Chapter 1: Discrete Models and the Fundamental Theorem of Asset Pricing¶
This chapter builds the theory of arbitrage-free pricing in discrete-time, finite-state markets. Starting from simple one-period models and Arrow-Debreu securities, we develop the binomial option pricing framework through replication, delta hedging, and risk-neutral expectation. The chapter culminates in the Fundamental Theorem of Asset Pricing, which establishes the deep equivalence between no-arbitrage and the existence of an equivalent martingale measure.
Key Concepts¶
Discrete-Time Foundations¶
A single-period economy consists of \(N\) traded assets across \(S\) possible future states, organized into a payoff matrix \(\mathbf{X} \in \mathbb{R}^{N \times S}\). A portfolio \(\theta\) generates payoff \(\theta^\top \mathbf{X}\) across states. An arbitrage is a portfolio with zero or negative cost and non-negative payoff in all states, with strict positivity in at least one. Arrow-Debreu securities pay one unit in exactly one state and zero otherwise; their prices \(\psi_s > 0\) form state prices that encode the market's valuation of contingent claims:
State prices exist if and only if the market is arbitrage-free, and they are unique if and only if the market is complete. The stochastic discount factor (pricing kernel) \(m_s = \psi_s / p_s\) connects state prices to physical probabilities, enabling the pricing formula \(P_j = \mathbb{E}^{\mathbb{P}}[m \cdot X_j]\). The Breeden-Litzenberger result shows how state price densities can be extracted from European option prices via \(\phi(K) = e^{r_f T} \partial^2 C / \partial K^2\), providing a bridge between theory and market observables.
The Binomial Model¶
The Cox-Ross-Rubinstein (1979) model assumes the stock price moves by multiplicative factors \(u\) (up) or \(d\) (down) at each step. The no-arbitrage condition requires
Three equivalent approaches yield the same derivative price:
- Replicating portfolio: find \((\Delta, B)\) in stock and bond matching the payoff in both states, with the hedge ratio \(\Delta = \frac{H_u - H_d}{(u-d)S_0}\) and the state price formula \(V_0 = \psi_u H_u + \psi_d H_d\) where \(\psi_u = e^{-r\Delta t} q\) and \(\psi_d = e^{-r\Delta t}(1-q)\)
- Delta hedging: combine the option with \(\Delta\) shares of stock to eliminate risk; the risk-free portfolio must earn rate \(r\), and the risk-neutral probability emerges from the hedging argument without being assumed a priori
- Risk-neutral pricing: compute the discounted expectation under the risk-neutral probability \(q = \frac{e^{r\Delta t} - d}{u - d}\)
Under the risk-neutral measure, the discounted stock price is a martingale:
Risk-neutral pricing is linear in payoffs, enabling decomposition of complex structures (e.g., bull spreads, put-call parity \(C_0 - P_0 = S_0 - Ke^{-r\Delta t}\)) into simpler components.
Multi-Period Extension and American Options¶
The multi-period binomial tree prices options via backward induction, applying the one-period formula recursively from terminal nodes to time zero. At node \((n,j)\), the stock price is \(S_{n,j} = S_0 u^j d^{n-j}\), and the recombining property \(ud = 1\) ensures only \(n+1\) nodes at time \(n\) (yielding \(O(N^2)\) computational complexity). Dynamic delta hedging rebalances the hedge ratio \(\Delta_{n,j} = \frac{V_{n+1,j+1} - V_{n+1,j}}{S_{n,j}(u-d)}\) at each node, with the self-financing property ensuring no external cash is injected during rebalancing. American options introduce optimal stopping: at each node, the holder compares the continuation value with the immediate exercise payoff, leading to the recursion
The early exercise boundary identifies critical stock prices where immediate exercise becomes optimal. For American calls on non-dividend-paying stocks, early exercise is never optimal. The trinomial model, with three possible moves per step, illustrates incomplete markets where the risk-neutral measure is no longer unique.
Binomial to Black-Scholes Limit¶
As the number of time steps \(n \to \infty\) with \(\Delta t = T/n \to 0\), the binomial model converges to the Black-Scholes framework. With the CRR parameterization \(u = e^{\sigma\sqrt{\Delta t}}\) and \(d = e^{-\sigma\sqrt{\Delta t}}\), the scaled random walk converges to Brownian motion via Donsker's theorem, log-prices become normally distributed by the central limit theorem, and the risk-neutral probability approaches \(q \to 1/2\). The Ito correction term \(-\frac{1}{2}\sigma^2\) emerges from Jensen's inequality applied to the concave logarithm: the martingale condition constrains the arithmetic return \(\mathbb{E}^{\mathbb{Q}}[S_{i+1}|S_i] = S_i e^{r\Delta t}\), but \(\mathbb{E}[\ln R_i] \approx (r - \sigma^2/2)\Delta t\) due to the gap between \(\mathbb{E}[\ln X]\) and \(\ln \mathbb{E}[X]\). Taylor expansion of the backward recursion yields the Black-Scholes PDE:
The discrete delta converges to \(\partial V / \partial S = \Phi(d_1)\), and binomial option prices converge at rate \(O(1/n)\) with oscillations between odd and even steps that can be smoothed by Richardson extrapolation.
The Fundamental Theorem of Asset Pricing¶
The FTAP establishes two fundamental equivalences:
- First FTAP: The market is arbitrage-free if and only if there exists an equivalent martingale measure \(\mathbb{Q} \sim \mathbb{P}\) under which discounted prices \(\tilde{S}^i_t = S^i_t / S^0_t\) are martingales
- Second FTAP: The market is complete if and only if the equivalent martingale measure is unique
The proof in the finite-state case relies on the separating hyperplane theorem: if no arbitrage exists, the set of attainable portfolio gains \(\mathcal{V} = \operatorname{Im}(X)\) and the positive orthant \(\mathbb{R}^n_{++}\) are disjoint convex sets, and the separating functional defines the state prices (or equivalently, the Radon-Nikodym derivative \(d\mathbb{Q}/d\mathbb{P}\)). The key geometric insight is that the separating vector \(q\) satisfies \(X^T q = 0\) with \(q_i > 0\) for all states, which after normalization becomes the EMM. Related results include Farkas' lemma (the linear programming incarnation) and Minkowski's separation theorem for relative interiors. In continuous time, the condition strengthens to No Free Lunch with Vanishing Risk (NFLVR), and the Delbaen-Schachermayer theorem (1994) establishes the equivalence via the Kreps-Yan theorem (a functional-analytic extension of separating hyperplanes to \(L^\infty\) spaces). The connection to Girsanov theory provides the explicit measure change \(d\mathbb{Q}/d\mathbb{P} = \mathcal{E}(-\int_0^T \lambda_s\, dW_s)\) where \(\lambda_t = (\mu_t - r)/\sigma_t\) is the market price of risk.
Numeraire and Change of Measure¶
A numeraire is any strictly positive traded asset \(N_t\). The FTAP generalizes: for each numeraire there exists an associated martingale measure \(\mathbb{Q}^N\) under which prices normalized by \(N_t\) are martingales. The change of numeraire theorem relates these measures via the Radon-Nikodym derivative \(\frac{d\mathbb{Q}^M}{d\mathbb{Q}^N}\big|_{\mathcal{F}_t} = \frac{M_t/M_0}{N_t/N_0}\), and the master pricing formula \(V_t = N_t \cdot \mathbb{E}^{\mathbb{Q}^N}[\Phi_T / N_T \mid \mathcal{F}_t]\) unifies all standard pricing expressions. Prices are invariant under numeraire changes---different numeraires yield different measures but identical arbitrage-free prices. Key applications include the forward measure \(\mathbb{Q}^T\) (using zero-coupon bond \(P(t,T)\) as numeraire, which eliminates discounting from interest-rate derivative pricing) and Margrabe's formula for exchange options (using one asset as numeraire to reduce a two-dimensional problem to a one-dimensional Black-Scholes integral): \(V_0 = S^1_0 \Phi(d_1) - S^2_0 \Phi(d_2)\) with \(\sigma_R = \sqrt{\sigma_1^2 - 2\rho\sigma_1\sigma_2 + \sigma_2^2}\).
From Discrete to Continuous
The discrete-time framework developed here provides the conceptual foundation for continuous-time models. Chapter 2 introduces Brownian motion and martingale theory, Chapter 3 develops stochastic calculus, and Chapter 6 derives the Black-Scholes model as the continuous-time limit of the binomial framework.