Derivatives and Risk Management Techniques Practice Problem Set
1. A company enters into a long futures contract to buy 50,000 units of a commodity for 90 cents per unit. The initial margin is $12,000 and the maintenance margin is $11,000. What is the futures price per unit below which there will be a margin call?
2. On March 1 the price of oil is $55.00 and the July futures price is $54.00. On June 1 the price of oil is $64.00 and the July futures price is $61.50. A company entered into a futures contracts on March 1 to hedge the purchase of oil on June 1. It closed out its position on June 1. After taking account of the cost of hedging, what is the effective price paid by the company for the oil?
3. Suppose that the standard deviation of monthly changes in the price of jet fuel is $4.18. The standard deviation of monthly changes in a futures price for a contract on heating oil (which is similar to jet fuel) is $3.30. The correlation between the futures price and the commodity price is 0.95. What hedge ratio should be used when hedging an exposure to the price of jet fuel?
The next 3 questions are connected.
4. Consider a soybean farmer who wants to hedge his price risk using the CBOT soybean futures contract. He will harvest his crop of 7 bushels in 4 months from now and plans to sell it immediately afterwards. The soybean futures contracts that are currently available in the markets have maturities of 3 months (futures A), 6 months (futures B) and 9 months (futures C). If the farmer's main objective is to minimize his overall basis risk, what strategy should he follow? You may assume that all three futures contracts are equally liquid.
(a) Sell futures B now, close out the position after four months and sell the soybeans in the market.
(b) Sell futures A now and roll-over the position into futures B after three months. After four months, close out the futures position and sell the soybeans in the market.
(c) Sell futures C now, close out the position after four months and sell the soybeans in the market.
(d) Sell futures A now and keep the position until the contract's maturity and deliver the soybeans.
(e) None of the above is a suitable hedging strategy
5. The CBOT soybean futures contract has a contract size of 5,000 bushels, and it is not possible to trade fractions of a contract. One of the farmer's friends is a statistician who estimated a correlation of 0.85 between the relative changes in the soybean spot and futures prices as well as volatilities of 0.70 and 0.80. What is the hedge ratio and how many futures should the farmer use for hedging if his main objective is to minimize the risk of his position?
6. Assume that the initial futures and spot prices per bushel are $1.42 and $1.43 and after four months they are $1.23 and $1.18. What is the effective price per bushel that the farmer sold his crop for?
(a) The effective price is $1.42 per bushel.
(b) The effective price is $1.37 per bushel.
(c) The effective price is $1.18 per bushel.
(d) The effective price is $1.32 per bushel.
(e) None of the above is true.
7. A fund manager wants to hedge the exposure of her US equity portfolio using futures contracts on the S&P 500. Her portfolio has a value of 15 million USD and a beta of 1.80. The S&P 500 futures is currently trading at 1,425 (multiplier 250). Assume that the hedging maturity is the same as the contract maturity. What should she do to reduce portfolio risk to a beta of 0.70?
8. Consider a futures contract that matures in T years. The current price of the underlying asset is S0. The current futures price is F0. The size of contract is one unit of the underlying asset. Suppose the underlying asset is an index with known dividend yield, q. You have found that the market futures price, F0, is greater than the price implied by the spot price. You can take advantage of this situation by
(a) selling one futures contract, buying 1 share of the index and re-investing the dividends in the index, and borrowing S0 dollars at the risk-free rate.
(b) selling one futures contract, shorting 1 share of the index and reinvesting the dividends in the index, and lending S0 dollars at the risk-free rate.
(c) selling one futures contract, buying e-qT shares of the index and reinvesting the dividends in the index, and borrowing S0e-qT dollars at the risk-free rate.
(d) selling one futures contract, buying e-qT shares of the index and reinvesting the dividends in the index, and lending S0e-qT dollars at the risk-free rate.
(e) none of the above.
9. Consider a currency swap with a remaining life of two years. It involves exchanging interest of 4.50% on EUR 600 for interest at 7.50% on A$954 annually. Both interest rates are annually compounded.
The (continuously compounded) risk-free rate is 7.39% in Australia and 3.60% in Europe (and both term structures are flat). The spot exchange rate is S0 = 1.5900 AUD/EUR. What is the value of the swap agreement, V0, to the party receiving the AUD?
10. Company X has entered into a swap agreement in which it is to receive the (continuously compounded) 1-year LIBOR and pay a (continuously compounded) fixed rate of 6.00% (per annum) on a principal of $100. Cash flows are exchanged once a year and the agreement stretches over two years, i.e. over two cash flow exchanges with the first occuring in one year's time. The current (continously compounded) 1-year LIBOR rate is 4.00% (per annum). The term structure is flat. What are the cash flows for company X in one year's time?
(a) Receive an amount that is unknown today, and pay $6.18.
(b) Receive $4.08, and pay $6.18.
(c) Receive $4.00, and pay $6.00.
(d) Receive an amount that is unknown today, and pay $6.00.
(e) None of the above.
11. Company X has entered into a swap agreement in which it is to receive the (continuously compounded) 1-year LIBOR and pay a (continuously compounded) fixed rate of 5.00% (per annum) on a principal of $1,000. Cash flows are exchanged once a year and the agreement stretches over two years, i.e. over two cash flow exchanges with the first occuring in one year's time. The initial (continously compounded) 1-year LIBOR rate was 3.00% (per annum). The term structure is flat. 6 months into the contract, the term structure is still flat, with LIBOR now at 3.12%. What is the value of the swap agreement to company X at this point in time?
The next 6 questions are related. You are working on the currency swap desk at a major Australian bank. A corporate client approaches you to configure a swap for him. The client has issued a T = 5-year bond in USD for Pf = 100m USD paying cf = 6.00% of interest in semi-annual installments. He now would like you to swap this outflow into AUD paying a coupon ch on a notional of Ph (AUD). The USD term structure is flat at rf = 4.00% (continuously compounded), and AUD discount rates are rh = 6.50% for all maturities. The exchange rate is currently S0 = 1.2800 AUD/USD.
12. Compute the AUD price Bf,AUD = S0Bf,USD of the bond that will form the USD leg of the swap.
(a) PV of foreign CF = $108.79m USD ( $139.25m AUD).
13. Compute what the notional of the AUD should be set to if the client requests it to be equal to the USD notional under today's AUD/USD forward rate.
(a) Forward rate is 1.4504. Forward equivalent notional = 145.04m AUD.
14. Without plugging in numbers and using the fact that swaps are constructed to be of zero value at initiation as well as the short cut formula for geometric series (appendix), derive an expression for ch, the annual coupon paid (in 2 installments) on the AUD leg of the swap as function of S0, Bf,USD, rf , rh, T under the assumption made above.
(a) Start with the fundamental swap pricing identify: Bh = S0Bf. Plug in the formula for the bond Bh, use that we require Ph = PfS0e(r_h-r_f)T and solve for c to get:
ch = ((Bf/Pf)e-(r_f -r_h)T - e-r_h T/½[e-r_h(e-r_h T - 1/e-r_h -1)])
15. Compute this coupon.
(a) Coupon rate that prices the swap to zero is 5.66%.
16. Instead of setting the notional and computing the coupon, the client would like to choose the coupon. Derive a formula for the notional of the AUD leg as function of S0, Bf,USD, rh, T, ch, again without plugging in numbers.
(a) This is much more direct. Simply solve for notional and get
Ph = (S0Bf)/(c/2)[e-r_h(e-r_h T -1/e-r_h-1] + e-r_h T
17. Assuming the client wants to pay no coupons on the AUD leg, how large does the final payment on the AUD leg have to be?
(a) Assuming c = 0.00%, the notional has to be 192.73m AUD.
18. If a trader expects that a stock will either have a large up or down move in the near future, which of the following strategies would help him to take advantage of his expectation?
(a) Long a butterfly spread.
(b) Short a strangle.
(c) Long a straddle.
(d) All of the above.
(e) None of the above.
19. A trader buys 100 European call options (i.e., one contract) with a strike price of $60 and a time to maturity of 6 months. The price received for each option is $8.82. At maturity, the price of the underlying asset is $57. What is the traders gain or loss? Show a dollar amount and indicate whether it is a gain or a loss.
(a) The call option expires OTM.
(b) The buyer of 100 call options with price $8.82 experiences a nominal loss of $882.
20. The price of a stock is $53 and the price of a three-month call option on the stock with a strike price of $53 is $2.65. Suppose a trader has $5,300 to invest and is trying to choose between buying 2,000 options and 100 shares of stock. How high does the stock price have to rise for an investment in options to lead to the same profit as an investment in the stock?
21. In a 1-period binomial tree for a put option with S0 = $80 and X = $60, the two possibilities for ST are $120 and $50. The payoffs across the two states are ______ and ______; the hedge ratio is ______.
(a) $0 and $10; -0.143
(b) $0 and $10; 0.857
(c) $0 and $30; -0.857
(d) $0 and $30; -0.429
(e) $0 and $10; 0.143
22. If the hedge ratio for a put on a stock is -0.27, what would be the hedge ratio for a call with the same expiration date and exercise price as the put?
(a) -0.27
(b) 0.73
(c) -0.73
(d) 0.27
(e) None of the above.
STOCHASTIC PROCESSES
The following applies to the next two questions. A stochastic process S starts at 81 and follows a generalized Wiener process dx = adt + bdz . During the first two years, a = 5 and b = 1. During the following three years, a = -1 and b = 5.
1. What is the expected value of S after five years?
(a) 7
(b) 50
(c) 88
(d) 85
(e) None of the above
2. What is the standard deviation of the value of S after five years?
(a) 4.12
(b) 7.28
(c) 8.77
(d) 5.10
(e) None of the above
3. A company's cash position, currently at $3.7m, follows a generalized Wiener process falling on average $0.2m per quarter with volatility of $1.2m per quarter. With what probability does the company find itself with a negative cash balance at the end of 5 quarters?
(a) 0.3538
(b) 0.1572
(c) 0.2422
(d) 0.9601
(e) None of the above.
4. A company's cash position (measured in millions of dollars) follows a generalized Wiener process growing on average $0.2m per quarter with volatility of $1.0m per quarter. How high must the initial cash balance be for the probability of a negative balance after 2.00 years not to exceed 5%?
(a) $3.052m
(b) $2.452m
(c) $1.926m
(d) $4.252m
(e) None of the above.
5. The share price of company XYZ Inc. exhibits an instantaneous drift of 17% per year with a return volatility of 39%. What is the probability that XYZ shares exceed $151 after 14 months when they cost $85 today?
(a) Probability = 13.57%.
6. A stock price has an (continuously compounded) expected return of 5% per annum and a volatility of 20% per annum. Currently the stock price is $85. Assume 252 trading days per year. What is the 90% confidence interval for the stock price after 5 trading days?
(a) $81.20; $89.08
7. The price process of a dividend-paying asset S can be described by a Geometric Brownian Motion, i.e. dS = (µ-q)Sdt+σSdz . Using Ito's Lemma, derive the price process of the Futures contract on S. What would this look like under the risk-neutral measure? What is the intuitive explanation for the latter observation?
(a) Compare your results to slides of Week 5 (or so). As for intuition, in a risk-neutral world something that does not require any cash investment should not yield positive returns.
Attachment:- Assignment File.rar