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The models that we have created thus far, including the Black-Scholes option pricing model, are designed to price European options, which are options that 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.
Asset pricing involves the prediction of future events and as such relies very heavily on the mathematical theory of probability. In this chapter, we begin a discussion of basic probability. This discussion will continue in later chapters, as the need for more information arises in connection with subsequent topics to be covered in the book.
Risk is an inevitable side effect of the effort to make more money than the next guy. To be sure, money makes money and this process can be carried out without any significant measure of risk: All an investor needs to do is buy United States Treasury bonds, generally considered to be riskfree investments. The price paid for such an investment is generally a boring rate of return.
In preparation for our study of derivative pricing models, we need to discuss the basics of stock options. Readers who are familiar with these derivatives will want merely to skim through the chapters to synchronize the terminology, as it were.
In this chapter we discuss some concepts of the general theory of probability, without restriction to finite or discrete sample spaces. 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. On the other hand, in continuous-time models, changes can take place (at least theoretically) at any real time during the life of the model.
As a simple introduction to the concept of arbitrage and how to use the assumption of no arbitrage to price assets, let us briefly discuss the pricing of forward contracts and some simple issues related to option pricing.
In this chapter, we discuss a specific discrete-time model known as the Cox-Ross-Rubinstein model because it was first described by these gentlemen in 1979. We will abbreviate Cox-Ross-Rubinstein by CRR. The CRR model is also referred to in the literature as the binomial model for reasons that will become apparent as we proceed.
In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that is, how to choose the percentage (by value) of each asset in the portfolio so as to minimize the overall risk for a given expected return. The first lesson that we will learn is that the risks of each asset in a portfolio alone do not present enough information to understand...
The Mathematics of Finance has become a hot topic in applied mathematics ever since the discovery of the Black-Scholes option pricing formulas in 1973. Unfortunately, there are very few undergraduate textbooks in this area. This book is specifically written for upper division undergraduate or beginning graduate students in mathematics, finance or economics. With the exception of an optional chapter...
Continuing our work on fluorogenic substrates labeled with single fluorophores for nucleic acid modifying enzymes, here we describe the development of such substrates for DNA ligases and some base excision repair enzymes. These substrates are hairpin-type synthetic DNA molecules with a single fluorophore located on a base close to the 3′ ends, an arrangement that results in strong fluorescence quenching...
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