Deriving the Black-Scholes Model
Quantitative Finance Derivative Securities Pricing
Pricing Financial Derivatives
The majority of time in undergraduate quantitative finance coursework is spent in developing intuition and understanding for securities pricing models. In this article, I am going to summarize what I have learned across several courses on derivatives pricing, primarily the logic and mathematics behind the Black-Scholes model.
Risk-Neutral Pricing
The essence of pricing derivatives lies in risk-neutrality. Theoretically, the goal is to construct a temporary risk-less hedged portfolio, and by our no-arbitrage assumption, create a pricing model by setting the portfolio return to U.S. Treasury bills, notes, and bonds (an appropriate risk-free return proxy) instead of unknown market expectations.
Brownian Motion
Brownian motion is a continuous-time random variable where future outcomes are unpredictable from historic outcomes. Brownian motion is a Martingale and a Markov Process.
Itô’s Lemma
This lemma helps us find the change of a time-dependent function of a stochastic variable, in this case, Brownian motion.
