No-Arbitrage Option Pricing and the Binomial Asset Pricing Model

Financial markets often employ the use of securities, which are defined to be any kind of tradable financial asset. Common types of securities include stocks and bonds. A particular type of security, known as a derivative security (or simply, a derivative), are financial instruments whose value is derived from another underlying security or asset (such as a stock). A common kind of derivative is an option, which is a contract that gives the holder the right but not the obligation to go through with the terms of said contract. An example of an option is the European Option, which we will use commonly throughout the paper. A European Option is an option that can only be exercised at the specified maturity time. The objective of this paper is to explore the problem of finding a no-arbitrage price for options (namely the European Option) through the binomial asset pricing model. A no-arbitrage price means that an investor cannot come up with a strategy (involving the no-arbitrage price) where they begin with no money, are able to make a profit with positive probability, and have zero probability of taking a loss. The binomial asset-pricing model provides a simple environment to work through the pricing of options. We will also the basics regarding the completeness of markets, with examples, and the concept of risk-neutral probabilities. The paper concludes with the connection of option pricing with probability theory and martingales as well as the statement of the First Fundamental Theorem of Asset Pricing