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Unformatted text preview: The option is worthless if the price decreases. If the price increases, the value of the option per gallon is: Value with price increase = $2.19 – 2.05 Value with price increase = $0.14 Next, we need to find the risk neutral probability of a price increase or decrease, which will be: .06 / (12/3) = 0.26(Probability of rise) + –0.21(1 – Probability of rise) Probability of rise = 0.4751 And the probability of a price decrease is: Probability of decrease = 1 – 0.4751 Probability of decrease = 0.5249 The contract will not be exercised if gasoline prices fall, so the value of the contract with a price decrease is zero. So, the value per gallon of the call option contract will be: C = [0.4751($0.14) + 0.5249(0)] / [1 + 0.06/(12 / 3)] C = $0.066 This means the value of the entire contract is: Value of contract = $0.066(5,000,000) Value of contract = $327,553.79 467 4. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since the company will exercise its option to build if the value of an office building rises, the right to build the office building is similar to a call option. Second, an office building would be worth $18.5 million today. This amount can be viewed as the current price of the underlying asset (S). Third, it will cost $20 million to construct such an office building. This amount can be viewed as the strike price of a call option (K), since it is the amount that the firm must pay in order to ‘exercise’ its right to erect an office building. Finally, since the firm’s right to build on the land lasts only 1 year, the time to expiration (t) of the real option is one year. We can use the twostate model to value the option to build on the land. First, we need to find the return of the land if the value rises or falls. The return will be: RRise = ($22,400,000 – 18,500,000) / $18,500,000 RRise = .2108 or 21.08% RFall = ($17,500,000 – 18,500,000) / $18,500,000 RFall = –.0541 or –5.41% Now we can find the risk-neutral probability of a rise in the value of the building as: Value of building (millions) Today 1 year $22.4 $18.5 $17.5 ? $0 =Max(0, $17.5 – 20) Value of real call option with a strike of $20 (millions) Today 1 year $2.4 =Max(0, $22.4 – 20) Risk-free rate = (ProbabilityRise)(ReturnRise) + (ProbabilityFall)(ReturnFall) Risk-free rate = (ProbabilityRise)(ReturnRise) + (1 - ProbabilityRise)(ReturnFall) 0.048 = (ProbabilityRise)(0.2108) + (1 – ProbabilityRise)(–.0541) ProbabilityRise = 0.3853 So, a probability of a fall is: ProbabilityFall = 1 – ProbabilityRise ProbabilityFall = 1 – 0.3853 ProbabilityFall = 0.6147 Using these risk-neutral probabilities, we can determine the expected payoff of the real option at expiration. Expected payoff at expiration = (.3853)($2,400,000) + (.6147)($0) Expected payoff at expiration = $924,734.69 468 Since this payoff will occur 1 year from now, it must be discounted at the risk-free rate in order to find its present value, which is: PV = ($924,734.69 / 1.048) PV = $882,380.43 Therefore, the right to build an office building over the next year is worth $882,380.43 today. Since the offer to purchase the land is less than the value of the real option to build, the company should not accept the offer. 5. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since the company will only choose to drill and excavate if the price of oil rises, the right to drill on the land can be viewed as a call option. Second, since the land contains 375,000 barrels of oil and the current price of oil is $58 per barrel, the current price of the underlying asset (S) to be used in the Black-Scholes model is: “Stock” price = 375,000($58) “Stock” price = $21,750,000 Third, since the company will not drill unless the price of oil in one year will compensate its excavation costs, these costs can be viewed as the real option’s strike price (K). Finally, since the winner of the auction has the right to drill for oil in one year, the real option can be viewed as having a time to expiration (t) of one year. Using the Black-Scholes model to determine the value of the option, we find: d1 = [ln(S/K) + (R + σ 2/2)(t) ] / (σ 2t)1/2 d1 = [ln($21,750,000/$35,000,000) + (.04 + .502/2) × (1)] / (.50 × d2 = –.6215 – (.50 ×
1 ) = –1.1215 1 ) = –.6215 Find N(d1) and N(d2), the area under the normal curve from negative infinity to d 1 and negative infinity to d2, respectively. Doing so: N(d1) = N(–0.6215) = 0.2671 N(d2) = N(–1.1215) = 0.1310 Now we can find the value of call option, which will be: C = SN(d1) – Ke-–RtN(d2) C = $21,750,000(0.2671) – ($35,000,000e–.04(1))(0.1310) C = $1,403,711.65 This is the maximum bid the company should be willing to make at auction. 469 Intermediate 6. When solving a question dealing with real options, begin by identifying the option-like features of the situation. First, since Sardano will only choose to manufacture the steel rods if the price of steel falls, the lease, which gives the firm the ability to manufact...
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