Unformatted text preview: ical payoffs to the put option described above. In order to do this, we need to short shares of the stock and lend at the riskfree rate. The number of shares that should be shorted sell is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock) Since the put option will be worth $0 if the stock price rises and $25 if it falls, the swing of the call option is –$25 (= $0 – 25). Since the stock price will either be $60 or $15 at the time of the option’s expiration, the swing of the stock is $45 (= $60 – 15). Given this information, the delta of the put option is: Delta = (Swing of option) / (Swing of stock) Delta = (–$25 / $45) Delta = –0.56 Therefore, the first step in creating a synthetic put option is to short 0.56 of a share of stock. Since the stock is currently trading at $30 per share, the amount received will be $16.67 (= 0.56 × $30) as a result of the short sale. In order to determine the amount to lend, compare the payoff of the actual put option to the payoff of delta shares at expiration. Put option If the stock price rises to $60: If the stock price falls to $15: Delta shares If the stock price rises to $60: If the stock price falls to $15: Payoff = $0 Payoff = $25 Payoff = (–0.56)($60) = –$33.33 Payoff = (–0.56)($15) = –$8.33 The payoff of the synthetic put position should be identical to the payoff of an actual put option. However, shorting 0.56 of a share leaves us exactly $33.33 below the payoff at expiration, whether the stock price rises or falls. In order to increase the payoff at expiration by $33.33, we should lend the present value of $33.33 now. In six months, we will receive $33.33, which will increase the payoffs so that they exactly match those of an actual put option. So, the amount to lend is: Amount to lend = $33.33 / 1.081/2 Amount to lend = $32.08 c. Since the short sale results in a positive cash flow of $16.67 and we will lend $32.08, the total cost of the synthetic put option is: Cost of synthetic put = $32.08 – 16.67 Cost of synthetic put = $15.41 452 This is exactly the same price as an actual put option. Since an actual put option and a synthetic put option provide identical payoff structures, we should not expect to pay more for one than for the other. 28. a. The company would be interested in purchasing a call option on the price of gold with a strike price of $875 per ounce and 3 months until expiration. This option will compensate the company for any increases in the price of gold above the strike price and places a cap on the amount the firm must pay for gold at $875 per ounce. In order to solve a problem using the twostate option model, first draw a price tree containing both the current price of the underlying asset and the underlying asset’s possible values at the time of the option’s expiration. Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements. Price of gold Today 3 months $975 $815 $740 ? $0 =Max(0, $740 – 875) Call option price with a strike of $875 Today 3 months $100 =Max(0, $975 – 875) b. The price of gold is $815 per ounce today. If the price rises to $975, the company will exercise its call option for $875 and receive a payoff of $100 at expiration. If the price of gold falls to $740, the company will not exercise its call option, and the firm will receive no payoff at expiration. If the price of gold rises, its return over the period is 19.63 percent [= ($975 / $815) – 1]. If the price of gold falls, its return over the period is –9.20 percent [= ($740 / $815) –1]. Use the following expression to determine the riskneutral probability of a rise in the price of gold: Riskfree rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Riskfree rate = (ProbabilityRise)(ReturnRise) + (1  ProbabilityRise)(ReturnFall) The riskfree rate over the next three months must be used in the order to match the timing of the expected price change. Since the riskfree rate per annum is 6.50 percent, the riskfree rate over the next three months is 1.59 percent [= (1.0650)1/4 – 1], so: .0159 = (ProbabilityRise)(.1963) + (1 – ProbabilityRise)(–.0920) ProbabilityRise = .3742 or 37.42% And the riskneutral probability of a price decline is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 –.3742 ProbabilityFall = .6258 or 62.58% Using these riskneutral probabilities, we can determine the expected payoff to of the call option at expiration, which will be. 453 Expected payoff at expiration = (.3742)($100) + (.6258)($0) Expected payoff at expiration = $37.42 Since this payoff occurs 3 months from now, it must be discounted at the riskfree rate in order to find its present value. Doing so, we find: PV(Expected payoff at expiration) = [$37.42 / (1.0650)1/4 ] PV(Expected payoff at expiration) = $36.83 Therefore, given the information about gold’s price movements over the next three months, a European call option with a strike price of $875 and three months until expiration is worth $36.83 today. c. Yes, there is a way to create...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of TexasTyler.
 Spring '10
 eshmalwi
 Finance, Corporate Finance

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