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*Options, Futures, and Other Derivatives, 10th Edition
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CHAPTER 28

Martingales and Measures

Practice Questions

Problem 28.1.

How is the market price of risk defined for a variable that is not the price of an investment asset?

The market price of risk for a variable that is not the price of an investment asset is the market price of risk of an investment asset whose price is instantaneously perfectly positively correlated with the variable.

Problem 28.2.

Suppose that the market price of risk for gold is zero. If the storage costs are 1% per annum and the risk-free rate of interest is 6% per annum, what is the expected growth rate in the price of gold? Assume that gold provides no income.

If its market price of risk is zero, gold must, after storage costs have been paid, provide an expected return equal to the risk-free rate of interest. In this case, the expected return after

storage costs must be 6% per annum. It follows that the expected growth rate in the price of gold must be 7% per annum.

Problem 28.3.

Consider two securities both of which are dependent on the same market variable. The expected returns from the securities are 8% and 12%. The volatility of the first security is 15%. The instantaneous risk-free rate is 4%. What is the volatility of the second security?

The market price of risk is r μσ- This is the same for both securities. From the first security we know it must be 008004026667015.-.=.. The volatility, σ for the second security is given by 012004026667σ.-.=. The volatility is 30%.

Problem 28.4.

An oil company is set up solely for the purpose of exploring for oil in a certain small area of Texas. Its value depends primarily on two stochastic variables: the price of oil and the quantity of proven oil reserves. Discuss whether the market price of risk for the second of these two variables is likely to be positive, negative, or zero.

It can be argued that the market price of risk for the second variable is zero. This is because the

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