Browsing by Subject "Financial market model"

In this thesis we cover some fundamental topics in mathematical finance and construct market models for the option pricing. An option on an asset is a contract giving the owner the right, but not the obligation, to trade the underlying asset for a fixed price at a future date. Our main goal is to find a price for an option that will not allow the existence of an arbitrage, that is, a way to make a riskless profit. We will see that the hedging has an essential role in this pricing. Both the hedging and the pricing are very import tasks for an investor trading at constantly growing derivative markets. We begin our mission by assuming that the time parameter is a discrete variable. The advantage of this approach is that we are able to jump into financial concepts with just a small quantity of prerequisites. The proper understanding of these concepts in discrete time is crucial before moving to continuous-time market models, that is, models in which the time parameter is a continuous variable. This may seem like a minor transition, but it has a significant impact on the complexity of the mathematical theory. In discrete time, we review how the existence of an equivalent martingale measure characterizes market models. If such measure exists, then market model does not contain arbitrages and the price of an option is determined by this measure via the conditional expectation. Furthermore, if the measure also unique, then all the European options (ones that can be exercised only at a predetermined time) are hedgeable in the model, that is, we can replicate the payoffs of those options with strategies constructed from other assets without adding or withdrawing capital after initial investments. In this case the market model is called complete. We also study how the hedging can be done in incomplete market models, particularly how to build risk-minimizing strategies. After that, we derive some useful tools to the problems of finding optimal exercise and hedging strategies for American options (ones that can be exercised at any moment before a fixed time) and introduce the Cox-Ross-Rubinstein binomial model to use it as a testbed for the methods we have developed so far. In continuous time, we begin by constructing stochastic integrals with respect to the Brownian motion, which is a stochastic component in our models. We then study important properties of stochastic integrals extensively. These help us comprehend dynamics of asset prices and portfolio values. In the end, we apply the tools we have developed to deal with the Black-Scholes model. Particularly, we use the Itô’s lemma and the Girsanov’s theorem to derive the Black-Scholes partial differential equation and further we exploit the Feynman-Kac formula to get the celebrated Black-Scholes formula.