Problem 1. Charles Schwab offers investors an equity-linked Certificate of Deposit, issued by First Union

National Bank. This product guarantees to repay the invested amount after 5.5 years, plus 70% of the

simple appreciation in the S&P 500 over that time.

(a) Suppose the S&P 500 index is 1400 at the time you invest $200,000 in the equity-linked CD.

Show that the equity-linked CD is equivalent to a combination of a zero-coupon bond and certain

number of call options on the S&P 500 index. What is the face value of the bond? What is the

number of the options?

(b) Suppose the 5.5-year interest rate is 5%, the volatility of the index is 25%, and the dividend yield

is 2%. What is the value of the equity-linked CD that you invest?

(c) What is the implicit commission you pay to the bank as a percent of your total investment in the

equity-linked CD?

(d) Describe in detail how the First Union National Bank should hedge the exposure to the risk, assuming

it is possible to buy long-term index options from a broker-dealer.

(e) If the broker-dealer charges a fee, which is 3% of the fair value (i.e., Black-Scholes value) of the

options, how much commission (as a percent of your total investment) the bank can realize after

hedging?

Problem 2. Live cattle refers to cattle (cows) that have reached the requisite weight for slaughter. This is different from feeder cattle, which are the younger animals that are still in the process of being fed into fished ?fed? cattle. Live-cattle futures are traded on the Chicago Mercantile Exchange. Each contract of live-cattle futures is for the delivery of 40,000 pounds, but the price is quoted in cents per pound. The delivery months are February, April, June, August, October, and December. Live-cattle options are American options on the live-cattle futures price. Suppose your only position in live cattle is that you have already sold a call option contract on April live-cattle futures with a strike price of 130 cents per pound. Suppose the futures price is now 135 cents and the most recent settlement price is 133 cents. (a) What happens to your margin account if the option contract is exercised now? (b) What happens to your position in live cattle if the option contract is exercised now? (c) Suppose the option is exercised now, but you do not close out your positions in live cattle. In addition to the change to your margin account in (a), how much more do you gain or lose on the April live-cattle futures at the end of the day if the live-cattle futures price settles at 134?

Problem 3. A one-year American call option on silver futures has an exercise price of $9.00. The current futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of the futures price is 25% per annum. (a) Create a four-step tree for the silver futures price. (b) On each node of the tree, what is the risk-neutral probability for the silver futures price to go up? (c) What is the value of the call option on the silver futures if the option is priced in a four-step binomial model? (d) Is it optimal for this futures American call option to exercise early on some nodes of the price tree? Is your answer consistent or in conflict to the property of no early exercise of American call options without expected dividend for the underlying?

Problem 4. To answer the questions in this problem, you need to read the article, ?British Petroleum Company Ltd. (1987 Stock Offering).? The article contains all the data needed for solving this problem. (a) The repurchase plan of the Bank of England can be viewed as a put option on BP shares held by underwriters. Although this is an american option, let us treat it as a European option here so that you can value the option by Black-Scholes formula. If you want to value the option on October 30, 1987, what are the time to maturity and strike price of the put option? (b) According to many analysts at the time, the volatility of BP share price was 85% shortly after the 1987 crash of the market. If you use Black-Scholes formula, how much was the repurchase plan worth to each of the four U.S. underwriters at the end of day on October 30, 1987? (c) Based on the stock prices and beta given in the article, what is the excess daily return on the portfolio of the three publicly-traded underwriters from October 29 to October 30, 1987? (Use the CAPM to calculate the excess return and assume that one-day risk-free interest rate is zero.) (d) If you use the 85% volatility and Black-Scholes formula, what proportion in the excess daily return calculated in (c) can be explained by the repurchase plan value calculated in (b)?

Problem 5. The stock of Binomial Hedge Inc. is currently priced at $680 with volatility of 30%. The risk-free interest rate is 2%. Consider a binomial tree of this stock price with three-month time steps. Also consider a European call option that strikes at $700 and matures in six months with two-step tree below. (a) What are the stock prices and option values on the nodes of the tree? (b) What are the delta and share purchase/sale on each node of the tree? (c) What is the position in cash on each node of the tree? (d) What is the net value of the hedged portfolio on each node of the tree? (e) What is the average performance ratio of the hedging strategy? What is the standard deviation of the performance ratio?

PLEASE see chart for Q5 in attachment

Derivative Securities F421 (2017 Spring) Assignment 5 Due: 4/2/Sun Problem 1. Charles Schwab offers investors an equity-linked Certificate of Deposit, issued by First Union National Bank. This product guarantees to repay the invested amount after 5.5 years, plus 70% of the simple appreciation in the S&P 500 over that time. (a) Suppose the S&P 500 index is 1400 at the time you invest $200,000 in the equity-linked CD. Show that the equity-linked CD is equivalent to a combination of a zero-coupon bond and certain number of call options on the S&P 500 index. What is the face value of the bond? What is the number of the options? (b) Suppose the 5.5-year interest rate is 5%, the volatility of the index is 25%, and the dividend yield is 2%. What is the value of the equity-linked CD that you invest? (c) What is the implicit commission you pay to the bank as a percent of your total investment in the equity-linked CD? (d) Describe in detail how the First Union National Bank should hedge the exposure to the risk, assuming it is possible to buy long-term index options from a broker-dealer. (e) If the broker-dealer charges a fee, which is 3% of the fair value (i.e., Black-Scholes value) of the options, how much commission (as a percent of your total investment) the bank can realize after hedging? Problem 2. Live cattle refers to cattle (cows) that have reached the requisite weight for slaughter. This is different from feeder cattle, which are the younger animals that are still in the process of being fed into fished \"fed\" cattle. Live-cattle futures are traded on the Chicago Mercantile Exchange. Each contract of live-cattle futures is for the delivery of 40,000 pounds, but the price is quoted in cents per pound. The delivery months are February, April, June, August, October, and December. Live-cattle options are American options on the live-cattle futures price. Suppose your only position in live cattle is that you have already sold a call option contract on April live-cattle futures with a strike price of 130 cents per pound. Suppose the futures price is now 135 cents and the most recent settlement price is 133 cents. (a) What happens to your margin account if the option contract is exercised now? (b) What happens to your position in live cattle if the option contract is exercised now? (c) Suppose the option is exercised now, but you do not close out your positions in live cattle. In addition to the change to your margin account in (a), how much more do you gain or lose on the April live-cattle futures at the end of the day if the live-cattle futures price settles at 134? Problem 3. A one-year American call option on silver futures has an exercise price of $9.00. The current futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of the futures price is 25% per annum. (a) Create a four-step tree for the silver futures price. (b) On each node of the tree, what is the risk-neutral probability for the silver futures price to go up? (c) What is the value of the call option on the silver futures if the option is priced in a four-step binomial model? (d) Is it optimal for this futures American call option to exercise early on some nodes of the price tree? Is your answer consistent or in conflict to the property of no early exercise of American call options without expected dividend for the underlying? Problem 4. To answer the questions in this problem, you need to read the article, \"British Petroleum Company Ltd. (1987 Stock Offering).\" The article contains all the data needed for solving this problem. 1 (a) The repurchase plan of the Bank of England can be viewed as a put option on BP shares held by underwriters. Although this is an american option, let us treat it as a European option here so that you can value the option by Black-Scholes formula. If you want to value the option on October 30, 1987, what are the time to maturity and strike price of the put option? (b) According to many analysts at the time, the volatility of BP share price was 85% shortly after the 1987 crash of the market. If you use Black-Scholes formula, how much was the repurchase plan worth to each of the four U.S. underwriters at the end of day on October 30, 1987? (c) Based on the stock prices and beta given in the article, what is the excess daily return on the portfolio of the three publicly-traded underwriters from October 29 to October 30, 1987? (Use the CAPM to calculate the excess return and assume that one-day risk-free interest rate is zero.) (d) If you use the 85% volatility and Black-Scholes formula, what proportion in the excess daily return calculated in (c) can be explained by the repurchase plan value calculated in (b)? Problem 5. The stock of Binomial Hedge Inc. is currently priced at $680 with volatility of 30%. The risk-free interest rate is 2%. Consider a binomial tree of this stock price with three-month time steps. Also consider a European call option that strikes at $700 and matures in six months with two-step tree below. (a) What are the stock prices and option values on the nodes of the tree? (b) What are the delta and share purchase/sale on each node of the tree? (c) What is the position in cash on each node of the tree? (d) What is the net value of the hedged portfolio on each node of the tree? (e) What is the average performance ratio of the hedging strategy? What is the standard deviation of the performance ratio? Node: Price: Option: Delta: Trade: Cash: Net: Node: Price: Option: Delta: Trade: Cash: Net: (0,0) @ @ @ @ Node: Price: Option: @ Delta: Trade: Cash: Net: Node: Price: Option: Delta: Trade: (1,0) Cash: Net: QQ Q From: Price: Q Option: Q Q Delta: Trade: (1,1) Cash: Net: QQ Q 2 Node: Price: Q Option: Q Q Delta: Trade: Cash: Net: (2,0) (1,0) (2,2) (1,1)