w_exam_one_solutions

9253 n d2 08983 finally vc 0 s 0et n d1

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Unformatted text preview: of the above. 8 Solution: (c) In our usual notation, 1 1 S (0) + r − δ + σ2 T d1 = √ ln K 2 σT 1 1 1 50 = + 0.05 − 0.02 + × 0.342 × = 1.44, ln 40 2 4 0.34 1/4 √ d2 = d1 − σ T = 1.27. The standard normal tables give us N (d1 ) = 0.9253, N (d2 ) = 0.8983. Finally, VC (0) = S (0)e−δT N (d1 ) − Ke−rT N (d2 ) = 10.55. 5. In this problem, use the Black-Scholes pricing model. Consider a bear spread consisting of a 20−strike put and a 25−strike put. Suppose that σ = 0.30, r = 0.04, δ = 0, T = 1 and S (0) = 15. What is the price of this bear spread? (a) About $4.36 (b) About $4.80 (c) About $9.16 (d) About $13.96 (e) None of the above Solution: (a) Here, you sell a 20−strike put and buy a 25−strike put. For the 20−strike put, we have 1 15 1 d1 = [ln( ) + (0.04 + · 0.32 ) · 1] = −0.6756; 0.3 · 1 20 2 d2 = −0.9756. The price of the 20−strike put is VP (0, 20) = Ke−rT N (−d2 ) − S (0)e−δT N (−d1 ) = 4.79698. For the 25−strike put, we have 1 15 1 d1 = [ln( ) + (0.04 + · 0.32 ) · 1] = −1.4194; 0.3 · 1 25 2 d2 = −1.7194. The price of the 25−strike put is VP (0, 25) ≈ 25e−0.04 N (−d2 ) − 15e−0.02 N (−d1 ) = 9.16076 So, the price of the bear spread is VP (0, 25) − VP (0, 20) = 9.16076 − 4.79698 = 4.36378. 9 6. Assume the Black-Scholes framework. Let the current price of a share of non-dividendpaying stock be equal to S (0) = 100; let its volatility be σ = 0.3. Consider a gap call option with expiration date T = 1, with the trigger price 100 and the strike price 90. You are given that the continuously compounded risk-free interest rate equals r = 0.04 per annum. Let VGC (0) denote the price of the above gap option. Then, (a) VGC (0) < $13.20 (b) $13.20 ≤ VGC (0) < $15.69 (c) $15.69 ≤ VGC (0) < $17.04 (d) 17.04 ≤ VGC (0) < $21.25 (e) None of the above. Solution: (d) In our usual notation, the Black-Scholes formula for the price of a gap call option reads as VGC (0) = S (0)−δT N (d1 ) − K1 e−rT N (d2 ) where 1 1 d1 = √ [ln(S (0)/K2 ) + (r − δ + σ 2 )T ], 2 σT √ d2 = d1 − σ T . In the present problem, 1 0.09 [ln(100/100) + (0.04 + )] ≈ 0.28; 0.3 2 d2 = −0.02. d1 = So, the price equals VGC (0) = 100N (0.28) − 90e−0.04 N (−0.02) = 18.49. 7. Assume the Black-Scholes setting. Alice wagers to pay one share of stock to Bob if the price of non-dividend-paying stock in 1 year is above $100.00. Assume S (0) = 100.00, σ = 0.3, and r = 0.04. What is the time−0 value of this bet? (a) About $61.15 (b) About $81.15 (c) About $91.15 (d) About $100 (e) None of the above. Solution: (a) or (e) The Black-Scholes price of this asset call is V 3 [S (0), 0] = S (0)N (d1 ) 10 with d1 = 1 0.3 ln( 100 1 ) + 0.04 + 0.09 100 2 = 0.2833. So, V 3 [S (0), 0] = 100N (0.28) = 61.03. Note: If you use the standard normal “calculator”, you get 61.15....
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This note was uploaded on 08/04/2013 for the course M 339W taught by Professor Cudina during the Fall '12 term at University of Texas.

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