The Black-Scholes
Model.

Understanding the pricing problem.

A European call with strike \(K\) and maturity \(T\) pays \((S_T-K)^+\) at maturity. The question is: what is its fair price \(V(t,S_t)\) today? The Black-Scholes-Merton answer is that the option can be replicated by a dynamic position in the underlying and a risk-free bond, so its price is the cost of that replicating portfolio.

This study follows the full chain: dynamics of the underlying, Itô's lemma, the Black-Scholes PDE, risk-neutral valuation, the closed-form formula, the Greeks, and a numerical application.

Model framework.

Geometric Brownian motion.

Under the historical measure \(\mathbb{P}\), the price \(S_t\) satisfies:

Applying Itô's lemma to \(f(S)=\ln S\), with \((dS_t)^2=\sigma^2S_t^2\,dt\), gives \(d\ln S_t=(\mu-\tfrac{1}{2}\sigma^2)\,dt+\sigma\,dW_t\), hence the explicit solution:

\(S_T\) is therefore lognormal: \(\ln S_T\) is Gaussian with mean \(\ln S_t+(\mu-\tfrac{1}{2}\sigma^2)(T-t)\) and variance \(\sigma^2(T-t)\). Moreover \(\mathbb{E}[S_T\mid S_t]=S_t\,e^{\mu(T-t)}\): the correction \(-\tfrac{1}{2}\sigma^2\) exactly compensates the convexity of the exponential.

Derivation of the partial differential equation.

Let \(V(t,S)\) be the option price. Itô's lemma gives:

Consider the portfolio \(\Pi=V-\Delta S\), short \(\Delta\) units of the underlying. Choosing \(\Delta=\partial V/\partial S\) cancels the \(dW_t\) term, and the drift \(\mu\) disappears as well:

The portfolio is locally risk-free, so by absence of arbitrage \(d\Pi=r\Pi\,dt=r\left(V-S\,\partial_SV\right)dt\). We obtain the Black-Scholes equation:

The expected return \(\mu\) does not appear: the price is independent of the investors' risk preferences. The equation is a backward parabolic PDE, closed by the terminal condition \(V(T,S)=(S-K)^+\) for a call and \(V(T,S)=(K-S)^+\) for a put. Through the change of variables \(x=\ln S\), \(\tau=T-t\), it reduces to the heat equation, whose Green kernel yields the solution.

Risk-neutral valuation.

By Girsanov's theorem, \(W_t^{\mathbb{Q}}=W_t+\frac{\mu-r}{\sigma}t\) is a Brownian motion under the risk-neutral measure \(\mathbb{Q}\), and:

By the Feynman-Kac theorem, the solution of the PDE is the discounted expected payoff under \(\mathbb{Q}\), equivalently the statement that the discounted price \(e^{-rt}V_t\) is a \(\mathbb{Q}\)-martingale:

where \(\Phi\) is the payoff. Since \(S_T\) is lognormal under \(\mathbb{Q}\) with parameters \((r-\tfrac{1}{2}\sigma^2)(T-t)\) and \(\sigma^2(T-t)\), the price becomes a Gaussian integral, computed in the next section.

Price of the European call and put.

Write \(\tau=T-t\). Splitting the payoff, \[\mathbb{E}^{\mathbb{Q}}[(S_T-K)^+]=\mathbb{E}^{\mathbb{Q}}[S_T\mathbf{1}_{S_T>K}]-K\,\mathbb{Q}(S_T>K).\] The second term gives \(K\,N(d_2)\); the first, after a change of measure (numéraire \(S\)), gives \(S\,e^{r\tau}N(d_1)\). Discounting:

\(N(\cdot)\) is the standard normal cumulative distribution function. \(N(d_2)=\mathbb{Q}(S_T>K)\) is the risk-neutral probability of finishing in the money, and \(N(d_1)\) is the same probability under the share measure. The call is thus "asset-or-nothing minus cash-or-nothing": \(S\,N(d_1)\) is the present value of the asset received, \(Ke^{-r\tau}N(d_2)\) that of the strike paid.

Limiting behaviours: as \(\sigma\sqrt{\tau}\to0\), the call tends to its discounted intrinsic value \((S-Ke^{-r\tau})^+\); as \(\sigma\to\infty\), it tends to \(S\). Non-arbitrage bounds are \((S-Ke^{-r\tau})^+\le C\le S\).

The Greeks.

The Greeks are the partial derivatives of the price with respect to the parameters. Let \(N'(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\) be the standard normal density. For a European call:

For a put, \(\Delta_P=N(d_1)-1\), while Gamma and Vega are identical to those of the call (by parity). Delta is the hedge ratio, Gamma its rate of change (the convexity of the price, always positive for a long vanilla option), Vega the exposure to volatility, Theta the time decay and Rho the exposure to interest rates.

The Greeks are linked to each other through the PDE itself: substituting into the Black-Scholes equation gives

For a delta-neutral position (\(\Delta=0\)), this reads \(\Theta\approx-\tfrac{1}{2}\sigma^2S^2\Gamma\) (when \(rV\) is negligible): the time decay pays for the convexity gained. This is the core trade-off of gamma trading.

Numerical example.

Take an at-the-money option with \(S=100\), \(K=100\), \(r=5\%\), \(\sigma=20\%\), \(\tau=1\) year.

Call-put parity.

Buying a call and selling a put with the same strike and maturity replicates a forward, which yields, independently of any model:

Implied volatility and extensions.

The only unobservable input is \(\sigma\). Since the price \(C(\sigma)\) is strictly increasing in \(\sigma\) (Vega \(>0\)), each market price corresponds to a unique implied volatility \(\sigma_{\text{imp}}\), solving \(C_{\text{BS}}(\sigma_{\text{imp}})=C_{\text{market}}\). It is found numerically, typically with Newton-Raphson:

In practice, \(\sigma_{\text{imp}}\) is not constant across strikes and maturities: it draws a smile or skew, which contradicts the constant-volatility assumption. Other departures from the model include jumps, fat-tailed returns, discrete hedging and transaction costs.

From underlying dynamics to price.