Valuation of Financial Derivatives in Discrete-Time Models

Abstract: The core subject of financial mathematics concerns the issue of pricing financial assets such as complex financial derivatives. The pricing technique is pervaded by the concept of arbitrage: mis-pricing will be spotted and exploited, resulting in a risk free return for any arbitrageur. A mispriced financial asset will expose the issuer to be exploited by the market as a money-pump. To prevent arbitrage, when pricing one turns to mathematics. The no-arbitrage pricing is thus formalized as a mathematical problem and it is possible to prove a mathematical pricing relationship for a financial derivative. In some specific cases it is even possible to calculate an explicit price. This thesis will consider the pricing technique of a widely used financial derivative - the option. Black-Scholes theory is, since its introduction in 1973, the main tool used for option pricing. The theory that derives the famous Black-Scholes formula involves a great amount of financial and mathematical theory, however often ignored by the user. This thesis tries to bring key concepts into light, hopefully leaving the reader (and writer) with a deeper understanding. Finance, in general, involves a great amount of uncertainty. To be able to express this uncertainty in a mathematical manner, one introduces probability theory. There will be a go-trough of basic probability theory needed to fully adopt the concept of an equivalent martingale measure which is the essential tool in arbitrage-free pricing. By introducing the time-discrete Cox-Ross-Rubinstein model and prove existence and uniqueness of an equivalent martingale measure, one is able to state the arbitrage-free price of a European call option. The model is then compared to the continues-time Black-Scholes model and in conclusion it is proved and showed that the asymptotic price of the CRR model is the same as the price calculated by the Black-Scholes formula.