An Introduction to Financial Mathematics

Option Valuation

Hugo D. Junghenn

• Publisher: Chapman and Hall/CRC
• Year: 2019
• ISBN:
  9780429558962
• Edition: 2
• Topic: Business
• Subtopic: Finance
• Index: Business

Introduction to Financial Mathematics: Option Valuation, Second Edition is a well-rounded primer to the mathematics and models used in the valuation of financial derivatives.

The book consists of fifteen chapters, the first ten of which develop option valuation techniques in discrete time, the last five describing the theory in continuous time.

The first half of the textbook develops basic finance and probability. The author then treats the binomial model as the primary example of discrete-time option valuation. The final part of the textbook examines the Black-Scholes model.

The book is written to provide a straightforward account of the principles of option pricing and examines these principles in detail using standard discrete and stochastic calculus models. Additionally, the second edition has new exercises and examples, and includes many tables and graphs generated by over 30 MS Excel VBA modules available on the author's webpage https://home.gwu.edu/~hdj/.

Chapter 1

Basic Finance

In this chapter we consider assets whose future values are completely determined by a fixed interest rate. If the asset is guaranteed, as in the case of an insured savings account or a government bond (which, typically, has only a small likelihood of default), then the asset is said to be risk-free. By contrast, a risky asset, such as a stock or commodity, is one whose future values cannot be determined with certainty. As we shall see in later chapters, mathematical models that describe the values of a risky asset typically include a risk-free component. Our first goal then is to describe how risk-free assets are valued, which is the content of this chapter.

1.1 Interest

Interest is a fee paid by one party for the use of assets of another. The amount of interest is generally time dependent: the longer the outstanding balance, the more interest is accrued. A familiar example is the interest generated by a money market account. The bank pays the depositor an amount that is a predetermined fraction of the balance in the account, that fraction derived from a prorated annual percentage called the nominal rate, denoted typically by the symbol r. In the following subsections we describe various ways that interest determines the value of an account.

Simple Interest

Consider first an account that pays simple interest at an annual rate of r × 100%. If an initial deposit of A₀ is made at time zero, then after one year the account has value A₁ = A₀ + rA₀ = A₀(1 + r), after two years the account has value A₂ = A₀ + 2rA₀ = A₀(1 + 2r), and so forth. In general, after t years the account has value

$A_t=A_0\left(1+tr\right)$

(1.1)

which is the so-called simple interest formula. Notice that interest is paid only on the initial deposit.

Discrete-Time Compound Interest

Suppose now that an account pays the same annual rate r × 100% but with interest compounded m times per year, for example, monthly (m = 12) or daily (m = 365). The interest rate per period is then i ≔ r/m. In this setting, if an initial deposit of A₀ is made at time zero, then after the first period the value of the account is A₁ = A₀ + iA₀ = A₀(1 + i), after the second period the value is A₂ = A₁ + iA₁ = A₁(1 + i) = A₀(1 + i)², and so forth. In general, the value of the account at time n is A₀(1 + ...