This coming Tuesday, we will complete the probability and statistics tutorial by studying the binomial and normal probability distributions. On Thursday, we introduce financial derivatives, relying upon a combination of textbook chapters and teaching notes that I have authored. We will begin by defining financial derivatives and motivating their study with examples of forwards, futures, and options. Derivatives are so named because the prices of these instruments are derived from the prices of one or more underlying assets. The types of underlying assets upon which derivatives are based are often traded financial assets such as stocks, bonds, currencies, or other derivatives, but they can be pretty much anything. For example, the Chicago Mercantile Exchange (CME) offers exchange-traded weather futures contracts and options on such contracts (see “Market Futures: Introduction To Weather Derivatives“). There are also so-called “prediction” markets in which derivatives based upon the outcome of political events are actively traded (see “Prediction Markets“).
Besides introducing derivatives and discussing various institutional aspects of markets in which they are traded, we’ll consider various properties of forward and option contracts, since virtually all financial derivatives feature payoffs that are isomorphic to either or both of these schemes. For example, a futures contract is simply an exchange-traded version of a forward contract. Similarly, since swaps involve exchanges between counter-parties of payment streams over time, these instruments essentially represent a series of forward contracts. In the option space, besides traded stock options there are also embedded options in corporate securities; e.g., a convertible bond represents a combination of a non-convertible bond plus a call option on company stock. Similarly, when a company makes an investment, there may be embedded “real” options to expand or abandon the investment at some future date.
Perhaps the most important (pre-Midterm 1) idea that we’ll introduce is the concept of a so-called “arbitrage-free” price for a financial derivative. While details will follow, the basic idea is that one can replicate the payoffs on a forward or option by forming a portfolio consisting of the underlying asset and a riskless bond. This portfolio is called the “replicating” portfolio, since it is designed to perfectly replicate the payoffs on the forward or option. Since the forward or option and its replicating portfolio produce the same payoffs, then they must also have the same value. However, suppose the replicating portfolio (forward or option) is more expensive than the forward or option (replicating portfolio). It this occurs, then one can earn a riskless arbitrage profit by simply selling the replicating portfolio (forward or option) and buying the forward or option (replicating portfolio). However, competition will ensure that opportunities for riskless arbitrage profits vanish very quickly. Thus the forward or option will be priced such that one cannot earn arbitrage profit from playing this game.