The subject of this book is how to determine the v not alue of a financial asset, such as a share of stock or a bar of gold, sometime in the future. We begin with a discussion of the basic properties of stock options. Readers who are familiar with these types of derivatives will want merely to skim through the chapter to synchronize terminology.
An is an investment opportunity arbitrage opportunity that is guaranteed not to result in a loss and may with positive probability result in a gain. We have remarked that if an arbitrage opportunity exists, then prices will be adjusted to eliminate that opportunity.
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Financial asset pricing involves the prediction of future events and as such relies very heavily on the mathematical theory of probability. In this chapter and the next, we describe the basic probability required for the study of discrete-time pricing models.
An Introduction to the Mathematics of Financial Derivatives - Hugendubel Fachinformationen
The following concept will prove quite useful in picturing discrete-time financial models. As shown in Figure 4. We now have the necessary mathematical background to discuss discrete-time pricing models; that is, pricing models in which all transactions take place at one of a finite number of times. In this chapter, we discuss a specific discrete-time model known as the binomial model , first described in by John Cox, Stephen Ross and Mark Rubinstein.
In Chapter 10, we will use this model to derive the famous Black—Scholes option pricing formula.
Introduction to the Mathematics of Finance
In this chapter, we discuss the problem of pricing nonattainable alternatives in an incomplete discrete-time model. The models that we have created thus far are designed to price European options, which can only be exercised at the expiration time. However, in the real world, most stock options are of the American variety.
In this chapter, we want to take a look at the issue of pricing American options, which can be exercised at any time between the purchase date and the expiration date. In this chapter, we discuss some concepts of the general theory of probability, without restriction on the size of the sample space. This is in preparation for our discussion of the Black—Scholes derivative pricing model.
The models that we have been studying are discrete-time models, because changes take place only at discrete points in time. The idea of European option pricing is extended to a general finite market model in chapter 3, where the first and second fundamental theorems of asset pricing are formulated and proved. In chapter 4, the Black-Scholes model is introduced, for which pricing and hedging of European and American contingent claims are developed, including the famous Black-Scholes option pricing formula.
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In chapter 5, the multi-dimensional Black-Scholes model is defined, which is considered as a continuous analogue of the discrete finite market model, and the fundamental theorems of asset pricing are extended to a class of continuous models. In the appendices, basic results from conditioning, discrete and continuous time martingales and stochastic calculus for Brownian motion are reviewed.
The text is clearly written and well-arranged and most of the results are proved in detail. Each chapter is completed with exercises, which makes the textbook very comprehensive. Skip to main content.