M09_MCDO8122_01_ISM_C09

M09_MCDO8122_01_ISM_C09 - Chapter 9 Parity and Other Option...

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Chapter 9 Parity and Other Option Relationships n Question 9.1 This problem is an application of put-call-parity for a stock with a continuous dividend. We have: 0 0.06 0.5 0.04 0.5 (35, 0.5) (35, 0.5) 35 (35, 0.5) $2.27 32 35 $5.5227. T rT P C e S e P e e δ - - - × - × = - + = - + = n Question 9.2 This problem is an application of put-call-parity for a stock with a discrete dividend. We have: 0, 0 (Div) (30, 0.5) (30, 0.5) 30 32 4.29 2.64 29.406 $0.9440. rT T PV S C P e - = - + - = - + - = n Question 9.3 1. The initial cash flow is: 800 75 45 770. - + - = - This position has a cash flow of $815 after one year for sure, because, whichever option is in-the- money, we will sell the stock for the strike price. Therefore, we have a one-year effective rate of return of 5.8442% which is a continuous return of ln(815/770) 5.6798%. = 2. We have a risk-less position in (a) that pays more than the risk-free rate. Therefore, we should borrow money at 5%, and buy a large amount of the aggregate position of (a), yielding a sure continuous return of 0.6798%. 3. An initial investment of 0.05 815 775.2520 e - × = would yield $815 after one year, invested at the risk- less rate of return of 5%. Therefore, the difference between call and put prices should be equal to $24.748. One could also use put-call parity to determine C - P . 4. Using put-call parity we can obtain the four price differences. We demonstrate with the 820-strike options: .05 0 ( , ) ( , ) 800 820 19.99187 rT C K T P K T S Ke e - - - = - = - =
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102 McDonald • Fundamentals of Derivatives Markets The others follow with a similar calculation: Strike Price Call-Put 780 58.04105 800 39.01646 820 19.99187 840 0.967283 n Question 9.4 We can make use of the put-call-parity for currency options: 0 0.04 0.06 ( , ) ( , ) 0.95 0.0571 0.93 0.91275 0.0571 0.87584 0.0202. f r T rT P K T e x C K T e K e e - - - - = - + + = - + + = - + + = A $0.93 strike European put option has a value of $0.0202. n Question 9.5 The payoff of the one-year yen-denominated put on the euro is 1 max[0,100 ] x - yen, where x 1 is the future uncertain ¥/ exchange rate. The payoff of the corresponding one-year euro-denominated call on yen is 1 max[0,1/ 1/100]. x - Let C E be the call premium which is denominated in euros. We can replicate the yen-denominated put payoff by purchasing 100 euro-denominated calls for 100 E C × euros. We can verify this by noting, if the yen-denominated put is in-the-money, its payoff is 1 100 x - yen; the euro-denominated call will also be in-the-money; 100 calls gives a payoff of 1 100/ 100 x - euros which is worth 1 1 1 (100/ 1) 100 x x x - × = - yen. Hence 100 euro-denominated calls must cost the same as 1 yen-denominated put. Since these are in different currencies, we use the spot exchange rate to convert either of the options. Converting the euro denominated call option to yen at the current exchange rate 0 ( 95¥/ ) x = we have, 0 100 100 95 8.763 8.763/9500.00092242 euros E Y E E C x P C C × × = × × = = n Question 9.6
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M09_MCDO8122_01_ISM_C09 - Chapter 9 Parity and Other Option...

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