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Problem Set 2

Derivative Securities

Instructions: This problem set is due on 12/1 at 11:59 pm Central Time and is an individual assignment. All problems must be handwritten. Write full sentences to answer questions that require an explanation. Scan your work and submit a PDF file.

Problem 1 A non-dividend-paying stock trades for $60, has a volatility of 30% per year, and has an expected return of \mu = 12\% per year. The risk-free rate is 4% per year with continuous compounding. Consider a European call on the stock with strike $65 that expires in six months.

  1. Use the Black-Scholes model to compute the price of the call.
  2. Compute the risk-neutral probability that the call finishes in the money, and the probability under the real distribution of the stock price.
  3. The expected payoff of the call under the real distribution is \operatorname{E}\left(\max(S_{T} - K, 0)\right) = S e^{\mu T} \mathop{\Phi}(\hat{d}_{1}) - K \mathop{\Phi}(\hat{d}_{2}), where \hat{d}_{1} and \hat{d}_{2} are d_{1} and d_{2} with \mu in place of r. Compute this expected payoff and discount it at the risk-free rate. Is the result above or below the Black-Scholes price? Explain why the expected payoff of the call should not be discounted at the risk-free rate.

Problem 2 A stock index trades at 100 and pays no dividends, and the risk-free rate is 4% per year with continuous compounding. The table below gives Black-Scholes prices of a three-month at-the-money call with strike 100 and a three-month put with strike 85 for several volatilities.

\sigma 15% 18% 21% 24% 27% 30% 33%
Call (K = 100) 3.50 4.09 4.68 5.27 5.87 6.46 7.05
Put (K = 85) 0.03 0.09 0.20 0.37 0.59 0.85 1.15

In the market, the call trades for $4.09 and the put trades for $0.85.

  1. Find the implied volatility of each option.
  2. According to the Black-Scholes model, what should the two implied volatilities be? What does the gap tell you about how the market prices the risk of a large fall in the index?
  3. A dealer who prices the put with the implied volatility of the call would quote it at $0.09. Who would buy puts from this dealer, and what risk would the dealer end up holding?

Problem 3 A portfolio manager holds $67 million in a fund that tracks the S&P 500 index. The index stands at 6,700, has a dividend yield of 1.3% per year and a volatility of 18% per year, and the risk-free rate is 4% per year with continuous compounding. To protect the portfolio over the next three months, the manager buys a put spread on SPX: long puts with strike 6,500 and short puts with strike 6,000. Each SPX contract is written on 100 times the index.

  1. How many contracts of each put does the manager need to cover the portfolio?
  2. Compute the price of each put and the total cost of the put spread.
  3. What is the largest payoff the spread can deliver, and what is the risk-neutral probability that it pays this maximum?
  4. Explain what protection the manager gives up by selling the puts with strike 6,000, and why the manager might accept that.

Problem 4 A German exporter will receive USD 10,000,000 from a US customer in one year. The EUR/USD spot rate is $1.16 per euro, the volatility of the exchange rate is 8% per year, and the interest rates are 3.75% per year in dollars and 2% per year in euros, both continuously compounded.

  1. Compute the one-year forward EUR/USD rate.
  2. Compute the prices of a one-year EUR call / USD put and a EUR put / USD call, both struck at the forward rate, per euro of notional. Explain why the two prices are equal.
  3. The exporter worries that the dollar will weaken. Which of the two options protects the exporter? The exporter buys that option with a strike of $1.20 per euro on the notional that converts USD 10,000,000 at the strike. Compute the notional in euros and the premium, in dollars and in euros at today’s spot rate.
  4. Show that the price of the call can be written as C = e^{-r T} \left[F \mathop{\Phi}(d_{1}) - K \mathop{\Phi}(d_{2})\right], where F is the forward rate and d_{1} = \dfrac{\ln(F/K) + \frac{1}{2} \sigma^{2} T}{\sigma \sqrt{T}}.

Problem 5 A farmer expects to sell 50,000 bushels of corn in three months. The December corn futures trades at $4.50 per bushel with a volatility of 25% per year, and the risk-free rate is 4% per year. To protect the sale, the farmer buys puts on the futures with strike $4.20 and sells calls on the futures with strike $4.90. The options expire in three months and each contract is written on 5,000 bushels. Assume the farmer sells the corn at the futures price on the expiration date of the options, ignoring the basis between the local cash price and the futures.

  1. Use Black’s model to compute the price of each option and the net cost of the collar for the 50,000 bushels.
  2. What is the range of prices per bushel, net of the cost of the collar, that the farmer locks in?
  3. The dealer who sold the puts to the farmer and bought the calls hedges with corn futures. Compute the delta of the dealer’s position per bushel and the number of futures contracts the dealer buys or sells.

Normal Distribution Table

The table below computes \Phi(z) = \operatorname{P}(Z \leq z) if Z \sim \mathcal{N}(0, 1) and z \geq 0. The rows denote the first decimal whereas the columns denote the second decimal. If you need \Phi(-z), you can always use \Phi(-z) = 1 - \Phi(z).

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990