PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

732 APPLICATION OF THE OPTION MARKET PARADIGMVariance of Hedged Portfolio at Expiry¾ 2 h =E(h2 ) ¡ E(h) 2=(b 2 + N(d (¹)0 ) ¢ (1 ¡ 2b)) ¢ E(x2 )¡ 2S ¢ N(d (¹)1) ¢ (1 ¡ b) ¢ E(x)+ S 2 ¢ N(d (¹)2) ¡ (E(x) ¢ (b ¡ N(d(¹)1 ))+ S ¢ N(d (¹)2 ))2=(b 2 + N(d (¹)0 ) ¢ (1 ¡ 2b)) ¢ E(x2 )¡ 2S ¢ N(d (¹)1) ¢ (1 ¡ b) ¢ E(x)+ S 2 ¢ N(d (¹)2 ) ¡ E(x)2 ¢ (b ¡ N(d (¹)1 ))2¡ 2SE(x) ¢ (b ¡ N(d (¹)1)) ¢ N(d(¹)2 )¡ S 2 ¢ N(d (¹)2 )2=E(x 2 ) ¢ (b 2 + N(d (¹)0 ) ¢ (1 ¡ 2b) ¡ E(x)2 ¢ (b ¡ N(d (¹)1 ))2¡ E(x) ¢ 2S ¢ (N(d (¹)1+ S 2 ¢ N(d (¹)2For the special case of b = N(d (¹)1 ),) ¢ (1 ¡ b)+(b ¡ N(d(¹)1)) ¢ N(d(¹)2 ))) ¢ (1 ¡ N(d(¹)2)): (A.3)¾h 2 =E(x2 ) ¢ ((N(d (¹)1 )2 + N(d (¹)0) ¢ (1 ¡ 2N(d(¹)1 )))¡ E(x) ¢ 2S ¢ (N(d (¹)1) ¢ (1 ¡ N(d(¹)1 ))+ S 2 ¢ N(d (¹)2In the example used in the paper,) ¢ (1 ¡ N(d(¹)2)): (A.3a)E(x 2 ) = 10178:28293, N(d (¹)0 )=0:58297E(x) = 100:71487, N(d (¹)1 )=0:56001