Black–Scholes–Merton pricing (European options)¶

The Black–Scholes–Merton (BSM) model is the classic closed-form benchmark for pricing European calls and puts. It combines:

a simple stock model (GBM with constant $\sigma$),
a frictionless market with continuous trading,
no-arbitrage replication (or equivalently, risk-neutral pricing).

Objectives¶

State the BSM assumptions and understand where they enter the derivation.
Derive the Black–Scholes PDE via delta hedging.
Present the closed-form European call/put formulas and interpret their terms.
Connect the PDE view to the risk-neutral expectation view.
Use put–call parity as a consistency check.

Assumptions (what the model needs)¶

A standard version assumes:

Trading is frictionless (no transaction costs, infinite divisibility, continuous trading).
A constant risk-free rate $r$ (continuous compounding).
The underlying follows GBM with constant volatility $\sigma$:

\[ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb{P}}. \]

No arbitrage, and enough traded instruments to replicate the claim (market completeness).
No dividends (add a dividend yield $q$ by replacing $r$ with $r-q$ in the stock drift under $\mathbb{Q}$).

Derivation 1: Replication $\Rightarrow$ PDE¶

Let $V(t,S)$ be the option value. Apply Itô’s lemma to $V(t,S_t)$:

\[ dV = V_t\,dt + V_S\,dS + \tfrac12 V_{SS}\,(dS)^2 = \left(V_t + \mu S V_S + \tfrac12\sigma^2 S^2 V_{SS}\right)dt + \sigma S V_S\,dW. \]

Form a self-financing hedged portfolio:

\[ \Pi = V - \Delta S. \]

Choose $\Delta = V_S$ (delta hedging) so the $dW$ term cancels:

\[ d\Pi = \left(V_t + \tfrac12\sigma^2 S^2 V_{SS}\right)dt. \]

The portfolio is locally riskless, so in an arbitrage-free market it must earn the risk-free rate:

\[ d\Pi = r\Pi\,dt = r(V - S V_S)dt. \]

Equating the two expressions yields the Black–Scholes PDE:

\[ \boxed{ V_t + \tfrac12\sigma^2 S^2 V_{SS} + r S V_S - rV = 0. } \]

Terminal conditions¶

For a call with strike $K$ and maturity $T$,

\[ V(T,S) = (S-K)^+. \]

For a put,

\[ V(T,S) = (K-S)^+. \]

Solving the PDE with these terminal conditions gives the closed-form formulas below.

Derivation 2: Risk-neutral expectation $\Rightarrow$ same answer¶

From risk-neutral pricing, under $\mathbb{Q}$ (no dividends),

\[ dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}. \]

GBM implies:

\[ \ln S_T \sim \mathcal N\!\left(\ln S_0 + (r-\tfrac12\sigma^2)T,\;\sigma^2 T\right). \]

So for a European payoff $g(S_T)$,

\[ V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[g(S_T)]. \]

Evaluating this expectation for $g(S_T)=(S_T-K)^+$ yields the same closed form as the PDE approach.

Closed-form prices¶

Let $\tau=T-t$ and define:

\[ d_1 = \frac{\ln(S/K) + (r + \tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}. \]

Let $N(\cdot)$ be the standard normal CDF.

European call¶

\[ \boxed{ C(t,S)= S\,N(d_1) - K e^{-r\tau} N(d_2). } \]

European put¶

\[ \boxed{ P(t,S)= K e^{-r\tau} N(-d_2) - S\,N(-d_1). } \]

Interpreting the formula¶

The call price can be read as:

$S\,N(d_1)$: “present value of receiving the stock”, weighted by a hedge ratio; $N(d_1)$ is the delta.
$K e^{-r\tau} N(d_2)$: present value of paying the strike, weighted by a risk-neutral exercise probability.

A useful identity: $N(d_2)$ is the risk-neutral probability that $S_T>K$ under the $T$-forward measure; in the simplest constant-rate setting it is commonly interpreted as the risk-neutral in-the-money probability.

Put–call parity¶

For European options on a non-dividend-paying stock,

\[ \boxed{ C - P = S - K e^{-r\tau}. } \]

This is a powerful consistency check and is often used for data cleaning.

Greeks (quick reference)¶

Under BSM (no dividends), the most common Greeks are:

Delta: $\Delta_C = N(d_1)$, $\Delta_P = N(d_1)-1$.
Gamma:

\[ \Gamma = \frac{\varphi(d_1)}{S\sigma\sqrt{\tau}}, \]

where $\varphi$ is the standard normal pdf.

Vega:

\[ \nu = S\sqrt{\tau}\,\varphi(d_1). \]

Theta and rho follow by differentiating the closed form.

Practical implementation notes¶

Use log-moneyness $\ln(S/K)$ and $\tau$ to avoid loss of precision.
For very small $\tau$ or extreme strikes, clamp $\sigma\sqrt{\tau}$ away from 0 when computing $d_1,d_2$.
To back out $\sigma$ from a market price, see Implied volatility.

Black-Scholes-Merton pricing for a digital option¶

Derivation under risk-neutral measure¶

Per reasons earlier stated, the option price at time 0 should equal the time discounted expectation value of the option payoff at expiry T under the risk-neutral measure. Which in the case of a digital option gives:

\[ V_0 = e^{-rT}\mathbb{E}^{\mathcal{Q}}[\phi(S_T)] \phi(S_T) = \mathbf{1}_{S_T \geq K} Q \]

Where Q is the payout of the digital option and K the strike price. Using the log-normal property of the BSM underlying stock $S_t$, we can reduce this to:

\[ V_0 = Qe^{-rT} \mathbb{P}(S_T \geq K) \]

\[ \mathbb{P}(S_T \geq K) \iff \mathbb{P}(\ln(S_T) \geq \ln K) \]

Substituting in $\ln(S_T) = \ln(S_0) + (r-q-\frac{1}{2})$ $$\ln(S_T) \geq \ln K$$