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More Exercises for Stochastic Calculus for Finance I; Steven ... - UFRJ

**More** **Exercises** **for** **Stochastic** **Calculus** **for****Finance** I; **Steven** Shreve VJun 02 2009Marco Cabralmapcabral@ufrj.brDepartment of MathematicsFederal University of Rio de JaneiroJuly 25, 20091 The Binomial No-Arbitrage Pricing Model1.1 Reading Comprehension (short questions and answers)1. Explain the meaning of the following phrases from the book:(a) “If d = u, the stock price at time one is not really random and the modelis uninteresting.” (p.2, second paragraph)(b) “[...] prices of derivative securities depend on the volatility of stock pricesbut not on their mean rates of growth.” (p.8, first paragraph)(c) “The arbitrage pricing theory approach to the option-pricing problemsis to replicate the option by trading in the stock and money markets.” (p.3, 6thparagraph)(d) “The mean rate of growth (with respect to the risk-neutral probabilities)of the stock is equal to the rate of growth of the money market account.”2. What is arbitrage (see p.2)?3. What prevents the persistency of arbitrage in a real market (p.2)?4. What is an stochastic process (p.11)?5. What is a complete market (p.14)?6. What is a path-dependent option (p.14)?7. On Example 1.3.2, p.17 the second term on the **for**mula **for** v n (s, m) isv n+1 (s/2, m). Is this correct? Why not v n+1 (s/2, max(m, s))?8. What are the main assumptions **for** pricing an option in the binomial model?(see p.5)9. What is the difference between the risk-neutral probabilities and the actualprobabilities?10. Compare (see p.7) the mean rate of growth of an investment in the moneymarket with the mean rate of growth of the stock under the:(a) risk-neutral probabilities; (b) actual probabilities.1

- Page 3 and 4: (a) S n under P; (b) S n under ˜P;
- Page 5 and 6: 4.2 Problems7. Determine if it is a