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FIN+4414+-+Options+-+Pricing+-+Chapters+9+and+13

FIN+4414+-+Options+-+Pricing+-+Chapters+9+and+13 - Options...

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Options: Pricing

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The binomial model assumes that pricing takes place over discrete units of time and that the value actually observed in the coming period will be one of two possibilities. Option Pricing - Binomial Model
Assumptions : S 0 = \$125.00 E = \$100.00 t = 3 months r = .08 r = .02 on a 3-month basis C 0 = \$25.00 at a minimum/share Option Pricing - Binomial Model

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Assumptions : S 1 > \$100.00 = E and S 1 = \$105.00 or S 1 = \$150.00 Option Pricing - Binomial Model
Buy Stock S* S S* S Investment Time Period 1 Time Period 0 Strategy S (and S>E) S* S* S* S S* S Investment Time Period 1 Time Period 0 Strategy Buy Call Buy T-Bill C E/(1+r) T S* - E S* - E + E + E C+E/(1+r) T S* S* < < > _ > _ Option Pricing - Binomial Model

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Strategy 1 Purchase \$12,500 (100 shares) of stock at a total price of SP 0 . Option Pricing - Binomial Model
Strategy 2 Purchase a T-Bill and one call option at a total price of OP 0 . C 0 = \$ 2,500 TB = \$10,000 OP 0 = \$12,500 Option Pricing - Binomial Model

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Outcome 1 : Stock price goes down to \$105: SP 1 (\$105)(100) = \$10,500 Option Pricing - Binomial Model
Outcome 1 : OP 1 C 1 = \$ 500 TB = \$10,000 Interest = \$ 200 Total \$10,700 Option Portfolio (OP) Dominates Stock Portfolio (SP) Option Pricing - Binomial Model

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Outcome 2 : Stock price goes up to \$150: SP 1 (\$150)(100) = \$15,000 Option Pricing - Binomial Model
Outcome 2 : OP 1 C 1 = \$ 5,000 TB = \$10,000 Interest = \$ 200 Total \$15,200 Option Portfolio (OP) Again Dominates Stock Portfolio (SP) Option Pricing - Binomial Model

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In a rational world, the option can not sell for \$25.00. In fact, with 3 months to expiration, it must sell for a price of \$26.96. This implies that the investor should purchase one call option (\$2,696) and one T-Bill (\$9,804). This is still a total investment of \$12,500. This would give us: Option Pricing - Binomial Model
Outcome 1: Stock price = \$105 SP 1 (\$105)(100) = \$10,500 OP 1 C 1 = \$ 500 TB = \$ 9,804 Interest = \$ 196 Total \$10,500 Option Pricing - Binomial Model

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Outcome 2: Stock price = \$150 SP 1 (\$150)(100) = \$15,000 OP 1 C 1 = \$ 5,000 TB = \$ 9,804 Interest = \$ 196 Total \$15,000 Option Pricing - Binomial Model
If the stock pays a \$1 dividend, then the price of the call will fall, and the amount of the T-Bill will increase, by the present value of the dividend. Option Pricing - Binomial Model

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C 0 = \$26.96 - (\$1.00/1.02) C 0 = \$25.98 \$2,598 TB = \$9,804 + (\$1.00/1.02)(100) TB = \$9,902 Option Pricing - Binomial Model
If the call option is correctly priced at \$25.98, then the payoff should be the same regardless of the state of the world that is actually observed. Option Pricing - Binomial Model

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Outcome 1: Stock price = \$105 SP 1 (\$105)(100) + \$100 = \$10,600 OP 1 C 1 = \$ 500 TB = \$ 9,902 Interest = \$ 198 Total \$10,600 Option Pricing - Binomial Model
Outcome 2 - Stock price = \$150 SP 1 (\$150)(100) + \$100 = \$15,100 OP 1 C 1 = \$ 5,000 TB = \$ 9,902 Interest = \$ 198 Total \$15,100 Option Pricing - Binomial Model

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FIN+4414+-+Options+-+Pricing+-+Chapters+9+and+13 - Options...

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