50. Lower and upper bounds for call and put options and put call parity.

50. Lower and upper bounds for call and put options and put call parity.

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Lower and upper bounds for the price of a European calls and puts A. Lower bound for the price of a European call: Time t = 0 Payoff at time t = 1 S 1 > E S 1 E Portfolio A : Buy 1 call - C S 1 - E 0 Cash (lend) - E 1+ r + E + E Total S 1 E Portfolio B : Buy 1 share - S 0 S 1 S 1 1
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B. Lower bound for the price of a European put: Time t = 0 Payoff at time t = 1 S 1 E S 1 < E Portfolio A : Buy 1 put - P 0 E - S 1 Buy 1 share - S 0 S 1 S 1 Total S 1 E Portfolio B : Cash (lend) - E 1+ r + E + E 2
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C. Upper bound for the price of a European call: No matter what happens, C S 0 If not, there will be an opportunity for a riskless profit by buying the stock and selling the call option. How? Suppose C > S 0 . Time t = 0 Payoff at time t = 1 S 1 > E S 1 E Sell 1 call C E - S 1 0 Buy 1 stock - S 0 S 1 S 1 Total C - S 0 E S 1 3
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D. Upper bound for the price of a European put: No matter what happens, P E 1+ r . If not, there will be an opportunity for a riskless profit by selling the put and investing
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Unformatted text preview: the proceeds at the risk free interest rate. How? Suppose P &gt; E 1+ r . Time t = 0 Payo at time t = 1 S 1 E S 1 &lt; E Sell 1 put P &gt; E 1+ r S 1-E 4 Put-Call Parity This is an important relationship between the price of a put and the price of the call. A put and the underlying stock can be combined in such a way that they have the same payo as a call at expiration. Consider the following two portfolios: Portfolio A : Buy the call and lend an amount of cash equal to E 1+ r . Portfolio B : Buy the stock, buy the put. This is shown on the table below: Time t = 0 Payo at time t = 1 S 1 &gt; E S 1 E Portfolio A : Buy 1 call-C S 1-E Lend cash-E 1+ r E E Total-C-E 1+ r S 1 E S 1 E S 1 &lt; E Portfolio B : Buy 1 put-P E-S 1 Buy 1 stock-S S 1 S 1 Total-P-S S 1 E 5...
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This note was uploaded on 06/02/2011 for the course STATS 183 taught by Professor Nicolas during the Spring '11 term at UCLA.

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50. Lower and upper bounds for call and put options and put call parity.

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