Trading Techniques

🧮 Option Pricing Breakdown

Explore the most important models for valuing options, along with assumptions and mathematical structure.

📌 1. Black-Scholes-Merton (BSM) Model

The most widely used closed-form formula for European call and put options on non-dividend-paying stocks.


  C = S·N(d₁) - K·e^-rT·N(d₂)
  
  where:
    d₁ = [ln(S/K) + (r + σ²/2)·T] / (σ√T)
    d₂ = d₁ - σ√T

Assumes log-normal distribution of prices
No early exercise (European style only)

📌 2. Binomial Tree Method

🌲 Binomial Tree: Option Price Simulation

The binomial tree approach discretizes price movements over time, allowing early exercise and American-style option valuation.

A discrete-time model useful for American options, modeling price movement as up/down over time steps.


  At each node:
    Option = max(Intrinsic Value, Discounted Expected Value)

Can handle early exercise features
Converges to BSM with more steps

📌 3. Monte Carlo Simulation

Used for complex derivatives and path-dependent options.

Simulates thousands of price paths
Averages discounted payoff
High flexibility, computationally intensive

📌 4. Finite Difference Methods (FDM)

Solves the partial differential equations (PDEs) directly.

Explicit, Implicit, and Crank-Nicolson schemes
Suited for exotic options and boundary conditions

🧠 Understanding the Black‑Scholes‑Merton Model

The Black-Scholes-Merton (BSM) model provides a closed-form solution to price European-style options. It’s a cornerstone of modern financial engineering, built on stochastic calculus and the assumption of a lognormally distributed asset price.

📐 The BSM Formula

The price of a European call option is given by:

C = S·N(d₁) - K·e^-rT·N(d₂)

Where:
d₁ = [ln(S/K) + (r + σ²/2)·T] / (σ√T)
d₂ = d₁ - σ√T

📊 Key Assumptions

Stock returns follow a geometric Brownian motion
No arbitrage opportunities
Constant risk-free interest rate r
No dividends during the option life
Perfect markets (no transaction costs)

🧮 Variables

S: Current price of the underlying asset
K: Strike price
r: Risk-free interest rate
σ: Volatility of the asset
T: Time to maturity
N(·): Cumulative normal distribution function

📈 Derivation Overview

The BSM formula is derived by modeling the stock price as a stochastic process:

dS = μSdt + σSdz

where dz is a Wiener process (Brownian motion). Using Itô’s Lemma and constructing a risk-free portfolio (a long stock and a short option), the portfolio's return is set equal to the risk-free rate.

Applying no-arbitrage arguments, this leads to the famous Black-Scholes partial differential equation (PDE):

∂C/∂t + (1/2)·σ²·S²·∂²C/∂S² + r·S·∂C/∂S - r·C = 0

Solving this PDE with final boundary condition C = max(S - K, 0) at maturity T, leads to the closed-form BSM formula shown above.

📚 Extensions

Black (1976) for options on futures
Merton (1973) included dividend yields
Garman-Kohlhagen for FX options