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INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS; STUDENT’S MANUAL Jakˇ sa Cvitani´ c and Fernando Zapatero September 2003 Table of Contents INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. FINANCIAL MARKETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. INTEREST RATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. MODELS OF SECURITIES PRICES IN FINANCIAL MARKETS . . . . . . . . 7 4. OPTIMAL CONSUMPTION/PORTFOLIO STRATEGIES . . . . . . . . . . . . . . . . 12 5. RISK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6. ARBITRAGE AND RISK-NEUTRAL PRICING . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7. OPTION PRICING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8. FIXED INCOME MARKET MODELS AND DERIVATIVES . . . . . . . . . . . . . 35 9. HEDGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 10. BOND HEDGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 11. NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 12. EQUILIBRIUM FUNDAMENTALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 13. CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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INTRODUCTION This manual contains the solutions to the end-of-the chapters problems in Introduction to the Economics and Mathematics of Financial Markets (MIT Press, 2003), that have a sign, as in 14 , when problem 14 is solved in the manual. For some problems for which the main part consists of writing a computer program and doing computations, we only give suggestions and not full solutions. For these, the book web page http://math.usc.edu/ cvitanic/book.html contains Microsoft Excel files that can be helpful. The files have names like “ch1.xls”. That particular file has all the figures from Chapter 1 of the book, and all the computations needed to produce them. We use only basic features of Excel, except for Monte Carlo simulation for which we use the Visual Basic programming language, incorporated in Excel. At a few places in the book we do give “Excel Tips” that point out what trickier commands have been used. Other, more mathematically oriented software may be more e ffi cient for longer computations such as Monte Carlo simulation, and we leave the choice of the software to be used with some of these to the instructor or the reader. Please report any errors or suggested corrections to [email protected] or zapa- [email protected] Corrections will be posted at the book web page. 1 FINANCIAL MARKETS 2. How “risk-free” are bonds as financial instruments? Solution: Bonds are called risk-free (or fixed income) securities because the stream of the cash-flows promised by the issuer until maturity is known in advance. That is, when investors buy bonds, they know the exact amounts and dates of payments. However, that does not make bonds completely risk-free. One reason is, there is a risk of default of the issuer. When the issuer is the U.S. government, the risk of default is considered non-existent. That is the reason why the denomination of “risk-free” securities is typically reserved for government bonds. Another reason is that if the investor decides to sell the bond in the market before maturity, the price will depend on prevailing interest rates at the time of the sale, which cannot be predicted. Still another reason is, even if the investor keeps the bond until maturity, only the nominal value of the payments is known, but not the real value.
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