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Finance 100 Problem Set Options

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Finance 100 Problem Set Options Finance 100 Problem Set Options 1. The Wall Street Journal reported the following prices for 3Com options for trading on Friday, February 7 1997. The stock itself closed at $50.75. Calls (Prices in $) Strike February March April July 45 6.625 7.5 8 10.5 50 2.8125 4.75 5.875 55 1.0625 2.625 3.75 6 60 0.4375 1.3125 1.875 4.25 65 0.1875 0.625 1.25 70 0.0625 Puts (Prices in $) Strike February March April July 45 1 1.8125 2.625 4.25 50 2.375 4 5 55 5.375 7.625 8.125 8.75 60 9.75 10.875 12.5 65 15.5 16.5 16.75 70 20.75 1.a Construct a payo® diagram for buying a $50 April call option on 3Com and for buying a $50 April put option. Construct the diagrams for net payo®s, i. e. after deducting the option premium. 1 1.b Construct a payo® diagram for writing a $50 April call option on 3Com and for writing a $50 April put option. Construct the diagrams for net payo®s, i. e. after adding the option premium. 1.c Construct payo® diagrams for the following options, all maturing in February. Construct net payo® diagrams only, i. e. include the premium paid/received for the contracts. It may help to also construct a table for this, with possible stock prices ranging from $40 to $80 in $5 intervals: i Buy one share and buy a put with X=$50. ii Buy a put with X=$50 and write a call with X=$70. iii Buy a call with X=$70 and buy a put with X=$50. iv Buy a call with X=$50 and buy a put with X=$70. v Sell one share and buy a call with X=$70. vi Buy one call with X=$50 and another one with X=$70. Write (sell) 2 calls with X=$60. vii Buy one put with X=$50 and another one with X=$70. Write (sell) 2 puts with X=$60. 1.d If the risk free rate of interest (continuously compounded) is 5%, then does put-call parity hold for the April options with X=$50, X=$60 and X=$65 (calculations based on 71 days to maturity, 365 day year)? 1.e Examine portfolios (iii) and (iv) from part 1.c. Which one is more expensive? Use put-call parity to show that portfolio (iii) should always cost less than portfolio (iv). 2 1.f Examine portfolios (vi) and (vii) in part 1.c. Can you establish a similar relationship as in 1.e? 1.g Compare portfolios (i) and (v) from part 1.c. In what sense are they "hedged" positions? Compare (i) with buying and (v) with selling the share, without the respective option position. Which risk does the option remove in each case? Would you prefer portfolio (i) or portfolio (v) if you believe that the stock price will go up (down)? 1.h Compare portfolios (iii) and (vi) from part 1.c. Which one do you prefer if you believe that volatility will go up (down)? 1.i Repeat the analysis under 1.h for portfolios (iv) and (vii). 1.j Use put call parity to establish the value of the following portfolios (Maturity = April): 1. Buy a put with X=$60, sell a call with X=$60. 2. Buy a call with X=$65, sell a put with X=$65. 1.k Reconsider the April option with $45 strike price in the table above. Use the Black-Scholes formula to price it, using volatilities between 20% and 50% in 5% increments. Chart the option values you receive against volatility. What do you observe? 1.l Which volatility do you need in (11) above in order to obtain exactly the quoted price for the $45 April call? Determine this either with trial and error, or use one of the EXCEL functions (GOALSEEK or SOLVER). 3 2. A Swiss company purchased machine tools from a US engineering ¯rm. The delivery date is in March (40 days from now), and the purchase price of the machine tools is SFR 5million. The US company is concerned about exchange rate movements between now and the delivery date and considers hedging its exposure. The futures price for March is $0.7021 USD/SFR. One futures contract is for SFR 125,000. The company could also use options. The following table gives the option prices on Swiss Francs: Strike Price Call premium Put premium ($US/SFR) (cents/SFR) (cents/SFR) 0.695 1.13 1.07 0.700 1.05 0.710 1.96 0.720 0.34 0.740 4.42 0.770 7.29 0.790 0.01 Option contracts are per SFR 62,500. Hence, the table reads as follows: If you buy one put option contract on Swiss Francs for March with a strike price of $0.71 today, you will have a right to sell SFR 62,500 at a price of 71 cents per SFR in March. Today you have to pay 1.96 cents per SFR you sell, i. e. the price of the contract is $0.0196*62,500=$1,225 today. 2.a Construct the optimal hedge using the futures contract described above. How many contracts does the US company have to buy/sell in order to hedge the exchange rate risk? 2.b Demonstrate that this hedge is perfect by looking at scenarios where the SFR is $0.65, $0.70 and $0.75. 2.c Construct a hedge using options. Compare alternative hedges, using at least 3 alternative strike prices. What type of contract do you buy/sell, and how 4 many? Compare against hedges for alternative scenarios for the exchange rate in March as in (2). Which option would you choose? 2.d Discuss the advantages and the disadvantages of a futures hedge you con- structed in 1.a versus the best option hedge you determined in 1.c, depending on alternative scenarios for the exchange rate. Which one would you choose? 2.e Suppose the US company has obtained an \exit clause" in the contract for the machine tools. This would allow the company not to sell to the Swiss buyer if the Swiss Franc fell below $0.695. They have determined that they make zero pro¯ts on the transactions if the receive $0.695 for each Swiss Franc, hence this is their break-even point (an example could be a price bid by a potential US buyer). Suppose that upon completion of the contract in March, the Swiss Franc falls to $0.69 and the exit clause comes in, and assume also that the company has used a perfect hedge using futures. The CEO argues: \We should complete the contract as agreed. The fall in the Swiss Franc does not concern us, since we are hedged in the futures market." The treasurer has a di®erent view: \We should use the clause in the contract and not sell to the Swiss buyer. We can take the pro¯ts from the hedge anyway." Who is right? 2.f When the company negotiated the exit clause discussed in (e), they had to pay a fee for the opportunity to opt out of the contract in the form of a collateral payment which they have to make today. This payment is due in addition to the price for the machine tools of SFR 5 million that is due in March. Hence this payment is like a fee the seller pays to the buyer, regardless of whether the contract is completed or not. What is the maximum fee the US company would be willing to pay for obtaining the exit clause in the contract? 3. European call and put options on Exxon are selling for $3.10 and $0.40 respectively. Both options are struck at $45 and have one month to maturity. 5 The current stock price is $46.60 and the one-month riskless rate is 5% p.a. Is there an arbitrage opportunity in this market? If so, demonstrate how to exploit it. 6.
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