Black-Scholes-Merton Model

The constant volatility Black-Scholes-Merton market model.
Author

John Robin Inston

Published

September 25, 2026

1 The Model

The Black-Scholes-Merton (BSM) model, introduced by Black and Scholes and by Merton in 1973, is a constant volatility model for a financial market. It is a cornerstone of Quantitative Finance, in particular of Option Pricing, where it describes the dynamics of the underlying assets of derivatives such as options.

The BSM model consists of two assets, a risk-free asset \(B\) and a risky asset (stock) \(S\) with price dynamics given by

\[ \begin{align} dB_t & = rB_t\,dt, \\ dS_t & = \mu S_t\,dt + \sigma S_t\,dW_t, \end{align} \]

where \(r\), \(\mu\) and \(\sigma\) are deterministic constants and \(W\) is a Brownian motion. Here \(r\) is the risk-free interest rate, \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient (the volatility).

The model assumes that

  • the interest rate, drift and volatility are constant,
  • the stock pays no dividends (relaxed in the section on dividends below),
  • trading is continuous and there are no transaction costs,
  • assets are infinitely divisible and short selling is allowed.

2 Solution and Properties

The risk-free asset solves \(B_t = B_0e^{rt}\), and the stock follows Geometric Brownian Motion with solution

\[ S_t = S_0\exp\left( \left( \mu - \frac{\sigma^2}{2} \right)t + \sigma W_t \right). \]

Here \(\mathcal{N}(m,v)\) denotes the normal distribution with mean \(m\) and variance \(v\) (not to be confused with the function \(N(x)\), the standard normal cumulative distribution function used in the pricing formulas in European Options). Since \(W_t \sim \mathcal{N}(0,t)\), the stock price is log-normally distributed,

\[ \log S_t \sim \mathcal{N}\left( \log S_0 + \left( \mu - \frac{\sigma^2}{2} \right)t,\; \sigma^2 t \right), \qquad \mathbb{E}[S_t] = S_0e^{\mu t}. \]

In particular the price stays positive, and \(\sigma\) is the standard deviation of the log-returns per unit of \(\sqrt{t}\).

3 Risk-Neutral Dynamics

Under the risk-neutral measure \(\mathbb{Q}\) the discounted stock price \(e^{-rt}S_t\) is a martingale, so the drift \(\mu\) is replaced by \(r\),

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}, \qquad S_T = S_0\exp\left( \left( r - \frac{\sigma^2}{2} \right)T + \sigma W_T^{\mathbb{Q}} \right). \]

Here \(W^{\mathbb{Q}}_t = W_t + \frac{\mu - r}{\sigma}t\) is a Brownian motion under \(\mathbb{Q}\). The change of measure, via Girsanov Theorem and the Radon-Nikodym Derivative, is constructed in Risk-Neutral Pricing. Note that the volatility \(\sigma\) is the same under \(\mathbb{P}\) and \(\mathbb{Q}\).

4 Black-Scholes PDE

Consider a derivative with payoff \(h(S_T)\) whose price at time \(t\) is \(V(t,S_t)\). The idea of Black-Scholes is to delta-hedge: trade the underlying stock to eliminate the randomness in the value of the option.

Hold one option together with \(\Delta_t\) shares of stock and \(\alpha_t\) units of the bond, with value

\[ \Pi_t = V(t,S_t) + \Delta_tS_t + \alpha_tB_t. \]

Choose the hedge to be self-financing, that is the value changes only through changes in the asset prices and no money is added or removed, so

\[ d\Pi_t = dV + \Delta_t\,dS_t + \alpha_t\,dB_t. \]

By Itô Formula, the option price satisfies

\[ dV = \left( \partial_t V + \mu S_t\,\partial_S V + \frac{1}{2}\sigma^2S_t^2\,\partial_{SS}V \right)dt + \sigma S_t\,\partial_S V\,dW_t. \]

Substituting the dynamics of \(S\) and \(B\),

\[ d\Pi_t = \left( \partial_t V + \mu S_t(\partial_S V + \Delta_t) + \frac{1}{2}\sigma^2S_t^2\,\partial_{SS}V + \alpha_trB_t \right)dt + \sigma S_t(\partial_S V + \Delta_t)\,dW_t. \]

We choose \(\Delta_t = -\partial_S V\) so that the coefficient of \(dW_t\) is zero. The drift of the stock \(\mu\) also cancels, leaving

\[ d\Pi_t = \left( \partial_t V + \frac{1}{2}\sigma^2S_t^2\,\partial_{SS}V + \alpha_trB_t \right)dt. \]

The portfolio \(\Pi_t = V - S_t\,\partial_S V + \alpha_tB_t\) is now deterministic over \(dt\), so to avoid arbitrage it must grow at the risk-free rate, \(d\Pi_t = r\Pi_t\,dt\). Equating the two expressions for \(d\Pi_t\), the \(\alpha_trB_t\) terms cancel and we obtain the Black-Scholes PDE.

Consider a derivative on a stock paying no dividends with maturity \(T\) and payoff \(h(S_T)\). Its price \(V(t,S)\) satisfies

\[ \begin{cases} \partial_t V + \dfrac{1}{2}\sigma^2S^2\,\partial_{SS}V + rS\,\partial_S V - rV = 0, \\[1ex] V(T,S) = h(S) \quad \text{for all } S, \end{cases} \]

where \(r\) is the risk-free interest rate and \(\sigma\) is the volatility of the stock.

Note that the drift \(\mu\) does not appear: the price of the option does not depend on the expected return of the stock. This is the same reason the risk-neutral measure replaces \(\mu\) by \(r\).

Interpretation. Rewrite the PDE as

\[ \partial_t V + \frac{1}{2}\sigma^2S^2\,\partial_{SS}V = r\left( V - S\,\partial_S V \right). \]

The right-hand side is the return, over an infinitesimal period, on the cash position: the option premium minus the amount spent on the hedging shares, times the interest rate. The left-hand side is the change in value of the delta-hedged option. The first term captures the effect of the shortening maturity (theta) and the second the effect of the curvature (gamma), which is the risk that remains after the delta is hedged. In terms of the Greeks (see Option Greeks),

\[ \Theta + \frac{1}{2}\sigma^2S^2\,\Gamma + rS\,\Delta - rV = 0. \]

The PDE can be reduced to the diffusion equation (heat equation) by a change of variables, and by the Feynman-Kac Formula its solution is the risk-neutral expectation \(e^{-r(T-t)}\mathbb{E}_{\mathbb{Q}}[h(S_T)\mid S_t]\).

5 Option Pricing

For an option with payoff \(h(S_T)\) at maturity \(T\) the model gives two equivalent ways to find the fair price, both developed in Option Pricing:

  • PDE approach: build a self-financing portfolio of stock and bond that replicates the payoff. No arbitrage then forces the price to solve the Black-Scholes PDE above.
  • Risk-neutral approach: the price at time \(0\) is the discounted expected payoff under \(\mathbb{Q}\), \[ e^{-rT}\,\mathbb{E}_{\mathbb{Q}}[h(S_T)]. \]

For European calls and puts these give closed-form prices. See European Options. Their sensitivities are the Option Greeks.

6 Dividend-Paying Stock

Suppose the stock pays a continuous dividend yield \(\delta\ge 0\), so that over \([t,t+dt]\) the holder receives the cash amount \(\delta S_t\,dt\) and the stock price drops accordingly. Equivalently, one share held at time \(0\) becomes \(e^{\delta t}\) shares at time \(t\) if dividends are reinvested.

Dividends reduce the growth of the stock, the opposite of the effect of the interest rate. Under \(\mathbb{Q}\) it is the dividend-reinvested value \(e^{-(r-\delta)t}S_t\) that is a martingale, so \(r\) is replaced by \(r-\delta\) in the dynamics,

\[ dS_t=(r-\delta)S_t\,dt+\sigma S_t\,dW_t^{\mathbb{Q}}. \]

For the PDE one must be careful about which occurrences of \(r\) change. The discounted option value \(e^{-rt}V_t\) is still a \(\mathbb{Q}\)-martingale, so the discount term \(-rV\) is unchanged. The drift term \(rS\,\partial_S V\) comes from the Itô Formula applied to \(V(t,S_t)\) and hence from the dynamics of \(S\) under \(\mathbb{Q}\), so it does change.

\[ \partial_tV+(r-\delta)S\,\partial_SV+\frac{1}{2}\sigma^2S^2\,\partial_{SS}V -rV=0,\qquad V(T,S)=h(S). \]

7 Multi-Asset Model

The model has a vector equivalent. Take \(d\) risky assets driven by an \(m\)-dimensional Brownian motion \(W = (W^1,\dots,W^m)\),

\[ dS_t^i = \mu_i S_t^i\,dt + S_t^i\sum_{j=1}^m \sigma_{ij}\,dW_t^j, \qquad i = 1,\dots,d, \]

with constant drift vector \(\mu \in \mathbb{R}^d\) and volatility matrix \(\sigma \in \mathbb{R}^{d\times m}\). The solution is

\[ S_t^i = S_0^i\exp\left( \left( \mu_i - \frac{1}{2}\sum_{j}\sigma_{ij}^2 \right)t + \sum_{j}\sigma_{ij}W_t^j \right). \]

The log-returns are jointly normal with covariance matrix \(\Sigma t\), where \(\Sigma = \sigma\sigma^\top\) has entries \(\Sigma_{ik}\) equal to the instantaneous covariance of \(\log S^i\) and \(\log S^k\), so correlation between assets enters through \(\Sigma\).

Under \(\mathbb{Q}\) every asset has drift \(r\). A risk-neutral measure exists if there is a market price of risk vector \(\theta \in \mathbb{R}^m\) with \(\sigma\theta = \mu - r\mathbf{1}\). The market is arbitrage-free if such a \(\theta\) exists, and complete if in addition \(\sigma\) is invertible (\(d = m\)). This model prices multi-asset options such as basket and exchange options.

8 Time Dependent Volatility

We replace the constant volatility in the Black-Scholes-Merton Model \(\sigma\) with a deterministic function in time \(\sigma_{t}\) then we can show that Black-Scholes still holds if \(\sigma^2\) is replaced with \(\frac{1}{T}\int _{0}^T\sigma^2_{s} \, ds\)

To see this consider the stock price dynamics under \(\mathbb{Q}\)

\[ dS_{t}=rS_{t}dt+\sigma_{t}S_{t}dW_{t}^\mathbb{Q}. \]

Applying the log transformation and Itô Formula we obtain

\[ S_{t}=S_{0}\exp\left\{ \int _{0}^t\left( r- \frac{\sigma^2_{s}}{2} \right) \, ds+\int _{0}^t\sigma_{s} \, dW_{s}^\mathbb{Q} \right\}. \]

As a result, we simply replace \(\sigma^2t\) with \(\int _{0}^t\sigma_{s}^2 \, ds\) and replace \(\sigma dW_{t}^\mathbb{Q}\) with \(\int _{0}^t \sigma_{s} \, dW_{s}^\mathbb{Q}\).

The implied volatility surface \(I(T,K)\) is not flat in \(T\) but still flat in \(K\). This is a subtle improvement on BS but is still far from ideal for practical purposes.

9 Limitations and Extensions

The assumption of a constant volatility is not consistent with market prices. Implied volatility varies with strike and maturity, giving the volatility smile or surface (see Volatility Surfaces). Models that relax it include

10 Key Results

11 References

  1. Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
  2. Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
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