# Lecture 21: Black-Scholes Formula, Risk Neutral Valuation Source: https://www.youtube.com/watch?v=2UCHztlWuZg Recap page: https://rapidrecap.app/video/2UCHztlWuZg Generated: 2025-12-03T16:12:24.207+00:00 --- ## 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 ($C - P = S_t - K e^{-r(T-t)}$). - 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. ## Detailed Analysis Vasily Strela explains that derivative pricing centers on risk-neutral valuation, showing how replicating a derivative's payoff with a portfolio of the underlying stock and cash allows for determining the derivative's current price without knowing real-world probabilities. In a discrete time example with zero interest rates, replicating a forward contract forces the strike price to equal the current stock price. When moving to continuous time and introducing interest rates ($r$), the core technique involves forming a portfolio combining the derivative $f$ and the stock $S$ such that $f - aS$ (where $a$ is the hedge ratio) grows deterministically at the risk-free rate, $r$. Differentiating this riskless portfolio using It {o}'s formula and equating the deterministic and stochastic parts yields the Black-Scholes partial differential equation (PDE). The professor emphasizes that this PDE is independent of the stock's real-world drift ($\mu$), depending only on volatility ($\sigma$) and $r$, which is the foundation of risk-neutral pricing, where $\mu$ is replaced by $r$ under the risk-neutral measure. The hedge ratio $a$ is found to be $\frac{\partial f}{\partial S}$. Finally, the lecture illustrates the power of replication by showing that the difference between a call and a put option perfectly replicates a forward contract ($C - P = S_t - K e^{-r(T-t)}$), a relationship independent of the stock's specific dynamics, although the non-constant implied volatility observed in Apple and IBM data suggests lognormal assumptions are insufficient for complex derivatives. ### Introduction to Derivative Pricing Concepts - Replicating payoffs using a portfolio of stock and cash - Hedging uncertainty shown via horse betting example - Focus shifts from real-world dynamics to market implications. ### Discrete Time Replication (Zero Rates) - Forward contract replication forces strike $K$ to equal current price $S_0$ when interest rates are zero - Call option pricing requires finding specific amounts of stock ($a$) and cash ($B_0$) to replicate the payoff $\max(S_T - K, 0)$. ### Continuous Time Derivation via It {o}'s Formula - Forming a riskless portfolio $f - aS$ that must grow at $e^{rt}$ allows differentiation - The resulting combination eliminates the stochastic $dW$ terms, leading to the Black-Scholes PDE. ### The Black-Scholes Equation and Its Implications - The PDE depends only on volatility ($\sigma$) and risk-free rate ($r$), not the real-world drift ($\mu$) - The hedging ratio $a$ equals the partial derivative of the option price with respect to the stock price, $a = \frac{\partial f}{\partial S}$. ### Risk-Neutral Measure - Under risk-neutral assumptions, the expected growth of the stock must equal the risk-free rate, meaning $\mu$ is replaced by $r$ - Pricing derivatives becomes calculating the expected payoff using this risk-neutral PDF. ### Call-Put Parity Replication - The portfolio (Long Call - Short Put) replicates a forward contract ($S_t - K e^{-r(T-t)}$) regardless of stock dynamics, demonstrating arbitrage opportunities if violated. ### Beyond Lognormal Dynamics - Observed implied volatility curves for Apple and IBM are not flat, indicating that the lognormal assumption underlying the basic Black-Scholes model is insufficient for real-world pricing.