Lecture 21: Black-Scholes Formula, Risk Neutral Valuation

Quick Overview

The lecture establishes that derivative pricing, exemplified by Black-Scholes, relies on risk-neutral valuation by constructing a replicating portfolio that eliminates uncertainty, leading to the Black-Scholes partial differential equation which surprisingly does not depend on the real-world drift ($\mu$) but only on volatility ($\sigma$) and the risk-free rate ($r$).

Key Points: The core idea is to price derivatives by replicating their payoff using a portfolio of the underlying asset and cash, which, in a no-arbitrage world, must have the same price as the derivative today. In the continuous time setting, forming a riskless portfolio of the option and stock leads to the Black-Scholes equation: $\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 = rf - r\frac{\partial f}{\partial S} S$. A crucial observation is that the derived Black-Scholes equation does not depend on the real-world drift $\mu$, meaning market participant preferences or real-world dynamics are irrelevant for pricing derivatives. The replicating portfolio amount of stock held, $a$, is determined by $a = \frac{\partial f}{\partial S}$, which forms the basis for hedging strategies. Risk-neutral pricing implies that under the risk-neutral measure, the underlying asset's expected growth rate must equal the risk-free rate ($r$), meaning $\mu$ is substituted by $r$ in the stock's dynamics. The relationship between call and put options demonstrates replication without dynamic assumptions: Long Call - Short Put = Stock Price - Discounted Strike Price ($C - P = St - K e^{-r(T-t)}$). The speaker challenges the audience to find a replicating portfolio for a digital option, suggesting it can be done without complex assumptions on stock dynamics.

Context: The lecture, titled "Lecture 21: Black-Scholes Formula, Risk-Neutral Valuation," begins by briefly analyzing recent yield curve movements compared to historical inverted curves preceding recessions, before transitioning into the main topic of derivative pricing. The professor introduces the concept using a simple horse betting analogy where setting odds based on market bets allows the bookie to hedge and break even, regardless of the outcome. This concept is then directly applied to financial derivatives like forward contracts, call options, and put options, setting the stage for deriving their present values using replication arguments.