Solution of Derivative

2210 question 66 a the forward prices reflect a

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Unformatted text preview: ls cannot be loaned. Then our cash and carry arbitrage becomes: Transaction Short forward Time 0 0 Time T=1 F0,T − S T Buy pencil borrow @ 0.1 Total -$0.20 $0.20 0 ST -$0.2210 F0,T − $0.2210 For there to be no arbitrage, F0,T ≤ $0.2210 Therefore, we have found the following no-arbitrage regions: 43 Part 2 Forwards, Futures, and Swaps loan=yes, short-sale=yes Lower bound on forward F0,T ≥ $0.2103 Upper bound on forward F0,T ≤ $0.2103 loan=no, short-sale=yes F0,T ≥ $0.2103 F0,T ≤ $0.2210 loan=yes, short-sale=no ---- F0,T ≤ $0.2103 loan=no, short-sale=no ---- F0,T ≤ $0.2210 Question 6.6. a) The forward prices reflect a market for widgets in which seasonality is important. Let us look at two examples, one with constant demand and seasonal supply, and another one with constant supply and seasonal demand. One possible explanation might be that widgets are extremely difficult to produce and that they need one key ingredient that is only available during July/August. However, the demand for the widget is constant throughout the year. In order to be able to sell the widgets throughout the year, widgets must be stored after production in August. The forward curve reflects the ever increasing storage costs of widgets until the next production cycle arrives. Once produced, widget prices fall again to the spot price. Another story that is consistent with the observed prices of widgets is that widgets are in particularly high demand during the summer months. The storage of widgets may be costly, which means that widget producers are reluctant to build up inventory long before the summer. Storage occurs slowly over the winter months and inventories build up sharply just before the highest demand during the summer months. The forward prices reflect those storage cycle costs. b) Let us take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate with our cash and carry arbitrage tableau: Transaction Short March forward Buy December Forward (=Buy spot) Pay storage cost Total Time 0 0 -3.00 Time T=3/12 3.075 − ST ST -3.00 -0.03 3.045 44 Chapter 6 Commodity Forwards and Futures We can calculate the annualized rate of return as: ⇔ 3.045 = e (r )×T 3.00 3.045 ln = r × 3 / 12 3.00 r = 0.05955 which is the prevailing risk-free interest rate of 0.06. This result seems to make sense. c) Let us again take the December 2001 forward price as a proxy for the spot price in December 2001. We can then calculate with our cash and carry arbitrage tableau: Transaction Short Sep forward Buy spot Pay storage cost Sep FV(Storage Jun) FV(Storage Mar) Total Time 0 0 -3.00 Time T=9/12 2.75 − ST ST -0.03 -0.0305 -0.0309 2.6586 -3.00 We can calculate the annualized rate of return as: 2.6586 = e (r )×T ⇔ 3.00 r = −0.16108 2.6586 ln = r × 9 / 12 3.00 This result does not seem to make sense. We would earn a negative annualized return of 16 % on such a cash and carry arbitrage. Therefore, it is likely that our naive calculations do not capture an important fact about the widget market. In particular, we will buy and hold the widget through a time where the forward curve indicates that there is a significant convenience yield attached to widgets. It is tempting, although premature, to conclude that a reverse cash and carry arbitrage may make a positive 16 % annualized return. Question 6.7. deals with this aspect. 45 Part 2 Forwards, Futures, and Swaps Question 6.8. a) The first possibility is a simple cash and carry arbitrage: Transaction Short March forward Buy December Forward (=Buy spot) Borrow @ 6% Pay storage cost Total Time 0 0 -3.00 Time T=3/12 3.10 − S T ST +3.00 -3.045 -0.03 +0.025 0 The second possibility involves using the June futures contract. It is a forward cash and carry strategy: Transaction Short March forward Buy June Forward Lend @ 6% Receive storage cost Total Time = T(1) = 3/12 3.10 − S T (1) 0 -3.10 0 Time = T(2) = 6/12 − ST ( 2 ) S T ( 2 ) − 3.152 3.1469 +0.03 +0.02485 We can use the June futures in our calculations and claim to receive storage costs, because it is easy to show that the value of it is reflecting the negative lease rate of the storage costs. b) It is not possible to undertake an arbitrage with the futures contracts that expire prior to September 2002. A decrease in the September futures value means that we would need to buy the September futures contract, and any arbitrage strategy would need some short position in the widget. However, the drop in the futures price in September indicates that there is a significant convenience yield factored into the futures price over the period June - September. As we have no information about it, it is not possible for us to guarantee that we find a lender of widgets at a favorable lease rate to follow through our arbitrage trading program. The decrease in the September futures may in fact reflect an increase in the opportunity costs of widget owners. Question 6.10. Our best bet for the current spot price is the first available forward price,...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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