# E s t 2 e rt 2 s 6 e s 1 t e s 1 e r σ 2 2 t σw t s

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E [ S T/ 2 ] = e rT/ 2 S 0 (6) E [ S - 1 T ] = E [ S - 1 0 e - ( r - σ 2 2 ) T - σW T ] = S - 1 0 e - rT e σ 2 T/ 2 E [ e - σW T ] = S - 1 0 e - rT e σ 2 T because e - σW T is lognormal LN (0 2 T ). (7) E bracketleftBigg parenleftbigg S T S 0 parenrightbigg k bracketrightBigg = E [ e ( r - σ 2 2 ) kT + - kσW T ] = e ( r - σ 2 2 ) kT E [ e - kσW T ] = e ( r - σ 2 2 ) kT e k 2 σ 2 T/ 2 = e rTk e σ 2 Tk 2 ( k - 1) (8) Direct computations using the Log Normal property of the stock price: E [ max ( S T ,G )] = E [ S T 1 S T >G ] + E [ G 1 S T G ] = E [ S T 1 S T >G ] + GP ( S T G ) = S 0 E [ e X T 1 e X T > G S 0 ] + GP parenleftbigg X T log parenleftbigg G S 0 parenrightbiggparenrightbigg = S 0 exp parenleftbigg m + V 2 parenrightbigg Φ parenleftBigg m + V - ln( G S 0 ) V parenrightBigg + GN ( - d 2 ) (1)

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where S T = S 0 e X T . And X T is a Normal with mean equal to m = ( r - σ 2 2 ) T and variance V = σ 2 T . d 2 is defined as follows: P parenleftbigg X T log parenleftbigg G S 0 parenrightbiggparenrightbigg = N log parenleftBig G S 0 parenrightBig - ( r - σ 2 2 ) T σ T = N ( - d 2 ) (9) Direct computations using the Log Normal property of the stock price: E [min( S T ,G )] = E [ S T 1 S T <G ] + E [ G 1 S T G ] = E [ S T 1 S T <G ] + GP ( S T G ) = S 0 E [ e X T 1 e X T < G S 0 ] + GP parenleftbigg X T log parenleftbigg G S 0 parenrightbiggparenrightbigg where S T = S 0 e X T . And X T is a Normal with mean equal to ( r - σ 2 2 ) T and variance σ 2 T . (10) The price is equal to E [2 1 S T < 110 + 10 1 S T > 110 ] which can also be written as: E [2 + 8 1 S T > 110 ] = 2 + 8 E [ 1 S T > 110 ] = 2 + 8 P [ S T > 110] = 2 + 8 P [log( S T /S 0 ) > log(110 /S 0 )] = 2 e - rT + 8 e - rT parenleftbigg 1 - N parenleftbigg log( S T /S 0 ) - ( r - σ 2 / 2) T σ T parenrightbiggparenrightbigg (11) S t = S 0 e ( r - σ 2 2 ) t + σW t
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