# mid1sol - x ( ) ( ) ( ) ( ) ( ) &amp;amp;amp; &amp;amp;amp;...

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2 First, from Black-Scholes pricing formula, we can get: delta = ( ) 1 d N Where ( ) ( ) , 2 / / ln 2 0 1 s T T r K S d + + = Second, since t S follows geometric Brownian motion, ( ) , , 0 ~ t N W t we have discrete time expression: ( ) t t t W W S t rS S - + D = D D + , which means ( ) . , ~ t S t rS N S D D D So, from delta normal valuation, ( ) ( ) ( ) . , ~ 1 1 t S t rS N d N S d N C D D D = D Then the 95% VaR and ES over 5 days for a long position are: ( ) ( ) ( ) ( ) . ) ( 20 , 1 95 . 0 1 1 95 . 0 1 t S d N Z d tN rS ES t S d N Z d tN rS VaR D - D - = D - D - = f For a short position are: ( ) ( ) ( ) ( ) . ) ( 20 , 1 95 . 0 1 1 95 . 0 1 t S d N Z d tN rS ES t S d N Z d tN rS VaR D - D = D - D = where 84 / 1 252 / 3 = = D t
3 Proof: ( ) ( ) ( ) ( ) ( ) ( ) ¢ F l ± ² ² L ³ - F - F + - - = - F ¥ - - ´ - x ds x s x s x C y 1 2 1 2 1 2 2 1 1 2 2 exp 1 2 1 r p Note that: ( ) ( ) ( ) l ± ² ² L ³ F = ¢ F - - 2 exp 2 2 1 1 x
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Unformatted text preview: x ( ) ( ) ( ) ( ) ( ) &amp; &amp; l L F &amp; &amp; l L -F-F +--= -F ---2 exp 2 1 2 2 exp 1 2 1 2 1 2 1 2 1 2 2 1 x ds x s x s x C y = ( ) ( ) ( ) ( ) ( ) ( ) &amp; &amp; l L -F-F F = &amp; &amp; l L -F-----F --2 1 1 2 2 1 2 1 1 2 exp 1 2 1 1 x y ds x s y Which gives us ( ) ( ) ( ) du x y y x C x &amp; &amp; l L -F-F F =--2 1 1 1 ,...
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## This note was uploaded on 10/30/2011 for the course AMS 517 taught by Professor Xinghaipeng during the Spring '11 term at SUNY Stony Brook.

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mid1sol - x ( ) ( ) ( ) ( ) ( ) &amp;amp;amp; &amp;amp;amp;...

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