3089 01 02403 365 n d1 n 031 06217 n d2

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Unformatted text preview: 365 r 172 d2 = 0.3089 0.1 = 0.2403 365 N (d1 ) = N (0.31) = 0.6217, N (d2 ) = N (0.24) = 0.5948 d1 = e rt e t ln 56 + (0.05 56 =e 0.05· 172 365 = 0.9767 =e 0.01· 172 365 = 0.9953 C = 56 · 0.9953 · 0.6217 56 · 0.9767 · 0.5948 = 2.119 Hence, the 10-day holding period profit is: 2.119 10 1.578e 0.05· 365 = 0.539 8 / 25 Profit for a portfolio of options. A bull spread. By definition, bull spread is the purchase of a call (or put) together with the sale of an otherwise identical higher-strike call (or put). 9 / 25 Example 11.2 A non-dividend paying stock has price 50 and volatility 30%. The continuously compounded risk-free rate is 10%. You purchase a bull spread consisting of buying a99 60-day European call with strike 960-day European call with strike price 55 price 50 and selling a (higher). Calculate: 1. for the spread 2. The profit on the spread after 30 days if the stock price is 52 then. 10 / 25 Example 11.2. Solution. 90 1. At issue (t = 365 ), let us calculate the value of the spread by calculating the values of each call and then subtracting one from another: The value of the 90-day 50-strike European call: 90 ln 50 + (0.1 + 0.5 · 0.32 ) 365 50 q = 0.2400 90 0.3 365 r 90 d2 = 0.24 0.3 = 0.0910 365 N (d1 ) = N (0.24) = 0.5948, N (d2 ) = N (0.09) = 0.5359 d1 = e rt =e 90 0.1· 365 C = 50 · 0.5948 90 (50) =e t = 0.9756 50 · 0.9756 · 0.5359 = 3.598 N (d1 ) = 0.5948 11 / 25 Example 11.2. Solution (cont.) The value of the 90-day 55-strike European call: 90 ln 50 + (0.1 + 0.5 · 0.32 ) 365 55 q = 0.3998 90 0.3 365 r 90 d2 = 0.3998 0.3 = 0.5488 365 N (d1 ) = N ( 0.40) = 0.3446, N (d2 ) = N ( 0.55) = 0.2912 d1 = e rt =e 90 0.1· 365 C = 50 · 0.3446 90 (55) =e t = 0.9756 55 · 0.9756 · 0.2912 = 1.6041 N (d1 ) = 0.3446 90-day value of the spread: 3.598 1.6041 = 1.994 for the portfolio: 0.5948 0.3446 = 0.2502 12 / 25 Example 11.2. Solution (cont.) After 30 days: The calls are now 60-day calls and S=52: The value of the 60-day 50-strike European call: 60 ln...
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