Key Concepts in Option Pricing Models

Why This Matters

Option pricing models form the backbone of derivatives valuation and risk management—two areas that dominate quantitative finance exams and real-world applications. You're being tested not just on whether you can plug numbers into formulas, but on whether you understand why different models exist and when each approach is appropriate. The core principles here—no-arbitrage pricing, risk-neutral valuation, volatility modeling, and numerical methods—appear repeatedly across financial mathematics, from basic derivative pricing to advanced portfolio hedging.

Don't just memorize the Black-Scholes formula or the structure of a binomial tree. Instead, focus on understanding what assumptions each model makes, what market behaviors it can (and can't) capture, and how models relate to one another. When an exam question asks you to choose an appropriate pricing method, you need to recognize whether the option is European or American, whether volatility is constant or stochastic, and whether the payoff depends on the entire price path. Know what problem each model solves, and you'll handle any question thrown at you.

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Closed-Form Analytical Models

These models provide exact mathematical solutions for option prices, making them computationally efficient and ideal for European-style options. The key assumption enabling closed-form solutions is typically that the underlying follows a specific stochastic process with known distributional properties.

Black-Scholes-Merton Model

• No-arbitrage pricing foundation—establishes that option prices must prevent risk-free profit opportunities in efficient markets
• Assumes constant volatility and interest rates, which simplifies the pricing PDE to yield the famous closed-form solution: $C = S_0 N(d_1) - Ke^{-rT}N(d_2)$
• Log-normal stock price distribution—models returns as normally distributed, which implies stock prices follow geometric Brownian motion

Garman-Kohlhagen Model

• Extends Black-Scholes to foreign exchange options by incorporating both domestic and foreign risk-free rates
• Interest rate differential replaces the dividend yield, reflecting the cost of carry between two currencies
• Closed-form solution for European FX options—essential for managing currency exposure in international portfolios

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Lattice-Based Models

Lattice models discretize time and price movements into a tree structure, allowing step-by-step backward induction. This discrete framework makes them particularly powerful for American options, where early exercise decisions must be evaluated at each node.

Binomial Option Pricing Model

• Discrete-time framework models stock prices moving up or down by fixed factors at each time step
• Handles American options by comparing continuation value against immediate exercise value at every node
• Converges to Black-Scholes as the number of time steps increases, providing intuition for the continuous-time limit

Cox-Ross-Rubinstein Model

• Recombining tree structure ensures the number of nodes grows linearly rather than exponentially with time steps
• Risk-neutral probabilities are derived directly from no-arbitrage conditions: $p = \frac{e^{r\Delta t} - d}{u - d}$
• Clear visual representation makes it an excellent teaching tool and a practical method for understanding option mechanics

Trinomial Tree Model

• Three possible movements per step—up, down, or unchanged—providing finer granularity than binomial models
• Greater pricing accuracy by capturing more potential outcomes with fewer time steps required
• Better convergence properties for pricing American options and options with complex early exercise features

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Stochastic Volatility Models

These models recognize that volatility itself is random and evolves over time. By treating volatility as a stochastic process, they capture real market phenomena like volatility clustering and the implied volatility smile.

Heston Model

• Volatility follows its own stochastic process, specifically a mean-reverting square-root diffusion (CIR process)
• Captures the volatility smile observed in market-implied volatilities across different strike prices
• Correlation parameter links asset returns to volatility changes, reflecting the leverage effect seen in equity markets

Stochastic Volatility Models (General Framework)

• Volatility as a random variable allows the model to adapt to changing market conditions dynamically
• Examples include Heston and SABR models, each with distinct volatility dynamics suited to different asset classes
• Improves pricing accuracy for longer-dated options and exotic derivatives where constant volatility assumptions break down

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Models for Discontinuous Price Movements

Markets don't always move smoothly—sudden jumps from earnings announcements, geopolitical events, or market crashes require models that incorporate discontinuities. Jump-diffusion models combine continuous Brownian motion with discrete Poisson-driven jumps.

Jump-Diffusion Models

• Incorporates sudden price jumps alongside continuous diffusion, capturing extreme market events
• Poisson process governs jump timing, with jump sizes typically drawn from a log-normal distribution
• Essential for assets with event risk—think earnings announcements, FDA approvals, or merger news where prices gap overnight

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Numerical Methods

When closed-form solutions don't exist—due to American exercise features, path dependence, or complex payoffs—numerical methods provide the answer. These techniques trade computational cost for flexibility in handling virtually any derivative structure.

Monte Carlo Simulation

• Random sampling of price paths generates thousands of possible scenarios to estimate expected payoffs
• Handles path-dependent options like Asian options, lookbacks, and barrier options where the entire price history matters
• Computationally intensive but highly flexible—can accommodate time-varying volatility, interest rates, and correlation structures

Finite Difference Methods

• Solves the pricing PDE numerically by discretizing both time and the underlying asset price on a grid
• Handles American options through free boundary conditions that identify optimal exercise regions
• Explicit, implicit, and Crank-Nicolson schemes offer different trade-offs between stability, accuracy, and computational speed

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Quick Reference Table

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Self-Check Questions

1. Which two models both provide closed-form solutions for European options but differ in their treatment of volatility—one assuming it's constant, the other modeling it as stochastic?

2. You need to price an American put option on a stock. Which models from this guide would be appropriate, and why can't you use Black-Scholes directly?

3. Compare and contrast Monte Carlo simulation with finite difference methods: what types of derivatives is each best suited for, and what are the computational trade-offs?

4. An FRQ presents implied volatility data showing higher volatilities for out-of-the-money puts than at-the-money options. Which model limitation does this reveal, and which alternative models address it?

5. The Garman-Kohlhagen model extends Black-Scholes for FX options. What specific modification does it make to the standard framework, and why is this necessary for currency derivatives?