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Unformatted text preview: 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: F ,T = S × e (r − δ l ) × T ⇔ F ,T S = e (r − δ l ) × T ⇔ ln µ F ,T S ¶ = (r − δ l ) × T ⇔ δ l = r − 1 T ln µ F ,T S ¶ 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 risk-free rate of 2%, we can again calculate the lease rate according to the formula: δ l = r − 1 T ln µ F ,T S ¶ Time to expiration Forward price Annualized lease rate 3 months $31.37 0.0995355 6 months $30.75 0.0996918 9 months $30.14 0.0998436 12 months $29.54 0.0999906 80 Chapter 6 Commodity Forwards and Futures The lease rate is higher than the risk-free interest rate. The forward curve is downward sloping, thus the prices of exercise 6.2. are an example of backwardation. Question 6.3. The question asks us to ±nd the lease rate such that F ,T = S . We take our pricing formula, F ,T = S × e (r − δ l ) × T , and immediately see that the sought equality is established if e (r − δ l ) × T = 1, which is guaranteed for any T > 0 if and only if r = δ . If the lease rate were 3.5%, the lease rate would be higher than the risk-free interest rate. Therefore, a graph of forward prices would be downward sloping, and thus there would be backwardation. Question 6.4. a) As we need to borrow a pencil to sell it short, we must pay the lender the lease rate for the time we borrow the asset, i.e., until expiration of the contract in one year. After one year, we have to pay back one pencil, which will cost us S T , the uncertain future pencil price, plus the leasing costs: Total payment = S T + S T × ( e . 05 − 1 ) = S T e . 05 = 1 . 05127 × S T . It does not make sense to store pencils in equilibrium, because even if we have an active lease market for pencils, the lease rate is smaller than the risk-free interest rate. Lending money at ten percent is more pro±table than lending pencils at ±ve percent. b) The equilibrium forward price is calculated according to our pricing formula: F ,T = S × e (r − δ l ) × T = $0 . 20 × e ( . 10 − . 05 ) × 1 = $0 . 20 × 1 . 05127 = $0 . 2103 , which is the price given in the exercise. c) Let us ±rst look at the different arbitrage strategies we can use in each case. c1) Pencils can be sold short. We can engage in our usual reverse cash and carry arbitrage: Transaction Time 0 Time T = 1 Long forward S T − F ,T Short-sell tailed pen-$0.19025 − S T cil position, @ 0.05 Lend short-sale − $0.19025 $0.2103 proceeds @ 0.1 Total $0 . 2103 − F ,T For there to be no arbitrage, F ,T ≥ $0 . 2103 c2) Suppose pencils cannot be sold short. Then we have no ability to create the short position necessary to offset the pencil price risk from the long forward. Consequently, we are not able to ±nd a lower boundary for the pencil forward in this case....
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- Spring '06
- Forward contract, ΔL, lease rate, $0.2210 0