Black-Scholes equation

The Black-Scholes model is one of the most celebrated results in mathematical finance. Published in 1973 by Fischer Black, Myron Scholes, and Robert Merton, it provides a closed-form formula for pricing European options on a financial asset. Scholes and Merton received the Nobel Prize in Economics in 1997 (Black had passed away in 1995).

Financial options

A European call option is a contract that gives its holder the right — but not the obligation — to buy an underlying asset at a fixed price \(K\) (the strike price) at a fixed future date \(T\) (the expiry). Its payoff at expiry is

\begin{equation} C(S_T, T) = \max(S_T - K,\, 0), \end{equation}

where \(S_T\) is the asset price at time \(T\). A European put option gives the right to sell at the strike, with terminal payoff

\begin{equation} P(S_T, T) = \max(K - S_T,\, 0). \end{equation}

The central question Black and Scholes answered is: what is the fair price of these contracts at any earlier time \(t < T\)?

The stochastic model: geometric Brownian motion

The fundamental assumption is that the asset price \(S_t\) follows geometric Brownian motion (GBM):

\begin{equation}\label{eq.gbm} dS = \mu S\, dt + \sigma S\, dW_t, \end{equation}

where \(\mu\) is the drift (expected instantaneous return), \(\sigma > 0\) is the volatility, and \(W_t\) is a standard Wiener process satisfying \(W_0 = 0\) with independent increments \(W_{t+\Delta t} - W_t \sim \mathcal{N}(0, \Delta t)\).

The multiplicative structure ensures \(S_t > 0\) for all \(t\) and captures the empirically observed fact that returns (not prices) are approximately normally distributed. Equation \eqref{eq.gbm} can be solved exactly. Writing \(S = e^Y\) and applying the change of variables we find that \(Y_t = \ln S_t\) follows arithmetic Brownian motion:

\begin{equation}\label{eq.lognormal} S_T = S_t \exp!\left[\left(\mu - \tfrac{1}{2}\sigma^2\right)(T-t) + \sigma\sqrt{T-t}\, Z\right], \quad Z \sim \mathcal{N}(0,1). \end{equation}

The correction \(-\tfrac{1}{2}\sigma^2\) is the Ito correction arising from the non-linearity of the logarithm when the driving noise is a Wiener process.

Ito’s lemma

Let \(V(S, t)\) be the value of any derivative security written on \(S\). Because \(S\) is a stochastic process, ordinary calculus does not apply to \(V\). The correct tool is Ito’s lemma: for any twice continuously differentiable function \(V(S,t)\),

\begin{equation}\label{eq.ito} dV = \frac{\partial V}{\partial t}\,dt + \frac{\partial V}{\partial S}\,dS + \frac{1}{2}\frac{\partial^2 V}{\partial S^2}(dS)^2. \end{equation}

The crucial result from stochastic calculus is \((dW_t)^2 = dt\) (in the Ito sense), so \((dS)^2 = \sigma^2 S^2\, dt\). Substituting \eqref{eq.gbm} into \eqref{eq.ito}:

\begin{equation}\label{eq.dV} dV = \underbrace{\left(\frac{\partial V}{\partial t} + \mu S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)}_{\text{drift}} dt

• \underbrace{\sigma S \frac{\partial V}{\partial S}}_{\text{diffusion}}\, dW_t. \end{equation}

The second-order term \(\frac{1}{2}\sigma^2 S^2 \partial^2 V/\partial S^2\), absent in classical calculus, is the hallmark of Ito’s lemma.

Delta hedging and the Black-Scholes PDE

The key insight is to construct a self-financing, instantaneously risk-free portfolio \(\Pi\) by holding one long option and shorting \(\Delta = \partial V/\partial S\) units of the underlying stock:

\begin{equation} \Pi = V - \frac{\partial V}{\partial S}\, S. \end{equation}

The instantaneous change in portfolio value is

\begin{equation} d\Pi = dV - \frac{\partial V}{\partial S}\, dS = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right) dt, \end{equation}

where the stochastic term \(dW_t\) has cancelled exactly — the portfolio is risk-free over the interval \(dt\). By the no-arbitrage principle it must earn the risk-free rate \(r\):

\begin{equation} d\Pi = r\Pi\, dt = r!\left(V - S\frac{\partial V}{\partial S}\right) dt. \end{equation}

Equating the two expressions for \(d\Pi\) and rearranging yields the Black-Scholes PDE:

\begin{equation}\label{eq.bspde} \boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}

• rS\frac{\partial V}{\partial S} - rV = 0.} \end{equation}

Remarkably, the drift \(\mu\) has vanished entirely. The fair price of any derivative is independent of the expected return on the stock — only its volatility \(\sigma\) and the risk-free rate \(r\) enter.

Boundary and terminal conditions

The PDE \eqref{eq.bspde} must be supplemented with conditions that identify the particular contract being priced.

For a European call: \begin{equation} C(S,T) = \max(S-K,0), \quad C(0,t) = 0, \quad C(S,t) \sim S - Ke^{-r(T-t)} \text{ as } S\to\infty. \end{equation}

For a European put: \begin{equation} P(S,T) = \max(K-S,0), \quad P(0,t) = Ke^{-r(T-t)}, \quad P(S,t)\to 0 \text{ as } S\to\infty. \end{equation}

Solution: reduction to the heat equation

The Black-Scholes PDE is equivalent to the classical heat equation, allowing its explicit solution.

Step 1: log-price variable. Set \(x = \ln(S/K)\), \(\tau = T - t\), and \(V(S,t) = K\, v(x,\tau)\). Substituting into \eqref{eq.bspde} (with the sign of the time derivative flipped because \(\tau\) increases backwards in time):

\begin{equation} \frac{\partial v}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 v}{\partial x^2}

• \left(r - \tfrac{1}{2}\sigma^2\right)\frac{\partial v}{\partial x} - rv. \end{equation}

Step 2: remove first-order and zeroth-order terms. Write \(v(x,\tau) = e^{\alpha x + \beta\tau} u(x,\tau)\) with

\begin{equation} \alpha = -\frac{r - \frac{1}{2}\sigma^2}{\sigma^2}, \qquad \beta = -\frac{\left(r-\frac{1}{2}\sigma^2\right)^2}{2\sigma^2} - r. \end{equation}

The equation for \(u\) becomes the heat equation:

\begin{equation}\label{eq.heat} \frac{\partial u}{\partial \tau} = \frac{\sigma^2}{2}\frac{\partial^2 u}{\partial x^2}. \end{equation}

Step 3: solve by convolution. The solution to \eqref{eq.heat} with initial data \(u(x,0) = u_0(x)\) is

\begin{equation} u(x,\tau) = \frac{1}{\sigma\sqrt{2\pi\tau}} \int_{-\infty}^{\infty} u_0(s)\, e^{-(x-s)^2/(2\sigma^2\tau)}\, ds. \end{equation}

Step 4: transform the call payoff. The terminal condition \(C(S,T)=\max(S-K,0)\) translates to

Evaluating the Gaussian integral by completing the square twice recovers the celebrated closed-form formula.

The Black-Scholes formula

Define the time to expiry \(\tau = T-t\) and the quantities

\begin{equation}\label{eq.d1d2} d_1 = \frac{\ln(S/K) + \left(r + \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau} = \frac{\ln(S/K) + \left(r - \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}. \end{equation}

The price of a European call is

\begin{equation}\label{eq.call} \boxed{C(S,t) = S\,\Phi(d_1) - K e^{-r\tau}\,\Phi(d_2),} \end{equation}

and the price of a European put is

\begin{equation}\label{eq.put} \boxed{P(S,t) = K e^{-r\tau}\,\Phi(-d_2) - S\,\Phi(-d_1),} \end{equation}

where \(\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-s^2/2}\,ds\) is the standard normal CDF.

Interpretation. Under the risk-neutral measure (see below):

• \(\Phi(d_2)\) is the probability that the call expires in-the-money, i.e.\ \(\mathbb{Q}[S_T > K]\);
• \(S\,\Phi(d_1)\) is the expected present value of receiving \(S_T\) conditional on finishing in-the-money, discounted at rate \(r\).

Thus the call price is simply the discounted expected value of the payoff \(\max(S_T-K,0)\).

Put-call parity

A model-independent no-arbitrage relation connects call and put prices with the same strike and expiry:

\begin{equation}\label{eq.pcp} C - P = S - Ke^{-r\tau}. \end{equation}

To prove it, observe that a portfolio long one call and short one put replicates a forward contract: its payoff is \((S_T-K)\) regardless of the direction of the market. Discounting that payoff gives the right-hand side. The Black-Scholes formulas \eqref{eq.call} and \eqref{eq.put} are automatically consistent with \eqref{eq.pcp} via the identity \(\Phi(x) + \Phi(-x) = 1\).

The Greeks

The partial derivatives of the option price quantify its sensitivities and are collectively called the Greeks. Let \(\phi(x) = \Phi'(x) = e^{-x^2/2}/\sqrt{2\pi}\) denote the standard normal PDF.

Delta \((\Delta)\) — sensitivity to the underlying price, and the hedge ratio: \begin{equation} \Delta_C = \frac{\partial C}{\partial S} = \Phi(d_1), \qquad \Delta_P = \frac{\partial P}{\partial S} = \Phi(d_1) - 1. \end{equation} Delta lies in \([0,1]\) for calls and \([-1,0]\) for puts. An at-the-money option has \(\Delta \approx \pm 0.5\).

Gamma \((\Gamma)\) — rate of change of delta; curvature of the price surface: \begin{equation} \Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{\tau}}. \end{equation} Gamma is identical for calls and puts (by put-call parity). It is largest for at-the-money options near expiry.

Theta \((\Theta)\) — time decay; how much value is lost per unit time: \begin{equation} \Theta_C = \frac{\partial C}{\partial t} = -\frac{S\phi(d_1)\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}\Phi(d_2). \end{equation} \(\Theta < 0\) for long option positions — the option loses value as expiry approaches, all else equal.

Vega \(({\cal V})\) — sensitivity to volatility: \begin{equation} \mathcal{V} = \frac{\partial V}{\partial\sigma} = S\phi(d_1)\sqrt{\tau}. \end{equation} Vega is the same for calls and puts and is always positive: higher volatility increases the probability of a large move, benefiting the option holder.

Rho \((\rho)\) — sensitivity to the risk-free interest rate: \begin{equation} \rho_C = \frac{\partial C}{\partial r} = K\tau\, e^{-r\tau}\Phi(d_2), \qquad \rho_P = -K\tau\, e^{-r\tau}\Phi(-d_2). \end{equation}

The Greeks satisfy a linear identity that is simply a restatement of the Black-Scholes PDE \eqref{eq.bspde}: \begin{equation} \Theta + \tfrac{1}{2}\sigma^2 S^2\,\Gamma + rS\,\Delta - rV = 0. \end{equation} This means that the time decay \(\Theta\) of a delta-hedged position is financed exactly by the gamma profit \(\frac{1}{2}\sigma^2 S^2 \Gamma\).

Risk-neutral pricing

A deeper, measure-theoretic derivation of the Black-Scholes formula uses the theory of risk-neutral (martingale) measures. By the fundamental theorem of asset pricing, the absence of arbitrage is equivalent to the existence of an equivalent probability measure \(\mathbb{Q}\) under which every discounted asset price is a martingale.

By Girsanov’s theorem, under \(\mathbb{Q}\) the Brownian motion is shifted by the market price of risk \(\lambda = (\mu - r)/\sigma\): \(\widetilde{W}_t = W_t + \lambda t,\) and the stock dynamics become

\begin{equation} dS = r S\, dt + \sigma S\, d\widetilde{W}_t. \end{equation}

The drift is replaced by \(r\) — under \(\mathbb{Q}\), all assets grow at the risk-free rate. Using \eqref{eq.lognormal} with \(\mu\to r\), the stock at expiry is

\begin{equation} S_T = S\exp!\left[\left(r - \tfrac{1}{2}\sigma^2\right)\tau + \sigma\sqrt{\tau}\,Z\right], \quad Z\overset{\mathbb{Q}}{\sim}\mathcal{N}(0,1). \end{equation}

The no-arbitrage price of any derivative with bounded payoff \(f(S_T)\) is then

\begin{equation}\label{eq.rn} V(S,t) = e^{-r\tau}\,\mathbb{E}^{\mathbb{Q}}!\left[f(S_T)\,\big|\,S_t=S\right]. \end{equation}

For a call, \(f(S_T)=\max(S_T-K,0)\). Splitting the expectation into the region \(S_T>K\) and completing the square in the exponent recovers \eqref{eq.call} exactly. The risk-neutral framework is more general than the PDE approach: it applies to path-dependent, American, and multi-asset derivatives where no closed-form PDE exists.

Dividends

When the underlying pays a continuous dividend yield \(q\), shareholders receive a cash flow \(qS\,dt\) per unit time. The stock dynamics become

\begin{equation} dS = (\mu - q)S\, dt + \sigma S\, dW_t. \end{equation}

Under the risk-neutral measure the drift becomes \((r-q)\), and the Black-Scholes formula is modified via Merton’s extension (1973):

\begin{equation} d_1 = \frac{\ln(S/K) + \left(r - q + \frac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}, \end{equation}

with the call price becoming \(C = S e^{-q\tau}\Phi(d_1) - Ke^{-r\tau}\Phi(d_2)\). Currency options follow the same formula with the foreign risk-free rate playing the role of \(q\) (the Garman-Kohlhagen model).

Assumptions and limitations

The Black-Scholes model rests on several idealizing assumptions:

1. Constant volatility. In practice, implied volatility varies with strike and maturity — the volatility smile or skew — a systematic deviation from the model. Extensions include Dupire’s local volatility model (1994), where \(\sigma = \sigma(S,t)\), and stochastic volatility models such as Heston (1993), where \(\sigma\) itself follows a mean-reverting diffusion.

2. Log-normal returns. Real returns exhibit heavy tails (excess kurtosis) and negative skewness. This leads to systematic mis-pricing of deep out-of-the-money options, which is precisely the origin of the volatility smile observed after the 1987 crash.

3. Continuous, frictionless trading. Continuous delta hedging is impossible in practice. Discrete rebalancing introduces hedging error proportional to the gamma of the position.

4. No transaction costs or liquidity constraints. Incorporating costs leads to non-linear PDEs (Leland 1985) and option price intervals rather than unique prices.

5. Constant risk-free rate. Interest rate risk is non-trivial for long-dated options. Term-structure models (Vasicek, Hull-White, CIR) address this.

Despite these limitations, Black-Scholes remains the lingua franca of options markets. Practitioners quote prices in implied volatility \(\sigma_{\mathrm{imp}}\) — the value of \(\sigma\) that equates \eqref{eq.call} to the observed market price — because the formula provides an invertible, universal mapping between prices and volatilities. In this sense, implied volatility is not a prediction but a convenient re-parameterization of price that facilitates comparison across strikes, expiries, and underlyings.