5 032 365 50 q 05184 60 03 365 r 60 d2 05184 03

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Unformatted text preview: 52 + (0.1 + 0.5 · 0.32 ) 365 50 q = 0.5184 60 0.3 365 r 60 d2 = 0.5184 0.3 = 0.3968 365 N (d1 ) = N (0.52) = 0.6985, N (d2 ) = N (0.40) = 0.6554 d1 = e rt =e 60 0.1· 365 C = 52 · 0.6985 = 0.9837 50 · 0.9837 · 0.6554 = 4.086 13 / 25 Example 11.2. Solution (cont.) The value of the 60-day 55-strike European call: 60 ln 52 + (0.1 + 0.5 · 0.32 ) 365 55 q = 0.2652 60 0.3 365 r 60 d2 = 0.2652 0.3 = 0.3868 365 N (d1 ) = N ( 0.27) = 0.3936, N (d2 ) = N ( 0.39) = 0.3483 d1 = e rt =e 60 0.1· 365 C = 52 · 0.3936 = 0.9837 55 · 0.9837 · 0.3483 = 1.623 The portfolio is worth: 4.086 The profit is: 2.463 1.994e 1.623 = 2.463 30 0.1· 365 = 0.453 14 / 25 Calendar spreads buy A calendar spread consists of selling a call and being another call with the same strike price on the same stock but a later expiry date. This is a bet on volatility. 15 / 25 Example 11.3 For a calendar spread consisting of selling a 30-day European call on a stock and buying a 90-day European call on the same stock, both with strike price 45, we are given: (i) The initial price of the stock is 50 (ii) = 0.3 (iii) r = 0.05 (iv) =0 the 1. Calculate the profit on a 30th day if the stock price goes up to 50. the 2. Calculate the profit on a 30th day if the stock price stays the same at 45. 16 / 25 Example 11.3. Solution. The investment in the spread is the di↵erence in the premiums of the options. The premium of the 30-day call is: 30 ln 50 + (0.05 + 0.5 · 0.32 ) 365 45 q = 1.3158 30 0.3 365 r 30 d2 = 1.3158 0.3 = 1.2298 365 N (d1 ) = N (1.32) = 0.9066, N (d2 ) = N (1.23) = 0.8907 d1 = C = 50 · 0.9066 45 · e 30 0.1· 365 · 0.8907 = 5.413 17 / 25 Example 11.3. Solution (cont.) The premium of the 90-day call is: 90 ln 50 + (0.05 + 0.5 · 0.32 ) 365 45 q = 0.8645 90 0.3 365 r 90 d2 = 0.8645 0.3 = 0.7155 365 N (d1 ) = N (0.86) = 0.8051, N (d2 ) = N (0.72) = 0.7642 d1 = C = 50 · 0.8051 45 · e 90 0.05· 365 · 0.7642 = 6.287 The initial investment is: 6.287 5.413 = 0.874 18 / 25 Example 11.3. Solution (cont.) After 30 days, if the stock price is 50, the value of the 90-day call, which is now a 60-day call,...
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This document was uploaded on 02/26/2014.

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