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Chapter 7 Derivatives markets Manual for SOA Exam FM CAS Exam 2 Chapter 7 Derivatives markets Section 7 4 Call options c 2009 Miguel A Arcones All rights reserved Extract from Arcones Manual for the SOA Exam FM CAS Exam 2 Financial Mathematics Fall 2009 Edition available at http www actexmadriver com 1 112 c 2009 Miguel A Arcones All rights reserved Manual for SOA Exam FM CAS Exam 2 Chapter 7 Derivatives markets Section 7 4 Call options Minimums and maximums Definition 1 Given two real numbers a and b i min a b denotes the minimum smallest of the two numbers ii max a b denotes the maximum biggest of the two numbers Example 1 min 10 5 5 max 10 5 10 min 1 5 1 max 1 5 5 min 2 100 100 max 2 100 2 2 112 c 2009 Miguel A Arcones All rights reserved Manual for SOA Exam FM CAS Exam 2 Chapter 7 Derivatives markets Section 7 4 Call options Definition 2 Given real numbers a1 an i min a1 an denotes the minimum smallest of these numbers ii max a1 an denotes the maximum biggest of these numbers Example 2 min 1 5 3 6 6 max 1 5 3 6 5 min 2 100 50 100 and max 2 100 50 2 3 112 c 2009 Miguel A Arcones All rights reserved Manual for SOA Exam FM CAS Exam 2 Chapter 7 Derivatives markets Section 7 4 Call options Theorem 1 For each a b c R and each 0 I min a b min b a I max a b max b a I min min a b c min a min b c min a b c I max max a b c max a max b c max a b c I min a c b c min a b c I max a c b c max a b c I min a b min a b I max a b max a b I min a b max a b I max a b min a b 4 112 c 2009 Miguel A Arcones All rights reserved Manual for SOA Exam FM CAS Exam 2 Chapter 7 Derivatives markets Section 7 4 Call options Definition 3 Given a real number a a a if a 0 and a a if a 0 Example 3 23 23 4 4 Theorem 2 For each a b R min a b max a b a b Proof min a b and max a b are a and b in some order Hence min a b max a b a b Theorem 3 For each a R a max a 0 min a 0 Proof If a 0 then max a 0 a min a 0 0 and max a 0 min a 0 a a If a 0 then max a 0 0 min a 0 a and max a 0 min a 0 a a 5 112