Chapter 22: Valuing Options: Binomial & Black-Scholes

Valuing Options: Binomial & Black-Scholes introduces the fundamental concept of the replicating portfolio, demonstrating how investors can mimic the payoff of a call option by combining a leveraged position in the underlying stock with risk-free borrowing. This framework supports the Law of One Price and leads to the risk-neutral valuation method, which simplifies pricing by assuming investors are indifferent to risk, allowing expected payoffs to be discounted at the risk-free interest rate. The text details the Binomial Method for valuing options, visualizing asset price movements through decision trees that break time into discrete up or down steps based on volatility. As these steps become infinitesimally small, the distribution of price changes approaches lognormality, forming the basis of the Black-Scholes formula. This continuous-time model calculates option value using five key variables: current stock price, exercise price, time to maturity, the risk-free interest rate, and the standard deviation of returns. A critical concept explored is the option delta, or hedge ratio, which measures the sensitivity of the option's price to changes in the underlying stock and defines the number of shares required to maintain a risk-free hedge. The summary explains that options possess higher betas and volatility than their underlying assets due to implicit leverage. Practical applications are discussed, including the valuation of executive stock options, warrants, and portfolio insurance, as well as the concept of implied volatility (VIX) derived from market prices. Finally, the chapter distinguishes between American and European options, analyzing the impact of dividends and the conditions under which early exercise of American options might be optimal.