midterm_1_-_questions_Winter_2008

# Assume the stock pays no dividend 121 ր 110 ր ց

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ously compounded). Assume the stock pays no dividend. 121 ր 110 ր ց 100 99 ց ր 90 ց 81 (a) [6 pts] Find the value at time 0 of a European derivative that pays max( S T 98 , 0) at a maturity of 6 months. (b) [2 pts] Find the value at time 0 of this derivative if it can be exercised after 3 or 6 months.

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5 of 10 3. [total=4 points] A company sells equity-linked contracts with a maturity T = 1 year for which an initial investment of \$1000 yields a payoff given by 1000 max parenleftbigg 1 , 1 + 0 . 6 parenleftbigg S T S 0 1 parenrightbiggparenrightbigg at maturity, where S T is the value of the index at time T . Assume the interest rate is equal to 5% (continuously coumpounded). a) [2 pts] Express this product as a portfolio of call options and zero-coupon bonds. Precise the strike of the call options. b) [2 pts] Represent the profit and the payoff of this product on a graph.
6 of 10 4. [Total=16 points] FORMULA SHEET at the END - See page 10 If S follows the geometric Brownian process as follows: dS t S t = μdt + σdW a) [8 pts] Using Ito’s lemma, write d ( te S t ) as u ( t, S t ) dt + v ( t, S t ) dW t . b) [1pt] Express integraltext s u d ( te S t ) as a function of u , s , S u and S t :

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7 of 10 c) [5pts] Assume X = e 3 t +2 W t where W is a standard Brownian motion.
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