — Merton (1976): underlying has occasional Poisson jumps; captures fat tails and short-term smile.
Local volatility — Dupire (1994): $\sigma(S, t)$ calibrated from the entire implied-vol surface.
American options — Binomial tree (Cox-Ross-Rubinstein 1979) or Longstaff-Schwartz Monte Carlo (2001) handle early exercise.
Exotic options — Asian (path average), barrier (knock-in/out), lookback, digital; usually priced via Monte Carlo or PDE with exotic boundary.
Stochastic interest rates — combine BSM with HJM or short-rate models (Vasicek / CIR).
Transaction costs — Leland (1985) adjusts effective volatility; broader utility-based hedging models exist.

Key references

Seminal papers, extensions, textbooks

Black, F. & Scholes, M. (1973).

The Pricing of Options and Corporate Liabilities.

Journal of Political Economy, 81(3), 637–654. doi:10.1086/260062

Merton, R. C. (1973).

Theory of Rational Option Pricing.

The Bell Journal of Economics and Management Science, 4(1), 141–183. doi:10.2307/3003143

Cox, J. C., Ross, S. A. & Rubinstein, M. (1979).

Option Pricing: A Simplified Approach.

Journal of Financial Economics, 7(3), 229–263. doi:10.1016/0304-405X(79)90015-1

Merton, R. C. (1976).

Option Pricing When Underlying Stock Returns Are Discontinuous.

Journal of Financial Economics, 3(1–2), 125–144. doi:10.1016/0304-405X(76)90022-2

Heston, S. L. (1993).

A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.

The Review of Financial Studies, 6(2), 327–343. doi:10.1093/rfs/6.2.327

Dupire, B. (1994).

Pricing with a Smile.

Risk, 7(1), 18–20.

Leland, H. E. (1985).

Option Pricing and Replication with Transactions Costs.

The Journal of Finance, 40(5), 1283–1301. doi:10.1111/j.1540-6261.1985.tb02383.x

Hull, J. C. (2021).

Options, Futures, and Other Derivatives, 11th ed.

Pearson. ISBN 978-0-13-693997-9.

Shreve, S. E. (2004).

Stochastic Calculus for Finance II: Continuous-Time Models.

Springer. ISBN 978-0-387-40101-0.

Björk, T. (2019).

Arbitrage Theory in Continuous Time, 4th ed.

Oxford University Press. ISBN 978-0-19-885161-5.

Wilmott, P. (2006).

Paul Wilmott on Quantitative Finance, 2nd ed.

Wiley. ISBN 978-0-470-01870-5.

Related sub-applications

Three standard numerical methods to price options when Black-Scholes fails — Americans, path-dependent payoffs, stochastic volatility, or jumps.

Finance landscape Binomial tree Monte Carlo Rate models

Reminder — educational purpose only

Implied volatility, Greeks, and option prices computed here assume constant $\sigma$ and $r$, continuous trading, no transaction costs, and no dividends — systematically violated in real markets. Not investment, hedging, or regulatory advice. Historical model performance never implies future results.