Lecture29-2004 - Lecture XXIX Option Pricing using...

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1 Lecture XXIX Option Pricing using Black-Scholes I. The European Call Option A. Material adapted from C.-f. Huang and R. H. Litzenberger Foundations for Financial Economics New York: North–Holland, 1988. B. First, we construct the payoff function for a security on which the option is written as: () :0 0otherwise jj j xk ∋− > = where j x k is the payoff a share of security j purchased an exercise price of k . The price of the European call option is then defined as ,m a x , 0 1 j f k pSk S r ≥− + C. Consider the strategy of selling one share of the security to buy one European call option written on the security. 1. The initial cost of the strategy is: ( ) , 1 j f k pSk S r −+ + The possible payoffs of this strategy are 0 i f if j x kx k x k −+= −+> < 2. In the first case, you exercise the option buying back the stock at the original price while in the second case the investor makes money because the stock price decreased (you make profit equal to the decrease in the stock price). 3. Therefore, to avoid a risk-less profit (you can’t make something for nothing) ( ) ,0 1 , 1 j f j f k r k r +> + ⇒> + D. Starting with a two-period economy, we start by assuming a power utility function for the representative agent: () () 11 00 0 1 BB uz uz z z ρ += + −− where is the time preference parameter. The arbitrage condition (selling short a share of stock and purchasing a call option) then implies 0 a x , 0 B j C E C   =−
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Lecture29-2004 - Lecture XXIX Option Pricing using...

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