Solutions for Mcdonald Chapter 6 Essays

2816 Words Jan 19th, 2013 12 Pages
Chapter 6
Commodity Forwards and Futures
Question 6.1. The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of 5%, we can calculate the annualized lease rates according to the formula: F0,T = S0 × e(r−δl )×T ⇔ F0,T S0 = e(r−δl )×T S0 = (r − δl ) × T F0,T 1 ln T S0

⇔ ln

F0,T

⇔ δl = r −

Time to expiration Forward price Annualized lease rate 3 months $70.70 0.0101987 6 months $71.41 0.0101147 9 months $72.13 0.0100336 12 months $72.86 0.0099555 The lease rate is less than the risk-free interest rate. The forward curve is upward sloping, thus the prices of exercise 6.1. are an example of contango. Question 6.2. The spot price of oil is $32.00 per barrel. With a continuously compounded annual
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a) The spot price of gold is $300.00 per ounce. With a continuously compounded annual riskfree rate of 5%, and at a one-year forward price of 310.686, we can calculate the lease rate according to the formula: δl = r − b) F0,T 1 ln T S0 = 0.05 − ln 310.686 300 = 0.015 Lower bound on forward F0,T ≥ $0.2103 F0,T ≥ $0.2103 — — Upper bound on forward F0,T ≤ $0.2103 F0,T ≤ $0.2210 F0,T ≤ $0.2103 F0,T ≤ $0.2210

Suppose gold cannot be loaned. Then our cash and carry “arbitrage” is: Transaction Short forward Buy gold Borrow @ 0.05 Total Time 0 Time T = 1 0 310.686 − ST −$300 ST $300 −$315.38 0 −4.6953
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Chapter 6 Commodity Forwards and Futures

The forward price bears an implicit lease rate. Therefore, if we try to engage in a cash and carry arbitrage, but if we do not have access to the gold loan market, and thus to the additional revenue on our long gold position, it is not possible for us to replicate the forward price. We incur a loss. c) If gold can be loaned, we engage in the following cash and carry arbitrage: Transaction Time 0 Time T = 1 Short forward 0 310.686 − ST Buy tailed gold −$295.5336 ST position, lend

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