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Chapter 6: The Black-Scholes Model

This chapter develops the Black-Scholes option pricing framework from first principles. Starting from geometric Brownian motion and self-financing portfolios, we derive the Black-Scholes PDE through four independent approaches, analyze its structure and boundary conditions, solve it using a range of analytic techniques, and arrive at the celebrated closed-form formula for European options.

Key Concepts

The Black-Scholes Model

The model, introduced by Black, Scholes (1973) and Merton (1973), assumes the underlying asset follows geometric Brownian motion with constant drift and volatility:

\[dS_t = \mu S_t\, dt + \sigma S_t\, dW_t\]

Six core assumptions -- GBM dynamics, frictionless markets, constant risk-free rate, no dividends, continuous trading, and no arbitrage -- make the mathematics tractable and enable closed-form solutions. By Ito's lemma, asset prices are log-normally distributed with explicit solution

\[S_t = S_0 \exp\bigl((\mu - \tfrac{1}{2}\sigma^2)t + \sigma W_t\bigr)\]

The model emerges as the continuous-time limit of the binomial framework, inheriting the key insights of dynamic hedging and risk-neutral valuation. The self-financing portfolio condition constrains admissible trading strategies and underpins the hedging arguments that follow.

Four Derivations of the Black-Scholes PDE

The pricing PDE

\[\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV\]

is derived via four conceptually distinct routes:

  • Delta hedging: construct a self-financing portfolio \(\Pi = V - \Delta S\) that eliminates stochastic risk by setting \(\Delta = \partial V / \partial S\), then apply no-arbitrage to require \(d\Pi = r\Pi\, dt\)
  • Risk-neutral pricing: invoke the fundamental theorem of asset pricing and Girsanov's theorem so that the discounted price \(e^{-rt}V\) is a \(\mathbb{Q}\)-martingale, and set its drift to zero
  • Change of numeraire: use the stock as numeraire with its associated measure \(\mathbb{Q}^S\) via the Radon-Nikodym derivative \(Z_t = S_t e^{-rt}/S_0\), and derive the PDE from pricing invariance
  • Equilibrium: derive the PDE from a representative-agent economy with CRRA preferences, where market clearing yields the equilibrium risk premium \(\mu - r = \gamma \sigma^2\) and the stochastic discount factor \(M_t = e^{-\rho t} S_t^{-\gamma}\)

PDE Structure and Conditions

The killing term \(-rV\) encodes continuous discounting and connects to the Feynman-Kac probabilistic representation

\[V(t,x) = \mathbb{E}[e^{-r(T-t)}\Phi(X_T) \mid X_t = x]\]

The discounted value process \(M_s = e^{-r(s-t)}V(s, X_s)\) is a martingale precisely when \(V\) solves the pricing PDE. The Greeks \(\Delta\), \(\Gamma\), \(\Theta\), \(\mathcal{V}\), and \(\rho\) emerge as partial derivatives of the PDE solution, linked by the theta-gamma relation

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

Terminal conditions \(V(T,S) = \Phi(S)\) specify the contract payoff (calls, puts, digitals, straddles), while boundary conditions -- Dirichlet, Neumann, or Robin -- select the unique solution from the family of PDE solutions. The PDE smooths non-smooth and discontinuous terminal data for \(t < T\).

Analytic Solution Methods

The Black-Scholes PDE is solved through multiple techniques:

  • Heat equation reduction: variable substitutions (\(\tau = T - t\), \(x = \ln S + (r - \tfrac{1}{2}\sigma^2)\tau\), and an exponential scaling) transform the PDE into the classical heat equation, whose fundamental solution is known
\[\frac{\partial u}{\partial \tau} = \tfrac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2}\]
  • Separation of variables: assume a product form \(u(x,t) = X(x)T(t)\) to reduce the PDE to independent ODEs; on the semi-infinite stock-price domain this yields a continuous spectrum and Fourier transforms rather than discrete eigenvalues
  • Similarity solutions: exploit scale invariance and the Buckingham Pi theorem to reduce to dimensionless groups \(S/K\), \(\sigma\sqrt{\tau}\), and \(r\tau\), giving the general form
\[V = K \cdot f(S/K, \sigma\sqrt{\tau}, r\tau)\]
  • Integral transforms: Fourier, Mellin, and Laplace transforms convert the PDE into first-order ODEs in the transform variable, solved explicitly via characteristic exponents and inverted to recover option prices
  • Feynman-Kac formula: bridge PDE and probability theory by representing the solution as conditional expectations under the risk-neutral measure, reducing the pricing problem to computing conditional expectations
\[u(x,t) = \mathbb{E}[e^{-r(T-t)}\Phi(X_T) \mid X_t = x]\]
  • Change of numeraire: provide alternative derivations using forward measures and stock-numeraire techniques, with the Radon-Nikodym derivative connecting different pricing measures
  • Viscosity solutions: handle non-smooth payoffs (digital options with \(\Phi(S) = \mathbf{1}_{S>K}\)) and free-boundary problems (American options) where classical \(C^2\) solutions fail, using test-function-based sub/supersolution definitions

The Black-Scholes Formula

For a European call with strike \(K\) and maturity \(T\):

\[C = S_0\,\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\]

where

\[d_{1,2} = \frac{\ln(S_0/K) + (r \pm \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\]

The term \(\mathcal{N}(d_2)\) is the risk-neutral probability of exercise, while \(\mathcal{N}(d_1)\) is the exercise probability under the stock-numeraire measure \(\mathbb{Q}^S\).

\[\begin{array}{lll} \mathcal{N}(d_1) &=& \mathbb{Q}^S(S_T > K)\\ \mathcal{N}(d_2) &=& \mathbb{Q}(S_T > K) \end{array}\]

The formula can be derived via direct integration of the log-normal density, via Girsanov's theorem and measure change, or as the solution to the heat equation. Put-call parity

\[C - P = S - Ke^{-r(T-t)}\]

relates call and put prices as a model-independent no-arbitrage condition derived from replicating portfolios. Option prices satisfy fundamental bounds, e.g.,

\[\max(S - Ke^{-rT}, 0) \leq C \leq S\]

monotonicity, and convexity properties. Asymptotic analysis in the limits \(S \to 0, \infty\) and \(\sigma \to 0, \infty\) confirms financial intuition: deep ITM calls behave like forward contracts

\[C \to S - Ke^{-rT}\]

and deep OTM calls become worthless. Digital option pricing

\[D_0 = e^{-rT}\Phi(d_2)\]

illustrates the framework applied to discontinuous payoffs.

Role in the Book

This chapter unifies the stochastic calculus tools from earlier chapters into a complete pricing framework. The four PDE derivations highlight how no-arbitrage, martingale theory, numeraire invariance, and general equilibrium all converge on the same equation. The analytic solution methods -- from heat equation reduction to viscosity solutions -- form the mathematical toolkit extended in later chapters to local volatility, stochastic volatility, jump-diffusion models, and numerical PDE methods.