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Unformatted text preview: Expected payoff at expiration = (.4286)($13) + (.5714)($0) Expected payoff at expiration = $5.57 449 Since this payoff occurs 1 year from now, we must discount it back to the value today. Since we are using riskneutral probabilities, we can use the riskfree rate, so: PV(Expected payoff at expiration) = $5.57 / 1.025 PV(Expected payoff at expiration) = $5.44 b. Yes, there is a way to create a synthetic call option with identical payoffs to the call option described above. In order to do this, we will need to buy shares of stock and borrow at the riskfree rate. The number of shares to buy is based on the delta of the option, where delta is defined as: Delta = (Swing of option) / (Swing of stock) Since the call option will be worth $13 if the stock price rises and $0 if it falls, the delta of the option is $13 (= 13 – 0). Since the stock price will either be $98 or $70 at the time of the option’s expiration, the swing of the stock is $28 (= $98 – 70). With this information, the delta of the option is: Delta = $13 / $28 Delta = 0.46 Therefore, the first step in creating a synthetic call option is to buy 0.46 of a share of the stock. Since the stock is currently trading at $80 per share, this will cost $31.71 [= (0.46)($70)/(1 + . 025)]. In order to determine the amount that we should borrow, compare the payoff of the actual call option to the payoff of delta shares at expiration. Call Option If the stock price rises to $98: If the stock price falls to $70: Delta Shares If the stock price rises to $98: If the stock price falls to $80: Payoff = $13 Payoff = $0 Payoff = (0.46)($98) = $45.50 Payoff = (0.46)($70) = $32.50 The payoff of his synthetic call position should be identical to the payoff of an actual call option. However, owning 0.46 of a share leaves us exactly $32.50 above the payoff at expiration, regardless of whether the stock price rises or falls. In order to reduce the payoff at expiration by $32.50, we should borrow the present value of $32.50 now. In one year, the obligation to pay $32.50 will reduce the payoffs so that they exactly match those of an actual call option. So, purchase 0.46 of a share of stock and borrow $31.71 (= $32.50 / 1.025) in order to create a synthetic call option with a strike price of $85 and 1 year until expiration. c. Since the cost of the stock purchase is $37.15 to purchase 0.46 of a share and $31.71 is borrowed, the total cost of the synthetic call option is: Cost of synthetic option = $37.15 – 31.71 Cost of synthetic option = $5.44 450 This is exactly the same price as an actual call option. Since an actual call option and a synthetic call option provide identical payoff structures, we should not expect to pay more for one than for the other. 27. a. In order to solve a problem using the twostate option model, we first draw a stock price tree containing both the current stock price and the stock’s possible values at the time of the option’s expiration. Next, we can 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 stock Today 6 months $60 $30 $15 ? $25 =Max(0, $40 – 15) Today Put option price with a strike of $40 6 months $0 =Max(0, $40 – 60) The stock price today is $30. It will either decrease to $15 or increase to $60 in six months. If the stock price falls to $15, the put will be exercised and the payoff will be $25. If the stock price rises to $60, the put will not be exercised, so the payoff will be zero. If the stock price rises, its return over the period is 100% [= (60/30) – 1]. If the stock price falls, its return over the period is –50% [= (15/30) –1]. Use the following expression to determine the riskneutral probability of a rise in the price of the stock: Riskfree rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Riskfree rate = (ProbabilityRise)(ReturnRise) + (1 – ProbabilityRise)(ReturnFall) The riskfree rate over the next six months must be used in the order to match the timing of the expected stock price change. Since the riskfree rate per annum is 8 percent, the riskfree rate over the next six months is 3.92 percent [= (1.08)1/2 –1], so. 0.392 = (ProbabilityRise)(1) + (1 – ProbabilityRise)(–.50) ProbabilityRise = .3595 or 35.95% Which means the riskneutral probability of a decrease in the stock price is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – .3595 ProbabilityFall = .6405 or 64.05% Using these riskneutral probabilities, we can determine the expected payoff to put option at expiration as: Expected payoff at expiration = (.3595)($0) + (.6405)($25) Expected payoff at expiration = $16.01 451 Since this payoff occurs 6 months from now, we must discount it at the riskfree rate in order to find its present value, which is: PV(Expected payoff at expiration) = $16.01 / (1.08)1/2 PV(Expected payoff at expiration) = $15.41 b. Yes, there is a way to create a synthetic put option with ident...
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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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