Solution of Derivative

Therefore we will sell the forward at 1120 and create

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Unformatted text preview: e time to expiration in years into the valuation formula and notice that the time to expiration is 6 months, or 0.5 years. We have: F0,T = S 0 × e (r −δ )×T = $1,100 × e (0.05 −0.02 )×0.5 = $1,100 × 1.01511 = $1,116.62 a) If we observe a forward price of 1,120, we know that the forward is too expensive, relative to the fair value we have determined. Therefore, we will sell the forward at 1,120, and create a synthetic forward for 1,116.82, making a sure profit of $3.38. As we sell the real forward, we engage in cash and carry arbitrage: 36 Chapter 5 Financial Forwards and Futures Description Today Short forward 0 Buy tailed position in − $1,100 × .99 index = −$1,089.055 $1,089.055 Borrow $1,089.055 TOTAL 0 In 9 months $1,120.00 − ST ST − $1,116.62 $3.38 This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. b) If we observe a forward price of 1,110, we know that the forward is too cheap, relative to the fair value we have determined. Therefore, we will buy the forward at 1,110, and create a synthetic short forward for 1116.62, thus making a sure profit of $6.62. As we buy the real forward, we engage in a reverse cash and carry arbitrage: Description Today Long forward 0 Sell short tailed position in $1,100 × .99 index = $1,089.055 − $1,089.055 0 Lend $1,089.055 TOTAL In 9 months ST − $1,110.00 − ST $1,116.62 $6.62 This position requires no initial investment, has no index price risk, and has a strictly positive payoff. We have exploited the mispricing with a pure arbitrage strategy. Question 5.10. a) We plug the continuously compounded interest rate, the forward price, the initial index level and the time to expiration in years into the valuation formula and solve for the dividend yield: F0,T = S 0 × e (r −δ )×T ⇔ F0,T S0 = e (r −δ )×T ⇔ F ln 0,T S 0 = (r − δ ) × T ⇔ δ =r− 1 F0,T ln T S0 ⇒ δ = 0.05 − 1 1129.257 ln = 0.05 − 0.035 = 0.015 0.75 1100 37 Part 2 Forwards, Futures, and Swaps Remark: Note that this result is consistent with exercise 5.6., in which we had the same forward prices, time to expiration etc. b) With a dividend yield of only 0.005, the fair forward price would be: F0,T = S 0 × e (r −δ )×T = 1,100 × e (0.05−0.005 )×0.75 = 1,100 × 1.0343 = 1,137.759 Therefore, if we think the dividend yield is 0.005, we consider the observed forward price of 1,129.257 to be too cheap. We will therefore buy the forward and create a synthetic short forward, capturing a certain amount of $8.502. We engage in a reverse cash and carry arbitrage: Description Today Long forward 0 Sell short tailed position in $1,100 × .99626 index = $1,095.88 Lend $1,095.88 TOTAL c) − $1,095.88 0 In 9 months ST − $1,129.257 − ST $1,137.759 $8.502 With a dividend yield of 0.03, the fair forward price would be: F0,T = S 0 × e (r −δ )×T = 1,100 × e (0.05−0.03 )×0.75 = 1,100 × 1.01511 = 1,116.62 Therefore, if we think the dividend yield is 0.03, we consider the observed forward price of 1,129.257 to be too expensive. We will therefore sell the forward and create a synthetic long forward, capturing a certain amount of $12.637. We engage in a cash and carry arbitrage: Description Today In 9 months Short forward 0 $1,129.257 − ST Buy tailed position in − $1,100 × .97775 ST index = −$1,075.526 $1,075.526 $1,116.62 Borrow $1,075.526 TOTAL 0 $12.637 Question 5.12. a) The notional value of 10 contracts is 10 × $250 × 950 = $2,375,000 , because each index point is worth $250, we buy 10 contracts and the S&P 500 index level is 950. With an initial margin of 10% of the notional value, this results in an initial dollar margin of $2,375,000 × 0.10 = $237,500 . 38 Chapter 5 Financial Forwards and Futures b) We first obtain an approximation. Because we have a 10% initial margin, a 2% decline in the futures price will result in a 20% decline in margin. As we will receive a margin call after a 20% decline in the initial margin, the smallest futures price that avoids the maintenance margin call is 950 × .98 = 931 . However, this calculation ignores the interest that we are able to earn in our margin account. Let us now calculate the details. We have the right to earn interest on our initial margin position. As the continuously compounded interest rate is currently 6%, after one week, our initial margin has grown to: $237,500e 0.06× 1 52 = $237,774.20 We will get a margin call if the initial margin falls by 20%. We calculate 80% of the initial margin as: $237,500 × 0.8 = $190,000 10 long S&P 500 futures contracts obligate us to pay $2,500 times the forward price at expiration of the futures contract. Therefore, we have to solve the following equation: $237,774.20 + (F1W − 950) × $2,500 ≥ $190,000 ⇔ $47774.20 ≥ −(F1W − 950) × $2,500 ⇔ 19.10968 − 950 ≥ − F1W ⇔ F1W ≥ 930.89 Therefore, the greatest S&P 500 index futures price at which we will receive a margin call is 930.88. Question 5.14. An arbitrageur believing that the observed forward price, F(0,T), is too low will undertake a reverse cash and carry arbitrage: Buy the forward, short sell the stock and lend out the proceeds from the short sale. The relevant prices are therefore the bid price of the...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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