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### Lecture_16_student0

Course: FINA 4200, Spring 2011
School: UGA
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Word Count: 708

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16: Lecture Options and Corporate Finance: II Topics Option Pricing: Black-Scholes Formula Application: Equity Value as Option Black-Scholes Model The binomial approach provide good intuition but they do not work effectively in the real world. There are many more than two possibilities for future stock prices. Black and Scholes Model Nobel-Price winning theory developed by Fisher Black and Myron Scholes....

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16: Lecture Options and Corporate Finance: II Topics Option Pricing: Black-Scholes Formula Application: Equity Value as Option Black-Scholes Model The binomial approach provide good intuition but they do not work effectively in the real world. There are many more than two possibilities for future stock prices. Black and Scholes Model Nobel-Price winning theory developed by Fisher Black and Myron Scholes. Based on replicating portfolio that duplicates a call over an infinitesimal horizon. Major tool used by the finance industry to price options Black-Scholes Model C0 = S N(d1 ) Ee rT N(d 2 ) Where C0 = the value of a European option at time t = 0 r = the risk-free interest rate. T=the maturity of option N(d) = Probability that a 2 ln(S / E ) + (r + )T standardized, normally 2 d1 = distributed, random variable T d 2 = d1 T will be less than or equal to d. The Black-Scholes Model allows us to value options in the real world. 4 Black-Scholes Model: Example Find the value of a six-month call option on Microsoft with an exercise price of \$150. The current value of a share of Microsoft is \$160. The interest rate available in the U.S. is r = 5%. The option maturity is 6 months (half of a year). The volatility of Microsoft stock return is 30% per annum. Before we start, note that the intrinsic value of the option is ____our answer must be at least that amount. Black-Scholes Model: Example First calculate d1 and d2 ln(S / E ) + (r + .5 2 )T d1 = T ln(160 / 150) + (.05 + .5(0.30) 2 ).5 d1 = = 0.52815 0.30 .5 Then, d 2 = d1 T = 0.52815 0.30 .5 = 0.31602 Black-Scholes Model: Example C0 = S N(d1 ) Ee rT N(d 2 ) d1 = 0.52815 d 2 = 0.31602 N(d1) = N(0.52815) = 0.7013 N(d2) = N(0.31602) = 0.62401 C0 = \$160 0.7013 150e .05.5 0.62401 C0 = \$20.92 In-Class Practice Find the value of a six-month call option on Ashley Co. with an exercise price of \$80. The current value of a share of Ashley is \$85. The interest rate available in the U.S. is r = 6%. The option is maturity 18 months (one and half years). The volatility of Ashley stock is 15% per annum. Equity Value as Call Option Recall: call option payoff Call options gives holder the right to purchase underlying asset at exercise price E. On the maturity date: Zero payoff if S, price of the underlying asset, is lower than E. Payoff equals S-E if S is higher than E. Equity Value as Call Option Payoff of equity: If at the maturity of the debt, the asset value is lower than the debt level, then payoff is zero (bankruptcy). If at the maturity of their debt, the assets of the firm (A) are greater in value than the debt (D), then payoff is A-D. Levered equity is actually a call option. The underlying asset comprises the assets of the firm. The strike price is the payoff of the bond. Equity Value as Call Option: Example Popov Company has been awarded the concessions at the Olympic Games in Antarctica. Popov has \$800 debt due after the games. The companys assets will be worth \$1,000 if the games are successful and \$500 if the games are unsuccessful. Shareholders payoffs If the games are successful, the shareholders will pay the bondholders \$800. Their payoff is \$200. If the games are unsuccessful, the shareholders will declare bankruptcy. Their payoff is \$0. The equity has the same payoff as a call option on the firms assets with exercise price \$800. Practice: Mapping Equity to Option Ashley Company has \$20 million debt due in one year. The companys assets will be worth \$25million if they are doing well and \$15million they are doing poorly. Map Ashleys equity to an option Exercise price is __________ Underlying asset is ____________________ It is a ____ option In Class Practice Tony Corporations market value of assets are \$40 million. It has debt with face value of \$18 million due in 4 years. Tony Corporations asset return standard deviation is 40%. Risk free rate is 4%. Calculate the market values of Tonys Corporations equity and debt using Black-Scholes Formula.
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UGA - STAT - 4260
UGA - STAT - 4260
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UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
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UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
UGA - STAT - 4260
Tree123456789101112averageVarianceBasal ar0.30.50.40.90.70.20.60.50.80.40.80.6Volume6971915512920918130.5611.83yi-r*xi-0.36-1.6-1.48-0.070.160.76-0.72-1.63.040.521.040.28tau-ytau-xMu-xrNnVar(ta
UGA - STAT - 4260
UGA - STAT - 4260
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UGA - STAT - 4260