Mathematics for Finance: An Introduction to Financial Engineering


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1.1 Basic Notions and Assumptions Suppose that two assets are traded: one risk-free and one risky security. The former can be thought of as a bank deposit or a bond issued by a government, a financial institution, or a company. The risky security will typically be some stock. It may also be a foreign currency, gold, a commodity or virtually any asset whose future price is unknown today.

Throughout the introduction we restrict the time scale to two instants only: today, t = 0, and some future time, say one year from now, t = 1. More refined and realistic situations will be studied in later chapters. The position in risky securities can be specified as the number of shares of stock held by an investor.

The price of one share at time t will be denoted by S(t). The current stock price S(0) is known to all investors, but the future price S(1) remains uncertain: it may go up as well as down. The difference S(1) − S(0) as a fraction of the initial value represents the so-called rate of return, or briefly return: KS = S(1) − S(0) S(0) , which is also uncertain.

The dynamics of stock prices will be discussed in Chapter 3. The risk-free position can be described as the amount held in a bank account. As an alternative to keeping money in a bank, investors may choose to invest in bonds. The price of one bond at time t will be denoted by A(t). The

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