ABSTRACT
In this first chapter, we introduce the reader to the main concepts of finance that we will use throughout the book. In particular, we discuss about the main types of derivatives, as forwards, futures, and options. Moreover, we introduce the concepts of arbitrage-free prices, replicating portfolios, complete markets, and risk-neutral probabilities. We see that, in complete markets and under the assumption of constant volatility, option prices have to satisfy the so-called Black-Scholes partial differential equation, whose solution is given by the famous Black-Scholes formula. Then, by the Feynman-Kack lemma, we will find the solution of this partial differential equation in terms of the risk-neutral expectation of the payoff assuming asset prices follow a geometric Brownian motion. We also introduce the concept of the Black-Scholes implied volatility, as the volatility that fits observed option prices, and we discuss the main properties of the empirical implied volatility surface, as well as its
