Mathematical Option Pricing

Construct the pricing framework by blending economics and math to price options. Apply no-arbitrage, Black-Scholes assumptions, and the risk-neutral measure to price European options and explore market completeness.

Explore the fundamental theorem of asset pricing by defining a market and self-financing portfolios. Learn how a risk-neutral measure yields arbitrage-free and complete markets for pricing european options by replication.

Explore the Black-Scholes pricing framework through its seven canonical assumptions, enabling risk-neutral pricing of calls, with stock dynamics following a normal distribution driven by the risk-free rate.

Demonstrates, through two no-arbitrage scenarios, that the forward price must align with the discounted expectation of future stock prices, using a forward contract and Black–Scholes assumptions (no dividends, no repo).

Derives the stock dynamics under Black-Scholes, including dividends and repos, and explains the forward pricing through no-arbitrage, linking drift, volatility, and risk-free rate.

Pricing a call option using risk-neutral probability under no-arbitrage, this lecture explains how to correct a naive expectation with a forward-based probability and determine the option's value.

Define vanilla call and put options within European style, detailing their payoffs, intrinsic value, and the premium, and explain how volatility, time to expiry, and interest rates shape prices.

Derive the European call price under Black-Scholes with risk-neutral pricing and Girsanov change of measure, using indicator functions and discussing dividend adjustments.

Explore vanilla options and their practical pricing, hedging, and risk management, including volatility smile, call spreads, butterfly structure, call-put parity, synthetic forward, and delta hedging.

Derive the Black-Scholes equation from a self-financing, delta-hedged portfolio to price any European claim under no arbitrage, with volatility that may vary.

Explore exchange options under Black-Scholes with two correlated stocks, showing the payoff reduces to a call on the ratio S1/S2 with strike 1, via change of measure and Ito calculus.

Explore the chooser option and its pricing by transforming the max into a call-put parity combination, revealing a closed formula price as a blend of call and put components.

Price forward start options under constant volatility and rates using risk-neutral valuation. Show how the payoff reduces to a standard call with unit spot and discuss volatility smile effects.

Explore local volatility and the volatility surface to price exotic options while preserving vanilla option prices, using Monte Carlo methods within the Black-Scholes framework.

Derive the Breeden-Litzenberger formula from vanilla option prices to the stock distribution, using Lebesgue's differentiation under the integral sign, linking strike derivatives to distribution and density for local volatility.

Derive the local volatility formula by linking call option prices to the stock distribution via the Fokker–Planck framework, yielding sigma squared as a function of option prices and strikes.

Derive the Fokker-Planck equation, also called the Kamogawa forward equation, linking stock distribution to drift mu and volatility sigma, and explain its role in local volatility and option pricing.

Explore barrier options like down-and-in, down-and-out, and rebates; use the reflection principle to relate max Brownian motion to barrier events, enabling closed formulas for vanilla and barrier options.

Price barrier options with a crude formula for buyer options under constant volatility using the reflection principle, drifted Brownian motion, and the joint distribution of stock value and its maximum.

Dive into the Ornstein-Uhlenbeck mean-reverting process, its role in equity hybrid models and stochastic volatility, and the classic trick to simplify its dynamics in a one-factor model.