Options Pricing Models

Options pricing is fundamental to quantitative finance, and understanding the various models is crucial for any quant role. This guide covers the essential models used in practice.

Black-Scholes Model

The most famous options pricing model, developed by Fischer Black, Myron Scholes, and Robert Merton.

Call Option Price:

C = S₀N(d₁) - Ke^(-rT)N(d₂)

Put Option Price:

P = Ke^(-rT)N(-d₂) - S₀N(-d₁)

Where:

d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T

Assumptions:

• Constant volatility and risk-free rate
• No dividends
• European exercise only
• Efficient markets with no transaction costs

Binomial Model

A discrete-time model that's more flexible than Black-Scholes and can handle American options.

Steps:

1. Build a binomial tree for stock prices
2. Calculate option values at expiration
3. Work backward using risk-neutral valuation

Key Formulas:

u = e^(σ√Δt)  (up factor)
d = 1/u       (down factor)
p = (e^(rΔt) - d) / (u - d)  (risk-neutral probability)

Monte Carlo Methods

Used for complex derivatives where analytical solutions don't exist.

Applications:

• Path-dependent options (Asian, lookback)
• Multiple underlying assets
• Complex payoff structures
• American options (with Longstaff-Schwartz method)

Interview Problems

Problem 1: Black-Scholes Calculation

Question: Stock price $100, strike $110, 3 months, 20% volatility, 5% risk-free rate. Calculate the call option price.

Solution: Using Black-Scholes formula → Answer: $1.63

Problem 2: Binomial Tree

Question: Build a 2-period binomial tree for the same parameters and compare to Black-Scholes.

Problem 3: Model Selection

Question: When would you use each model in practice?

Answer:

• Black-Scholes: European options with constant parameters
• Binomial: American options, dividend-paying stocks
• Monte Carlo: Complex payoffs, multiple assets

Related Problems

These models are tested extensively at top quant firms like Jane Street, Citadel, and Two Sigma.