solmid2

# solmid2 - ACTSC/STAT 446/846 – Midterm 2 – Fall 2011...

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Unformatted text preview: ACTSC/STAT 446/846 – Midterm 2 – Fall 2011 – Solutions 1. [14 points] A stock with current price S (0) = 100 is modeled by a binomial model with two time-steps of length h = 0 . 5 year each, using u = e ( r-δ ) h + σ √ h and d = e ( r-δ ) h-σ √ h . The stock pays a continuous dividend at an annual rate of δ = 8% and has volatility σ = 0 . 25. The current continuously compounded annual risk-free rate is r = 10%. (a) Determine the value of an American call option on this stock with strike price K = 95, where exercise is allowed at time t = 0 , . 5 or 1 year. Solution: [12pts] We have that u = e ( r-δ ) h + σ √ h = 1 . 2054, d = e ( r-δ ) h-σ √ h = 0 . 8464, and q = e ( r-δ ) h-d u-d = 0 . 4559 . Hence we have the tree (with payoﬀ in [ · ]): 100[5] → 120 . 54[25 . 54] → 145 . 30[50 . 30] & 84 . 64[0] & → 102 . 03[7 . 03] & → 71 . 64[0] So H (1 , 1) = e-. 10 * . 5 (50 . 30 q + 7 . 03(1-q )) = 25 . 45 E (1 , 1) =25 . 54 V (1 , 1) =25 . 54 H (1 , 0) = e-. 10 * . 5 (7 . 03 q ) = 3 . 05 H (0 , 0) = e-. 10 * . 5 (25 . 54 q + 3 . 05(1-q )) = 12 . 65 V (0 , 0) =12 . 65 (b) Brieﬂy explain how to use your work in (a) to determine if the corresponding European call option would be cheaper, without actually computing its price. Solution: [2pts] The European option would be cheaper since in (a) we found out that E (1 , 1) > H (1 , 1), i.e., we would early exercise in the node (1,1). 2. [11 points] A single-period market model includes two securities with initial price ~ S (0) = [1 , 4] and possible values at time 1 given by S (1 , Ω) = 0 5 2 4...
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solmid2 - ACTSC/STAT 446/846 – Midterm 2 – Fall 2011...

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